%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: bcm3.dvi %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips bcm3 -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.24:1230 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 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b(of)h(a)g(c)n(harged)e(particle)h (elastically)g(in)n(teracting)g(with)h(an)g(in\014nite)h(hard)e(core)g (gas.)456 5010 y(When)c(the)g(hard)f(core)g(particles)g(are)f(\014xed)i (\(Loren)n(tz)f(gas\))g(a)g(nonph)n(ysical)g(phenomenon)h(of)p 456 5122 499 4 v 555 5216 a Fu(W)-6 b(ork)24 b(partially)g(supp)r (orted)g(b)n(y)g(the)h(GNFM-IND)n(AM)e(and)h(the)g(Italian)h(Ministry)d (of)i(the)g(Univ)n(ersit)n(y)-6 b(.)1933 5315 y Fo(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fo(2)709 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(heating)g(arises:)33 b(the)23 b(a)n(v)n(erage)d (kinetic)j(energy)e(of)i(the)g(c)n(harged)e(particle)h(is)h(increased)f (b)n(y)g(the)456 550 y(electric)31 b(\014eld,)i(while)g(it)f(cannot)f (b)r(e)i(dissipated)e(b)n(y)h(the)g(elastic)g(collisions)f(with)h(the)g (\014xed)456 649 y(obstacles.)59 b(A)n(t)36 b(the)f(same)g(time)h (there)g(is)f(not)g(an)g(asymptotic)g(drift)h(v)n(elo)r(cit)n(y)f([22)o (].)60 b(The)456 749 y(more)24 b(realistic)h(case)f(of)h(mo)n(ving)g (hard)f(core)g(particles)h(with)h(a)f(\014nite)g(mass)g(seems)g(to)g(b) r(e)h(to)r(o)456 849 y(di\016cult.)36 b(Another)25 b(situation)g(whic)n (h)g(has)f(b)r(een)h(considered)f(is)h(the)g(motion)g(along)f(one)g (axis)456 948 y(of)g(a)g(c)n(harged)f(particle)g(with)i(an)f(hard)g (core)f(and)i(elastically)e(in)n(teracting)g(with)i(a)f(gas)f(of)i (free)456 1048 y(particles.)33 b(In)21 b(this)g(case)f(it)h(is)f(p)r (ossible)g(to)h(pro)n(v)n(e)e(the)i(existence)f(of)g(a)h(limit)g(drift) g(v)n(elo)r(cit)n(y)f(and)456 1147 y(to)k(in)n(v)n(estigate)f(some)h (prop)r(erties)f(of)h(the)h(limit)g(state.)36 b(Ho)n(w)n(ev)n(er)22 b(the)j(tec)n(hnical)f(assumption)456 1247 y(that)j(the)g(free)f(gas)g (is)g(in)h(a)g(Gibbs)g(state)f(with)h(a)g(cuto\013)g(on)f(the)h(slo)n (w)f(transv)n(ersal)e(v)n(elo)r(cities)456 1347 y(is)33 b(necessary)e([5,)i(6)o(].)54 b(\(On)33 b(related)g(topics)g(see)f (also)g([2,)h(3,)g(4,)g(13)o(,)g(21)o(,)g(25)o(,)h(26)o(]\).)54 b(In)33 b(other)456 1446 y(w)n(orks,)d(di\013eren)n(t)i(media)f(are)g (considered,)g(where,)h(in)g(general,)f(the)h(goal)e(is)h(to)h(describ) r(e)f(a)456 1546 y(microscopic)g(mo)r(del)i(for)g(linear)f(friction.)53 b(W)-7 b(e)33 b(only)g(recall)f(the)h(recen)n(t)f(pap)r(er)h([7)o(])h (on)e(this)456 1646 y(argumen)n(t)26 b(and)i(address)e(the)i(reader)e (to)h(the)h(references)f(quoted)g(there)h(in.)555 1745 y(W)-7 b(e)24 b(\014nally)f(remark)g(that)g(man)n(y)g(ph)n(ysical)g (problems)g(only)g(concern)f(the)i(b)r(eha)n(vior)f(of)g(the)456 1845 y(system)30 b(on)f(a)h(long)g(time)g(scale)g(rather)f(than)h(its)g (gen)n(uine)g(steady)g(state.)44 b(Mathematically)456 1944 y(it)c(corresp)r(onds)d(to)j(study)g(a)f(transien)n(t)g(situation) h(during)f(a)g(long)g(but)h(\014nite)h(time.)73 b(In)456 2044 y(this)25 b(case)f(for)g(b)r(ounded)h(in)n(teractions)f(also)f (Ohm's)i(la)n(w)f(could)h(appro)n(ximativ)n(ely)d(hold)j(un)n(til)456 2144 y(the)k(time)h(in)g(whic)n(h)f(the)h(escap)r(e)f(e\013ects)h(b)r (ecome)f(relev)-5 b(an)n(t.)42 b(Concerning)28 b(these)h(e\013ects)h(w) n(e)456 2243 y(recall)d(for)g(instance)h(the)g(phenomenon)f(of)h(the)h (so)e(called)g Fp(runaway)i Fr(electrons)e(discussed)g(in)456 2343 y([16)o(].)42 b(In)30 b(this)f(pap)r(er)g(w)n(e)g(actually)g (presen)n(t)g(an)g(Hamiltonian)g(mo)r(del)h(whic)n(h)f(exhibits)h(suc)n (h)456 2443 y(a)d(b)r(eha)n(vior.)555 2542 y(In)32 b(the)h(next)f (section)g(w)n(e)f(describ)r(e)h(the)g(mo)r(del)h(and)e(w)n(e)h(state)g (the)g(main)g(result)g(of)g(the)456 2642 y(pap)r(er.)k(The)28 b(pro)r(ofs)e(will)i(b)r(e)g(giv)n(en)f(in)h(Sects.)g(3)f(and)g(4.)1025 2868 y(2.)41 b Fq(The)32 b(model)f(and)g(st)-6 b(a)g(tement)33 b(of)f(the)f(resul)-6 b(ts)555 3017 y Fr(W)f(e)24 b(consider)e(the)i (follo)n(wing)e(Hamiltonian)h(system)h(of)f(in\014nitely)h(man)n(y)f (particles)f(in)i(one)456 3117 y(dimension.)57 b(A)35 b(c)n(harged)e(particle)h(of)h(mass)e Fn(M)44 b Fr(and)34 b(c)n(harge)f Fn(q)38 b Fr(is)c(sub)5 b(ject)35 b(to)f(a)g(constan)n(t) 456 3217 y(electric)25 b(\014eld)i Fn(E)h(>)23 b Fr(0,)j(and)g(it)g(is) g(coupled)g(with)h(an)f(in\014nite)g(system)g(of)g(neutral)g(particles) g(of)456 3316 y(unit)e(mass)g(b)n(y)f(means)h(of)g(a)g(non-negativ)n (e,)f(symmetric,)h(t)n(wice)g(di\013eren)n(tiable,)h(short-range,)456 3416 y(t)n(w)n(o-b)r(o)r(dy)30 b(p)r(oten)n(tial)i(\011.)49 b(Without)32 b(loss)f(of)h(generalit)n(y)e(w)n(e)h(assume)g(that)h Fn(q)s(=)-5 b(M)38 b Fr(=)30 b(1)h(\(it)i(is)456 3515 y(just)28 b(a)f(rede\014nition)g(of)h(the)g(electric)f(\014eld\).)555 3615 y(The)37 b(neutral)f(particles)g(in)n(teract)g(among)g(themselv)n (es)g(b)n(y)h(means)f(of)h(a)f(non-negativ)n(e,)456 3715 y(symmetric,)23 b(t)n(wice)g(di\013eren)n(tiable,)h(short-range,)e(t)n (w)n(o-b)r(o)r(dy)g(p)r(oten)n(tial)h(\010.)35 b(W)-7 b(e)24 b(also)e(require)456 3814 y(\010\(0\))h Fn(>)g Fr(0,)28 b(whic)n(h)f(implies)h(that)g(the)h(in)n(teraction)d(\010)i (is)g(sup)r(erstable)f([24)o(].)38 b(Without)29 b(loss)e(of)456 3914 y(generalit)n(y)f(w)n(e)j(further)f(assume)g(that)g(b)r(oth)h (\011)f(and)g(\010)h(ha)n(v)n(e)e(range)g(not)i(greater)d(than)j(one,) 456 4014 y(i.e.)1293 4133 y(\010\()p Fn(s)p Fr(\))24 b(=)e(0)p Fn(;)97 b Fr(\011\()p Fn(s)p Fr(\))23 b(=)g(0)165 b(if)84 b Fm(j)p Fn(s)p Fm(j)23 b Fn(>)f Fr(1)p Fn(:)667 b Fr(\(2.1\))555 4270 y(W)-7 b(e)33 b(denote)g(b)n(y)g(\()p Fn(x;)14 b(v)s Fr(\))32 b Fm(2)g Fl(R)1461 4240 y Fo(2)1538 4270 y Fr([resp.)g(\()p Fn(x)1844 4282 y Fk(i)1872 4270 y Fn(;)14 b(v)1949 4282 y Fk(i)1977 4270 y Fr(\))32 b Fm(2)g Fl(R)2182 4240 y Fo(2)2225 4270 y Fr(])h(the)g(p)r(osition)g (and)f(v)n(elo)r(cit)n(y)g(of)h(the)456 4369 y(c)n(harged)d(particle)i ([resp.)g(neutral)g(particles].)50 b(The)32 b(state)g Fn(X)37 b Fr(=)31 b Fm(f)p Fr(\()p Fn(x)2698 4381 y Fk(i)2726 4369 y Fn(;)14 b(v)2803 4381 y Fk(i)2830 4369 y Fr(\))p Fm(g)2904 4381 y Fk(i)p Fj(2)p Fi(N)3051 4369 y Fr(is)32 b(assumed)456 4469 y(to)26 b(ha)n(v)n(e)e(a)i(lo)r(cally)f(\014nite)i (densit)n(y)f(and)g(energy)-7 b(.)35 b(In)26 b(particular,)f(for)h(an)n (y)f Fn(\026)e Fm(2)h Fl(R)32 b Fr(and)25 b Fn(R)f(>)f Fr(0,)456 4569 y(it)28 b(is)f(w)n(ell)g(de\014ned)h(the)g(quan)n(tit)n (y:)917 4772 y Fn(Q)p Fr(\()p Fn(X)7 b Fr(;)14 b Fn(\026;)g(R)q Fr(\))1354 4725 y Fn(:)1333 4772 y Fr(=)1421 4693 y Fh(X)1469 4870 y Fk(i)1554 4772 y Fn(\037)1606 4784 y Fk(i)1634 4772 y Fr(\()p Fn(\026;)g(R)q Fr(\))1863 4655 y Fh(\032)1935 4715 y Fn(v)1978 4685 y Fo(2)1975 4737 y Fk(i)p 1935 4753 81 4 v 1955 4829 a Fr(2)2044 4772 y(+)2137 4715 y(1)p 2137 4753 42 4 v 2137 4829 a(2)2220 4693 y Fh(X)2202 4871 y Fk(j)s Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)2371 4772 y Fr(\010\()p Fn(x)2510 4784 y Fk(i)2557 4772 y Fm(\000)k Fn(x)2687 4784 y Fk(j)2722 4772 y Fr(\))h(+)f(1)2898 4655 y Fh(\033)2960 4772 y Fn(;)291 b Fr(\(2.2\))456 5016 y(where)19 b Fn(\037)740 5028 y Fk(i)767 5016 y Fr(\()p Fn(\026;)14 b(R)q Fr(\))23 b(=)g Fn(\037)p Fr(\()p Fm(j)p Fn(x)1247 5028 y Fk(i)1278 5016 y Fm(\000)r Fn(\026)p Fm(j)g(\024)f Fn(R)q Fr(\))e(and)f Fn(\037)p Fr(\()p Fn(A)p Fr(\))i(denotes)e(the)h(c)n(haracteristic)d(function)k(of)e(the) 456 5116 y(set)25 b Fn(A)p Fr(.)36 b(In)25 b(order)e(to)i(consider)f (con\014gurations)f(whic)n(h)i(are)f(t)n(ypical)g(for)h(the)g(thermo)r (dynamic)456 5216 y(states,)g(w)n(e)f(allo)n(w)g(initial)i(data)e(with) i(logarithmic)e(div)n(ergences)f(in)i(the)h(v)n(elo)r(cities)e(and)h (lo)r(cal)p eop %%Page: 3 3 3 2 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)308 b(3)456 450 y Fr(densities.)37 b(More)26 b(precisely)-7 b(,)27 b(b)n(y)g(de\014ning)1262 653 y Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))1511 606 y Fn(:)1490 653 y Fr(=)23 b(sup)1620 719 y Fk(\026)1890 653 y Fr(sup)1717 726 y Fk(R)p Fo(:)p Fk(R>)p Fo(log)q(\()p Fk(e)p Fo(+)p Fj(j)p Fk(\026)p Fj(j)p Fo(\))2212 597 y Fn(Q)p Fr(\()p Fn(X)7 b Fr(;)14 b Fn(\026;)g(R)q Fr(\))p 2212 634 394 4 v 2356 710 a(2)p Fn(R)2615 653 y(;)636 b Fr(\(2.3\))456 881 y(the)28 b(set)f Fm(X)843 834 y Fn(:)823 881 y Fr(=)22 b Fm(f)p Fn(X)29 b Fr(:)23 b Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))23 b Fn(<)f Fr(+)p Fm(1g)27 b Fr(has)g(a)g(full)i(measure)d(w.r.t.)i(an)n(y)f (Gibbs)h(state)f([14)o(].)555 981 y(The)h(time)g(ev)n(olution)f(is)g (de\014ned)h(b)n(y)g(the)f(follo)n(wing)g(system)g(of)h(in\014nite)g (equations:)681 1072 y Fh(8)681 1146 y(>)681 1171 y(>)681 1196 y(>)681 1221 y(>)681 1246 y(<)681 1396 y(>)681 1420 y(>)681 1445 y(>)681 1470 y(>)681 1495 y(:)801 1152 y Fr(\177)-47 b Fn(x)p Fr(\()p Fn(t)p Fr(\))24 b(=)f Fm(\000)p Fn(M)1204 1118 y Fj(\000)p Fo(1)1306 1074 y Fh(X)1350 1250 y Fk(j)1439 1152 y Fm(r)p Fr(\011\()p Fn(x)p Fr(\()p Fn(t)p Fr(\))d Fm(\000)e Fn(x)1896 1164 y Fk(j)1931 1152 y Fr(\()p Fn(t)p Fr(\)\))i(+)e Fn(E)5 b(;)801 1357 y Fr(\177)-47 b Fn(x)843 1369 y Fk(i)871 1357 y Fr(\()p Fn(t)p Fr(\))24 b(=)e Fm(\000r)p Fr(\011\()p Fn(x)1354 1369 y Fk(i)1382 1357 y Fr(\()p Fn(t)p Fr(\))d Fm(\000)f Fn(x)p 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Fk(n)p Fo(\))2392 2339 y Fk(i)2493 2316 y Fr(\()p Fn(t)p Fr(\)\))p Fm(g)2661 2328 y Fk(i)p Fj(2)p Fk(I)2758 2336 y Fg(n)2804 2316 y Fr(,)26 b(is)e(de\014ned)i(b)n(y)e(the)456 2416 y(solution)j(of)g(the)h(di\013eren)n(tial)g(system:)514 2509 y Fh(8)514 2584 y(>)514 2609 y(>)514 2633 y(>)514 2658 y(>)514 2683 y(<)514 2833 y(>)514 2858 y(>)514 2882 y(>)514 2907 y(>)514 2932 y(:)634 2587 y Fr(\177)-47 b Fn(x)676 2553 y Fo(\()p Fk(n)p Fo(\))774 2587 y Fr(\()p Fn(t)p Fr(\))23 b(=)g Fm(\000)p Fn(M)1134 2553 y Fj(\000)p Fo(1)1249 2508 y Fh(X)1236 2686 y Fk(j)s Fj(2)p Fk(I)1340 2694 y Fg(n)1396 2587 y Fm(r)p Fr(\011\()p Fn(x)1609 2553 y Fo(\()p Fk(n)p Fo(\))1706 2587 y Fr(\()p Fn(t)p Fr(\))c Fm(\000)f Fn(x)1949 2544 y Fo(\()p Fk(n)p Fo(\))1949 2610 y Fk(j)2047 2587 y Fr(\()p Fn(t)p Fr(\)\))h(+)f Fn(E)5 b(;)634 2793 y Fr(\177)-47 b Fn(x)676 2750 y Fo(\()p Fk(n)p Fo(\))676 2816 y Fk(i)774 2793 y Fr(\()p Fn(t)p Fr(\))23 b(=)g Fm(\000r)p Fr(\011\()p Fn(x)1257 2750 y Fo(\()p Fk(n)p Fo(\))1257 2816 y Fk(i)1354 2793 y Fr(\()p Fn(t)p Fr(\))c Fm(\000)f Fn(x)1597 2759 y Fo(\()p Fk(n)p Fo(\))1695 2793 y Fr(\()p Fn(t)p Fr(\)\))h Fm(\000)1998 2714 y Fh(X)1923 2893 y Fk(j)s Fj(2)p Fk(I)2027 2901 y Fg(n)2069 2893 y Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)2207 2793 y Fm(r)p Fr(\010\()p Fn(x)2415 2750 y Fo(\()p Fk(n)p Fo(\))2415 2816 y Fk(i)2513 2793 y Fr(\()p Fn(t)p Fr(\))g Fm(\000)f Fn(x)2756 2750 y Fo(\()p Fk(n)p Fo(\))2756 2816 y Fk(j)2853 2793 y Fr(\()p Fn(t)p Fr(\)\))p Fn(;)98 b(i)23 b Fm(2)g Fn(I)3266 2805 y Fk(n)3312 2793 y Fn(;)629 2986 y Fr(\()p Fn(x)708 2956 y Fo(\()p Fk(n)p Fo(\))806 2986 y Fr(\(0\))p Fn(;)14 b(v)992 2956 y Fo(\()p Fk(n)p Fo(\))1089 2986 y Fr(\(0\)\))24 b(=)e(\(0)p Fn(;)14 b Fr(0\))p Fn(;)97 b(X)1719 2956 y Fo(\()p Fk(n)p Fo(\))1815 2986 y Fr(\(0\))23 b(=)g Fm(f)p Fr(\()p Fn(x)2153 2998 y Fk(i)2181 2986 y Fn(;)14 b(v)2258 2998 y Fk(i)2286 2986 y Fr(\))p Fm(g)2360 2998 y Fk(i)p Fj(2)p Fk(I)2457 3006 y Fg(n)2502 2986 y Fn(:)3274 3087 y Fr(\(2.5\))456 3186 y(Then,)27 b(for)h(an)n(y)e Fn(t)d Fm(\025)g Fr(0,)k(the)h(follo)n (wing)f(limits)h(exist:)957 3348 y(lim)902 3398 y Fk(n)p Fj(!)p Fo(+)p Fj(1)1141 3348 y Fn(x)1188 3314 y Fo(\()p Fk(n)p Fo(\))1285 3348 y Fr(\()p Fn(t)p Fr(\))c(=)e Fn(x)p Fr(\()p Fn(t)p Fr(\))p Fn(;)250 b Fr(lim)1849 3398 y Fk(n)p Fj(!)p Fo(+)p Fj(1)2087 3348 y Fn(v)2130 3314 y Fo(\()p Fk(n)p Fo(\))2228 3348 y Fr(\()p Fn(t)p Fr(\))23 b(=)g Fn(v)s Fr(\()p Fn(t)p Fr(\))p Fn(;)957 3500 y Fr(lim)902 3550 y Fk(n)p Fj(!)p Fo(+)p Fj(1)1141 3500 y Fn(x)1188 3457 y Fo(\()p Fk(n)p Fo(\))1188 3523 y Fk(i)1285 3500 y Fr(\()p Fn(t)p Fr(\))h(=)e Fn(x)1537 3512 y Fk(i)1566 3500 y Fr(\()p Fn(t)p Fr(\))p Fn(;)221 b Fr(lim)1849 3550 y Fk(n)p Fj(!)p Fo(+)p Fj(1)2087 3500 y Fn(v)2130 3457 y Fo(\()p Fk(n)p Fo(\))2127 3523 y Fk(i)2228 3500 y Fr(\()p Fn(t)p Fr(\))23 b(=)g Fn(v)2473 3512 y Fk(i)2501 3500 y Fr(\()p Fn(t)p Fr(\))p Fn(;)166 b(i)23 b Fm(2)h Fl(N)t Fn(:)3274 3439 y Fr(\(2.6\))456 3696 y(Moreo)n(v)n(er,)d(the)j (\015o)n(w)f Fn(t)g Fm(7!)g(f)p Fr(\()p Fn(x)p Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(v)s Fr(\()p Fn(t)p Fr(\)\);)g Fn(X)7 b Fr(\()p Fn(t)p Fr(\))p Fm(g)p Fr(,)25 b Fn(X)7 b Fr(\()p Fn(t)p Fr(\))23 b(=)g Fm(f)p Fr(\()p Fn(x)2427 3708 y Fk(i)2455 3696 y Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(v)2626 3708 y Fk(i)2654 3696 y Fr(\()p Fn(t)p Fr(\)\))p Fm(g)2822 3708 y Fk(i)p Fj(2)p Fi(N)2937 3696 y Fr(,)24 b(is)f(the)h(unique)456 3795 y(\(global\))j(solution)g(to)g(Eqs.)g (\(2.4\))g(suc)n(h)h(that)f Fn(X)7 b Fr(\()p Fn(t)p Fr(\))23 b Fm(2)h(X)12 b Fr(.)555 3895 y(W)-7 b(e)32 b(are)e(in)n(terested)h(in) g(the)h(asymptotic)f(b)r(eha)n(vior)f(of)h(the)g(c)n(harged)f(particle) h(when)g(the)456 3994 y(bac)n(kground)26 b(particle)j(system)f(is)h (initially)g(in)g(a)f(thermo)r(dynamic)g(state.)40 b(The)29 b(full)h(system)456 4094 y([c)n(harged)21 b(particle])h(+)g([bac)n (kground)f(particles])h(is)h(not)f(in)h(equilibrium,)h(and)f(its)f (time)i(ev)n(olu-)456 4194 y(tion)j(is)h(describ)r(ed)f(b)n(y)h(the)g (statistical)f(solutions)g(of)h(the)g(Newton)f(equations)g(\(2.4\))h (relativ)n(e)456 4293 y(to)f(the)h(corresp)r(onding)e(random)g(initial) i(data)g Fn(X)7 b Fr(.)36 b(Our)27 b(main)g(result)h(is)f(the)h(con)n (ten)n(t)f(of)h(the)456 4393 y(follo)n(wing)e(theorems)h(whic)n(h)h (will)f(b)r(e)h(pro)n(v)n(ed)e(in)i(Sect.)g(4.)456 4557 y Fv(Theorem)i(2.1.)40 b Fp(Given)30 b Fn(Q)23 b(>)g Fr(0)29 b Fp(and)h Fn(L)23 b(>)f Fr(0)30 b Fp(let:)1018 4716 y Fm(B)1073 4728 y Fk(Q;L)1238 4669 y Fn(:)1217 4716 y Fr(=)22 b Fm(f)p Fn(X)29 b Fm(2)24 b(X)49 b Fr(:)37 b Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))23 b Fm(\024)f Fn(Q;)99 b Fm(j)p Fn(x)2265 4728 y Fk(i)2293 4716 y Fm(j)23 b(\025)g Fn(L)84 b Fm(8)14 b Fn(i)21 b Fm(2)j Fl(N)t Fm(g)p Fn(:)397 b Fr(\(2.7\))456 4875 y Fp(Then,)28 b(for)g(e)l(ach)g Fn(E)g(>)23 b Fr(0)j Fp(and)i Fn(Q)23 b(>)f Fr(0)27 b Fp(ther)l(e)g(exists)g Fn(L)2144 4887 y Fo(0)2203 4875 y Fn(>)c Fr(0)k Fp(such)g(that,)h(for)g(any)f Fn(L)22 b Fm(\025)h Fn(L)3249 4887 y Fo(0)3313 4875 y Fp(and)456 4975 y Fn(X)29 b Fm(2)23 b(B)687 4987 y Fk(Q;L)809 4975 y Fp(,)1707 5151 y Fr(lim)1660 5200 y Fk(t)p Fj(!)p Fo(+)p Fj(1)1892 5094 y Fn(v)s Fr(\()p Fn(t)p Fr(\))p 1892 5131 138 4 v 1946 5207 a Fn(t)2063 5151 y Fr(=)g Fn(E)5 b(:)1034 b Fr(\(2.8\))p eop %%Page: 4 4 4 3 bop 456 251 a Fo(4)709 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fv(Theorem)30 b(2.2.)40 b Fp(F)-6 b(or)30 b(any)g Fn(E)e Fm(\025)23 b Fr(0)29 b Fp(and)h Fn(X)g Fm(2)23 b(X)12 b Fp(,)1662 628 y Fr(lim)i(inf)1672 678 y Fk(t)p Fj(!)p Fo(+)p Fj(1)1915 572 y Fn(v)s Fr(\()p Fn(t)p Fr(\))p 1915 609 138 4 v 1969 685 a Fn(t)2086 628 y Fm(\025)23 b Fr(0)p Fn(:)1035 b Fr(\(2.9\))555 806 y(An)28 b(easy)f(consequence)g(of)g(these)h(theorems)f(is:)456 952 y Fv(Theorem)34 b(2.3.)42 b Fp(F)-6 b(or)33 b(any)g(Gibbs)g(state)g Fm(h\001i)g Fp(of)h(the)e(b)l(ackgr)l(ound)i(p)l(article)g(system)e (and)h(for)456 1051 y(any)d(p)l(ositive)h(ele)l(ctric)f(\014eld)h Fn(E)5 b Fp(,)1629 1229 y Fr(lim)14 b(inf)1640 1279 y Fk(t)p Fj(!)p Fo(+)p Fj(1)1883 1173 y Fm(h)p Fn(v)s Fr(\()p Fn(t)p Fr(\))p Fm(i)p 1883 1210 203 4 v 1969 1286 a Fn(t)2119 1229 y(>)22 b Fr(0)p Fn(:)961 b Fr(\(2.10\))456 1411 y Fp(Pr)l(o)l(of.)46 b Fr(By)30 b(w)n(ell)f(kno)n(wn)h(prop)r(erties)f (of)h(the)g(DLR)h(states,)f(see)f(e.g.)h([14)o(],)h(the)f(subset)g Fm(B)3323 1423 y Fk(Q;L)456 1511 y Fr(has)e(p)r(ositiv)n(e)g(measure)f (w.r.t)h(the)h(state)g Fm(h\001i)g Fr(for)f Fn(Q)g Fr(large)f(enough)h (\(dep)r(ending)h(on)f Fm(h\001i)p Fr(\))i(and)456 1611 y(for)d(an)n(y)f Fn(L)d Fm(\025)g Fr(0.)36 b(Then,)28 b(b)n(y)f(\014xing)h Fn(Q)f Fr(large)f(enough)h(and)h(then)g Fn(L)f Fr(as)g(in)h(Theorem)e(2.1,)722 1790 y(lim)14 b(inf)733 1840 y Fk(t)p Fj(!)p Fo(+)p Fj(1)976 1734 y Fm(h)p Fn(v)s Fr(\()p Fn(t)p Fr(\))p Fm(i)p 976 1771 V 1062 1847 a Fn(t)1271 1790 y Fm(\025)1419 1673 y Fh(\034)1481 1790 y Fr(lim)g(inf)1492 1840 y Fk(t)p Fj(!)p Fo(+)p Fj(1)1735 1734 y Fn(v)s Fr(\()p Fn(t)p Fr(\))p 1735 1771 138 4 v 1789 1847 a Fn(t)1883 1673 y Fh(\035)1271 2023 y Fr(=)1419 1906 y Fh(\034)1481 2023 y Fn(\037)p Fr(\()p Fm(B)1620 2035 y Fk(Q;L)1742 2023 y Fr(\))60 b(lim)1788 2073 y Fk(t)p Fj(!)p Fo(+)p Fj(1)2020 1967 y Fn(v)s Fr(\()p Fn(t)p Fr(\))p 2020 2004 V 2074 2080 a Fn(t)2168 1906 y Fh(\035)2248 2023 y Fr(+)2331 1906 y Fh(\034)2394 2023 y Fn(\037)p Fr(\()p Fm(B)2536 1989 y Fk(c)2533 2043 y(Q;L)2654 2023 y Fr(\))14 b(lim)g(inf)2711 2073 y Fk(t)p Fj(!)p Fo(+)p Fj(1)2954 1967 y Fn(v)s Fr(\()p Fn(t)p Fr(\))p 2954 2004 V 3008 2080 a Fn(t)3102 1906 y Fh(\035)1271 2197 y Fm(\025)83 b(h)p Fn(\037)p Fr(\()p Fm(B)1590 2209 y Fk(Q;L)1712 2197 y Fr(\))p Fm(i)14 b Fn(E)28 b(>)23 b Fr(0)p Fn(;)456 2332 y Fr(where)i(in)h(the)g(\014rst)g(inequalit)n(y) f(w)n(e)h(used)g(F)-7 b(atou's)25 b(lemma.)36 b(Eq.)26 b(\(2.10\))f(is)g(th)n(us)h(pro)n(v)n(ed.)82 b Ff(\003)555 2466 y Fr(W)-7 b(e)34 b(remark)d(that)j(this)f(result)g(is)g(in)g(con)n (trast)f(with)h(the)h(classical)e(di\013usion)h(theory)f(in)456 2566 y(whic)n(h)38 b(the)g(motion)g(of)g(the)h(c)n(harged)d(particle,)k (in)f(the)f(di\013usiv)n(e)g(space-time)g(scaling,)i(is)456 2665 y(describ)r(ed)g(as)g(the)h(sum)g(of)g(a)f(drift)h(term)g Fn(d)p Fr(\()p Fn(E)5 b Fr(\))41 b(plus)g(a)g(di\013usion)g(of)f (di\013usivit)n(y)h Fn(\033)s Fr(\()p Fn(E)5 b Fr(\).)456 2765 y(Moreo)n(v)n(er)37 b(b)r(oth)k Fn(d)p Fr(\()p Fn(E)5 b Fr(\))42 b(and)e Fn(\033)s Fr(\()p Fn(E)5 b Fr(\))42 b(are)d(con)n(tin)n(uous)h(functions)g(of)h Fn(E)k Fr(and)c(the)g (Einstein)456 2865 y(relation)26 b(holds,)i(i.e.)1615 3013 y(lim)1597 3066 y Fk(E)s Fj(!)p Fo(0)1772 2957 y Fn(d)p Fr(\()p Fn(E)5 b Fr(\))p 1772 2994 174 4 v 1806 3070 a Fn(q)s(E)1979 3013 y Fr(=)2076 2957 y Fn(\033)2126 2927 y Fo(2)2164 2957 y Fr(\(0\))p 2076 2994 194 4 v 2084 3070 a(2)p Fn(K)h(T)2280 3013 y(;)929 b Fr(\(2.11\))456 3169 y(where)40 b Fn(T)51 b Fr(is)41 b(the)g(absolute)e(temp)r(erature) i(and)f Fn(K)46 b Fr(the)41 b(Boltzmann)f(constan)n(t.)75 b(On)40 b(the)456 3269 y(con)n(trary)-7 b(,)25 b(from)j(\(2.10\))e(w)n (e)i(obtain)f(here)g(that:)1240 3404 y Fn(d)p Fr(\()p Fn(E)5 b Fr(\))23 b(=)70 b(lim)1524 3453 y Fk(t)p Fj(!)p Fo(+)p Fj(1)1732 3404 y Fm(h)p Fn(v)s Fr(\()p Fn(t)p Fr(\))p Fm(i)25 b Fr(=)e(+)p Fm(1)165 b(8)14 b Fn(E)27 b(>)c Fr(0)p Fn(:)456 3581 y Fv(Remark)43 b(2.1.)k Fr(By)39 b(exploiting)f(the)h(pro)r(of)g(of)g(Theorem)f(2.1)g(it)h(is)g(easy)e (to)i(c)n(hec)n(k)f(that)456 3681 y(w)n(e)f(actually)h(pro)n(v)n(e)e (also)h(the)h(follo)n(wing)f(statemen)n(t.)68 b(Assume)38 b(the)h(c)n(harged)d(particle)i(is)456 3780 y(initially)32 b(in)h(the)g(p)r(osition)g Fn(x)f Fr(=)f(0)h(with)h(a)g(p)r(ositiv)n(e) f(v)n(elo)r(cit)n(y)g Fn(v)2521 3792 y Fo(0)2558 3780 y Fr(.)52 b(Then)33 b(for)f(eac)n(h)g Fn(X)38 b Fm(2)32 b(X)456 3880 y Fr(and)g Fn(E)j(>)c Fr(0)g(there)h(exists)g(a)g (threshold)i(\026)-45 b Fn(v)1821 3892 y Fo(0)1891 3880 y Fr(suc)n(h)32 b(that)g(Eq.)g(\(2.8\))g(holds)g(for)f(an)n(y)h Fn(v)3181 3892 y Fo(0)3249 3880 y Fm(\025)h Fr(\026)-45 b Fn(v)3384 3892 y Fo(0)3421 3880 y Fr(.)456 3980 y(This)27 b(Hamiltonian)h(mo)r(del)f(is)h(th)n(us)f(an)h(example)f(of)g Fp(runaway)i Fr(c)n(harged)d(particle.)456 4126 y Fv(Remark)h(2.2.)37 b Fr(By)24 b(symmetry)-7 b(,)25 b(the)f(limit)i(on)e(the)g(left-hand)h (side)f(of)31 b(\(2.9\))24 b(is)g(non-p)r(ositiv)n(e)456 4226 y(in)j(the)g(case)f Fn(E)i Fm(\024)23 b Fr(0.)36 b(In)27 b(particular)e(the)j(gro)n(wth)d(of)i(the)g(v)n(elo)r(cit)n(y)f (is)h(sub-linear)f(in)h(time)g(for)456 4325 y Fn(E)i Fr(=)24 b(0)k(\(i.e.)39 b Fn(v)s Fr(\()p Fn(t)p Fr(\))p Fn(=t)24 b Fm(!)h Fr(0)j(as)f Fn(t)d Fm(!)h Fr(+)p Fm(1)p Fr(\).)39 b(As)28 b(w)n(e)g(shall)g(see)g(this)g(sub-linear)g(gro)n (wth)f(will)h(b)r(e)456 4425 y(the)g(k)n(ey)e(p)r(oin)n(t)i(in)g(the)g (pro)r(of.)456 4571 y Fv(Remark)33 b(2.3.)41 b Fr(It)30 b(is)f(clear)f(that)i(the)g(limit)g(\(2.10\))f(holds)g(not)g(only)g (for)g(Gibbs)h(states)f(but)456 4671 y(for)e(an)n(y)f(reasonable)g (equilibrium)i(or)e(non-equilibrium)i(thermo)r(dynamic)f(state.)456 4817 y Fv(Remark)k(2.4.)40 b Fr(Un)n(til)28 b(no)n(w)f(w)n(e)g(ha)n(v)n (e)g(discussed)g(a)g(one-dimensional)g(mo)r(del.)37 b(It)28 b(w)n(ould)f(b)r(e)456 4917 y(in)n(teresting)19 b(to)i(in)n(v)n (estigate)e(also)g(mo)r(dels)h(in)h(higher)f(dimensions,)i(in)e(whic)n (h)h(the)f(bac)n(kground)456 5016 y(is)27 b(comp)r(osed)f(b)n(y)h (in\014nitely)h(man)n(y)e(in)n(teracting)g(particles)h(in)g Fl(R)2507 4986 y Fo(2)2578 5016 y Fr(or)f(in)h Fl(R)2829 4986 y Fo(3)2872 5016 y Fr(,)h(but)f(this)h(seems)456 5116 y(to)35 b(b)r(e)i(a)e(prett)n(y)h(hard)f(task.)62 b(In)36 b(fact)g(in)g(this)h(case)e(w)n(e)g(cannot)h(mime)g(the)h(pro)r (of)e(of)h(the)456 5216 y(presen)n(t)25 b(pap)r(er,)h(essen)n(tially)g (b)r(ecause)f(w)n(e)h(cannot)g(con)n(trol)f(the)i(gro)n(wth)e(of)h(the) g(bac)n(kground)p eop %%Page: 5 5 5 4 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)308 b(5)456 450 y Fr(particles)27 b(v)n(elo)r(cit)n(y)g(in)h(a)g (sub-linear)f(w)n(a)n(y)-7 b(.)37 b(Actually)28 b(there)g(are)f(man)n (y)g(pap)r(ers)h(on)g(the)g(time)456 550 y(ev)n(olution)e(of)h (in\014nite)h(systems)f([1)o(,)h(11)o(,)f(12)o(,)g(14)o(,)g(15,)g(17)o (,)g(18)o(,)g(19)o(,)h(20)o(,)f(23)o(])g(but)h(the)g(estimates)456 649 y(b)r(ecome)h(quite)h(bad)g(for)f(v)n(ery)g(long)g(times)h(\(as)f (w)n(e)g(need)h(in)g(the)g(presen)n(t)g(pap)r(er\))f(when)h(the)456 749 y(space)c(dimension)i(is)f(larger)f(than)i(one.)555 849 y(It)k(is)f(not)g(hop)r(eless)f(to)h(extend)g(some)g(results)f(of)h (this)h(pap)r(er)e(to)h(t)n(w)n(o)f(di\013eren)n(t)i(mo)r(dels.)456 948 y(Firstly)-7 b(,)27 b(w)n(e)h(could)f(in)n(v)n(estigate)g(the)h(v)n (ery)f(simple)h(case)f(in)h(whic)n(h)f(the)i(bac)n(kground)c(is)j(com-) 456 1048 y(p)r(osed)20 b(b)n(y)g(free)g(particles)f(in)i Fl(R)1411 1018 y Fo(3)1474 1048 y Fr(distributed)g(according)e(to)h(a)g (Gibbs)g(measure.)34 b(The)20 b(c)n(harged)456 1147 y(particle)29 b(is)g(initially)h(at)f(rest)g(in)h(the)g(origin)e(and)i(in)n(teracts)e (with)i(a)f(constan)n(t)g(electric)g(\014eld)456 1247 y(and)g(with)i(the)f(bac)n(kground)e(particles)i(via)f(a)h(p)r(oten)n (tial)g(\011.)43 b(As)30 b(in)h(this)f(pap)r(er)g(w)n(e)f(assume)456 1347 y(\011)e(to)h(b)r(e)g(smo)r(oth,)g(non-negativ)n(e,)e(and)i(with)h (\014nite)f(range.)36 b(It)29 b(is)e(reasonable)f(to)i(obtain)g(an)456 1446 y(a)n(v)n(erage)21 b(drift)j(v)n(elo)r(cit)n(y)g(whic)n(h)g (increases)e(linearly)i(in)g(time.)36 b(Of)24 b(course)f(the)i(time)f (ev)n(olution)456 1546 y(of)36 b(the)g(bac)n(kground)e(is)i(trivial)f (but)i(the)f(set)g(of)g(initial)g(data)g(considered)f(in)h(the)g (presen)n(t)456 1646 y(pap)r(er)24 b(is)g(to)r(o)h(wide)g(to)f(exclude) h(big)f(densities)h(close)f(to)g(the)h(c)n(harged)e(particle.)36 b(W)-7 b(e)25 b(hop)r(e)g(to)456 1745 y(o)n(v)n(ercome)h(this)k (di\016cult)n(y)f(b)n(y)g(using)f(the)i(explicit)f(prop)r(erties)f(of)h (the)g(free)g(particles)f(Gibbs)456 1845 y(measure.)555 1944 y(Secondly)-7 b(,)21 b(w)n(e)e(could)g(try)g(to)g(relax)g(the)g (strict)h(one-dimensionalit)n(y)d(of)j(the)g(presen)n(t)e(mo)r(del.)456 2044 y(In)29 b(fact,)i(here)e(w)n(e)g(hea)n(vily)g(rely)g(on)g(the)h (fact)g(that)g(the)g(motion)f(tak)n(es)g(place)g(on)g(a)h(straigh)n(t) 456 2144 y(line)21 b(and)f(so)g(the)h(e\013ect)g(of)g(the)g(collision)f (b)r(et)n(w)n(een)g(t)n(w)n(o)g(particles)g(is)h(v)n(ery)e(small,)j(b)r (ecause)e(of)h(a)456 2243 y(comp)r(ensation)g(b)r(efore)h(and)f(after)h (the)g(sho)r(c)n(k.)35 b(Just)21 b(this)i(prop)r(ert)n(y)e(allo)n(ws)f (us)i(to)g(extend)g(the)456 2343 y(result)h(of)g([9])g(to)g(the)h(case) f(of)g(small)g(electric)g(\014elds.)35 b(By)23 b(using)g(a)g(sligh)n (tly)g(di\013eren)n(t)h(strategy)-7 b(,)456 2443 y(w)n(e)35 b(could)g(in)n(v)n(estigate)f(a)h(mo)r(del)g(in)h(whic)n(h)f(the)h (in\014nitely)f(man)n(y)g(particles)g(of)g(the)h(bac)n(k-)456 2542 y(ground)24 b(are)h(con\014ned)g(in)h Fl(R)1346 2512 y Fo(3)1415 2542 y Fr(close)f(to)g(the)h Fn(x)p Fr(-axis)f(b)n(y)g(a)g(non-negativ)n(e)f(t)n(wice)i(di\013eren)n (tiable)456 2642 y(external)d(p)r(oten)n(tial)g Fn(V)c Fr(\()p Fn(r)r Fr(\),)26 b(whic)n(h)e(is)g(de\014ned)g(in)g(the)g (follo)n(wing)f(w)n(a)n(y:)33 b(assuming)23 b(cylindrical)456 2742 y(co)r(ordinates)k(\()p Fn(x;)14 b(r)n(;)g(\022)r Fr(\),)32 b(there)d(exist)g Fn(r)1663 2754 y Fo(0)1727 2742 y Fn(>)c Fr(0)k(and)g(0)c Fn(<)h(\013)g Fm(\024)g Fr(1)j(suc)n(h)g(that)g Fn(V)19 b Fr(\()p Fn(r)r Fr(\))27 b(=)f(Const)p Fn(:)13 b(r)3396 2711 y Fk(\013)456 2841 y Fr(for)33 b(an)n(y)f Fn(r)k Fm(\025)c Fn(r)958 2853 y Fo(0)996 2841 y Fr(.)54 b(W)-7 b(e)34 b(ha)n(v)n(e)e(to)i(sho)n(w)e (that)i(the)g(v)n(elo)r(cities)f(of)g(the)h(bac)n(kground)d(particles) 456 2941 y(ma)n(y)26 b(gro)n(w)g(at)i(most)f(sub-linearly)g(in)g(time.) 555 3094 y(W)-7 b(e)32 b(conclude)g(the)g(section)g(with)g(a)f (notation)h(w)n(arning:)44 b(in)32 b(the)g(sequel,)g(if)h(not)f (further)456 3194 y(sp)r(eci\014ed,)f(w)n(e)f(shall)g(denote)g(b)n(y)g Fn(C)37 b Fr(a)30 b(generic)f(p)r(ositiv)n(e)h(constan)n(t)f(whose)h(n) n(umerical)g(v)-5 b(alue)456 3294 y(ma)n(y)28 b(c)n(hange)h(from)g (line)h(to)g(line)f(and)h(it)g(ma)n(y)f(p)r(ossibly)g(dep)r(end)h(only) f(on)h(the)g(in)n(teractions)456 3393 y(\010)d(and)h(\011.)749 3584 y(3.)42 b Fq(New)31 b(bounds)g(on)g(the)h(gr)n(o)n(wth)g(of)f(the) h(p)-6 b(ar)g(ticles)33 b(velocity)555 3733 y Fr(If)28 b Fn(E)33 b Fr(is)28 b(large)e(enough)h(\(w.r.t.)h(the)g(Gibbs)g (state\))g(the)g(limit)h(\(2.10\))e(has)g(b)r(een)h(pro)n(v)n(ed)e(in) 456 3833 y([9)o(].)41 b(The)29 b(pro)r(of)f(is)h(based)f(on)h(a)f (result,)h(\014rstly)g(pro)n(v)n(ed)e(in)i([10)o(],)h(on)e(the)h(gro)n (wth)f(in)h(time)h(of)456 3933 y(the)j(particle)g(v)n(elo)r(cit)n(y)-7 b(.)54 b(More)32 b(precisely)-7 b(,)34 b(it)g(can)f(b)r(e)h(pro)n(v)n (ed)e(that)h(in)h(one)f(dimension)g(the)456 4032 y(particle)26 b(v)n(elo)r(cit)n(y)h(of)g(an)g(in\014nite)h(system)f(ma)n(y)f(gro)n(w) g(at)h(most)g(linearly)f(in)i(time.)37 b(T)-7 b(o)27 b(pro)n(v)n(e)456 4132 y(Theorem)34 b(2.3)h(w)n(e)g(need)g(to)g(impro)n (v)n(e)f(this)i(result,)h(b)n(y)e(sho)n(wing)f(that)i(the)g(gro)n(wth)e (of)h(the)456 4232 y(bac)n(kground)25 b(particle)i(v)n(elo)r(cit)n(y)g (is)g(in)h(fact)g(sub-linear)f(in)g(time.)555 4331 y(The)e(argumen)n(t) f(leading)g(to)h(this)g(new)g(estimate)g(is)g(not)g(really)e (a\013ected)i(b)n(y)g(the)g(presence)456 4431 y(of)35 b(the)h(c)n(harged)e(particle.)61 b(Then,)38 b(to)d(simplify)i (notation)e(and)g(to)h(mak)n(e)f(more)f(clear)h(the)456 4530 y(global)22 b(strategy)-7 b(,)24 b(w)n(e)g(giv)n(e)f(a)g(detailed) h(pro)r(of)g(of)g(this)g(b)r(ound)h(in)f(the)g(case)f(when)i(the)f(c)n (harged)456 4630 y(particle)j(is)g(absen)n(t.)36 b(The)28 b(equations)f(of)g(motion)h(then)g(reads:)1115 4700 y Fh(8)1115 4775 y(<)1115 4925 y(:)1236 4784 y Fr(\177)-47 b Fn(x)1278 4796 y Fk(i)1306 4784 y Fr(\()p Fn(t)p Fr(\))23 b(=)g Fm(\000)1607 4705 y Fh(X)1590 4884 y Fk(j)s Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)1758 4784 y Fm(r)p Fr(\010\()p Fn(x)1966 4796 y Fk(i)1995 4784 y Fr(\()p Fn(t)p Fr(\))c Fm(\000)f Fn(x)2238 4796 y Fk(j)2273 4784 y Fr(\()p Fn(t)p Fr(\)\))p Fn(;)98 b(i)23 b Fm(2)g Fl(N)t Fn(;)1231 4973 y(X)7 b Fr(\(0\))22 b(=)h Fn(X)r(:)3274 4871 y Fr(\(3.1\))555 5116 y(W)-7 b(e)32 b(recall)e(t)n(w)n(o)h(basic)g(estimates)g(on)g(the) h(in\014nite)g(dynamics.)48 b(Let)31 b Fn(t)f Fm(7!)f Fn(X)7 b Fr(\()p Fn(t)p Fr(\),)33 b Fn(X)7 b Fr(\()p Fn(t)p Fr(\))29 b(=)456 5216 y Fm(f)p Fr(\()p Fn(x)577 5228 y Fk(i)604 5216 y Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(v)775 5228 y Fk(i)804 5216 y Fr(\()p Fn(t)p Fr(\)\))p Fm(g)972 5228 y Fk(i)p Fj(2)p Fi(N)1087 5216 y Fr(,)36 b(b)r(e)e(the)h(solution)e(to)h(Eqs.)g(\(3.1\).)56 b(F)-7 b(or)34 b(an)n(y)f(initial)i(data)e Fn(X)41 b Fm(2)34 b(X)47 b Fr(w)n(e)p eop %%Page: 6 6 6 5 bop 456 251 a Fo(6)709 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(ha)n(v)n(e:)548 593 y Fm(j)p Fn(v)611 605 y Fk(i)638 593 y Fr(\()p Fn(t)p Fr(\))p Fm(j)i(\024)f Fn(C)946 500 y Fh(h)985 518 y(p)p 1068 518 960 4 v 75 x Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))14 b(log\()p Fn(e)k Fr(+)g Fm(j)p Fn(x)1637 605 y Fk(i)1665 593 y Fm(j)h Fr(+)f Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2365 500 y Fh(i)2584 593 y Fm(8)14 b Fn(i)21 b Fm(2)j Fl(N)93 b Fm(8)14 b Fn(t)22 b Fm(\025)g Fr(0)p Fn(;)92 b Fr(\(3.2\))456 756 y(and,)27 b(for)g(an)n(y)g Fn(\026)c Fm(2)g Fl(R)34 b Fr(and)28 b Fn(R)23 b(>)g Fr(log\()p Fn(e)c Fr(+)f Fm(j)p Fn(\026)p Fm(j)p Fr(\),)534 895 y Fn(Q)p Fr(\()p Fn(X)7 b Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g(R)q Fr(\))23 b Fm(\024)f Fn(C)6 b(Q)p Fr(\()p Fn(X)h Fr(\))1417 828 y Fh(\002)1451 895 y Fn(R)20 b Fr(+)e(log\()p Fn(e)g Fr(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g(\(1)h(+)f Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)\))p Fn(t)2679 861 y Fo(2)2716 828 y Fh(\003)2930 895 y Fm(8)14 b Fn(t)22 b Fm(\025)h Fr(0)p Fn(:)78 b Fr(\(3.3\))555 1034 y(W)-7 b(e)36 b(brie\015y)g(discuss)f(these)h(estimates)f(in)h (the)g(App)r(endix.)62 b(The)36 b(inequalit)n(y)h(\(3.2\))e(has)456 1133 y(b)r(een)27 b(\014rstly)h(pro)n(v)n(ed)d(in)j([10)o(].)37 b(The)28 b(b)r(ound)f(\(3.3\))h(is)f(actually)g(an)g(easy)f (consequence)h(of)g(the)456 1233 y(same)e(pro)r(of)g(of)h(the)g (existence)f(of)h(the)g(dynamics,)g(but)g(it)g(has)f(not)h(explicitly)g (pro)n(v)n(ed)e(in)i(the)456 1332 y(quoted)h(pap)r(ers.)36 b(W)-7 b(e)28 b(no)n(w)f(state)h(the)g(new)f(result.)456 1484 y Fv(Theorem)38 b(3.1.)44 b Fp(Ther)l(e)37 b(exist)e Fn(K)40 b Fm(\025)34 b Fr(1)h Fp(and)i Fn(a)2006 1496 y Fo(0)2077 1484 y Fm(2)d Fr(\(0)p Fn(;)14 b Fr(1\))36 b Fp(such)g(that)g(for)h(any)f Fn(a)e Fm(2)h Fr(\(0)p Fn(;)14 b(a)3385 1496 y Fo(0)3421 1484 y Fr(])456 1584 y Fp(and)28 b Fn(X)h Fm(2)23 b(X)40 b Fp(the)28 b(fol)t(lowing)i (holds.)39 b(L)l(et)27 b Fn(t)c Fm(7!)g Fn(X)7 b Fr(\()p Fn(t)p Fr(\))23 b(=)g Fm(f)p Fr(\()p Fn(x)2323 1596 y Fk(i)2350 1584 y Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(v)2521 1596 y Fk(i)2550 1584 y Fr(\()p Fn(t)p Fr(\))p Fm(g)2686 1596 y Fk(i)p Fj(2)p Fi(N)2828 1584 y Fp(b)l(e)27 b(the)h(solution)f (to)456 1683 y(Eqs.)j Fr(\(3.1\))p Fp(.)39 b(Then,)30 b(for)h(any)f Fn(i)23 b Fm(2)g Fl(N)40 b Fp(and)30 b Fn(t)23 b Fm(\025)g Fr(0)p Fp(,)1143 1837 y Fm(j)p Fn(v)1206 1849 y Fk(i)1234 1837 y Fr(\()p Fn(t)p Fr(\))p Fm(j)h(\024)f Fn(a)1507 1803 y Fj(\000)p Fk(K)t Fo([1+)p Fk(Q)p Fo(\()p Fk(X)5 b Fo(\))1885 1778 y Fe(3)1917 1803 y Fo(])p Fk(=a)2006 1778 y Fe(2)2042 1762 y Fh(p)p 2125 1762 434 4 v 75 x Fr(log\()p Fn(e)19 b Fr(+)f Fm(j)p Fn(x)2475 1849 y Fk(i)2503 1837 y Fm(j)p Fr(\))h(+)f Fn(at:)517 b Fr(\(3.4\))555 1989 y(The)20 b(pro)r(of)g(is)f(rather)g(tec)n(hnical)h(so)f(let)h(us)g (\014rstly)g(giv)n(e)f(an)g(outline)h(of)g(it.)35 b(W)-7 b(e)20 b(demonstrate)456 2088 y(the)40 b(b)r(ound)f(\(3.4\))h(b)n(y)f (con)n(tradiction:)59 b(w)n(e)39 b(assume)g(that)h(at)f(time)h Fn(T)51 b Fr(large)38 b(enough)h(the)456 2188 y(v)n(elo)r(cit)n(y)g (absolute)g(v)-5 b(alue)40 b(of)g(a)f(particle)g(is)h(greater)e(than)i Fn(aT)51 b Fr(and)40 b(w)n(e)g(then)g(sho)n(w)f(this)456 2288 y(particle)20 b(do)r(es)h(not)g(c)n(hange)f(v)n(ery)g(m)n(uc)n(h)h (its)g(v)n(elo)r(cit)n(y)f(during)h(the)g(bac)n(kw)n(ard)f(motion,)i (so)e(that)456 2387 y(the)25 b(latter)f(is)h(initially)f(larger)f(than) i Fn(aT)9 b(=)p Fr(2.)35 b(F)-7 b(or)24 b Fn(T)36 b Fr(large)23 b(enough)h(this)h(fact)g(con)n(tradicts)e(the)456 2487 y(assumptions)i(on)h(the)g(initial)h(data.)36 b(T)-7 b(o)25 b(understand)h(the)h(general)d(idea)i(let)h(us)f(start)f(with)i (a)456 2587 y(particular)g(situation:)39 b(at)29 b(a)f(large)g(time)h Fn(T)40 b Fr(there)28 b(is)h(only)g(one)f(fast)h(particle)f(with)h(v)n (elo)r(cit)n(y)456 2686 y Fn(aT)12 b Fr(,)33 b(while)h(the)f(other)g (particles)f(ha)n(v)n(e)g(v)n(elo)r(cities)h(smaller)f(than)i Fn(aT)9 b(=)p Fr(4)31 b(during)i(the)h(whole)456 2786 y(time)24 b(in)n(terv)-5 b(al)23 b([0)p Fn(;)14 b(T)e Fr(].)35 b(Then,)24 b(the)h(a)n(v)n(erage)c(force)i(acting)g(on)h(the)g (fast)g(particle)f(is)h(v)n(ery)f(small)456 2885 y(for)28 b(t)n(w)n(o)h(reasons:)38 b(\014rstly)-7 b(,)30 b(the)f(fast)h (particle)e(in)n(teracts)h(with)g(a)g(slo)n(w)g(one)g(for)f(a)h(v)n (ery)f(short)456 2985 y(time)d(\(in)n(v)n(ersely)f(prop)r(ortional)f (to)i(the)g(v)n(elo)r(cit)n(y)f(gap\);)h(secondly)-7 b(,)25 b(there)g(is)g(a)f(comp)r(ensation)456 3085 y(e\013ect)d(in)g (the)g(action)f(of)h(the)g(forces)f(during)g(a)h(collision)f(of)g(t)n (w)n(o)g(particle)h(\(later)f(on)h(this)g(fact)g(is)456 3184 y(pro)n(v)n(ed)j(b)n(y)i(a)f(\\p)r(erturbativ)n(e)g(argumen)n (t"\).)35 b(The)26 b(fast)g(particle)f(th)n(us)h(returns)f(to)h(the)g (initial)456 3284 y(time)31 b(essen)n(tially)e(with)j(the)f(same)f(v)n (elo)r(cit)n(y)f Fn(aT)42 b Fr(and)30 b(this)h(fact)g(giv)n(es)f(an)g (absurd)g(for)g(large)456 3384 y Fn(T)12 b Fr(.)57 b(Of)35 b(course)e(the)j(assumption)e(that)h(the)g(fast)g(particle)f(is)h (alone)f(and)g(the)h(bac)n(kground)456 3483 y(do)r(es)24 b(not)h(increase)f(its)h(v)n(elo)r(cit)n(y)f(is)h(to)r(o)g(drastic.)35 b(W)-7 b(e)25 b(no)n(w)g(mak)n(e)f(an)h(essen)n(tial)f(observ)-5 b(ation:)456 3583 y(Eq.)32 b(\(3.3\))g(implies)g(that,)i(for)e(eac)n(h) g(time)h Fn(t)e Fm(2)h Fr([0)p Fn(;)14 b(T)e Fr(],)32 b(the)h(total)f(n)n(um)n(b)r(er)g(of)h(fast)f(particles)456 3682 y(whic)n(h)c(are)g(initially)h(in)g(a)g(region)e(of)i(order)f Fn(T)1937 3652 y Fo(2)2002 3682 y Fr(\(i.e.)h(whic)n(h)g(can)g(in)n (teract)f(with)h(the)h(tagged)456 3782 y(one\))24 b(do)r(es)g(not)g (dep)r(end)h(on)g Fn(T)35 b Fr(\(it)25 b(is)f(prop)r(ortional)f(to)h Fn(a)2256 3752 y Fj(\000)p Fo(2)2345 3782 y Fr(\).)37 b(F)-7 b(or)23 b Fn(T)36 b Fr(large)23 b(enough)h(this)g(fact)456 3882 y(imp)r(oses)j(that)h(there)f(exists)h(a)f(v)n(elo)r(cit)n(y)g (gap)g(b)r(et)n(w)n(een)g(the)h(fast)g(and)f(slo)n(w)g(particles.)36 b(Then)456 3981 y(w)n(e)22 b(can)h(\014nd)h(an)f Fn(")g Fr(small)g(enough)f(suc)n(h)h(that)h(in)f(the)h(in)n(terv)-5 b(al)22 b([\(1)10 b Fm(\000)g Fn(")p Fr(\))p Fn(T)d(;)14 b(T)e Fr(])21 b(the)i(bac)n(kground)456 4081 y(do)r(es)29 b(not)g(increase)f(v)n(ery)g(m)n(uc)n(h)h(its)g(v)n(elo)r(cit)n(y)g (and)g(the)h(fast)f(particles)f(remain)h(suc)n(h)g(\(as)g(w)n(e)456 4181 y(will)f(see)g(in)g(Lemma)g(3.1)g(b)r(elo)n(w,)g(the)h(con)n(trol) d(on)i(the)h(bac)n(kground)d(is)i(not)h(trivial\).)38 b(So)28 b(the)456 4280 y(e\013ect)33 b(of)g(the)g(bac)n(kground)e(on)i (the)g(fast)g(particles)f(is)h(small.)52 b(W)-7 b(e)33 b(m)n(ust)g(no)n(w)g(con)n(trol)e(the)456 4380 y(m)n(utual)f(in)n (teractions)f(among)g(the)h(fast)g(particles:)41 b(it)31 b(is)f(p)r(ossible)g(to)g(sho)n(w)f(that)h(eac)n(h)f(fast)456 4479 y(particle)34 b(after)h(some)f(collisions)g(either)g(remains)g (alone)g(\(and)i(so)e(it)h(do)r(es)g(not)g(c)n(hange)e(its)456 4579 y(v)n(elo)r(cit)n(y\))h(or)h(it)h(remains)e(in)i(a)f(small)g (cluster)g(\(in)h(momen)n(tum)f(space\),)i(whose)e(cen)n(ter)g(of)456 4679 y(mass)28 b(is)h(unc)n(hanged.)41 b(In)30 b(conclusion,)e(the)i(v) n(elo)r(cit)n(y)e(of)i(eac)n(h)e(fast)h(particle)g(in)g(the)h(in)n (terv)-5 b(al)456 4778 y([\(1)18 b Fm(\000)f Fn(")p Fr(\))p Fn(T)7 b(;)14 b(T)e Fr(])26 b(is)i(almost)e(unc)n(hanged.)36 b(Rep)r(eating)28 b Fn(")2158 4748 y Fj(\000)p Fo(1)2274 4778 y Fr(times)g(this)g(estimate,)f(w)n(e)g(arriv)n(e)f(to)456 4878 y(an)h(absurd.)555 4978 y(W)-7 b(e)28 b(no)n(w)f(giv)n(e)g(the)h (rigorous)d(pro)r(of)i(of)h(the)g(ab)r(o)n(v)n(e)e(heuristic)h(argumen) n(t.)456 5116 y Fp(Pr)l(o)l(of)35 b(of)g(The)l(or)l(em)g(3.1.)53 b Fr(By)33 b(\(3.2\))f(the)h(pro)r(of)f(of)38 b(\(3.4\))32 b(reduces)g(to)g(pro)n(v)n(e)f(the)i(follo)n(wing)456 5216 y(statemen)n(t:)42 b(there)30 b(are)g Fn(K)j Fm(\025)28 b Fr(1)i(and)g Fn(a)1713 5228 y Fo(0)1778 5216 y Fm(2)e Fr(\(0)p Fn(;)14 b Fr(1\))30 b(suc)n(h)g(that,)i(for)e(an)n(y)g Fn(a)d Fm(2)h Fr(\(0)p Fn(;)14 b(a)3072 5228 y Fo(0)3109 5216 y Fr(],)32 b Fn(X)i Fm(2)28 b(X)p eop %%Page: 7 7 7 6 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)308 b(7)456 450 y Fr(and)27 b Fn(i)c Fm(2)g Fl(N)t Fr(,)1391 580 y Fm(j)p Fn(v)1454 592 y Fk(i)1482 580 y Fr(\()p Fn(t)p Fr(\))p Fm(j)g(\024)g Fn(at)166 b Fm(8)14 b Fn(t)22 b Fm(\025)g Fn(T)2199 592 y Fk(a)2239 580 y Fr(\()p Fn(X)r(;)14 b(x)2426 592 y Fk(i)2454 580 y Fr(\))p Fn(;)765 b Fr(\(3.5\))456 727 y(where,)27 b(for)g Fn(w)f Fm(2)d Fl(R)p Fr(,)1181 903 y Fn(T)1230 915 y Fk(a)1269 903 y Fr(\()p Fn(X)r(;)14 b(w)r Fr(\))1547 856 y Fn(:)1526 903 y Fr(=)23 b Fn(a)1658 869 y Fj(\000)p Fk(K)t Fo([1+)p Fk(Q)p Fo(\()p Fk(X)5 b Fo(\))2036 844 y Fe(3)2068 869 y Fo(])p Fk(=a)2157 844 y Fe(2)2193 828 y Fh(p)p 2276 828 420 4 v 75 x Fr(log)q(\()p Fn(e)18 b Fr(+)g Fm(j)p Fn(w)r Fm(j)p Fr(\))q Fn(:)555 b Fr(\(3.6\))555 1068 y(Let)25 b(us)f(pro)n(v)n(e)f(\(3.5\))o(.)36 b(In)25 b(the)g(sequel)f(w)n(e)g(alw)n(a)n(ys)e(assume)i Fn(a)f Fm(2)g Fr(\(0)p Fn(;)14 b(a)2704 1080 y Fo(0)2741 1068 y Fr(])24 b(and)h(the)f(statemen)n(t)456 1168 y Fn(a)500 1180 y Fo(0)572 1168 y Fp(smal)t(l)37 b(enough)f Fr(means)e(that)h(it)h(can)e(b)r(e)i(\014xed)f(small)f(enough)h(indep)r (enden)n(tly)g(of)g Fn(X)42 b Fm(2)456 1267 y(X)50 b Fr(and)37 b Fn(i)i Fm(2)h Fl(N)t Fr(.)73 b(Analogously)-7 b(,)38 b(the)g(parameter)e Fn(K)45 b Fm(\025)40 b Fr(1)d(will)g(b)r(e)h (c)n(hosen)f(large)f(enough)456 1367 y(indep)r(enden)n(tly)28 b(of)f Fn(X)j Fm(2)23 b(X)40 b Fr(and)28 b Fn(i)22 b Fm(2)i Fl(N)t Fr(.)555 1466 y(W)-7 b(e)28 b(\014rst)g(assume)e Fn(a)1200 1478 y Fo(0)1265 1466 y Fr(small)h(enough)g(that,)h(for)f (all)h Fn(K)g Fm(\025)23 b Fr(1,)909 1599 y Fh(p)p 992 1599 640 4 v 76 x Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))14 b(log)o(\()p Fn(e)19 b Fr(+)f Fm(j)p Fn(w)r Fm(j)p Fr(\))24 b Fn(<)1752 1618 y(aT)1845 1630 y Fk(a)1884 1618 y Fr(\()p Fn(X)r(;)14 b(w)r Fr(\))p 1752 1655 367 4 v 1914 1731 a(8)2294 1675 y Fm(8)g Fn(w)25 b Fm(2)e Fl(R)89 b Fm(8)14 b Fn(X)28 b Fm(2)c(X)12 b Fn(:)456 1867 y Fr(This)27 b(implies,)h(b)n(y)f(the)h (de\014nition)g(\(2.3\),)1223 2080 y Fm(j)p Fn(v)1286 2092 y Fk(i)1314 2080 y Fm(j)23 b Fn(<)1457 2024 y(aT)1550 2036 y Fk(a)1590 2024 y Fr(\()p Fn(X)r(;)14 b(x)1777 2036 y Fk(i)1805 2024 y Fr(\))p 1457 2061 380 4 v 1626 2137 a(4)2013 2080 y Fm(8)g Fn(i)22 b Fm(2)h Fl(N)93 b Fm(8)14 b Fn(X)28 b Fm(2)c(X)12 b Fn(:)597 b Fr(\(3.7\))456 2273 y(W)-7 b(e)29 b(next)h(pro)r(ceed)e(b)n(y)h(con)n(tradiction.)41 b(W)-7 b(e)29 b(assume)g(there)g(are)f Fn(i)e Fm(2)g Fl(N)39 b Fr(and)29 b Fn(T)37 b(>)25 b(T)3157 2285 y Fk(a)3197 2273 y Fr(\()p Fn(X)r(;)14 b(x)3384 2285 y Fk(i)3412 2273 y Fr(\))456 2372 y(suc)n(h)27 b(that)i Fm(j)p Fn(v)887 2384 y Fk(i)915 2372 y Fr(\()p Fn(T)12 b Fr(\))p Fm(j)23 b Fn(>)h(aT)12 b Fr(.)38 b(W)-7 b(e)28 b(shall)g(pro)n(v)n(e)e(that,)j(for)f Fn(a)2279 2384 y Fo(0)2344 2372 y Fr(small)g(enough)f(and)h Fn(K)i Fm(\025)23 b Fr(1)28 b(large)456 2472 y(enough,)37 b(this)f(implies)g Fm(j)p Fn(v)1298 2484 y Fk(i)1326 2472 y Fr(\(0\))p Fm(j)g Fn(>)h(aT)9 b(=)p Fr(4)34 b(for)h(all)h Fn(a)g Fm(2)h Fr(\(0)p Fn(;)14 b(a)2399 2484 y Fo(0)2436 2472 y Fr(].)61 b(By)37 b(\(3.7\))e(w)n(e)h(th)n(us)f(get)h(an)456 2571 y(absurd)26 b(since)i Fn(v)972 2583 y Fk(i)1000 2571 y Fr(\(0\))23 b(=)f Fn(v)1256 2583 y Fk(i)1312 2571 y Fr(and)27 b Fn(T)35 b(>)22 b(T)1693 2583 y Fk(a)1733 2571 y Fr(\()p Fn(X)r(;)14 b(x)1920 2583 y Fk(i)1948 2571 y Fr(\).)555 2704 y(By)27 b(\(3.2\),)f(the)g Fn(j)5 b Fr(-th)26 b(particle)f(ma)n(y)g(en)n(ter)g(in)n(to)h(the)g(in)n(terv) -5 b(al)25 b([)p Fn(x)2577 2716 y Fk(i)2620 2704 y Fm(\000)15 b Fn(L;)f(x)2841 2716 y Fk(i)2883 2704 y Fr(+)g Fn(L)p Fr(])26 b(during)f(the)456 2803 y(time)j(in)n(terv)-5 b(al)27 b([0)p Fn(;)14 b(t)p Fr(])27 b(only)g(if:)853 3018 y Fm(j)p Fn(x)923 3030 y Fk(j)977 3018 y Fm(\000)18 b Fn(x)1107 3030 y Fk(i)1135 3018 y Fm(j)23 b(\024)g Fn(L)18 b Fr(+)g Fn(C)6 b(t)1536 2901 y Fh(\024)1580 2920 y(q)p 1663 2920 967 4 v 98 x Fn(Q)p Fr(\()p Fn(X)h Fr(\))14 b(log)o(\()p Fn(e)19 b Fr(+)f Fm(j)p Fn(x)2232 3030 y Fk(j)2267 3018 y Fm(j)h Fr(+)f Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2967 2901 y Fh(\025)3024 3018 y Fn(;)227 b Fr(\(3.8\))456 3228 y(whic)n(h)27 b(implies:)831 3393 y Fm(j)p Fn(x)901 3405 y Fk(j)936 3393 y Fm(j)19 b(\000)f Fr(log\()p Fn(e)g Fr(+)g Fm(j)p Fn(x)1410 3405 y Fk(j)1446 3393 y Fm(j)p Fr(\))23 b Fm(\024)g Fn(C)1691 3326 y Fh(\002)1726 3393 y Fn(L)18 b Fr(+)g Fm(j)p Fn(x)1954 3405 y Fk(i)1982 3393 y Fm(j)g Fr(+)g(log\()p Fn(e)h Fr(+)f Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2961 3359 y Fo(2)2998 3326 y Fh(\003)3046 3393 y Fn(;)456 3558 y Fr(whence:)1102 3688 y Fm(j)p Fn(x)1172 3700 y Fk(j)1207 3688 y Fm(j)23 b(\024)g Fn(C)1420 3620 y Fh(\002)1455 3688 y Fn(L)18 b Fr(+)g Fm(j)p Fn(x)1683 3700 y Fk(i)1711 3688 y Fm(j)g Fr(+)g(log\()p Fn(e)h Fr(+)f Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2690 3653 y Fo(2)2727 3620 y Fh(\003)2775 3688 y Fn(:)456 3835 y Fr(Inserting)26 b(the)h(last)g(b)r(ound)g(in)g(\(3.8\))g(w)n(e)f(conclude)h(that,)g (for)g(an)n(y)f Fn(L)c Fm(\025)h Fr(1,)k(the)g Fn(j)5 b Fr(-th)27 b(particle)456 3935 y(ma)n(y)f(en)n(ter)i(in)n(to)f(the)h (in)n(terv)-5 b(al)27 b([)p Fn(x)1527 3947 y Fk(i)1574 3935 y Fm(\000)18 b Fn(L;)c(x)1798 3947 y Fk(i)1844 3935 y Fr(+)k Fn(L)p Fr(])27 b(during)g(the)h(time)g(in)n(terv)-5 b(al)27 b([0)p Fn(;)14 b(t)p Fr(])27 b(only)g(if)622 4101 y Fm(j)p Fn(x)692 4113 y Fk(j)746 4101 y Fm(\000)18 b Fn(x)876 4113 y Fk(i)904 4101 y Fm(j)24 b(\024)e Fn(L)c Fr(+)g(log\()p Fn(e)h Fr(+)f Fn(L)p Fr(\))g(+)g(log\()p Fn(e)h Fr(+)f Fm(j)p Fn(x)2016 4113 y Fk(i)2044 4101 y Fm(j)p Fr(\))h(+)f Fn(C)2280 4033 y Fh(\002)2314 4101 y Fr(log)q(\()p Fn(e)g Fr(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p Fn(t)3169 4066 y Fo(2)3206 4033 y Fh(\003)3255 4101 y Fn(;)456 4265 y Fr(so)27 b(that,)g(recalling)g(the)h(de\014nition)g(\(3.6\))o(,) 554 4431 y(inf)497 4489 y Fk(s)p Fj(2)p Fo([0)p Fk(;T)9 b Fo(])726 4431 y Fm(j)p Fn(x)796 4443 y Fk(j)831 4431 y Fr(\()p Fn(s)p Fr(\))16 b Fm(\000)f Fn(x)1077 4443 y Fk(i)1105 4431 y Fm(j)23 b(\024)f Fn(L)106 b Fr(=)-14 b Fm(\))106 b(j)p Fn(x)1711 4443 y Fk(j)1761 4431 y Fm(\000)15 b Fn(x)1888 4443 y Fk(i)1916 4431 y Fm(j)23 b(\024)g Fn(L)15 b Fr(+)g(log\()p Fn(e)g Fr(+)g Fn(L)p Fr(\))g(+)f Fn(C)6 b Fr([1)16 b(+)e Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)3173 4397 y Fo(2)3209 4431 y Fn(:)42 b Fr(\(3.9\))456 4647 y(Moreo)n(v)n(er,)25 b(b)n(y)j(\(3.2\),)1334 4813 y(sup)1290 4887 y Fk(s)p Fj(2)p Fo([0)p Fk(;T)9 b Fo(])1518 4813 y Fm(j)p Fn(x)1588 4825 y Fk(i)1616 4813 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)1868 4825 y Fk(i)1896 4813 y Fm(j)23 b(\024)g Fn(C)6 b Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)2551 4779 y Fo(2)2587 4813 y Fn(:)456 5051 y Fr(By)28 b(\(3.9\))f(and)h(the)g(last)f (inequalit)n(y)-7 b(,)27 b(setting)h Fn(L)1978 5063 y Fk(T)2053 5051 y Fr(=)22 b Fn(C)2199 5063 y Fj(\003)2238 5051 y Fr([1)c(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)2694 5021 y Fo(2)2758 5051 y Fr(and)27 b(de\014ning:)686 5216 y Fm(P)795 5169 y Fn(:)774 5216 y Fr(=)c Fm(f)o Fn(j)28 b Fm(2)23 b Fl(N)47 b Fr(:)37 b Fm(j)p Fn(x)1270 5228 y Fk(j)1324 5216 y Fm(\000)18 b Fn(x)1454 5228 y Fk(i)1482 5216 y Fm(j)23 b(\024)g Fr(4)p Fn(L)1715 5228 y Fk(T)1766 5216 y Fm(g)14 b Fn(;)180 b Fm(P)2083 5228 y Fo(1)2164 5169 y Fn(:)2143 5216 y Fr(=)22 b Fm(f)p Fn(j)28 b Fm(2)23 b Fl(N)47 b Fr(:)37 b Fm(j)p Fn(x)2639 5228 y Fk(j)2693 5216 y Fm(\000)18 b Fn(x)2823 5228 y Fk(i)2851 5216 y Fm(j)23 b(\024)g Fr(6)p Fn(L)3084 5228 y Fk(T)3135 5216 y Fm(g)14 b Fn(;)p eop %%Page: 8 8 8 7 bop 456 251 a Fo(8)709 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(w)n(e)27 b(can)g(c)n(ho)r(ose)f Fn(C)1052 462 y Fj(\003)1114 450 y Fm(\025)d Fr(1)k(so)g(large)f(that)i(the)g (follo)n(wing)e(holds:)1249 612 y(sup)1204 685 y Fk(s)p Fj(2)p Fo([0)p Fk(;T)9 b Fo(])1432 612 y Fm(j)p Fn(x)1502 624 y Fk(i)1530 612 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)1782 624 y Fk(i)1810 612 y Fm(j)24 b(\024)e Fn(L)2001 624 y Fk(T)2053 612 y Fn(;)1156 b Fr(\(3.10\))1261 804 y(inf)1204 861 y Fk(s)p Fj(2)p Fo([0)p Fk(;T)9 b Fo(])1432 804 y Fm(j)p Fn(x)1502 816 y Fk(j)1538 804 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)1790 816 y Fk(i)1818 804 y Fm(j)23 b(\024)g Fr(3)p Fn(L)2051 816 y Fk(T)2208 804 y Fr(=)-14 b Fm(\))106 b Fn(j)28 b Fm(2)23 b(P)7 b Fn(;)556 b Fr(\(3.11\))1261 979 y(inf)1204 1037 y Fk(s)p Fj(2)p Fo([0)p Fk(;T)9 b Fo(])1432 979 y Fr(min)1437 1033 y Fk(r)r Fj(2P)1584 979 y Fm(j)p Fn(x)1654 991 y Fk(j)1690 979 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)1942 991 y Fk(r)1979 979 y Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\024)e Fr(1)106 b(=)-14 b Fm(\))106 b Fn(j)28 b Fm(2)23 b(P)2802 991 y Fo(1)2839 979 y Fn(:)370 b Fr(\(3.12\))456 1198 y(\(to)26 b(get)h(\(3.12\))f(\014rst)g(observ)n(e)f(that)i Fm(j)p Fn(x)1679 1210 y Fk(r)1717 1198 y Fr(\()p Fn(s)p Fr(\))17 b Fm(\000)f Fn(x)1965 1210 y Fk(i)1993 1198 y Fm(j)23 b(\024)g Fr(4)p Fn(L)2226 1210 y Fk(T)2294 1198 y Fr(+)16 b(log)q(\()p Fn(e)g Fr(+)h Fn(L)2709 1210 y Fk(T)2760 1198 y Fr(\))g(+)g Fn(C)6 b Fr([1)16 b(+)h Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)3409 1167 y Fo(2)456 1297 y Fr(for)30 b(all)h Fn(s)e Fm(2)h Fr([0)p Fn(;)14 b(T)e Fr(])30 b(and)h Fn(r)h Fm(2)d(P)7 b Fr(,)32 b(and)f(then)h(use)f (\(3.9\))o(\).)49 b(In)31 b(particular)f Fm(P)38 b Fr(con)n(tains)30 b(all)h(the)456 1397 y(particles)h(whic)n(h)g(can)h(in)n(teract)f(with) h(the)g Fn(i)p Fr(-th)g(particle)f(during)g(the)h(time)h(in)n(terv)-5 b(al)32 b([0)p Fn(;)14 b(T)e Fr(],)456 1496 y(and)35 b Fm(P)683 1508 y Fo(1)756 1496 y Fr(con)n(tains)g(all)h(the)h (particles)e(whic)n(h)h(can)f(in)n(teract)h(with)g(those)g(in)g Fm(P)43 b Fr(during)35 b(the)456 1596 y(same)25 b(time)i(in)n(terv)-5 b(al.)36 b(Moreo)n(v)n(er,)24 b(b)n(y)j(\(3.2\))f(and)g(the)g (de\014nition)h(\(3.6\),)f(there)g(is)g Fn(\032)d Fm(\025)g Fr(1)j(suc)n(h)456 1696 y(that:)1326 1822 y(max)1326 1876 y Fk(j)s Fj(2P)1448 1884 y Fe(1)1539 1822 y Fr(sup)1494 1896 y Fk(s)p Fj(2)p Fo([0)p Fk(;T)9 b Fo(])1723 1822 y Fm(j)p Fn(v)1786 1834 y Fk(j)1821 1822 y Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\024)e Fn(\032)p Fr([1)c(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)g(:)657 b Fr(\(3.13\))456 2029 y(W)-7 b(e)28 b(next)f(de\014ne)h(the)g(subset)g(of)f(the)h Fp(fast)g Fr(particles,)1413 2236 y Fm(J)1469 2248 y Fk(t)1543 2189 y Fn(:)1522 2236 y Fr(=)1610 2119 y Fh(\032)1672 2236 y Fn(j)g Fm(2)23 b(P)44 b Fr(:)37 b Fm(j)p Fn(v)2037 2248 y Fk(j)2072 2236 y Fr(\()p Fn(t)p Fr(\))p Fm(j)24 b Fn(>)2310 2180 y(aT)p 2310 2217 104 4 v 2342 2293 a Fr(4)2424 2119 y Fh(\033)3232 2236 y Fr(\(3.14\))456 2448 y(\(clearly)j Fn(i)22 b Fm(2)i(J)940 2460 y Fk(T)993 2448 y Fr(\).)38 b(By)27 b(the)h(energy)f(conserv)-5 b(ation)26 b(these)i(particles)f(are)g(v)n(ery)f(rare)h(for)g(large)456 2547 y Fn(T)12 b Fr(.)36 b(In)27 b(fact,)h(b)n(y)h(\(2.2\))o(,)f(for)f (an)n(y)g Fn(t)c Fm(2)g Fr([0)p Fn(;)14 b(T)e Fr(],)755 2762 y Fm(jJ)834 2774 y Fk(t)863 2762 y Fm(j)23 b(\024)1055 2706 y Fr(32)p 1007 2743 179 4 v 1007 2819 a Fn(a)1051 2795 y Fo(2)1088 2819 y Fn(T)1149 2795 y Fo(2)1213 2683 y Fh(X)1209 2861 y Fk(j)s Fj(2P)1361 2706 y Fn(v)1401 2718 y Fk(j)1436 2706 y Fr(\()p Fn(t)p Fr(\))1530 2676 y Fo(2)p 1361 2743 208 4 v 1443 2819 a Fr(2)1601 2762 y Fm(\024)1746 2706 y Fr(32)p 1698 2743 179 4 v 1698 2819 a Fn(a)1742 2795 y Fo(2)1779 2819 y Fn(T)1840 2795 y Fo(2)1887 2762 y Fn(Q)p Fr(\()p Fn(X)7 b Fr(\()p Fn(t)p Fr(\);)14 b Fn(x)2239 2774 y Fk(i)2266 2762 y Fn(;)g Fr(4)p Fn(L)2402 2774 y Fk(T)2472 2762 y Fr(+)k Fn(\032)p Fr([1)g(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)3054 2728 y Fo(2)3090 2762 y Fr(\))p Fn(;)456 3014 y Fr(where)35 b(w)n(e)h(also)g(used)g(\(3.13\))o(.)63 b(Then,)39 b(b)n(y)e(\(3.3\))f (and)g(the)h(de\014nition)f(\(3.6\),)j(there)d(exists)456 3113 y Fn(\032)499 3125 y Fj(\003)560 3113 y Fm(\025)22 b Fr(1)28 b(suc)n(h)f(that:)1574 3289 y Fm(jJ)1653 3301 y Fk(t)1682 3289 y Fm(j)d Fn(<)e(\032)1859 3301 y Fj(\003)1907 3232 y Fr(1)c(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2256 3202 y Fo(2)p 1907 3270 386 4 v 2060 3346 a Fn(a)2104 3322 y Fo(2)2303 3289 y Fn(:)906 b Fr(\(3.15\))456 3461 y(F)-7 b(rom)32 b(\(3.15\))h(it)g(follo)n(ws)f(the)i(existence)f(of)g (gaps)f(in)h(the)h(spatial)f(and)g(momen)n(tum)g(distri-)456 3561 y(bution)f(of)f(fast)h(particles.)48 b(More)31 b(precisely)-7 b(,)31 b(there)h(exist)f Fn(y)2402 3526 y Fj(\000)2399 3582 y Fk(t)2488 3561 y Fm(2)f Fr([)p Fn(x)2643 3573 y Fk(i)2692 3561 y Fm(\000)21 b Fr(2)p Fn(L)2877 3573 y Fk(T)2928 3561 y Fn(;)14 b(x)3012 3573 y Fk(i)3061 3561 y Fm(\000)21 b Fn(L)3204 3573 y Fk(T)3256 3561 y Fr(])32 b(and)456 3662 y Fn(y)500 3626 y Fo(+)497 3682 y Fk(t)577 3662 y Fm(2)24 b Fr([)p Fn(x)726 3674 y Fk(i)769 3662 y Fr(+)14 b Fn(L)905 3674 y Fk(T)957 3662 y Fn(;)g(x)1041 3674 y Fk(i)1084 3662 y Fr(+)g(2)p Fn(L)1262 3674 y Fk(T)1314 3662 y Fr(],)26 b(suc)n(h)f(that)i(there)e(are)g(no)g(particles)g(in)h Fm(J)2692 3674 y Fk(t)2748 3662 y Fr(whose)f(p)r(ositions)g(\(at)456 3761 y(time)j Fn(t)p Fr(\))g(lie)f(in)h(the)g(set:)965 3859 y Fh(\024)1009 3976 y Fn(y)1053 3941 y Fj(\000)1050 3997 y Fk(t)1127 3976 y Fm(\000)1320 3920 y Fn(C)1379 3932 y Fj(\003)1418 3920 y Fn(a)1462 3890 y Fo(2)1499 3920 y Fn(T)1560 3890 y Fo(2)p 1220 3957 476 4 v 1220 4033 a Fn(\032)1263 4045 y Fj(\003)1301 4033 y Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])1706 3976 y Fn(;)14 b(y)1787 3941 y Fj(\000)1784 3997 y Fk(t)1843 3859 y Fh(\025)1905 3976 y Fm([)1979 3859 y Fh(\024)2022 3976 y Fn(y)2066 3941 y Fo(+)2063 3997 y Fk(t)2121 3976 y Fn(;)g(y)2202 3941 y Fo(+)2199 3997 y Fk(t)2275 3976 y Fr(+)2468 3920 y Fn(C)2527 3932 y Fj(\003)2566 3920 y Fn(a)2610 3890 y Fo(2)2647 3920 y Fn(T)2708 3890 y Fo(2)p 2368 3957 V 2368 4033 a Fn(\032)2411 4045 y Fj(\003)2449 4033 y Fr([1)k(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])2854 3859 y Fh(\025)2912 3976 y Fn(:)456 4197 y Fr(W)-7 b(e)22 b(p)r(oin)n(t)h(out)f(that,)i(b)n(y)f(\(3.10\))o(,)h Fn(x)1563 4209 y Fk(i)1591 4197 y Fr(\()p Fn(t)p Fr(\))g Fm(2)f Fr(\()p Fn(y)1863 4161 y Fj(\000)1860 4217 y Fk(t)1919 4197 y Fn(;)14 b(y)2000 4161 y Fo(+)1997 4217 y Fk(t)2055 4197 y Fr(\))23 b(for)e(all)h Fn(t)i Fm(2)f Fr([0)p Fn(;)14 b(T)e Fr(].)34 b(Analogously)20 b(there)j(is)456 4297 y Fn(\015)499 4309 y Fk(t)551 4297 y Fm(2)g Fr(\(1)p Fn(=)p Fr(4)p Fn(;)14 b Fr(1)p Fn(=)p Fr(2\))25 b(suc)n(h)h(that)h (there)f(are)g(no)h(particles)f(in)g Fm(P)34 b Fr(with)27 b(v)n(elo)r(cities)f(whose)g(absolute)456 4396 y(v)-5 b(alue)27 b(\(at)h(time)g Fn(t)p Fr(\))g(lies)f(in)h(the)g(in)n(terv)-5 b(al:)1358 4494 y Fh(\024)1401 4611 y Fn(\015)1444 4623 y Fk(t)1474 4611 y Fn(aT)7 b(;)14 b(\015)1654 4623 y Fk(t)1682 4611 y Fn(aT)29 b Fr(+)2104 4555 y Fn(a)2148 4525 y Fo(3)2185 4555 y Fn(T)p 1897 4592 555 4 v 1897 4668 a Fr(4)p Fn(\032)1982 4680 y Fj(\003)2020 4668 y Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2392 4644 y Fo(2)2429 4668 y Fr(])2462 4494 y Fh(\025)2519 4611 y Fn(:)690 b Fr(\(3.16\))555 4828 y(No)n(w)38 b(let)h Fn(")925 4798 y Fj(\000)p Fo(1)1052 4828 y Fr(b)r(e)g(a)e(p)r(ositiv)n (e)h(in)n(teger)g(to)g(b)r(e)h(\014xed)f(later,)i(and)e(set)h Fn(T)2898 4840 y Fk(k)3000 4781 y Fn(:)2979 4828 y Fr(=)h Fn(k)s("T)12 b Fr(,)40 b Fn(k)k Fr(=)456 4928 y(0)p Fn(;)14 b(:)g(:)g(:)f(;)h(")721 4897 y Fj(\000)p Fo(1)809 4928 y Fr(.)37 b(W)-7 b(e)28 b(next)g(de\014ne,)g(for)f(all)g Fn(k)f Fr(=)d(0)p Fn(;)14 b(:)g(:)g(:)f(;)h(")2126 4897 y Fj(\000)p Fo(1)2215 4928 y Fr(,)805 5089 y Fm(V)856 5101 y Fk(k)1001 5042 y Fn(:)980 5089 y Fr(=)1128 5022 y Fh(\010)1176 5089 y Fn(j)28 b Fm(2)23 b(P)44 b Fr(:)37 b Fm(j)p Fn(v)1541 5101 y Fk(j)1576 5089 y Fr(\()p Fn(T)1657 5101 y Fk(k)1698 5089 y Fr(\))p Fm(j)23 b Fn(>)g(\015)1907 5101 y Fk(k)1948 5089 y Fn(aT)7 b(;)96 b(y)2211 5054 y Fj(\000)2208 5114 y Fk(k)2290 5089 y Fm(\024)22 b Fn(x)2424 5101 y Fk(j)2460 5089 y Fr(\()p Fn(T)2541 5101 y Fk(k)2581 5089 y Fr(\))i Fm(\024)e Fn(y)2768 5054 y Fo(+)2765 5114 y Fk(k)2823 5022 y Fh(\011)2885 5089 y Fn(;)324 b Fr(\(3.17\))756 5216 y Fm(M)856 5228 y Fk(k)1001 5169 y Fn(:)980 5216 y Fr(=)83 b Fm(f)o Fn(j)28 b Fm(2)23 b(P)44 b Fr(:)37 b Fm(j)p Fn(v)1534 5228 y Fk(j)1569 5216 y Fr(\()p Fn(T)1650 5228 y Fk(k)1691 5216 y Fr(\))p Fm(j)23 b(\024)g Fn(\015)1900 5228 y Fk(k)1941 5216 y Fn(aT)12 b Fm(g)g Fn(;)1109 b Fr(\(3.18\))p eop %%Page: 9 9 9 8 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)308 b(9)456 450 y Fr(where)32 b(w)n(e)g(shorthanded)g Fn(y)1353 415 y Fj(\006)1350 475 y Fk(T)1389 484 y Fg(k)1460 450 y Fr(=)f Fn(y)1600 415 y Fj(\006)1597 475 y Fk(k)1689 450 y Fr(and)h Fn(\015)1898 462 y Fk(T)1937 471 y Fg(k)2009 450 y Fr(=)f Fn(\015)2148 462 y Fk(k)2222 450 y Fr(\(observ)n(e)g(also) g(that)i Fn(i)e Fm(2)h(V)3109 466 y Fk(")3140 449 y Fd(\000)p Fe(1)3222 450 y Fr(\).)52 b(W)-7 b(e)456 554 y(note)27 b(that,)h(b)n(y)h(\(3.13\))o(,)e(c)n(ho)r(osing)g(e.g.)1490 761 y Fn(")1529 726 y Fj(\000)p Fo(1)1641 761 y Fm(\025)1738 704 y Fr(10)p Fn(\032\032)1908 716 y Fj(\003)1945 704 y Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])2340 674 y Fo(2)p 1738 742 639 4 v 1968 818 a Fn(C)2027 830 y Fj(\003)2066 818 y Fn(a)2110 794 y Fo(2)2387 761 y Fn(;)822 b Fr(\(3.19\))456 955 y(the)35 b(particles)f(in)h Fm(J)1106 967 y Fk(T)1145 976 y Fg(k)1221 955 y Fr(can)f(\014ll)i(up)f (during)f(the)h(time)h(in)n(terv)-5 b(al)34 b([)p Fn(T)2631 967 y Fk(k)q Fj(\000)p Fo(1)2756 955 y Fn(;)14 b(T)2842 967 y Fk(k)2883 955 y Fr(])35 b(no)f(more)g(than)456 1056 y(one)k(third)h(of)g(the)g(spatial)g(gaps)f(around)f([)p Fn(y)1938 1020 y Fj(\000)1935 1081 y Fk(k)1994 1056 y Fn(;)14 b(y)2075 1020 y Fo(+)2072 1081 y Fk(k)2130 1056 y Fr(].)71 b(Then,)42 b(during)c(this)i(time)f(in)n(terv)-5 b(al,)456 1155 y(the)38 b(particles)f(in)i Fm(V)1111 1167 y Fk(k)1190 1155 y Fr(do)f(not)g(in)n(teract)f(with)i(those)f(in)g Fm(J)2382 1167 y Fk(T)2421 1176 y Fg(k)2488 1155 y Fm(n)25 b(V)2606 1167 y Fk(k)2684 1155 y Fr(and,)41 b(b)n(y)e(\(3.11\))o(,)i (they)456 1255 y(ma)n(y)30 b(in)n(teract)h(only)g(with)h(particles)e (in)i Fm(P)7 b Fr(.)48 b(Th)n(us,)32 b(at)f(eac)n(h)f(time)i Fn(T)2677 1267 y Fk(k)2718 1255 y Fr(,)g(w)n(e)f(ha)n(v)n(e)f(isolated) h(a)456 1355 y(\014nite)k(set)g Fm(V)863 1367 y Fk(k)938 1355 y Fr(\(whose)g(cardinalit)n(y)e(is)i(indep)r(enden)n(t)g(of)g Fn(T)12 b Fr(\))34 b(of)h(fast)f(particles,)i(whic)n(h)f(can)456 1454 y(only)j(in)n(teract)g(among)g(themselv)n(es)h(and)f(with)i(the)f (bac)n(kground)e Fm(M)2760 1466 y Fk(k)2839 1454 y Fr(of)i(slo)n(w)f (particles)456 1554 y(during)i(the)h(time)h(in)n(terv)-5 b(al)40 b([)p Fn(T)1476 1566 y Fk(k)q Fj(\000)p Fo(1)1601 1554 y Fn(;)14 b(T)1687 1566 y Fk(k)1728 1554 y Fr(].)76 b(W)-7 b(e)42 b(next)f(study)g(their)g(ev)n(olution)f(during)g(this)456 1653 y(time)f(in)n(terv)-5 b(al,)42 b(b)n(y)c(getting)h(su\016cien)n (tly)g(sharp)f(b)r(ounds)i(on)e(the)i(v)n(elo)r(cit)n(y)e(v)-5 b(ariations)38 b(to)456 1753 y(conclude)29 b(that)i(the)f(\(v)n(ery)f (fast\))i Fn(i)p Fr(-th)e(particle)h(sta)n(ys)f(in)h Fm(V)2367 1765 y Fk(k)2438 1753 y Fr(also)f(for)g(an)n(y)h Fn(k)f(<)e(")3099 1723 y Fj(\000)p Fo(1)3188 1753 y Fr(.)44 b(Then)456 1853 y Fm(j)p Fn(v)519 1865 y Fk(i)547 1853 y Fr(\(0\))p Fm(j)23 b Fn(>)f(\015)829 1865 y Fo(0)867 1853 y Fn(aT)34 b(>)22 b(aT)9 b(=)p Fr(4)27 b(and)g(the)h(theorem)f(is) h(pro)n(v)n(ed.)555 1952 y(Recalling)g(\(3.16\))o(,)g(if)g Fn(a)1300 1964 y Fo(0)1365 1952 y Fr(is)g(small)f(enough,)g(then:)1194 2106 y Fm(9)-42 b Fn(=)14 b(j)28 b Fm(2)c(P)43 b Fr(:)37 b Fn(\015)1599 2118 y Fk(k)1640 2106 y Fn(aT)d Fm(\024)23 b(j)p Fn(v)1918 2118 y Fk(j)1953 2106 y Fr(\()p Fn(T)2034 2118 y Fk(k)2075 2106 y Fr(\))p Fm(j)g(\024)g Fr(\()p Fn(\015)2316 2118 y Fk(k)2375 2106 y Fr(+)18 b(3)p Fn(\014)t Fr(\))p Fn(aT)7 b(;)526 b Fr(\(3.20\))456 2259 y(where)1655 2411 y Fn(\014)1750 2364 y(:)1729 2411 y Fr(=)1979 2354 y Fn(a)2023 2324 y Fo(3)p 1827 2391 386 4 v 1827 2468 a Fr(1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2176 2444 y Fo(2)2222 2411 y Fn(:)987 b Fr(\(3.21\))456 2591 y(First)35 b(of)h(all)f(w)n(e)g(con)n(trol)g(the)h(ev)n(olution)f(of)g(the)h(bac)n (kground)e Fm(M)2638 2603 y Fk(k)2679 2591 y Fr(,)j(b)n(y)f(sho)n(wing) e(that)i(in)456 2690 y(the)27 b(time)g(in)n(terv)-5 b(al)27 b([)p Fn(T)1158 2702 y Fk(k)q Fj(\000)p Fo(1)1283 2690 y Fn(;)14 b(T)1369 2702 y Fk(k)1410 2690 y Fr(])27 b(the)g(gap)f(in)h (momen)n(tum)h(space)e(whic)n(h)h(separates)e Fm(M)3208 2702 y Fk(k)3276 2690 y Fr(from)456 2790 y Fm(V)507 2802 y Fk(k)577 2790 y Fr(cannot)30 b(b)r(e)g(saturated)f(b)n(y)h(the)g(slo) n(w)f(particles.)43 b(This)30 b(is)g(the)g(con)n(ten)n(t)f(of)h(the)h (follo)n(wing)456 2890 y(lemma.)456 3050 y Fv(Lemma)e(3.1.)40 b Fp(If)30 b Fn(a)1115 3062 y Fo(0)1182 3050 y Fp(is)g(smal)t(l)h (enough)f(and)1647 3204 y Fn(")1686 3169 y Fj(\000)p Fo(1)1798 3204 y Fr(=)23 b(In)n(t)2006 3136 y Fh(\002)2041 3204 y Fn(\014)2092 3169 y Fj(\000)p Fo(6)2181 3136 y Fh(\003)2229 3204 y Fn(;)980 b Fr(\(3.22\))456 3362 y Fp(then,)30 b(for)g(any)g Fn(k)c Fr(=)d(0)p Fn(;)14 b(:)g(:)g(:)f(;)h (")1379 3332 y Fj(\000)p Fo(1)1468 3362 y Fp(,)1280 3516 y Fr(max)1262 3570 y Fk(j)s Fj(2M)1415 3579 y Fg(k)1582 3516 y Fr(sup)1466 3590 y Fk(s)p Fj(2)p Fo([)p Fk(T)1600 3599 y Fg(k)q Fd(\000)p Fe(1)1709 3590 y Fk(;T)1768 3599 y Fg(k)1805 3590 y Fo(])1837 3516 y Fm(j)p Fn(v)1900 3528 y Fk(j)1936 3516 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fr(\()p Fn(\015)2248 3528 y Fk(k)2307 3516 y Fr(+)18 b(2)p Fn(\014)t Fr(\))p Fn(aT)7 b(:)594 b Fr(\(3.23\))456 3750 y Fp(Pr)l(o)l(of.)39 b Fr(Fix)28 b Fn(k)e Fm(2)d(f)p Fr(0;)14 b Fn(:)g(:)g(:)f Fr(;)h Fn(")1315 3720 y Fj(\000)p Fo(1)1404 3750 y Fm(g)27 b Fr(and)g(let:)935 3954 y Fn(t)965 3966 y Fo(0)1046 3907 y Fn(:)1025 3954 y Fr(=)c(sup)1252 3837 y Fh(\032)1314 3954 y Fn(t)g Fm(2)g Fr([)p Fn(T)1517 3966 y Fk(k)q Fj(\000)p Fo(1)1643 3954 y Fn(;)14 b(T)1729 3966 y Fk(k)1769 3954 y Fr(])37 b(:)55 b(max)1889 4007 y Fk(j)s Fj(2M)2042 4016 y Fg(k)2093 3954 y Fm(j)p Fn(v)2156 3966 y Fk(j)2191 3954 y Fr(\()p Fn(t)p Fr(\))p Fm(j)24 b Fn(>)e Fr(\()p Fn(\015)2494 3966 y Fk(k)2554 3954 y Fr(+)c(2)p Fn(\014)t Fr(\))p Fn(aT)2867 3837 y Fh(\033)2942 3954 y Fn(;)456 4152 y Fr(setting)29 b Fn(t)760 4164 y Fo(0)823 4152 y Fr(=)c Fn(T)962 4164 y Fk(k)q Fj(\000)p Fo(1)1117 4152 y Fr(if)30 b(the)f(ab)r(o)n(v)n(e)f(set)i(is)f(empt)n(y) -7 b(.)42 b(W)-7 b(e)29 b(ha)n(v)n(e)f(to)h(sho)n(w)g(that)g Fn(t)2936 4164 y Fo(0)2999 4152 y Fr(=)d Fn(T)3139 4164 y Fk(k)q Fj(\000)p Fo(1)3264 4152 y Fr(.)42 b(W)-7 b(e)456 4252 y(argue)23 b(b)n(y)i(con)n(tradiction)f(and)h(assume)f Fn(t)1768 4264 y Fo(0)1828 4252 y Fn(>)f(T)1965 4264 y Fk(k)q Fj(\000)p Fo(1)2090 4252 y Fr(.)36 b(Then)26 b(there)f(exists)f Fn(q)i Fm(2)e(M)3042 4264 y Fk(k)3107 4252 y Fr(suc)n(h)h(that)456 4352 y Fm(j)p Fn(v)519 4364 y Fk(q)556 4352 y Fr(\()p Fn(t)618 4364 y Fo(0)655 4352 y Fr(\))p Fm(j)e Fr(=)g(\()p Fn(\015)896 4364 y Fk(k)956 4352 y Fr(+)18 b(2)p Fn(\014)t Fr(\))p Fn(aT)12 b Fr(.)36 b(Setting)1143 4505 y Fn(t)1173 4517 y Fo(1)1254 4458 y Fn(:)1233 4505 y Fr(=)23 b(inf)e Fm(f)o Fn(t)i Fm(2)h Fr([)p Fn(t)1662 4517 y Fo(0)1699 4505 y Fn(;)14 b(T)1785 4517 y Fk(k)1825 4505 y Fr(])37 b(:)g Fm(j)p Fn(v)2008 4517 y Fk(q)2045 4505 y Fr(\()p Fn(t)p Fr(\))p Fm(j)24 b Fn(<)e Fr(\()p Fn(\015)2348 4517 y Fk(k)2408 4505 y Fr(+)c Fn(\014)t Fr(\))p Fn(aT)12 b Fm(g)h Fn(;)456 4659 y Fr(w)n(e)21 b(get)i(an)e(absurd)h(if)h(w)n(e)f(sho)n(w)f(that)i Fn(t)1676 4671 y Fo(1)1736 4659 y Fr(=)f Fn(T)1872 4671 y Fk(k)1913 4659 y Fr(.)35 b(F)-7 b(rom)22 b(the)g(equations)g(of)g (motion)g(and)g(\(3.12\))o(,)906 4800 y Fh(\014)906 4849 y(\014)934 4870 y Fm(j)p Fn(v)997 4882 y Fk(q)1034 4870 y Fr(\()p Fn(t)1096 4882 y Fo(1)1133 4870 y Fr(\))p Fm(j)d(\000)f Fr(\()p Fn(\015)1365 4882 y Fk(k)1425 4870 y Fr(+)g(2)p Fn(\014)t Fr(\))p Fn(aT)1738 4800 y Fh(\014)1738 4849 y(\014)1788 4870 y Fm(\024)k(kr)p Fr(\010)p Fm(k)2088 4882 y Fj(1)2172 4757 y Fh(Z)2255 4778 y Fk(t)2280 4786 y Fe(1)2218 4946 y Fk(t)2243 4954 y Fe(0)2316 4870 y Fn(ds)2510 4791 y Fh(X)2426 4970 y Fk(j)s Fj(2P)2548 4978 y Fe(1)2581 4970 y Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(q)2728 4870 y Fn(\037)2780 4882 y Fk(q)r(;j)2867 4870 y Fr(\()p Fn(s)p Fr(\))p Fn(;)239 b Fr(\(3.24\))456 5116 y(where)33 b Fn(\037)754 5128 y Fk(q)r(;j)841 5116 y Fr(\()p Fn(s)p Fr(\))999 5069 y Fn(:)978 5116 y Fr(=)g Fn(\037)p Fr(\()p Fm(j)p Fn(x)1230 5128 y Fk(q)1268 5116 y Fr(\()p Fn(s)p Fr(\))23 b Fm(\000)f Fn(x)1528 5128 y Fk(j)1564 5116 y Fr(\()p Fn(s)p Fr(\))p Fm(j)34 b(\024)f Fr(1\),)i(i.e.)g(the)f(c)n (haracteristic)e(function)i(of)g(the)g(set)456 5216 y Fm(f)p Fn(s)27 b Fm(2)h Fr([)p Fn(t)700 5228 y Fo(0)737 5216 y Fn(;)14 b(t)804 5228 y Fo(1)841 5216 y Fr(])28 b(:)f Fm(j)p Fn(x)1012 5228 y Fk(q)1050 5216 y Fr(\()p Fn(s)p Fr(\))20 b Fm(\000)g Fn(x)1305 5228 y Fk(j)1341 5216 y Fr(\()p Fn(s)p Fr(\))p Fm(j)28 b(\024)f Fr(1)p Fm(g)p Fr(.)44 b(W)-7 b(e)31 b(lo)r(ok)f(for)f(an)h(upp)r(er)h(b)r (ound)g(to)f(the)h(righ)n(t-hand)p eop %%Page: 10 10 10 9 bop 456 251 a Fo(10)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(side)32 b(of)39 b(\(3.24\))o(:)48 b(if)33 b(w)n(e)f(sho)n(w)g(it)h(can)f(b)r(e)h(done)g(smaller)f(than)g(e.g.)h Fn(\014)t(aT)9 b(=)p Fr(2,)33 b(then)g Fn(t)3167 462 y Fo(1)3236 450 y Fr(=)e Fn(T)3381 462 y Fk(k)3421 450 y Fr(.)456 550 y(W)-7 b(e)28 b(decomp)r(ose:)1089 698 y Fm(f)p Fn(s)22 b Fm(2)i Fr([)p Fn(t)1324 710 y Fo(0)1361 698 y Fn(;)14 b(t)1428 710 y Fo(1)1465 698 y Fr(])23 b(:)g Fm(j)p Fn(x)1627 710 y Fk(q)1665 698 y Fr(\()p Fn(s)p Fr(\))c Fm(\000)f Fn(x)1917 710 y Fk(j)1952 698 y Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\024)e Fr(1)p Fm(g)g Fr(=)h(\001)2452 662 y Fo(+)2452 721 y Fk(q)r(;j)2558 698 y Fm([)c Fr(\001)2701 662 y Fj(\000)2701 721 y Fk(q)r(;j)2788 698 y Fn(;)456 846 y Fr(with)873 959 y(\001)942 924 y Fo(+)942 982 y Fk(q)r(;j)1073 912 y Fn(:)1053 959 y Fr(=)j Fm(f)p Fn(s)h Fm(2)g Fr([)p Fn(t)1375 971 y Fo(0)1412 959 y Fn(;)14 b(t)1479 971 y Fo(1)1517 959 y Fr(])37 b(:)f Fm(j)p Fn(v)1699 971 y Fk(j)1735 959 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b Fn(>)g(\015)2015 971 y Fk(k)2056 959 y Fn(aT)7 b(;)96 b Fm(j)p Fn(x)2345 971 y Fk(q)2382 959 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)2634 971 y Fk(j)2670 959 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fr(1)p Fm(g)13 b Fn(;)873 1118 y Fr(\001)942 1082 y Fj(\000)942 1141 y Fk(q)r(;j)1073 1070 y Fn(:)1053 1118 y Fr(=)22 b Fm(f)p Fn(s)h Fm(2)g Fr([)p Fn(t)1375 1130 y Fo(0)1412 1118 y Fn(;)14 b(t)1479 1130 y Fo(1)1517 1118 y Fr(])37 b(:)f Fm(j)p Fn(v)1699 1130 y Fk(j)1735 1118 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fn(\015)2015 1130 y Fk(k)2056 1118 y Fn(aT)7 b(;)96 b Fm(j)p Fn(x)2345 1130 y Fk(q)2382 1118 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)2634 1130 y Fk(j)2670 1118 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fr(1)p Fm(g)13 b Fn(;)456 1263 y Fr(so)27 b(that)g Fn(\037)789 1275 y Fk(q)r(;j)877 1263 y Fr(\()p Fn(s)p Fr(\))c(=)g Fn(\037)p Fr(\(\001)1244 1227 y Fo(+)1244 1286 y Fk(q)r(;j)1331 1263 y Fr(\)\()p Fn(s)p Fr(\))d(+)e Fn(\037)p Fr(\(\001)1722 1227 y Fj(\000)1722 1286 y Fk(q)r(;j)1809 1263 y Fr(\)\()p Fn(s)p Fr(\).)38 b(W)-7 b(e)28 b(observ)n(e)e(that:) 1090 1425 y Fm(j)p Fn(v)1153 1437 y Fk(q)1190 1425 y Fr(\()p Fn(s)p Fr(\))p Fm(j)19 b(\000)f(j)p Fn(v)1481 1437 y Fk(j)1516 1425 y Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\025)f Fn(\014)t(aT)177 b Fm(8)14 b Fn(s)22 b Fm(2)h Fr(\001)2344 1389 y Fj(\000)2344 1448 y Fk(q)r(;j)2515 1425 y Fm(8)14 b Fn(j)27 b Fm(2)c(P)2773 1437 y Fo(1)456 1578 y Fr(and,)k(b)n(y)i(\(3.13\))o(,)859 1656 y Fh(\014)859 1706 y(\014)887 1727 y Fm(j)p Fn(v)950 1739 y Fk(q)987 1727 y Fr(\()p Fn(s)p Fr(\))p Fm(j)19 b(\000)f(j)p Fn(v)1278 1739 y Fk(j)1313 1727 y Fr(\()p Fn(s)p Fr(\))p Fm(j)1439 1656 y Fh(\014)1439 1706 y(\014)1490 1727 y Fm(\024)23 b Fr(2)p Fn(\032T)12 b Fr([1)k(+)i Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])166 b Fm(8)14 b Fn(s)22 b Fm(2)h Fr(\001)2552 1691 y Fo(+)2552 1750 y Fk(q)r(;j)2723 1727 y Fm(8)14 b Fn(j)27 b Fm(2)c(P)2981 1739 y Fo(1)3018 1727 y Fn(:)456 1889 y Fr(Then,)31 b(letting)g Fn(\022)1004 1901 y Fo(0)1090 1842 y Fn(:)1069 1889 y Fr(=)c(inf)7 b Fm(f)p Fr(\001)1373 1854 y Fo(+)1373 1913 y Fk(q)r(;j)1481 1889 y Fm([)20 b Fr(\001)1625 1854 y Fj(\000)1625 1913 y Fk(q)r(;j)1713 1889 y Fm(g)p Fr(,)31 b Fn(\022)1848 1901 y Fo(1)1934 1842 y Fn(:)1913 1889 y Fr(=)c(sup)p Fm(f)p Fr(\001)2241 1854 y Fo(+)2241 1913 y Fk(q)r(;j)2349 1889 y Fm([)20 b Fr(\001)2493 1854 y Fj(\000)2493 1913 y Fk(q)r(;j)2581 1889 y Fm(g)p Fr(,)31 b(and)f Fn(D)r(v)s Fr(\()p Fn(s)p Fr(\))3107 1842 y Fn(:)3087 1889 y Fr(=)d Fn(v)3219 1901 y Fk(q)3256 1889 y Fr(\()p Fn(s)p Fr(\))21 b Fm(\000)456 1994 y Fn(v)496 2006 y Fk(j)531 1994 y Fr(\()p Fn(s)p Fr(\),)28 b(w)n(e)f(get:)864 2208 y(2)82 b Fm(\025)1136 2137 y Fh(\014)1136 2187 y(\014)1163 2208 y Fn(x)1210 2220 y Fk(q)1247 2208 y Fr(\()p Fn(\022)1318 2220 y Fo(1)1356 2208 y Fr(\))19 b Fm(\000)f Fn(x)1537 2220 y Fk(j)1572 2208 y Fr(\()p Fn(\022)1643 2220 y Fo(1)1681 2208 y Fr(\))g Fm(\000)g Fn(x)1861 2220 y Fk(q)1899 2208 y Fr(\()p Fn(\022)1970 2220 y Fo(0)2007 2208 y Fr(\))h(+)f Fn(x)2188 2220 y Fk(j)2223 2208 y Fr(\()p Fn(\022)2294 2220 y Fo(0)2332 2208 y Fr(\))2364 2137 y Fh(\014)2364 2187 y(\014)2415 2208 y Fr(=)2502 2087 y Fh(\014)2502 2137 y(\014)2502 2187 y(\014)2502 2237 y(\014)2544 2095 y(Z)2627 2115 y Fk(\022)2659 2123 y Fe(1)2590 2283 y Fk(\022)2622 2291 y Fe(0)2695 2208 y Fn(ds)c(D)r(v)s Fr(\()p Fn(s)p Fr(\))3008 2087 y Fh(\014)3008 2137 y(\014)3008 2187 y(\014)3008 2237 y(\014)988 2445 y Fr(=)1136 2325 y Fh(\014)1136 2374 y(\014)1136 2424 y(\014)1136 2474 y(\014)1177 2332 y(Z)1223 2521 y Fo(\001)1278 2494 y Fd(\000)1278 2540 y Fg(q)q(;j)1361 2445 y Fn(ds)g(D)r(v)s Fr(\()p Fn(s)p Fr(\))19 b(+)1776 2332 y Fh(Z)1822 2521 y Fo(\001)1877 2494 y Fe(+)1877 2540 y Fg(q)q(;j)1960 2445 y Fn(ds)14 b(D)r(v)s Fr(\()p Fn(s)p Fr(\))2273 2325 y Fh(\014)2273 2374 y(\014)2273 2424 y(\014)2273 2474 y(\014)988 2694 y Fm(\025)1136 2581 y Fh(Z)1182 2770 y Fo(\001)1237 2743 y Fd(\000)1237 2789 y Fg(q)q(;j)1319 2694 y Fn(ds)g Fm(j)p Fn(D)r(v)s Fr(\()p Fn(s)p Fr(\))p Fm(j)20 b(\000)1781 2581 y Fh(Z)1827 2770 y Fo(\001)1882 2743 y Fe(+)1882 2789 y Fg(q)q(;j)1964 2694 y Fn(ds)14 b Fm(j)p Fn(D)r(v)s Fr(\()p Fn(s)p Fr(\))p Fm(j)988 2901 y(\025)83 b Fn(\014)t(aT)12 b Fm(j)p Fr(\001)1384 2866 y Fj(\000)1384 2924 y Fk(q)r(;j)1470 2901 y Fm(j)19 b(\000)f Fr(2)p Fn(\032T)12 b Fr([1)k(+)i Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fm(j)p Fr(\001)2226 2866 y Fo(+)2226 2924 y Fk(q)r(;j)2313 2901 y Fm(j)p Fn(;)456 3064 y Fr(where)25 b(w)n(e)h(used)g(that)g Fn(D)r(v)j Fr(has)d(zero)f(a)n(v)n(erage)e(on)j([)p Fn(\022)2114 3076 y Fo(0)2151 3064 y Fn(;)14 b(\022)2227 3076 y Fo(1)2264 3064 y Fr(])h Fm(n)g Fr(\(\001)2460 3028 y Fo(+)2460 3087 y Fk(q)r(;j)2563 3064 y Fm([)h Fr(\001)2703 3028 y Fj(\000)2703 3087 y Fk(q)r(;j)2790 3064 y Fr(\))27 b(and)e(that)i(its)f(sign)456 3178 y(is)h(constan)n(t)g(on)g(\001)1058 3142 y Fj(\000)1058 3201 y Fk(q)r(;j)1146 3178 y Fr(.)37 b(W)-7 b(e)28 b(th)n(us)f(obtain:)1293 3387 y Fm(j)p Fr(\001)1385 3352 y Fj(\000)1385 3411 y Fk(q)r(;j)1472 3387 y Fm(j)c(\024)1673 3331 y Fr(2)p 1616 3368 156 4 v 1616 3444 a Fn(\014)t(aT)1800 3387 y Fr(+)1893 3331 y(2)p Fn(\032)p Fr([1)17 b(+)h Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p 1893 3368 480 4 v 2085 3444 a Fn(\014)t(a)2382 3387 y Fm(j)p Fr(\001)2474 3352 y Fo(+)2474 3411 y Fk(q)r(;j)2561 3387 y Fm(j)p Fn(:)456 3597 y Fr(It)28 b(follo)n(ws)e(there)i(is)f(a)g (decomp)r(osition)g(\001)1799 3562 y Fj(\000)1799 3620 y Fk(q)r(;j)1910 3597 y Fr(=)22 b(\001)2066 3567 y Fj(\003)2066 3619 y Fk(q)r(;j)2172 3597 y Fm([)d Fr(\001)2315 3557 y Fj(y)2315 3620 y Fk(q)r(;j)2430 3597 y Fr(with:)705 3807 y(\001)774 3772 y Fj(\003)774 3827 y Fk(q)r(;j)880 3807 y Fm(\\)g Fr(\001)1023 3767 y Fj(y)1023 3830 y Fk(q)r(;j)1133 3807 y Fr(=)k Fm(;)p Fn(;)180 b Fm(j)p Fr(\001)1558 3772 y Fj(\003)1558 3827 y Fk(q)r(;j)1645 3807 y Fm(j)23 b(\024)1846 3750 y Fr(2)p 1789 3788 156 4 v 1789 3864 a Fn(\014)t(aT)1954 3807 y(;)180 b Fm(j)p Fr(\001)2249 3767 y Fj(y)2249 3830 y Fk(q)r(;j)2336 3807 y Fm(j)23 b(\024)2480 3750 y Fr(2)p Fn(\032)p Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p 2480 3788 480 4 v 2672 3864 a Fn(\014)t(a)2969 3807 y Fm(j)p Fr(\001)3061 3771 y Fo(+)3061 3830 y Fk(q)r(;j)3149 3807 y Fm(j)p Fn(:)456 4016 y Fr(Then)27 b Fn(\037)p Fr(\(\001)825 3981 y Fj(\000)825 4040 y Fk(q)r(;j)913 4016 y Fr(\)\()p Fn(s)p Fr(\))d(=)e Fn(\037)p Fr(\(\001)1312 3986 y Fj(\003)1312 4038 y Fk(q)r(;j)1400 4016 y Fr(\)\()p Fn(s)p Fr(\))d(+)f Fn(\037)p Fr(\(\001)1790 3977 y Fj(y)1790 4040 y Fk(q)r(;j)1878 4016 y Fr(\)\()p Fn(s)p Fr(\))28 b(and)g(therefore:)727 4123 y Fh(Z)810 4143 y Fk(t)835 4151 y Fe(1)773 4311 y Fk(t)798 4319 y Fe(0)872 4236 y Fn(ds)14 b(\037)1020 4248 y Fk(q)r(;j)1107 4236 y Fr(\()p Fn(s)p Fr(\))83 b Fm(\024)1441 4123 y Fh(Z)1524 4143 y Fk(t)1549 4151 y Fe(1)1487 4311 y Fk(t)1512 4319 y Fe(0)1585 4236 y Fn(ds)14 b(\037)p Fr(\(\001)1834 4201 y Fj(\003)1834 4256 y Fk(q)r(;j)1922 4236 y Fr(\)\()p Fn(s)p Fr(\))19 b(+)2159 4119 y Fh(\024)2203 4236 y Fr(1)f(+)2356 4179 y(2)p Fn(\032)p Fr([1)g(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p 2356 4217 V 2548 4293 a Fn(\014)t(a)2845 4119 y Fh(\025)2903 4236 y Fm(j)p Fr(\001)2995 4200 y Fo(+)2995 4259 y Fk(q)r(;j)3082 4236 y Fm(j)1293 4481 y Fr(=)1441 4368 y Fh(Z)1524 4389 y Fk(t)1549 4397 y Fe(1)1487 4557 y Fk(t)1512 4565 y Fe(0)1585 4481 y Fn(ds)1695 4364 y Fh(\024)1739 4481 y Fn(\037)p Fr(\(\001)1892 4447 y Fj(\003)1892 4502 y Fk(q)r(;j)1980 4481 y Fr(\)\()p Fn(s)p Fr(\))19 b(+)2227 4425 y(3)p Fn(\032)p Fr([1)e(+)h Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p 2227 4462 V 2419 4538 a Fn(\014)t(a)2716 4481 y(\037)p Fr(\(\001)2869 4446 y Fo(+)2869 4504 y Fk(q)r(;j)2957 4481 y Fr(\)\()p Fn(s)p Fr(\))3092 4364 y Fh(\025)3150 4481 y Fn(:)59 b Fr(\(3.25\))456 4684 y(F)-7 b(rom)27 b(\(3.24\))g(and)g(\(3.25\))g(w)n(e)g(obtain)g(the)h (follo)n(wing)f(estimate:)1130 4761 y Fh(\014)1130 4811 y(\014)1158 4832 y Fm(j)p Fn(v)1221 4844 y Fk(q)1258 4832 y Fr(\()p Fn(t)1320 4844 y Fo(1)1357 4832 y Fr(\))p Fm(j)19 b(\000)f Fr(\()p Fn(\015)1589 4844 y Fk(k)1649 4832 y Fr(+)g(2)p Fn(\014)t Fr(\))p Fn(aT)1962 4761 y Fh(\014)1962 4811 y(\014)2012 4832 y Fm(\024)k(kr)p Fr(\010)p Fm(k)2312 4844 y Fj(1)2382 4832 y Fr(\()p Fn(A)2476 4844 y Fo(1)2532 4832 y Fr(+)c Fn(A)2677 4844 y Fo(2)2714 4832 y Fr(\))p Fn(;)463 b Fr(\(3.26\))456 4979 y(with)1170 5129 y Fn(A)1232 5141 y Fo(1)1293 5129 y Fr(=)1380 5016 y Fh(Z)1463 5037 y Fk(t)1488 5045 y Fe(1)1427 5205 y Fk(t)1452 5213 y Fe(0)1525 5129 y Fn(ds)1719 5050 y Fh(X)1635 5229 y Fk(j)s Fj(2P)1757 5237 y Fe(1)1790 5229 y Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(q)1937 5129 y Fn(\037)p Fr(\(\001)2090 5095 y Fj(\003)2090 5150 y Fk(q)r(;j)2177 5129 y Fr(\)\()p Fn(s)p Fr(\))24 b Fm(\024)2491 5073 y Fr(2)p 2434 5110 156 4 v 2434 5186 a Fn(\014)t(aT)2623 5108 y Fr(\026)2599 5129 y Fn(N)2666 5141 y Fk(k)2707 5129 y Fn(;)502 b Fr(\(3.27\))p eop %%Page: 11 11 11 10 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(11)456 450 y Fr(where)717 429 y(\026)693 450 y Fn(N)760 462 y Fk(k)826 450 y Fr(denotes)24 b(the)i(n)n(um)n(b)r(er)e(of)h (particles)f(whic)n(h)h(can)g(in)n(teract)f(with)i(the)f Fn(q)s Fr(-th)g(particle)456 550 y(during)i(the)h(time)g([)p Fn(T)1123 562 y Fk(k)q Fj(\000)p Fo(1)1248 550 y Fn(;)14 b(T)1334 562 y Fk(k)1374 550 y Fr(],)28 b(and)g(\(recall)f Fn(t)1894 562 y Fo(1)1949 550 y Fm(\000)18 b Fn(t)2062 562 y Fo(0)2123 550 y Fm(\024)k Fn("T)12 b Fr(\))1062 743 y Fn(A)1124 755 y Fo(2)1245 743 y Fr(=)1402 687 y(3)p Fn(\032)p Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p 1402 724 480 4 v 1594 800 a Fn(\014)t(a)1905 630 y Fh(Z)1988 651 y Fk(t)2013 659 y Fe(1)1951 819 y Fk(t)1976 827 y Fe(0)2050 743 y Fn(ds)2244 664 y Fh(X)2160 843 y Fk(j)s Fj(2P)2282 851 y Fe(1)2315 843 y Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(q)2461 743 y Fn(\037)p Fr(\(\001)2614 708 y Fo(+)2614 766 y Fk(q)r(;j)2702 743 y Fr(\)\()p Fn(s)p Fr(\))395 b(\(3.28\))1245 1014 y Fm(\024)1402 958 y Fr(3)p Fn(\032)p Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p 1402 995 V 1594 1071 a Fn(\014)t(a)1905 1014 y("T)141 b Fr(sup)2018 1088 y Fk(s)p Fj(2)p Fo([)p Fk(T)2152 1097 y Fg(k)q Fd(\000)p Fe(1)2261 1088 y Fk(;T)2320 1097 y Fg(k)2356 1088 y Fo(])2389 1014 y Fn(N)2465 980 y Fk(>)2456 1038 y(k)2521 1014 y Fr(\()p Fn(s)p Fr(\))p Fn(;)585 b Fr(\(3.29\))456 1223 y(where)30 b Fn(N)775 1190 y Fk(>)766 1248 y(k)831 1223 y Fr(\()p Fn(s)p Fr(\))i(denotes)f(the)h(n)n(um)n(b)r (er)f(of)g(particles)f(of)h Fn(X)7 b Fr(\()p Fn(s)p Fr(\))32 b(whic)n(h)f(are)f(con)n(tained)h(in)g(the)456 1327 y(in)n(terv)-5 b(al)752 1260 y Fh(\002)786 1327 y Fn(x)833 1339 y Fk(q)870 1327 y Fr(\()p Fn(s)p Fr(\))10 b Fm(\000)g Fr(1)p Fn(;)k(x)1184 1339 y Fk(q)1221 1327 y Fr(\()p Fn(s)p Fr(\))c(+)g(1)1451 1260 y Fh(\003)1509 1327 y Fr(and)24 b(whose)f(v)n(elo)r(cit)n(y)f(is)i (bigger)e(than)h Fn(\015)2767 1339 y Fk(k)2808 1327 y Fn(aT)12 b Fr(.)35 b(W)-7 b(e)24 b(need)f(no)n(w)456 1432 y(to)k(b)r(ound)h(the)g(quan)n(tities)1363 1411 y(\026)1339 1432 y Fn(N)1406 1444 y Fk(k)1475 1432 y Fr(and)f Fn(N)1712 1399 y Fk(>)1703 1457 y(k)1768 1432 y Fr(\()p Fn(s)p Fr(\).)37 b(By)29 b(\(3.13\))e(w)n(e)g(ha)n(v)n(e:) 1043 1553 y(\026)1019 1574 y Fn(N)1086 1586 y Fk(k)1149 1574 y Fm(\024)c Fn(N)1313 1506 y Fh(\000)1351 1574 y Fn(X)7 b Fr(\()p Fn(T)1508 1586 y Fk(k)q Fj(\000)p Fo(1)1633 1574 y Fr(\);)14 b Fn(x)1749 1586 y Fk(q)1786 1574 y Fr(\()p Fn(T)1867 1586 y Fk(k)q Fj(\000)p Fo(1)1993 1574 y Fr(\))p Fn(;)g Fr(1)k(+)g(2)p Fn("\032T)2390 1539 y Fo(2)2425 1574 y Fr([1)g(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])2820 1506 y Fh(\001)2858 1574 y Fn(;)351 b Fr(\(3.30\))456 1710 y(where)23 b Fn(N)9 b Fr(\()p Fn(X)e Fr(;)14 b Fn(\026;)g(R)q Fr(\))23 b(denotes)g(the)h(n)n(um)n(b)r(er)g(of)f(particles)g(of)h(the) g(con\014guration)e Fn(X)30 b Fr(whic)n(h)23 b(are)456 1809 y(con)n(tained)g(in)h(the)h(in)n(terv)-5 b(al)23 b([)p Fn(\026)11 b Fm(\000)g Fn(R)q(;)j(\026)d Fr(+)g Fn(R)q Fr(].)35 b(W)-7 b(e)25 b(next)f(observ)n(e)e(that,)j(since)f (the)g(in)n(teraction)456 1909 y(is)j(sup)r(erstable)g(with)h(range)e (b)r(ounded)i(b)n(y)g(1,)702 2045 y Fn(N)9 b Fr(\()p Fn(X)e Fr(;)14 b Fn(\026;)g(R)q Fr(\))1106 2010 y Fo(2)1166 2045 y Fm(\024)22 b Fn(C)6 b(R)q(Q)p Fr(\()p Fn(X)h Fr(;)14 b Fn(\026;)g(R)q Fr(\))165 b Fm(8)14 b Fn(X)29 b Fm(2)23 b(X)95 b(8)14 b Fn(\026)22 b Fm(2)i Fl(R)89 b Fm(8)14 b Fn(R)22 b Fm(\025)h Fr(1)p Fn(:)246 b Fr(\(3.31\))456 2181 y(By)32 b(\(3.21\))e(and)h(\(3.22\))f(it)i(follo)n(ws)e Fn("T)1705 2150 y Fo(2)1770 2181 y Fm(\025)e Fr(1)j(for)f(all)h Fn(T)40 b Fm(\025)28 b Fn(T)2416 2193 y Fk(a)2456 2181 y Fr(\()p Fn(X)r(;)14 b(x)2643 2193 y Fk(i)2671 2181 y Fr(\),)32 b(so)f(that)g(w)n(e)g(can)g(use)456 2280 y(\(3.31\))37 b(to)h(b)r(ound)g(the)g(righ)n(t-hand)f(side)h(of)44 b(\(3.30\))o(.)68 b(Since)38 b Fm(j)p Fn(x)2544 2292 y Fk(q)2581 2280 y Fr(\()p Fn(T)2662 2292 y Fk(k)q Fj(\000)p Fo(1)2788 2280 y Fr(\))p Fm(j)i(\024)g(j)p Fn(x)3058 2292 y Fk(i)3086 2280 y Fm(j)26 b Fr(+)f Fn(C)6 b Fr([1)25 b(+)456 2380 y Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)746 2350 y Fo(2)781 2380 y Fr(,)28 b(b)n(y)h(\(3.3\))e(and)g(the)h (de\014nitions)g(\(3.6\),)g(\(3.21\))o(,)f(and)h(\(3.22\))f(w)n(e)g (conclude)g(that:)1514 2496 y(\026)1490 2517 y Fn(N)1557 2529 y Fk(k)1621 2517 y Fm(\024)22 b Fn(C)1773 2453 y Fm(p)p 1843 2453 39 4 v 1843 2517 a Fn(")13 b(T)1956 2483 y Fo(2)1992 2517 y Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(:)822 b Fr(\(3.32\))456 2653 y(Recalling)27 b(that)g Fn(\015)1039 2665 y Fk(k)1103 2653 y Fn(>)c Fr(1)p Fn(=)p Fr(4,)j(the)i(quan)n(tit)n(y)f Fn(N)1917 2620 y Fk(>)1908 2678 y(k)1973 2653 y Fr(\()p Fn(s)p Fr(\))h(can)f(b)r(e)h(b)r(ounded)g(as)f Fm(jJ)2889 2665 y Fk(t)2919 2653 y Fm(j)h Fr(in)g(\(3.15\))o(:)1461 2846 y(sup)1416 2920 y Fk(s)p Fj(2)p Fo([0)p Fk(;T)9 b Fo(])1644 2846 y Fn(N)1720 2812 y Fk(>)1711 2870 y(k)1776 2846 y Fr(\()p Fn(s)p Fr(\))24 b Fm(\024)e Fn(C)2065 2790 y Fr(1)c(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2414 2760 y Fo(2)p 2065 2827 386 4 v 2218 2903 a Fn(a)2262 2879 y Fo(2)2461 2846 y Fn(:)748 b Fr(\(3.33\))456 3049 y(F)-7 b(rom)27 b(\(3.27\))o(,)h(\(3.32\))o(,)g(and)f(recalling)f(the)i (de\014nition)g(\(3.21\))f(of)h Fn(\014)t Fr(,)g(w)n(e)f(th)n(us)h (get:)1455 3238 y Fn(A)1517 3250 y Fo(1)1577 3238 y Fm(\024)23 b Fn(C)1730 3175 y Fm(p)p 1799 3175 39 4 v 63 x Fn(")1862 3182 y Fr(1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2211 3152 y Fo(5)p 1862 3219 386 4 v 2014 3295 a Fn(a)2058 3271 y Fo(8)2271 3238 y Fn(\014)t(aT)g(:)787 b Fr(\(3.34\))456 3402 y(Analogously)-7 b(,)26 b(from)h(\(3.28\))g(and)g(\(3.33\))o(,) 1489 3591 y Fn(A)1551 3603 y Fo(2)1612 3591 y Fm(\024)c Fn(C)6 b(")1827 3535 y Fr(1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2176 3505 y Fo(7)p 1827 3572 V 1963 3648 a Fn(a)2007 3624 y Fo(10)2237 3591 y Fn(\014)t(aT)g(:)821 b Fr(\(3.35\))456 3755 y(F)-7 b(rom)38 b(\(3.26\))o(,)j(\(3.34\))o(,)g(\(3.35\),)g(and)d (\(3.22\))o(,)j(b)n(y)d(c)n(ho)r(osing)f Fn(a)2480 3767 y Fo(0)2556 3755 y Fr(small)h(enough)g(w)n(e)g(obtain)456 3784 y Fh(\014)456 3834 y(\014)483 3855 y Fm(j)p Fn(v)546 3867 y Fk(q)583 3855 y Fr(\()p Fn(t)645 3867 y Fo(1)683 3855 y Fr(\))p Fm(j)19 b(\000)f Fr(\()p Fn(\015)915 3867 y Fk(k)974 3855 y Fr(+)g(2)p Fn(\014)t Fr(\))p Fn(aT)1287 3784 y Fh(\014)1287 3834 y(\014)1337 3855 y Fm(\024)23 b Fn(\014)t(aT)9 b(=)p Fr(2,)26 b(hence)i Fn(t)1972 3867 y Fo(1)2032 3855 y Fr(=)23 b Fn(T)2169 3867 y Fk(k)2237 3855 y Fr(and)k(the)h(lemma)g(is)f(pro)n(v)n(ed.)221 b Ff(\003)555 3990 y Fr(Let)33 b(us)f(go)f(bac)n(k)g(to)h(the)h(pro)r (of)f(of)g(Theorem)f(3.1.)50 b(T)-7 b(o)32 b(analyze)f(the)h(time)h(ev) n(olution)f(of)456 4090 y(the)d(particles)g(in)h Fm(V)1085 4102 y Fk(k)1155 4090 y Fr(during)f(the)h(time)g(in)n(terv)-5 b(al)29 b([)p Fn(T)2130 4102 y Fk(k)q Fj(\000)p Fo(1)2255 4090 y Fn(;)14 b(T)2341 4102 y Fk(k)2382 4090 y Fr(],)30 b(w)n(e)f(shall)g(use)g(a)g(p)r(erturbativ)n(e)456 4190 y(argumen)n(t,)36 b(based)e(on)h(the)g(existence)g(of)g(gaps)f(in)h (the)h(v)n(elo)r(cit)n(y)e(distribution)h(of)g(the)g(fast)456 4289 y(particles.)47 b(T)-7 b(o)31 b(implemen)n(t)g(this,)i(for)e(eac)n (h)f Fn(t)f Fm(2)h Fr([)p Fn(T)2096 4301 y Fk(k)q Fj(\000)p Fo(1)2221 4289 y Fn(;)14 b(T)2307 4301 y Fk(k)2348 4289 y Fr(])31 b(w)n(e)g(divide)g Fm(V)2829 4301 y Fk(k)2901 4289 y Fr(in)n(to)h Fp(clusters)f Fr(of)456 4389 y(particles,)26 b(according)e(to)i(the)h(follo)n(wing)f(rule.)36 b(Giv)n(en)26 b(a)g(parameter)f Fn(\016)h(>)d Fr(0,)j(t)n(w)n(o)g(particles)f Fn(j)456 4489 y Fr(and)d Fn(j)651 4458 y Fj(0)697 4489 y Fr(b)r(elong)g(to)g(the)h(same)f(\()p Fn(\016)o(;)14 b(t)p Fr(\)-)p Fp(cluster)22 b Fr(if)h(there)g(is)f(a)g(p)r(ositiv)n(e) g(in)n(teger)g Fn(p)g Fr(and)h(a)f(sequence)456 4588 y(of)27 b(particles)g Fn(j)917 4600 y Fo(0)954 4588 y Fn(;)14 b(:)g(:)g(:)g(;)g(j)1173 4600 y Fk(p)1234 4588 y Fm(2)24 b(V)1364 4600 y Fk(k)1405 4588 y Fr(,)j Fn(j)1489 4600 y Fo(0)1550 4588 y Fr(=)22 b Fn(j)5 b Fr(,)28 b Fn(j)1761 4600 y Fk(p)1823 4588 y Fr(=)22 b Fn(j)1949 4558 y Fj(0)1973 4588 y Fr(,)27 b(suc)n(h)h(that:)1183 4724 y Fm(j)p Fn(v)1246 4736 y Fk(j)1273 4744 y Fg(s)p Fd(\000)p Fe(1)1383 4724 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1619 4736 y Fk(j)1646 4744 y Fg(s)1683 4724 y Fr(\()p Fn(t)p Fr(\))p Fm(j)24 b Fn(<)f(\016)s("T)176 b Fm(8)14 b Fn(s)22 b Fr(=)h(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(p:)456 4860 y Fr(W)-7 b(e)21 b(denote)h(b)n(y)f(\005\()p Fn(\016)o(;)14 b(t)p Fr(\))21 b(the)h(partition)f(of)g Fm(V)1829 4872 y Fk(k)1892 4860 y Fr(in)n(to)g(\()p Fn(\016)o(;)14 b(t)p Fr(\)-clusters)20 b(and)h(b)n(y)g(\()p Fn(x)2886 4872 y Fj(C)2930 4860 y Fn(;)14 b(v)3007 4872 y Fj(C)3050 4860 y Fr(\))22 b(the)g(phase)456 4960 y(p)r(oin)n(t)27 b(of)h(the)g(cen)n(ter)f(of)g(mass)g(of)h(the)g(cluster)f Fm(C)5 b Fr(:)1298 5130 y Fn(x)1345 5142 y Fj(C)1412 5130 y Fr(=)1536 5074 y(1)p 1510 5111 95 4 v 1510 5187 a Fm(jC)g(j)1628 5051 y Fh(X)1631 5229 y Fk(j)s Fj(2C)1762 5130 y Fn(x)1809 5142 y Fk(j)1844 5130 y Fn(;)180 b(v)2087 5142 y Fj(C)2154 5130 y Fr(=)2278 5074 y(1)p 2251 5111 V 2251 5187 a Fm(jC)5 b(j)2370 5051 y Fh(X)2372 5229 y Fk(j)s Fj(2C)2503 5130 y Fn(v)2543 5142 y Fk(j)2579 5130 y Fn(:)630 b Fr(\(3.36\))p eop %%Page: 12 12 12 11 bop 456 251 a Fo(12)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(W)-7 b(e)35 b(next)g(in)n(tro)r(duce)f(a)h(sequence)f (of)h(decomp)r(ositions)f(of)g Fm(V)2445 462 y Fk(k)2521 450 y Fr(in)n(to)g(\()p Fn(\014)2779 420 y Fk(`)2812 450 y Fn(;)14 b(\034)2885 462 y Fk(`)2917 450 y Fr(\)-clusters,)36 b Fn(`)f Fr(=)456 550 y(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(`)717 520 y Fj(\003)754 550 y Fr(,)28 b Fn(\014)k Fr(as)26 b(in)i(\(3.21\))o(,)g(where)e(the)i(times)g Fn(\034)1985 562 y Fo(1)2045 550 y Fm(\025)23 b Fn(\034)2169 562 y Fo(2)2230 550 y Fm(\025)f Fn(:)14 b(:)g(:)23 b Fm(\025)g Fn(\034)2561 562 y Fk(`)2589 546 y Fd(\003)2655 550 y Fr(and)k(the)h(in)n(teger)e Fn(`)3268 520 y Fj(\003)3333 550 y Fr(are)456 649 y(de\014ned)i(b)n(y:)516 712 y Fh(8)516 787 y(>)516 812 y(>)516 837 y(<)516 986 y(>)516 1011 y(>)516 1036 y(:)631 779 y Fn(\034)667 791 y Fo(1)748 732 y Fn(:)728 779 y Fr(=)22 b Fn(T)864 791 y Fk(k)905 779 y Fn(;)631 935 y(\034)667 947 y Fk(`)p Fo(+1)827 888 y Fn(:)806 935 y Fr(=)h(inf)1009 818 y Fh(\032)1071 935 y Fn(t)g Fm(2)h Fr([)p Fn(T)1275 947 y Fk(k)q Fj(\000)p Fo(1)1400 935 y Fn(;)14 b(\034)1473 947 y Fk(`)1505 935 y Fr(])37 b(:)g Fn(\030)1661 947 y Fk(`)1693 935 y Fr(\()p Fn(t)p Fr(\))24 b Fn(<)1909 879 y(\014)1960 849 y Fk(`)1992 879 y Fn("T)p 1909 916 183 4 v 1979 992 a Fr(8)2101 935 y Fn(;)41 b(\021)2206 947 y Fk(`)2238 935 y Fr(\()p Fn(t)p Fr(\))24 b Fn(<)2454 879 y(\014)2505 849 y Fk(`)2537 879 y Fn("T)p 2454 916 V 2524 992 a Fr(8)2646 818 y Fh(\033)2708 935 y Fn(;)180 b(`)22 b Fm(\025)h Fr(1)p Fn(;)631 1084 y(`)666 1054 y Fj(\003)727 1084 y Fr(=)g(min)p Fm(f)p Fn(`)f Fm(2)h Fl(N)47 b Fr(:)37 b Fn(\034)1323 1096 y Fk(`)p Fo(+1)1463 1084 y Fr(=)22 b Fn(T)1599 1096 y Fk(k)q Fj(\000)p Fo(1)1725 1084 y Fm(g)p Fn(;)3232 932 y Fr(\(3.37\))456 1215 y(where:)824 1350 y Fn(\030)860 1362 y Fk(`)893 1350 y Fr(\()p Fn(t)p Fr(\))1091 1303 y Fn(:)1070 1350 y Fr(=)129 b(sup)1218 1423 y Fk(s)p Fj(2)p Fo([)p Fk(t;\034)1389 1432 y Fg(`)1417 1423 y Fo(])1450 1350 y Fr(max)1618 1283 y Fh(\010)1666 1350 y Fm(j)p Fn(v)1729 1362 y Fj(C)1773 1350 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)2018 1362 y Fj(C)2061 1350 y Fr(\()p Fn(\034)2129 1362 y Fk(`)2162 1350 y Fr(\))p Fm(j)37 b Fr(:)g Fm(C)27 b(2)d Fr(\005\()p Fn(\014)2609 1316 y Fk(`)2642 1350 y Fn(;)14 b(\034)2715 1362 y Fk(`)2747 1350 y Fr(\))2779 1283 y Fh(\011)2841 1350 y Fn(;)819 1554 y(\021)860 1566 y Fk(`)893 1554 y Fr(\()p Fn(t)p Fr(\))1091 1507 y Fn(:)1070 1554 y Fr(=)129 b(sup)1218 1628 y Fk(s)p Fj(2)p Fo([)p Fk(t;\034)1389 1637 y Fg(`)1417 1628 y Fo(])1450 1554 y Fr(max)1618 1487 y Fh(\010)1666 1554 y Fm(j)p Fn(v)1729 1566 y Fk(j)1765 1554 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)2010 1566 y Fj(C)2053 1554 y Fr(\()p Fn(s)p Fr(\))p Fm(j)37 b Fr(:)g Fn(j)28 b Fm(2)c(C)5 b Fn(;)41 b Fm(C)28 b(2)23 b Fr(\005\()p Fn(\014)2825 1520 y Fk(`)2858 1554 y Fn(;)14 b(\034)2931 1566 y Fk(`)2963 1554 y Fr(\))2995 1487 y Fh(\011)3057 1554 y Fn(;)456 1765 y Fr(and)22 b(setting)h Fn(\034)916 1777 y Fk(`)p Fo(+1)1056 1765 y Fr(=)f Fn(\034)1179 1777 y Fk(`)1234 1765 y Fr(if)i Fn(\021)1347 1777 y Fk(`)1379 1765 y Fr(\()p Fn(\034)1447 1777 y Fk(`)1479 1765 y Fr(\))g Fm(\025)e Fn(\014)1673 1734 y Fk(`)1706 1765 y Fn("T)9 b(=)p Fr(8.)34 b(Our)22 b(goal)f(is)i(no)n(w)f(to)h(sho)n(w)f(that,)i (b)n(y)f(c)n(ho)r(osing)456 1864 y Fn(K)28 b Fm(\025)23 b Fr(1)k(large)f(enough)h(in)h(the)g(de\014nition)g(\(3.6\))f(and)h Fn(a)2214 1876 y Fo(0)2278 1864 y Fr(small)g(enough,)f(w)n(e)g(ha)n(v)n (e:)1479 2052 y Fn(`)1514 2018 y Fj(\003)1575 2052 y Fm(\024)c(jV)1737 2064 y Fk(k)1777 2052 y Fm(j)g Fn(<)g(\032)1954 2064 y Fj(\003)2002 1996 y Fr(1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2351 1966 y Fo(2)p 2002 2033 386 4 v 2154 2109 a Fn(a)2198 2085 y Fo(2)2398 2052 y Fn(;)811 b Fr(\(3.38\))456 2215 y(\(the)26 b(second)g(inequalit)n(y)g(follo)n(ws)f(from)g (\(3.15\))h(since)g Fm(V)2237 2227 y Fk(k)2301 2215 y Fm(\032)c(J)2444 2227 y Fk(T)2483 2236 y Fg(k)2524 2215 y Fr(\).)37 b(W)-7 b(e)26 b(\014rst)g(note)g(that)h(if)f Fn(a)3407 2227 y Fo(0)456 2315 y Fr(is)h(small)g(enough)g(then,)h(for)g (an)n(y)e Fn(t)d Fm(2)h Fr([)p Fn(\034)1729 2327 y Fk(`)p Fo(+1)1845 2315 y Fn(;)14 b(\034)1918 2327 y Fk(`)1950 2315 y Fr(],)698 2506 y(min)851 2438 y Fh(\010)899 2506 y Fm(j)p Fn(v)962 2518 y Fk(j)997 2506 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1233 2518 y Fk(q)1270 2506 y Fr(\()p Fn(t)p Fr(\))p Fm(j)38 b Fr(:)f Fm(C)27 b(2)d Fr(\005\()p Fn(\014)1780 2471 y Fk(`)1813 2506 y Fn(;)14 b(\034)1886 2518 y Fk(`)1918 2506 y Fr(\))p Fn(;)42 b(j)28 b Fm(2)23 b(C)5 b Fn(;)41 b(q)26 b Fm(2)e(V)2461 2518 y Fk(k)2520 2506 y Fm(n)18 b(C)2629 2438 y Fh(\011)2700 2506 y Fm(\025)2797 2449 y Fn(\014)2848 2419 y Fk(`)2881 2449 y Fn("T)p 2797 2487 183 4 v 2868 2563 a Fr(2)3232 2506 y(\(3.39\))456 2664 y(and)739 2821 y(min)892 2754 y Fh(\010)940 2821 y Fm(j)p Fn(v)1003 2833 y Fk(j)1038 2821 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(v)1274 2833 y Fk(q)1311 2821 y Fr(\()p Fn(t)p Fr(\))p Fm(j)38 b Fr(:)f Fm(C)27 b(2)d Fr(\005\()p Fn(\014)1821 2787 y Fk(`)1854 2821 y Fn(;)14 b(\034)1927 2833 y Fk(`)1959 2821 y Fr(\))p Fn(;)42 b(j)28 b Fm(2)23 b(C)5 b Fn(;)41 b(q)26 b Fm(2)e(M)2551 2833 y Fk(k)2591 2754 y Fh(\011)2663 2821 y Fm(\025)2760 2765 y Fn(\014)t(aT)p 2760 2802 156 4 v 2817 2878 a Fr(2)2925 2821 y Fn(:)284 b Fr(\(3.40\))456 2984 y(T)-7 b(o)27 b(obtain)h(\(3.40\))f(w)n(e)h(ha)n(v)n(e)f(also)g(used)h(Lemma)f(3.1.) 38 b(In)28 b(fact,)g(b)n(y)h(\(3.20\))e(and)h(\(3.23\))o(,)h(if)f Fn(a)3407 2996 y Fo(0)456 3084 y Fr(is)f(small)g(enough,)g(for)g(an)n (y)g Fm(C)h(2)23 b Fr(\005\()p Fn(\014)1644 3053 y Fk(`)1677 3084 y Fn(;)14 b(\034)1750 3096 y Fk(`)1782 3084 y Fr(\))28 b(and)g Fn(t)23 b Fm(2)g Fr([)p Fn(\034)2194 3096 y Fk(`)p Fo(+1)2310 3084 y Fn(;)14 b(\034)2383 3096 y Fk(`)2416 3084 y Fr(],)652 3304 y(min)663 3358 y Fk(j)s Fj(2C)804 3304 y Fm(j)p Fn(v)867 3316 y Fk(j)902 3304 y Fr(\()p Fn(t)p Fr(\))p Fm(j)19 b(\000)37 b Fr(max)1121 3358 y Fk(q)r Fj(2M)1276 3367 y Fg(k)1327 3304 y Fm(j)p Fn(v)1390 3316 y Fk(q)1426 3304 y Fr(\()p Fn(t)p Fr(\))p Fm(j)24 b(\025)f Fn(\014)t(aT)30 b Fm(\000)1912 3200 y Fk(`)1940 3175 y Fd(\003)1974 3200 y Fj(\000)p Fo(1)1925 3225 y Fh(X)1929 3404 y Fk(`)p Fo(=1)2083 3248 y Fn(\014)2134 3218 y Fk(`)2166 3248 y Fn("T)p 2083 3285 183 4 v 2153 3361 a Fr(4)2298 3304 y Fm(\025)23 b Fn(\014)t(aT)29 b Fm(\000)2727 3248 y Fn(\014)t("T)p 2652 3285 301 4 v 2652 3361 a Fr(4\(1)18 b 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Fk(`)1555 4158 y Fr(\()p Fn(\034)1623 4170 y Fk(`)p Fo(+1)1740 4158 y Fr(\))23 b(=)1893 4102 y Fn(\014)1944 4072 y Fk(`)1976 4102 y Fn("T)p 1893 4139 V 1963 4215 a Fr(8)2251 4158 y Fm(8)14 b Fn(`)21 b Fm(\024)i Fr(min)p Fm(f)p Fn(`)2671 4124 y Fj(\003)2708 4158 y Fn(;)14 b Fr(2)p Fm(jV)2861 4170 y Fk(k)2902 4158 y Fm(jg)p Fn(:)242 b Fr(\(3.42\))456 4321 y(W)-7 b(e)32 b(c)n(ho)r(ose)e Fn(a)914 4333 y Fo(0)981 4321 y Fn(<)f Fr(1)p Fn(=)p Fr(\(8)p Fn(\032)1276 4333 y Fj(\003)1313 4321 y Fr(\))j(with)g Fn(\032)1613 4333 y Fj(\003)1683 4321 y Fr(as)f(in)h(\(3.15\))f(so)g (that,)i(b)n(y)e(the)h(second)f(inequalit)n(y)g(in)456 4420 y(\(3.38\))o(,)1302 4520 y Fn(\014)1353 4490 y Fk(`)1385 4520 y Fn("T)p 1302 4557 V 1303 4633 a Fr(8)p Fm(jV)1419 4645 y Fk(k)1460 4633 y Fm(j)1517 4576 y(\025)1615 4520 y Fn(\014)1666 4490 y Fk(`)1698 4520 y Fn("T)p 1615 4557 V 1685 4633 a Fr(8)1929 4520 y Fn(\032)1972 4532 y Fj(\003)2010 4520 y Fn(a)2054 4490 y Fo(2)p 1817 4557 386 4 v 1817 4633 a Fr(1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2166 4609 y Fo(2)2236 4576 y Fn(>)22 b(\014)2374 4542 y Fk(`)p Fo(+1)2491 4576 y Fn("T)7 b(:)456 4742 y Fr(Then,)27 b(from)h(\(3.42\))f(it)h(follo)n(ws)e(that:)925 4881 y Fm(j)p Fr(\005\()p Fn(\014)1093 4847 y Fk(`)p Fo(+1)1210 4881 y Fn(;)14 b(\034)1283 4893 y Fk(`)p Fo(+1)1399 4881 y Fr(\))p Fm(j)24 b(\025)f(j)p Fr(\005\()p Fn(\014)1734 4847 y Fk(`)1766 4881 y Fn(;)14 b(\034)1839 4893 y Fk(`)1872 4881 y Fr(\))p Fm(j)k Fr(+)g(1)166 b Fm(8)14 b Fn(`)22 b Fm(\024)g Fr(min)q Fm(f)p Fn(`)2657 4847 y Fj(\003)2694 4881 y Fn(;)14 b Fr(2)p Fm(jV)2847 4893 y Fk(k)2887 4881 y Fm(jg)p Fn(;)456 5016 y Fr(so)22 b(that)h(there)g(is)f(only)h(one)f (particle)h(in)g(eac)n(h)f(cluster)g(of)h(\005\()p Fn(\014)2407 4986 y Fk(`)2440 5016 y Fn(;)14 b(\034)2513 5028 y Fk(`)2545 5016 y Fr(\))24 b(if)f Fn(`)g Fm(\025)f(jV)2891 5028 y Fk(k)2932 5016 y Fm(j)p Fr(.)35 b(Th)n(us)23 b Fn(\021)3260 5028 y Fk(`)3315 5016 y Fm(\021)g Fr(0)456 5116 y(for)31 b Fn(`)e Fm(\025)h(jV)820 5128 y Fk(k)861 5116 y Fm(j)p Fr(,)i(whic)n(h)g(implies,)h(b)n(y)g(\(3.41\))e(and)g(the)h (de\014nition)g(\(3.37\),)g(that)g Fn(\034)3029 5128 y Fk(`)p Fo(+1)3176 5116 y Fr(=)d Fn(T)3319 5128 y Fk(k)q Fj(\000)p Fo(1)456 5216 y Fr(for)e Fn(`)22 b Fm(\025)h(jV)802 5228 y Fk(k)843 5216 y Fm(j)p Fr(,)28 b(whence)f(the)h(\014rst)f (inequalit)n(y)h(in)f(\(3.38\))g(follo)n(ws.)p eop %%Page: 13 13 13 12 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(13)555 450 y Fr(T)-7 b(o)36 b(pro)n(v)n(e)f(\(3.41\))h(w)n(e)g(apply) h(the)g(p)r(erturbativ)n(e)e(argumen)n(t)h(\014rstly)g(used)h(in)f([10) o(].)64 b(Let)456 550 y Fm(C)27 b(2)d Fr(\005\()p Fn(\014)751 520 y Fk(`)783 550 y Fn(;)14 b(\034)856 562 y Fk(`)889 550 y Fr(\))28 b(and)f(de\014ne:)915 745 y Fn(p)957 757 y Fj(C)1000 745 y Fr(\()p Fn(t)p Fr(\))1198 698 y Fn(:)1177 745 y Fr(=)83 b Fn(v)1365 757 y Fj(C)1408 745 y Fr(\()p Fn(t)p Fr(\))19 b(+)1641 689 y(1)p 1614 726 95 4 v 1614 802 a Fm(jC)5 b(j)1732 666 y Fh(X)1735 844 y Fk(j)s Fj(2C)1920 666 y Fh(X)1866 848 y Fk(q)r Fj(2V)1985 857 y Fg(k)2022 848 y Fj(nC)2108 632 y Fh(Z)2191 652 y Fk(t)2155 820 y Fo(0)2221 745 y Fn(ds)14 b Fm(r)p Fr(\010\()p Fn(x)2525 757 y Fk(j)2561 745 y Fr(\()p Fn(s)p Fr(\))k Fm(\000)g Fn(x)2812 757 y Fk(q)2850 745 y Fr(\()p Fn(s)p Fr(\)\))1325 1022 y(+)1440 966 y(1)p 1413 1003 V 1413 1079 a Fm(jC)5 b(j)1532 944 y Fh(X)1534 1122 y Fk(j)s Fj(2C)1701 944 y Fh(X)1665 1122 y Fk(q)r Fj(2M)1820 1131 y Fg(k)1881 966 y Fr(\010\()p Fn(x)2020 978 y Fk(j)2055 966 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(x)2298 978 y Fk(q)2335 966 y Fr(\()p Fn(t)p Fr(\)\))p 1881 1003 582 4 v 1950 1079 a Fn(v)1990 1091 y Fk(j)2025 1079 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(v)2261 1091 y Fk(q)2298 1079 y Fr(\()p Fn(t)p Fr(\))2472 1022 y Fn(:)737 b Fr(\(3.43\))456 1245 y(F)-7 b(rom)27 b(the)h(equations)f(of)g(motion)g(and)h(recalling)e (that)i(\010)g(is)f(symmetric)g(w)n(e)h(ha)n(v)n(e:)761 1426 y(_)-39 b Fn(p)787 1438 y Fj(C)829 1426 y Fr(\()p Fn(t)p Fr(\))84 b(=)1191 1369 y(1)p 1165 1407 95 4 v 1165 1483 a Fm(jC)5 b(j)1283 1347 y Fh(X)1286 1525 y Fk(j)s Fj(2C)1453 1347 y Fh(X)1417 1525 y Fk(q)r Fj(2M)1572 1534 y Fg(k)1644 1347 y Fh(X)1622 1525 y Fk(s)p Fo(:)p Fk(s)p Fj(6)p Fo(=)p Fk(j)1809 1369 y Fr(\010\()p Fn(x)1948 1381 y Fk(j)1984 1369 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(x)2227 1381 y Fk(q)2264 1369 y Fr(\()p Fn(t)p Fr(\)\))p Fm(r)p Fr(\010\()p Fn(x)2598 1381 y Fk(j)2635 1369 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(x)2878 1381 y Fk(s)2914 1369 y Fr(\()p Fn(t)p Fr(\)\))p 1809 1407 1232 4 v 2153 1483 a(\()p Fn(v)2225 1495 y Fk(j)2260 1483 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(v)2496 1495 y Fk(q)2533 1483 y Fr(\()p Fn(t)p Fr(\)\))2659 1459 y Fo(2)1155 1697 y Fm(\000)1270 1641 y Fr(1)p 1243 1678 95 4 v 1243 1754 a Fm(jC)5 b(j)1361 1618 y Fh(X)1364 1796 y Fk(j)s Fj(2C)1531 1618 y Fh(X)1495 1796 y Fk(q)r Fj(2M)1650 1805 y Fg(k)1723 1618 y Fh(X)1701 1797 y Fk(s)p Fo(:)p Fk(s)p Fj(6)p Fo(=)p Fk(q)1889 1641 y Fr(\010\()p Fn(x)2028 1653 y Fk(j)2064 1641 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(x)2307 1653 y Fk(q)2344 1641 y Fr(\()p Fn(t)p Fr(\)\))p Fm(r)p Fr(\010\()p Fn(x)2678 1653 y Fk(q)2717 1641 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(x)2960 1653 y Fk(s)2996 1641 y Fr(\()p Fn(t)p Fr(\)\))p 1889 1678 1234 4 v 2234 1754 a(\()p Fn(v)2306 1766 y Fk(j)2341 1754 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(v)2577 1766 y Fk(q)2614 1754 y Fr(\()p Fn(t)p Fr(\)\))2740 1730 y Fo(2)3132 1697 y Fn(:)456 1925 y Fr(Then,)27 b(recalling)h(\(3.40\))o(,)700 2101 y Fm(j)16 b Fr(_)-39 b Fn(p)765 2113 y Fj(C)807 2101 y Fr(\()p Fn(t)p Fr(\))p Fm(j)24 b(\024)f(k)p Fr(\010)p Fm(k)1180 2113 y Fj(1)1249 2101 y Fm(kr)p Fr(\010)p Fm(k)1462 2113 y Fj(1)1649 2045 y Fr(4)p 1542 2082 258 4 v 1542 2158 a(\()p Fn(\014)t(aT)12 b Fr(\))1762 2134 y Fo(2)1845 2045 y Fr(1)p 1819 2082 95 4 v 1819 2158 a Fm(jC)5 b(j)1937 2022 y Fh(X)1940 2200 y Fk(j)s Fj(2C)2071 2101 y Fn(N)2138 2113 y Fk(j)2172 2101 y Fr(\()p Fn(t)p Fr(\))2266 2067 y Fo(2)2470 2101 y Fm(8)14 b Fn(t)22 b Fm(2)i Fr([)p Fn(\034)2721 2113 y Fk(`)p Fo(+1)2837 2101 y Fn(;)14 b(\034)2910 2113 y Fk(`)2942 2101 y Fr(])p Fn(:)244 b Fr(\(3.44\))456 2328 y(where)24 b Fn(N)760 2340 y Fk(j)794 2328 y Fr(\()p Fn(t)p Fr(\))i(is)e(the)h(n)n(um)n(b)r(er)g(of)f (particles)g(whic)n(h)h(are)f(con)n(tained)g(at)g(time)h Fn(t)g Fr(in)g(the)g(in)n(terv)-5 b(al)456 2428 y([)p Fn(x)526 2440 y Fk(j)561 2428 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fr(2)p Fn(;)c(x)883 2440 y Fk(j)918 2428 y Fr(\()p Fn(t)p Fr(\))19 b(+)f(2].)37 b(Then,)27 b(for)g(an)n(y)g Fn(t)c Fm(2)h Fr([)p Fn(\034)1953 2440 y Fk(`)p Fo(+1)2069 2428 y Fn(;)14 b(\034)2142 2440 y Fk(`)2174 2428 y Fr(],)777 2615 y Fm(j)p Fn(p)842 2627 y Fj(C)885 2615 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(p)1123 2627 y Fj(C)1166 2615 y Fr(\()p Fn(\034)1234 2627 y Fk(`)1266 2615 y Fr(\))p Fm(j)24 b(\024)1538 2559 y Fn(C)p 1442 2596 258 4 v 1442 2672 a Fr(\()p Fn(\014)t(aT)12 b Fr(\))1662 2648 y Fo(2)1746 2559 y Fr(1)p 1719 2596 95 4 v 1719 2672 a Fm(jC)5 b(j)1838 2536 y Fh(X)1840 2714 y Fk(j)s Fj(2C)2088 2615 y Fr(sup)1971 2689 y Fk(s)p Fj(2)p Fo([)p Fk(T)2105 2698 y Fg(k)q Fd(\000)p Fe(1)2215 2689 y Fk(;T)2274 2698 y Fg(k)2310 2689 y Fo(])2343 2615 y Fn(N)2410 2627 y Fk(j)2445 2615 y Fr(\()p Fn(s)p Fr(\))2590 2502 y Fh(Z)2673 2523 y Fk(\034)2704 2532 y Fg(`)2636 2691 y Fk(\034)2667 2700 y Fg(`)p Fe(+1)2770 2615 y Fn(dt)2843 2581 y Fj(0)2881 2615 y Fn(N)2948 2627 y Fk(j)2982 2615 y Fr(\()p Fn(t)3044 2581 y Fj(0)3068 2615 y Fr(\))p Fn(:)456 2842 y Fr(By)28 b(\(3.31\))o(:)1085 2946 y Fn(N)1152 2958 y Fk(j)1187 2946 y Fr(\()p Fn(t)p Fr(\))1281 2912 y Fo(2)1342 2946 y Fm(\024)22 b Fn(C)6 b(Q)p Fr(\()p Fn(X)h Fr(\()p Fn(t)p Fr(\);)14 b Fn(x)1846 2958 y Fk(j)1882 2946 y Fr(\()p Fn(t)p Fr(\))p Fn(;)g Fr(2\))23 b Fm(\024)g Fn(C)6 b Fr([1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))2635 2912 y Fo(2)2672 2946 y Fr(])p Fn(T)2756 2912 y Fo(2)2792 2946 y Fn(;)456 3066 y Fr(where)26 b(in)i(the)f(last)g(inequalit)n(y)g(w)n(e)g(used)g(\(3.3\))g(with)g Fm(j)p Fn(x)2243 3078 y Fk(j)2279 3066 y Fr(\()p Fn(t)p Fr(\))p Fm(j)c(\024)g(j)p Fn(x)2577 3078 y Fk(i)2605 3066 y Fm(j)18 b Fr(+)f Fn(C)6 b Fr([1)17 b(+)h Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])p Fn(T)3248 3036 y Fo(2)3311 3066 y Fr(and)456 3166 y(the)30 b(de\014nition)g(\(3.6\))o(.)43 b(W)-7 b(e)30 b(next)g(observ)n(e)e(that,)j(during)e(the)h(time)g([)p Fn(\034)2706 3178 y Fk(`)p Fo(+1)2822 3166 y Fn(;)14 b(\034)2895 3178 y Fk(`)2927 3166 y Fr(],)31 b(the)f(distance)456 3265 y(b)r(et)n(w)n(een)d(the)h(particle)g Fn(j)g Fm(2)23 b(C)33 b Fr(and)27 b(a)h(particle)f Fn(q)f Fm(2)e(M)2210 3277 y Fk(k)2278 3265 y Fr(can)j(b)r(e)h(smaller)f(than)h(2)f(only)h (for)f(a)456 3366 y(time)e(not)g(longer)f(than)h(4)p Fn(=)p Fr(\()p Fn(\014)t(aT)12 b Fr(\).)35 b(Moreo)n(v)n(er,)22 b(the)k(total)e(n)n(um)n(b)r(er)2632 3345 y(~)2608 3366 y Fn(N)2675 3378 y Fk(k)2740 3366 y Fr(of)h(particles)f Fn(q)i Fm(2)e(M)3404 3378 y Fk(k)456 3466 y Fr(suc)n(h)31 b(that)i Fm(j)p Fn(x)902 3478 y Fk(j)937 3466 y Fr(\()p Fn(t)p Fr(\))22 b Fm(\000)f Fn(x)1186 3478 y Fk(q)1223 3466 y Fr(\()p Fn(t)p Fr(\))p Fm(j)31 b(\024)f Fr(2)i(for)f(some)h Fn(t)e Fm(2)h Fr([)p Fn(T)2102 3478 y Fk(k)q Fj(\000)p Fo(1)2227 3466 y Fn(;)14 b(T)2313 3478 y Fk(k)2354 3466 y Fr(])32 b(can)g(b)r(e)g(b)r(ounded)g(similarly)g(to)480 3545 y(\026)456 3566 y Fn(N)523 3578 y Fk(k)591 3566 y Fr(in)c(\(3.32\))o(.)37 b(Then:)528 3653 y Fh(Z)611 3673 y Fk(\034)642 3682 y Fg(`)574 3841 y Fk(\034)605 3850 y Fg(`)p Fe(+1)709 3766 y Fn(ds)14 b(N)872 3778 y Fk(j)906 3766 y Fr(\()p Fn(s)p Fr(\))84 b(=)1258 3687 y Fh(X)1240 3865 y Fk(q)r Fj(2V)1359 3874 y Fg(k)1410 3653 y Fh(Z)1493 3673 y Fk(\034)1524 3682 y Fg(`)1456 3841 y Fk(\034)1487 3850 y Fg(`)p Fe(+1)1590 3766 y Fn(ds)24 b Fr(~)-52 b Fn(\037)1738 3778 y Fk(j;q)1822 3766 y Fr(\()p Fn(s)p Fr(\))19 b(+)2063 3687 y Fh(X)2027 3865 y Fk(q)r Fj(2M)2182 3874 y Fg(k)2232 3653 y Fh(Z)2315 3673 y Fk(\034)2346 3682 y Fg(`)2278 3841 y Fk(\034)2309 3850 y Fg(`)p Fe(+1)2413 3766 y Fn(ds)24 b Fr(~)-52 b Fn(\037)2561 3778 y Fk(j;q)2645 3766 y Fr(\()p Fn(s)p Fr(\))23 b Fm(\024)g(jV)2933 3778 y Fk(k)2974 3766 y Fm(j)p Fn("T)29 b Fr(+)3210 3709 y(4)3276 3688 y(~)3252 3709 y Fn(N)3319 3721 y Fk(k)p 3207 3747 156 4 v 3207 3823 a Fn(\014)t(aT)1093 4041 y Fm(\024)82 b Fn(C)1316 3985 y Fr(1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))1665 3955 y Fo(2)p 1316 4022 386 4 v 1468 4098 a Fn(a)1512 4074 y Fo(2)1711 4041 y Fn("T)30 b Fr(+)18 b Fn(C)1977 3978 y Fm(p)p 2046 3978 39 4 v 63 x Fn("T)2155 3985 y Fr(1)g(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p 2155 4022 349 4 v 2281 4098 a Fn(\014)t(a)2536 4041 y Fm(\024)23 b Fn(C)2689 3978 y Fm(p)p 2758 3978 39 4 v 63 x Fn("T)2867 3985 y Fr(1)18 b(+)g Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p 2867 4022 349 4 v 2994 4098 a Fn(\014)t(a)3226 4041 y(;)456 4223 y Fr(where)35 b(~)-52 b Fn(\037)746 4235 y Fk(j;q)830 4223 y Fr(\()p Fn(s)p Fr(\))23 b(=)g Fn(\037)p Fr(\()p Fm(j)p Fn(x)1198 4235 y Fk(j)1234 4223 y Fr(\()p Fn(s)p Fr(\))16 b Fm(\000)e Fn(x)1479 4235 y Fk(q)1517 4223 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fr(2\))j(and)g(w)n(e)f(used)h (the)h(second)e(inequalit)n(y)h(in)g(\(3.38\))o(.)456 4322 y(In)h(conclusion:)1557 4447 y Fm(j)p Fn(p)1622 4459 y Fj(C)1664 4447 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(p)1902 4459 y Fj(C)1945 4447 y Fr(\()p Fn(\034)2013 4459 y Fk(`)2046 4447 y Fr(\))p Fm(j)23 b(\024)2234 4390 y Fn(C)p 2222 4427 89 4 v 2222 4504 a(\014)2273 4480 y Fo(2)2320 4447 y Fn(;)456 4611 y Fr(where)18 b(w)n(e)h(used)g(the)h (de\014nitions)f(\(3.21\))g(and)g(\(3.22\))o(.)34 b(On)19 b(the)h(other)f(hand,)i(b)n(y)d(the)i(de\014nition)456 4710 y(\(3.43\))26 b(and)i(using)g(\(3.39\))f(and)h(\(3.40\))o(,)797 4897 y Fm(j)p Fn(v)860 4909 y Fj(C)904 4897 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(p)1142 4909 y Fj(C)1185 4897 y Fr(\()p Fn(t)p Fr(\))p Fm(j)83 b(\024)1569 4841 y Fr(1)p 1543 4878 95 4 v 1543 4954 a Fm(jC)5 b(j)1661 4819 y Fh(X)1664 4997 y Fk(j)s Fj(2C)1795 4780 y Fh(\024)1849 4841 y Fr(2)p Fm(kr)p Fr(\010)p Fm(k)2104 4853 y Fj(1)p 1849 4878 325 4 v 1920 4954 a Fn(\014)1971 4930 y Fk(`)2003 4954 y Fn("T)2183 4897 y Fm(jV)2257 4909 y Fk(k)2298 4897 y Fm(j)18 b Fr(+)2432 4841 y(2)p Fm(k)p Fr(\010)p Fm(k)2618 4853 y Fj(1)p 2432 4878 255 4 v 2482 4954 a Fn(\014)t(aT)2697 4897 y(N)2764 4909 y Fk(j)2799 4897 y Fr(\()p Fn(t)p Fr(\))2893 4780 y Fh(\025)1385 5170 y Fm(\024)83 b Fn(C)1612 5053 y Fh(\024)1778 5114 y Fr(1)p 1666 5151 267 4 v 1666 5227 a Fn(\014)1717 5203 y Fk(`)p Fo(+1)1833 5227 y Fn("T)1960 5170 y Fr(+)2077 5114 y(1)p 2053 5151 89 4 v 2053 5227 a Fn(\014)2104 5203 y Fo(2)2152 5053 y Fh(\025)2541 5170 y Fm(8)14 b Fn(t)23 b Fm(2)g Fr([)p Fn(T)2805 5182 y Fk(k)q Fj(\000)p Fo(1)2930 5170 y Fn(;)14 b(T)3016 5182 y Fk(k)3057 5170 y Fr(])p Fn(;)129 b Fr(\(3.45\))p eop %%Page: 14 14 14 13 bop 456 251 a Fo(14)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(where)27 b(w)n(e)g(again)f(used)i(\(3.21\))f(and)g (\(3.22\))o(.)37 b(Then,)28 b(for)f(an)n(y)g Fn(t)c Fm(2)g Fr([)p Fn(\034)2616 462 y Fk(`)p Fo(+1)2733 450 y Fn(;)14 b(\034)2806 462 y Fk(`)2838 450 y Fr(],)647 589 y Fm(j)p Fn(v)710 601 y Fj(C)753 589 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)989 601 y Fj(C)1033 589 y Fr(\()p Fn(\034)1101 601 y Fk(`)1133 589 y Fr(\))p Fm(j)84 b(\024)e(j)p Fn(v)1482 601 y Fj(C)1526 589 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(p)1764 601 y Fj(C)1806 589 y Fr(\()p Fn(t)p Fr(\))p Fm(j)i Fr(+)e Fm(j)p Fn(p)2091 601 y Fj(C)2133 589 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(p)2371 601 y Fj(C)2414 589 y Fr(\()p Fn(\034)2482 601 y Fk(`)2515 589 y Fr(\))p Fm(j)h Fr(+)f Fm(j)p Fn(p)2737 601 y Fj(C)2780 589 y Fr(\()p Fn(\034)2848 601 y Fk(`)2880 589 y Fr(\))h Fm(\000)f Fn(v)3054 601 y Fj(C)3097 589 y Fr(\()p Fn(\034)3165 601 y Fk(`)3198 589 y Fr(\))p Fm(j)1272 763 y(\024)82 b Fn(C)1498 646 y Fh(\024)1664 707 y Fr(1)p 1552 744 267 4 v 1552 820 a Fn(\014)1603 796 y Fk(`)p Fo(+1)1719 820 y Fn("T)1847 763 y Fr(+)1963 707 y(1)p 1940 744 89 4 v 1940 820 a Fn(\014)1991 796 y Fo(2)2038 646 y Fh(\025)2096 763 y Fn(:)1113 b Fr(\(3.46\))456 951 y(Recalling)30 b(the)i(de\014nitions)g(\(3.6\),)g(\(3.21\),)g(\(3.22\))o(,)h(and)e (the)h(second)f(inequalit)n(y)g(in)h(\(3.38\))o(,)456 1051 y(if)f Fn(K)k Fm(\025)28 b Fr(1)j(is)g(large)e(enough)i(and)g Fn(a)1599 1063 y Fo(0)1667 1051 y Fr(is)g(small)g(enough)f(the)i (inequalit)n(y)f(\(3.46\))g(implies)g(the)456 1151 y(b)r(ound)d (\(3.41\))o(.)555 1250 y(W)-7 b(e)29 b(can)e(no)n(w)h(conclude)g(the)g (pro)r(of)g(of)g(the)g(theorem.)38 b(F)-7 b(rom)28 b(the)g(previous)f (analysis)g(the)456 1350 y(v)n(elo)r(cit)n(y)d(v)-5 b(ariation)25 b(of)g(a)g(particle)g Fn(j)j Fm(2)c(V)1756 1362 y Fk(k)1822 1350 y Fr(in)i(the)g(time)g(in)n(terv)-5 b(al)25 b([)p Fn(T)2615 1362 y Fk(k)q Fj(\000)p Fo(1)2740 1350 y Fn(;)14 b(T)2826 1362 y Fk(k)2867 1350 y Fr(])25 b(is)h(b)r(ounded)g(b)n(y:)640 1512 y Fm(j)p Fn(v)703 1524 y Fk(j)738 1512 y Fr(\()p Fn(T)819 1524 y Fk(k)q Fj(\000)p Fo(1)945 1512 y Fr(\))18 b Fm(\000)g Fn(v)1118 1524 y Fk(j)1154 1512 y Fr(\()p Fn(T)1235 1524 y Fk(k)1275 1512 y Fr(\))p Fm(j)24 b(\024)1441 1433 y Fh(X)1487 1611 y Fk(`)1561 1445 y Fj(\003)1599 1512 y Fm(j)p Fn(v)1662 1524 y Fk(j)1698 1512 y Fr(\()p Fn(\034)1766 1524 y Fk(`)p Fo(+1)1882 1512 y Fr(\))19 b Fm(\000)f Fn(v)2056 1524 y Fk(j)2091 1512 y Fr(\()p Fn(\034)2159 1524 y Fk(`)2192 1512 y Fr(\))p Fm(j)806 1747 y(\024)893 1668 y Fh(X)939 1847 y Fk(`)1013 1680 y Fj(\003)1065 1747 y Fr(\()q Fm(j)p Fn(v)1161 1759 y Fk(j)1196 1747 y Fr(\()p Fn(\034)1264 1759 y Fk(`)p Fo(+1)1380 1747 y Fr(\))h Fm(\000)f Fn(v)1554 1759 y Fj(C)1590 1768 y Fg(`)1624 1747 y Fr(\()p Fn(\034)1692 1759 y Fk(`)p Fo(+1)1808 1747 y Fr(\))p Fm(j)h Fr(+)f Fm(j)p Fn(v)2028 1759 y Fj(C)2064 1768 y Fg(`)2097 1747 y Fr(\()p Fn(\034)2165 1759 y Fk(`)p Fo(+1)2282 1747 y Fr(\))h Fm(\000)f Fn(v)2456 1759 y Fj(C)2492 1768 y Fg(`)2525 1747 y Fr(\()p Fn(\034)2593 1759 y Fk(`)2626 1747 y Fr(\))p Fm(j)g Fr(+)g Fm(j)p Fn(v)2845 1759 y Fj(C)2881 1768 y Fg(`)2915 1747 y Fr(\()p Fn(\034)2983 1759 y Fk(`)3015 1747 y Fr(\))h Fm(\000)f Fn(v)3189 1759 y Fk(j)3224 1747 y Fr(\()p Fn(\034)3292 1759 y Fk(`)3325 1747 y Fr(\))p Fm(j)p Fr(\))806 2044 y Fm(\024)893 1941 y Fk(`)921 1916 y Fd(\003)956 1941 y Fj(\000)p Fo(1)907 1966 y Fh(X)911 2144 y Fk(`)p Fo(=1)1065 1988 y Fr(3)p Fn(\014)1158 1958 y Fk(`)1189 1988 y Fn("T)p 1065 2025 224 4 v 1156 2101 a Fr(8)1321 2044 y Fm(\024)1473 1988 y Fr(3)p Fn(\014)t("T)p 1419 2025 301 4 v 1419 2101 a Fr(8\(1)g Fm(\000)g Fn(\014)t Fr(\))1752 2044 y Fm(\024)23 b Fn(")1889 1988 y(aT)p 1889 2025 104 4 v 1920 2101 a Fr(2)2002 2044 y Fn(:)1207 b Fr(\(3.47\))456 2270 y(where)24 b Fm(C)737 2282 y Fk(`)793 2270 y Fr(is)h(the)g(cluster)g(of)g(\005\()p Fn(\014)1517 2240 y Fk(`)1549 2270 y Fn(;)14 b(\034)1622 2282 y Fk(`)1655 2270 y Fr(\))25 b(whic)n(h)f(con)n(tains)g(the)h Fn(j)5 b Fr(-th)25 b(particle)g(and)3035 2208 y Fh(P)3123 2295 y Fk(`)3155 2229 y Fj(\003)3218 2270 y Fr(means)456 2370 y(the)30 b(sum)g(o)n(v)n(er)e(those)i Fn(`)d Fm(2)g(f)p Fr(1;)14 b Fn(:)g(:)g(:)f Fr(;)h Fn(`)1626 2340 y Fj(\003)1683 2370 y Fm(\000)20 b Fr(1)p Fm(g)29 b Fr(for)h(whic)n(h)g Fn(\034)2287 2382 y Fk(`)p Fo(+1)2430 2370 y Fn(<)d(\034)2558 2382 y Fk(`)2620 2370 y Fr(\(for)j(the)g(last)g(inequalit)n(y)456 2470 y(recall)20 b(that)i Fn(a)889 2482 y Fo(0)949 2470 y Fn(<)h Fr(1)p Fn(=)p Fr(8)d(whence)i Fn(\014)28 b(<)22 b(a=)p Fr(8\).)34 b(In)22 b(particular,)g(the)g(previous)e(b)r(ound)j (guaran)n(tees)456 2569 y(that)h(the)h Fn(i)p Fr(-th)f(particle,)g (whose)g(v)n(elo)r(cit)n(y)f(at)h(time)h Fn(T)35 b Fr(is)24 b(assumed)g(bigger)f(than)h Fn(aT)12 b Fr(,)24 b(remains)456 2669 y Fp(fast)p Fr(,)i(i.e.)f(it)g(b)r(elongs)f(to)h Fm(V)1290 2681 y Fk(k)1356 2669 y Fr(for)f(all)h Fn(k)h Fr(=)c(0)p Fn(;)14 b(:)g(:)g(:)f(;)h(")2014 2639 y Fj(\000)p Fo(1)2128 2669 y Fr(\(in)25 b(fact)g(w)n(e)g(obtained)g Fm(j)p Fn(v)2937 2681 y Fk(i)2965 2669 y Fr(\(0\))p Fm(j)e Fn(>)f(aT)9 b(=)p Fr(2\).)456 2768 y(The)27 b(theorem)g(is)h(th)n(us)f (pro)n(v)n(ed.)1898 b Ff(\003)456 2920 y Fv(Remark)30 b(3.1.)39 b Fr(The)27 b(dep)r(endence)g(of)g Fn(T)1764 2932 y Fk(a)1804 2920 y Fr(\()p Fn(x)1883 2932 y Fk(i)1911 2920 y Fn(;)14 b(X)7 b Fr(\))27 b(on)f(the)h(small)g(parameter)e Fn(a)i Fr(is)g(v)n(ery)f(bad:)456 3020 y(w)n(e)18 b(ha)n(v)n(e)f(to)i (w)n(ait)f(a)g(sup)r(er-exp)r(onen)n(tially)g(large)f(\(w.r.t.)i Fn(a)2288 2990 y Fj(\000)p Fo(1)2377 3020 y Fr(\))g(time)g(to)g(catc)n (h)f(the)h(asymptotic)456 3120 y(estimate)h(on)h(the)g Fn(i)p Fr(-th)f(particle)h(v)n(elo)r(cit)n(y)-7 b(.)33 b(On)21 b(the)g(other)f(hand,)i(in)f(our)f(strategy)-7 b(,)21 b(this)g(c)n(hoice)456 3219 y(is)f(a)g(useful)g(mathematical)g (device)g(to)g(con)n(trol)f(the)i(e\013ect)f(of)g(the)h(m)n(utual)f(in) n(teraction)g(among)456 3319 y(the)31 b(fast)f(particles.)46 b(In)30 b(fact)h(it)g(guaran)n(tees)e(an)h Fc(apriori)f Fr(b)r(ound)i(on)g(the)g(maximal)f(n)n(um)n(b)r(er)456 3418 y(of)38 b(re\014nemen)n(ts)g(in)n(to)f(small)h(clusters)g(are)f (needed)h(to)g(follo)n(w)g(the)g(ev)n(olution)g(of)g(the)g(fast)456 3518 y(particles)26 b(during)i(the)g(time)g(in)n(terv)-5 b(al)27 b([)p Fn(T)1756 3530 y Fk(k)q Fj(\000)p Fo(1)1881 3518 y Fn(;)14 b(T)1967 3530 y Fk(k)2007 3518 y Fr(].)456 3670 y Fv(Remark)23 b(3.2.)34 b Fr(It)21 b(is)g(w)n(orth)n(while)f(to)g (emphasize)h(here)f(the)h(follo)n(wing)f(fact.)35 b(F)-7 b(or)20 b(an)n(y)g(in)n(teger)456 3769 y Fn(n)p Fr(,)27 b(b)n(y)g(de\014ning)1109 3920 y Fn(T)1158 3932 y Fk(a)1197 3920 y Fr(\()p Fn(X)r(;)14 b(w)r Fr(\))24 b(=)f Fn(a)1586 3886 y Fj(\000)p Fk(K)t Fo([1+)p Fk(Q)p Fo(\()p Fk(X)5 b Fo(\))1964 3860 y Fe(2)p Fg(n)p Fe(+1)2103 3886 y Fo(])p Fk(=a)2192 3860 y Fe(2)p Fg(n)2265 3845 y Fh(p)p 2348 3845 420 4 v 75 x Fr(log)q(\()p Fn(e)18 b Fr(+)g Fm(j)p Fn(w)r Fm(j)p Fr(\))q Fn(;)456 4058 y Fr(instead)31 b(of)38 b(\(3.6\))31 b(\(whic)n(h)g(corresp)r(onds)f(to)h(the)h(case)e Fn(n)g Fr(=)f(1\),)j(and)f(decomp)r(osing)g Fm(V)3232 4070 y Fk(k)3304 4058 y Fr(in)n(to)456 4158 y(\()p Fn(\016)528 4128 y Fk(`)560 4158 y Fn(;)14 b(\034)633 4170 y Fk(`)665 4158 y Fr(\)-clusters)23 b(with)i Fn(\016)h Fr(=)c Fn(\020)6 b(\014)1451 4128 y Fk(n)1521 4158 y Fr(\()p Fn(\020)30 b(>)23 b Fr(0)g(a)h(\014xed)f(parameter\),)h(w)n(e)f(analogously)f(pro) n(v)n(e)g(that,)456 4258 y(pro)n(vided)k Fn(a)841 4270 y Fo(0)906 4258 y Fr(small)h(enough)g(and)h Fn(K)33 b Fr(large)26 b(enough,)h(for)g(an)n(y)g Fn(j)h Fm(2)23 b(V)2662 4270 y Fk(k)2731 4258 y Fr(and)k Fn(t)c Fm(2)g Fr([)p Fn(T)3095 4270 y Fk(k)3136 4258 y Fn(;)14 b(T)3222 4270 y Fk(k)q Fj(\000)p Fo(1)3347 4258 y Fr(],)1321 4439 y Fm(j)p Fn(v)1384 4451 y Fk(j)1419 4439 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1655 4451 y Fk(j)1690 4439 y Fr(\()p Fn(T)1771 4451 y Fk(k)1812 4439 y Fr(\))p Fm(j)23 b(\024)2042 4383 y Fr(3)p Fn(\016)s("T)p 1988 4420 290 4 v 1988 4496 a Fr(8\(1)18 b Fm(\000)g Fn(\016)s Fr(\))2310 4439 y Fm(\024)2408 4383 y Fn(\016)s("T)p 2408 4420 139 4 v 2456 4496 a Fr(2)2556 4439 y Fn(:)1239 4664 y Fr(4.)41 b Fq(Pr)n(oof)31 b(of)h(Theorems)g(2.1)e(and)i(2.2)555 4813 y Fr(W)-7 b(e)28 b(start)e(b)n(y)h(recalling)f(the)i(estimates)e (analogous)f(to)i(Eqs.)g(\(3.2\))o({\(3.3\))g(for)f(the)i(system)456 4913 y([c)n(harged)c(particle])h(+)g([bac)n(kground)f(particles],)i (whose)f(pro)r(of)g(is)g(discussed)h(in)g(the)g(App)r(en-)456 5012 y(dix.)36 b(Let)28 b Fn(t)23 b Fm(7!)g(f)p Fr(\()p Fn(x)p Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(v)s Fr(\()p Fn(t)p Fr(\)\);)g Fn(X)7 b Fr(\()p Fn(t)p Fr(\))p Fm(g)p Fr(,)28 b Fn(X)7 b Fr(\()p Fn(t)p Fr(\))23 b(=)g Fm(f)p Fr(\()p Fn(x)2059 5024 y Fk(i)2086 5012 y Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(v)2257 5024 y Fk(i)2286 5012 y Fr(\()p Fn(t)p Fr(\)\))p Fm(g)2454 5024 y Fk(i)p Fj(2)p Fi(N)2569 5012 y Fr(,)27 b(b)r(e)g(the)h(solution)e(to)h(Eqs.)456 5112 y(\(2.4\))g(and)g(de\014ne)1566 5216 y Fn(Q)1632 5228 y Fk(E)1688 5216 y Fr(\()p Fn(X)7 b Fr(\))1871 5169 y Fn(:)1850 5216 y Fr(=)23 b Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))18 b(+)g Fn(E)5 b(:)940 b Fr(\(4.1\))p eop %%Page: 15 15 15 14 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(15)456 450 y Fr(F)-7 b(or)27 b(an)n(y)f Fn(X)k Fm(2)23 b(X)12 b Fr(,)28 b Fn(i)23 b Fm(2)g Fl(N)t Fr(,)34 b(and)27 b Fn(t)d Fm(\025)e Fr(0,)884 607 y Fm(j)p Fn(v)s Fr(\()p Fn(t)p Fr(\))p Fm(j)84 b(\024)f Fn(C)1378 515 y Fh(h)1417 532 y(p)p 1500 532 849 4 v 75 x Fn(Q)1566 619 y Fk(E)1622 607 y Fr(\()p Fn(X)7 b Fr(\))14 b(log\()p Fn(e)k Fr(+)g Fn(Q)2121 619 y Fk(E)2177 607 y Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g Fn(Q)2516 619 y Fk(E)2572 607 y Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2742 515 y Fh(i)2795 607 y Fn(;)456 b Fr(\(4.2\))859 790 y Fm(j)p Fn(v)922 802 y Fk(i)950 790 y Fr(\()p Fn(t)p Fr(\))p Fm(j)84 b(\024)f Fn(C)1378 698 y Fh(h)1417 715 y(p)p 1500 715 1072 4 v 75 x Fn(Q)1566 802 y Fk(E)1622 790 y Fr(\()p Fn(X)7 b Fr(\))14 b(log\()p Fn(e)k Fr(+)g Fm(j)p Fn(x)2125 802 y Fk(i)2153 790 y Fm(j)h Fr(+)f Fn(Q)2344 802 y Fk(E)2399 790 y Fr(\()p Fn(X)7 b Fr(\)\))19 b(+)f Fn(Q)2739 802 y Fk(E)2795 790 y Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2965 698 y Fh(i)3018 790 y Fn(;)233 b Fr(\(4.3\))456 952 y(and,)27 b(for)g(an)n(y)g Fn(\026)c Fm(2)g Fl(R)p Fr(,)34 b Fn(R)24 b(>)f Fr(log\()p Fn(e)18 b Fr(+)g Fm(j)p Fn(\026)p Fm(j)p Fr(\),)28 b(and)g Fn(t)23 b Fm(\025)f Fr(0,)654 1090 y Fn(Q)p Fr(\()p Fn(X)7 b Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g(R)q Fr(\))23 b Fm(\024)f Fn(C)6 b(Q)1383 1102 y Fk(E)1439 1090 y Fr(\()p Fn(X)h Fr(\))1593 1023 y Fh(\002)1627 1090 y Fn(R)20 b Fr(+)e(log\()p Fn(e)g Fr(+)g Fn(Q)2138 1102 y Fk(E)2194 1090 y Fr(\()p Fn(X)7 b Fr(\)\))19 b(+)f(\(1)g(+)g Fn(Q)2709 1102 y Fk(E)2765 1090 y Fr(\()p Fn(X)7 b Fr(\)\))p Fn(t)2967 1056 y Fo(2)3004 1023 y Fh(\003)3052 1090 y Fn(:)199 b Fr(\(4.4\))456 1266 y Fp(Pr)l(o)l(of)23 b(of)f(The)l(or)l(em)h(2.1.) 36 b Fr(W)-7 b(e)20 b(\014x)f Fn(E)5 b(;)14 b(Q)22 b(>)h Fr(0)c(and)g(de\014ne)g Fn(Q)2301 1278 y Fk(E)2400 1219 y Fn(:)2380 1266 y Fr(=)j Fn(Q)q Fr(+)q Fn(E)j Fr(so)18 b(that)h Fn(Q)3016 1278 y Fk(E)3072 1266 y Fr(\()p Fn(X)7 b Fr(\))23 b Fm(\024)g Fn(Q)3389 1278 y Fk(E)456 1366 y Fr(for)i(an)n(y)f Fn(X)30 b Fm(2)23 b(B)967 1378 y Fk(Q;L)1088 1366 y Fr(.)36 b(By)27 b(\(4.2\))e(and)g(\(4.3\),)h(a)f (necessary)f(condition)h(for)g(the)h Fn(i)p Fr(-th)f(particle)g(to)456 1466 y(satisfy)1578 1568 y(inf)1533 1625 y Fk(s)p Fj(2)p Fo([0)p Fk(;t)p Fo(])1738 1568 y Fm(j)p Fn(x)p Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)g Fn(x)2061 1580 y Fk(i)2089 1568 y Fr(\()p Fn(s)p Fr(\))p Fm(j)k(\024)g Fr(2)456 1734 y(is)k(that:)489 1875 y Fm(j)p Fn(x)559 1887 y Fk(i)587 1875 y Fm(j)19 b(\000)f Fn(C)6 b(t)821 1783 y Fh(h)860 1800 y(p)p 943 1800 792 4 v 75 x Fn(Q)1009 1887 y Fk(E)1079 1875 y Fr(log\()p Fn(e)18 b Fr(+)g Fm(j)p Fn(x)1428 1887 y Fk(i)1456 1875 y Fm(j)h Fr(+)f Fn(Q)1647 1887 y Fk(E)1703 1875 y Fr(\))g(+)g Fn(Q)1902 1887 y Fk(E)1958 1875 y Fn(t)1988 1783 y Fh(i)2050 1875 y Fm(\024)23 b Fr(2)18 b(+)g Fn(C)6 b(t)2390 1783 y Fh(h)2429 1800 y(p)p 2512 1800 570 4 v 75 x Fn(Q)2578 1887 y Fk(E)2648 1875 y Fr(log\()p Fn(e)19 b Fr(+)f Fn(Q)2994 1887 y Fk(E)3049 1875 y Fr(\))h(+)f Fn(Q)3249 1887 y Fk(E)3305 1875 y Fn(t)3335 1783 y Fh(i)3388 1875 y Fn(;)456 2032 y Fr(whic)n(h)27 b(implies:)1112 2169 y Fm(j)p Fn(x)1182 2181 y Fk(i)1210 2169 y Fm(j)19 b(\000)f Fr(log\()p Fn(e)g Fr(+)g Fm(j)p Fn(x)1684 2181 y Fk(i)1713 2169 y Fm(j)p Fr(\))23 b Fm(\024)g Fn(C)1958 2102 y Fh(\002)1992 2169 y Fr(log)q(\()p Fn(e)18 b Fr(+)g Fn(Q)2338 2181 y Fk(E)2394 2169 y Fr(\))g(+)g Fn(Q)2593 2181 y Fk(E)2649 2169 y Fn(t)2679 2135 y Fo(2)2716 2102 y Fh(\003)2765 2169 y Fn(:)456 2312 y Fr(Then)27 b(there)h(exists)f(a)g (constan)n(t)1537 2291 y(\026)1518 2312 y Fn(C)i Fm(\025)23 b Fr(1)k(suc)n(h)g(that:)655 2449 y(inf)610 2507 y Fk(s)p Fj(2)p Fo([0)p Fk(;t)p Fo(])815 2449 y Fm(j)p Fn(x)p Fr(\()p Fn(s)p Fr(\))20 b Fm(\000)e Fn(x)1138 2461 y Fk(i)1166 2449 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fr(2)105 b(=)-14 b Fm(\))106 b(j)p Fn(x)1860 2461 y Fk(i)1888 2449 y Fm(j)24 b(\024)e Fn(Y)2070 2461 y Fk(t)2143 2402 y Fn(:)2123 2449 y Fr(=)2229 2428 y(\026)2210 2449 y Fn(C)2289 2382 y Fh(\002)2324 2449 y Fr(log\()p Fn(e)d Fr(+)f Fn(Q)2670 2461 y Fk(E)2725 2449 y Fr(\))h(+)f Fn(Q)2925 2461 y Fk(E)2981 2449 y Fn(t)3011 2415 y Fo(2)3048 2382 y Fh(\003)3097 2449 y Fn(;)154 b Fr(\(4.5\))456 2643 y(from)27 b(whic)n(h)g(it)h(follo)n(ws)f(that)h(if)g Fm(j)p Fn(x)1571 2655 y Fk(i)1599 2643 y Fm(j)23 b(\025)g Fr(2)1793 2622 y(\026)1775 2643 y Fn(C)d Fr(log\()p Fn(e)e Fr(+)g Fn(Q)2199 2655 y Fk(E)2255 2643 y Fr(\))28 b(then:)1157 2867 y(inf)1112 2924 y Fk(s)p Fj(2)p Fo([0)p Fk(;t)p Fo(])1317 2867 y Fm(j)p Fn(x)p Fr(\()p Fn(s)p Fr(\))20 b Fm(\000)e Fn(x)1640 2879 y Fk(i)1668 2867 y Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\024)e Fr(2)106 b(=)-14 b Fm(\))106 b Fn(t)23 b Fm(\025)2433 2716 y Fh(s)p 2516 2716 249 4 v 2580 2811 a Fm(j)p Fn(x)2650 2823 y Fk(i)2678 2811 y Fm(j)p 2526 2848 229 4 v 2526 2926 a Fr(2)2587 2905 y(\026)2568 2926 y Fn(C)6 b(Q)2699 2938 y Fk(E)2765 2867 y Fn(:)486 b Fr(\(4.6\))555 3070 y(The)26 b(parameter)e Fn(L)1176 3082 y Fo(0)1235 3070 y Fr(=)f Fn(L)1380 3082 y Fo(0)1417 3070 y Fr(\()p Fn(E)5 b(;)14 b(Q)p Fr(\))25 b(for)g(whic)n(h)g(w)n(e)g(shall)g(pro)n(v)n(e)f(the)i(theorem)f(is)g (c)n(hosen)f(in)456 3169 y(the)k(follo)n(wing)e(w)n(a)n(y)-7 b(.)36 b(Let)1358 3350 y(0)22 b Fn(<)h(a)1554 3362 y Fo(0)1614 3350 y Fm(\024)1712 3294 y Fr(min)p Fm(f)p Fr(1;)14 b Fn(E)5 b Fm(g)p 1712 3331 366 4 v 1874 3407 a Fr(8)2088 3350 y Fn(;)180 b(K)28 b Fm(\025)23 b Fr(1)p Fn(;)456 3520 y Fr(b)r(e)34 b(t)n(w)n(o)g(parameters)f(to)h(b)r(e)h (\014xed)f(later.)57 b(Then)34 b Fn(L)2138 3532 y Fo(0)2209 3520 y Fm(\025)g Fr(2)2369 3499 y(\026)2350 3520 y Fn(C)20 b Fr(log\()p Fn(e)j Fr(+)g Fn(Q)2784 3532 y Fk(E)2839 3520 y Fr(\))35 b(is)f(c)n(hosen)g(large)456 3620 y(enough)27 b(that,)h(for)f(an)n(y)f Fn(a)d Fm(2)h Fr(\(0)p Fn(;)14 b(a)1529 3632 y Fo(0)1566 3620 y Fr(],)27 b Fn(K)i Fm(\025)23 b Fr(1,)k(and)g Fn(L)c Fm(\025)f Fn(L)2304 3632 y Fo(0)2341 3620 y Fr(,)1039 3773 y Fn(T)35 b Fm(\025)22 b Fn(a)1254 3738 y Fj(\000)p Fk(K)t Fo(\(1+)p Fk(Q)1528 3713 y Fe(3)1528 3755 y Fg(E)1576 3738 y Fo(\))p Fk(=a)1672 3713 y Fe(2)1709 3698 y Fh(p)p 1792 3698 551 4 v 75 x Fr(log)14 b([)p Fn(e)k Fr(+)g Fn(Y)2124 3785 y Fk(T)2195 3773 y Fr(+)g(1])166 b Fm(8)14 b Fn(T)33 b Fm(\025)22 b Fn(T)2788 3785 y Fk(L)2837 3773 y Fn(;)414 b Fr(\(4.7\))456 3910 y(where)1668 4075 y Fn(T)1717 4087 y Fk(L)1810 4028 y Fn(:)1790 4075 y Fr(=)1877 3927 y Fh(s)p 1960 3927 249 4 v 2056 4019 a Fn(L)p 1970 4056 229 4 v 1970 4134 a Fr(2)2030 4113 y(\026)2012 4134 y Fn(C)6 b(Q)2143 4146 y Fk(E)2209 4075 y Fn(:)1042 b Fr(\(4.8\))555 4259 y(No)n(w)27 b(let:)1272 4361 y Fn(U)9 b Fr(\()p Fn(t)p Fr(\))1476 4314 y Fn(:)1455 4361 y Fr(=)23 b(max)1608 4413 y Fk(i)1725 4361 y Fn(G)14 b Fr(\()p Fm(j)p Fn(x)1906 4373 y Fk(i)1934 4361 y Fm(j)19 b(\000)f Fn(Y)2107 4373 y Fk(t)2136 4361 y Fr(\))48 b(sup)2182 4434 y Fk(s)p Fj(2)p Fo([0)p Fk(;t)p Fo(])2388 4361 y Fm(j)p Fn(v)2451 4373 y Fk(i)2479 4361 y Fr(\()p Fn(s)p Fr(\))p Fm(j)p Fn(;)646 b Fr(\(4.9\))456 4548 y(with)25 b Fn(Y)690 4560 y Fk(t)744 4548 y Fr(as)f(in)h(\(4.5\))f(and)h Fn(G)e Fm(2)g Fn(C)6 b Fr(\()p Fl(R)q Fr(\))31 b(a)24 b(not)h(increasing)e(function)i(satisfying:)35 b Fn(G)p Fr(\()p Fn(x)p Fr(\))24 b(=)f(1)h(for)456 4647 y Fn(x)j Fm(\024)g Fr(0,)k Fn(G)p Fr(\()p Fn(x)p Fr(\))e(=)d(0)k(for)g Fn(x)e Fm(\025)f Fr(1.)44 b(By)31 b(\(4.5\),)g(the)f(con)n(tin)n(uous)f (and)h(non)g(decreasing)f(function)456 4747 y Fn(U)9 b Fr(\()p Fm(\001)p Fr(\))24 b(is)g(an)g(upp)r(er)g(b)r(ound)h(for)e (the)i(maximal)e(v)n(elo)r(cit)n(y)h(of)g(an)n(y)f(particle)h(whic)n(h) g(ma)n(y)f(in)n(teract)456 4847 y(with)28 b(the)g(c)n(harged)e(one)h (during)g(the)h(time)g([0)p Fn(;)14 b(t)p Fr(].)37 b(W)-7 b(e)28 b(next)f(de\014ne:)699 5033 y Fn(T)760 4999 y Fj(\003)841 4986 y Fn(:)820 5033 y Fr(=)22 b(sup)1046 4916 y Fh(\032)1109 5033 y Fn(t)h(>)f Fr(0)37 b(:)g(max)o Fm(f)p Fn(U)9 b Fr(\()p Fn(s)p Fr(\);)14 b Fm(j)p Fn(v)s Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(E)5 b(s)p Fm(jg)23 b(\024)2351 4977 y Fn(E)5 b(s)p 2351 5014 105 4 v 2383 5090 a Fr(4)2549 5033 y Fm(8)14 b Fn(s)22 b Fm(2)h Fr([0)p Fn(;)14 b(t)p Fr(])2904 4916 y Fh(\033)2966 5033 y Fn(;)243 b Fr(\(4.10\))456 5216 y(setting)27 b Fn(T)789 5185 y Fj(\003)849 5216 y Fr(=)c(0)k(if)h(the)g(ab)r(o)n(v)n(e)f(set)g(is)h (empt)n(y)-7 b(.)p eop %%Page: 16 16 16 15 bop 456 251 a Fo(16)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)555 450 y Fr(By)29 b(\(4.6\),)f(\(4.8\),)g(and)g(recalling)f(that)i Fm(j)p Fn(x)1877 462 y Fk(i)1905 450 y Fm(j)24 b(\025)f Fn(L)h Fm(\025)f Fr(2)2269 429 y(\026)2251 450 y Fn(C)d Fr(log\()p Fn(e)f Fr(+)f Fn(Q)2676 462 y Fk(E)2732 450 y Fr(\),)28 b(w)n(e)g(ha)n(v)n(e)f Fn(U)9 b Fr(\()p Fn(t)p Fr(\))24 b(=)g(0)456 550 y(and)35 b Fn(v)s Fr(\()p Fn(t)p Fr(\))h(=)f Fn(E)5 b(t)35 b Fr(for)g Fn(t)g Fm(\024)g Fn(T)1378 562 y Fk(L)1428 550 y Fr(,)i(whence)e Fn(T)1847 520 y Fj(\003)1920 550 y Fn(>)g(T)2069 562 y Fk(L)2118 550 y Fr(.)59 b(W)-7 b(e)36 b(next)f(pro)n(v)n(e)f(that)h(if)h Fn(a)3092 562 y Fo(0)3164 550 y Fr(is)f(small)456 649 y(enough)27 b(and)g Fn(K)33 b Fr(is)28 b(large)e(enough)h(then:)1094 885 y(max)p Fm(f)p Fn(U)9 b Fr(\()p Fn(t)p Fr(\);)14 b Fm(j)p Fn(v)s Fr(\()p Fn(t)p Fr(\))k Fm(\000)h Fn(E)5 b(t)p Fm(jg)22 b(\024)2031 828 y Fn(E)5 b(t)p 2031 866 97 4 v 2058 942 a Fr(8)2303 885 y Fm(8)14 b Fn(t)22 b Fm(2)h Fr([)p Fn(T)2566 897 y Fk(L)2616 885 y Fn(;)14 b(T)2714 850 y Fj(\003)2751 885 y Fr(\))p Fn(;)426 b Fr(\(4.11\))456 1105 y(whic)n(h)33 b(implies,)j(b)n(y)e(con)n(tin)n (uit)n(y)-7 b(,)35 b Fn(T)1611 1075 y Fj(\003)1681 1105 y Fr(=)f(+)p Fm(1)p Fr(.)55 b(Moreo)n(v)n(er,)33 b(since)g(w)n(e)h (actually)f(pro)n(v)n(e)f(that)456 1205 y Fm(j)p Fn(v)s Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(E)5 b(t)p Fm(j)23 b Fr(=)g Fm(O)r Fr(\(log)14 b Fn(t)p Fr(\))28 b(for)f Fn(t)d(<)e(T)1588 1175 y Fj(\003)1625 1205 y Fr(,)28 b(the)g(limit)g(\(2.8\))g(follo)n(ws.)555 1305 y(T)-7 b(o)34 b(b)r(ound)h Fn(U)9 b Fr(\()p Fn(t)p Fr(\))34 b(w)n(e)g(can)g(apply)g(in)h(the)f(presen)n(t)g(con)n(text)g(the)g (analysis)f(dev)n(elop)r(ed)h(in)456 1404 y(Sect.)28 b(3.)36 b(W)-7 b(e)28 b(\014rst)f(observ)n(e)f(that)i(b)n(y)h(\(4.7\))e (and)g(the)h(de\014nition)g(\(3.6\),)923 1602 y Fn(T)34 b Fm(\025)22 b Fn(T)1142 1614 y Fk(L)1381 1602 y Fr(=)-14 b Fm(\))189 b Fn(T)34 b Fm(\025)22 b Fn(T)1923 1614 y Fk(a)1963 1602 y Fr(\()p Fn(X)r(;)14 b(x)2150 1614 y Fk(i)2178 1602 y Fr(\))83 b Fm(8)14 b Fn(i)36 b Fr(:)h Fm(j)p Fn(x)2549 1614 y Fk(i)2577 1602 y Fm(j)23 b(\024)g Fn(Y)2759 1614 y Fk(T)2830 1602 y Fr(+)18 b(1)p Fn(:)456 1799 y Fr(Moreo)n(v)n(er,)31 b(since)h(for)g Fn(t)g(<)f(T)1409 1769 y Fj(\003)1479 1799 y Fr(the)i(c)n(harged)e(particle)h(is)g(m)n (uc)n(h)h(faster)f(than)g(the)h(particles)456 1899 y(it)27 b(meets,)h(the)f(in)n(teraction)f(with)i(this)f(particle)g(do)r(es)g (not)g(a\013ect)g(to)r(o)g(m)n(uc)n(h)g(the)g(v)n(elo)r(cit)n(y)g(of) 456 1999 y(eac)n(h)d(bac)n(kground)f(particle)i(up)h(to)f(this)g(time.) 37 b(More)24 b(precisely)-7 b(,)25 b(for)g(an)n(y)g(0)d Fm(\024)h Fn(\034)3014 2011 y Fo(1)3074 1999 y Fm(\024)g Fn(\034)3198 2011 y Fo(2)3259 1999 y Fn(<)f(T)3407 1968 y Fj(\003)456 2098 y Fr(and)27 b Fn(i)c Fm(2)g Fl(N)t Fr(,)1237 2186 y Fh(\014)1237 2236 y(\014)1237 2285 y(\014)1237 2335 y(\014)1278 2193 y(Z)1361 2214 y Fk(\034)1392 2222 y Fe(2)1324 2382 y Fk(\034)1355 2390 y Fe(1)1428 2306 y Fn(ds)14 b Fm(r)p Fr(\011\()p Fn(x)1737 2318 y Fk(i)1765 2306 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)p Fr(\()p Fn(s)p Fr(\)\))2152 2186 y Fh(\014)2152 2236 y(\014)2152 2285 y(\014)2152 2335 y(\014)2204 2306 y Fm(\024)2302 2250 y Fr(2)p Fm(kr)p Fr(\011)p Fm(k)2562 2262 y Fj(1)p 2302 2287 329 4 v 2384 2363 a Fn(E)5 b(T)2499 2375 y Fk(L)2640 2306 y Fn(:)569 b Fr(\(4.12\))456 2536 y(Note)35 b(in)h(fact)g(that)g(the)g Fn(i)p Fr(-th)g(particle)f(ma)n(y)g(in)n (teract)g(with)h(the)g(c)n(harged)e(one)i(only)f(after)456 2636 y(the)30 b(time)g Fn(T)841 2648 y Fk(L)890 2636 y Fr(,)g(and)f(hence)h(for)f(a)g(time)h(not)g(bigger)e(than)i(2)p Fn(=)p Fr(\()p Fn(E)5 b(\034)2545 2648 y Fo(1)2582 2636 y Fr(\))26 b Fm(\024)g Fr(2)p Fn(=)p Fr(\()p Fn(E)5 b(T)2962 2648 y Fk(L)3011 2636 y Fr(\).)43 b(F)-7 b(rom)29 b(the)456 2736 y(previous)34 b(estimate)h(the)h(strategy)e(used)h(for)g(pro)n (ving)g(\(3.5\))g(applies)g(in)h(this)g(case)e(almost)456 2835 y(unc)n(hanged,)26 b(getting:)1002 3033 y Fm(j)p Fn(v)1065 3045 y Fk(i)1093 3033 y Fr(\()p Fn(T)12 b Fr(\))p Fm(j)23 b(\024)f Fn(aT)177 b Fm(8)14 b Fn(T)33 b Fm(2)24 b Fr([)p Fn(T)1915 3045 y Fk(L)1964 3033 y Fn(;)14 b(T)2062 2999 y Fj(\003)2099 3033 y Fr(\))83 b Fm(8)14 b Fn(i)36 b Fr(:)h Fm(j)p Fn(x)2470 3045 y Fk(i)2498 3033 y Fm(j)23 b(\024)g Fn(Y)2680 3045 y Fk(T)2751 3033 y Fr(+)18 b(1)p Fn(;)333 b Fr(\(4.13\))456 3230 y(whic)n(h)27 b(in)h(particular)e 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Fk(q)820 4150 y Fr(\()p Fn(t)882 4162 y Fo(1)920 4150 y Fr(\))p Fm(j)19 b(\000)f Fr(\()p Fn(\015)1152 4162 y Fk(k)1211 4150 y Fr(+)g(2)p Fn(\014)t Fr(\))p Fn(aT)1524 4080 y Fh(\014)1524 4129 y(\014)1574 4150 y Fm(\024)23 b(kr)p Fr(\010)p Fm(k)1875 4162 y Fj(1)1958 4037 y Fh(Z)2041 4058 y Fk(t)2066 4066 y Fe(1)2004 4226 y Fk(t)2029 4234 y Fe(0)2103 4150 y Fn(ds)2297 4071 y Fh(X)2213 4250 y Fk(j)s Fj(2P)2335 4258 y Fe(1)2368 4250 y Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(q)2515 4150 y Fn(\037)2567 4162 y Fk(q)r(;j)2654 4150 y Fr(\()p Fn(s)p Fr(\))c(+)2869 4094 y(2)p Fm(kr)p Fr(\011)p Fm(k)3129 4106 y Fj(1)p 2869 4131 329 4 v 2951 4207 a Fn(E)5 b(T)3066 4219 y Fk(L)456 4438 y Fr(\(where)36 b(the)h(parameter)e Fn(\014)41 b Fr(is)c(no)n(w)f(de\014ned)h(as)f(in)h(\(3.21\))e(but)j (with)f Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))36 b(replaced)g(b)n(y)456 4538 y Fn(Q)522 4550 y Fk(E)577 4538 y Fr(\()p Fn(X)7 b Fr(\)\))28 b(so)f(that)h(Eq.)f(\(3.26\))g(b)r(ecomes:)905 4712 y Fh(\014)905 4762 y(\014)933 4783 y 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b(\(3.31\))f(and)h(\(4.4\))f(w)n(e)g(get:)681 5216 y Fn(N)9 b Fr(\()p Fn(s)p Fr(\))860 5181 y Fo(2)921 5216 y Fm(\024)22 b Fn(C)6 b(Q)p Fr(\()p Fn(X)h Fr(\()p Fn(s)p Fr(\);)14 b Fn(x)p Fr(\()p Fn(s)p Fr(\))p Fn(;)g Fr(2\))24 b Fm(\024)e Fn(C)6 b(Q)1890 5228 y Fk(E)1960 5148 y Fh(\002)1995 5216 y Fr(log\()p Fn(e)18 b Fr(+)g Fn(Q)2340 5228 y Fk(E)2396 5216 y Fr(\))h(+)f(\(1)g(+)g Fn(Q)2771 5228 y Fk(E)2827 5216 y Fr(\))p Fn(s)2898 5181 y Fo(2)2935 5148 y Fh(\003)2984 5216 y Fn(;)225 b Fr(\(4.17\))p eop %%Page: 18 18 18 17 bop 456 251 a Fo(18)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(where)28 b(w)n(e)g(used)h(that)g Fm(j)p Fn(x)p Fr(\()p Fn(s)p Fr(\))p Fm(j)c(\024)g Fn(C)6 b(s)p Fr([)1630 379 y Fh(p)p 1713 379 570 4 v 71 x Fn(Q)1779 462 y Fk(E)1849 450 y Fr(log\()p Fn(e)18 b Fr(+)g Fn(Q)2194 462 y Fk(E)2250 450 y Fr(\))h(+)g Fn(Q)2451 462 y Fk(E)2507 450 y Fn(s)p Fr(],)29 b(whic)n(h)g(follo)n(ws)e(b)n(y)j(\(4.2\))o(.)456 550 y(The)25 b(term)f([1)13 b(+)g Fn(N)c Fr(\()p Fn(s)p Fr(\)])p Fn(=)p Fr(\()p Fn(E)c(s)p Fr(\))25 b(in)g(\(4.16\))f(can)h(b)r (e)g(b)r(ounded)h(using)f(\(4.17\))o(;)h(b)n(y)f(the)g(de\014nitions) 456 649 y(of)i Fn(L)g Fr(and)h Fn(T)845 661 y Fk(L)922 649 y Fr(w)n(e)f(th)n(us)h(obtain,)f(for)g(an)n(y)g Fn(t)c Fm(\025)f Fn(T)1981 661 y Fk(L)2031 649 y Fr(,)776 842 y Fm(j)p Fn(p)p Fr(\()p Fn(t)p Fr(\))d Fm(\000)f Fn(p)p Fr(\()p Fn(T)1160 854 y Fk(L)1210 842 y Fr(\))p Fm(j)83 b(\024)g Fn(C)1571 786 y Fr(1)18 b(+)g Fn(Q)1780 798 y Fk(E)p 1571 823 265 4 v 1670 899 a Fn(E)1859 729 y Fh(Z)1942 749 y Fk(t)1905 918 y(T)1944 926 y Fg(L)1993 842 y Fn(ds)2099 786 y(N)9 b Fr(\()p Fn(s)p Fr(\))p 2099 823 180 4 v 2136 899 a Fn(E)c(s)1348 1087 y Fr(=)83 b Fn(C)1571 1031 y Fr(1)18 b(+)g Fn(Q)1780 1043 y Fk(E)p 1571 1068 265 4 v 1670 1144 a Fn(E)1859 1008 y Fh(X)1904 1185 y Fk(j)1993 974 y Fh(Z)2076 995 y Fk(t)2039 1163 y(T)2078 1171 y Fg(L)2126 1087 y Fn(ds)2264 1031 y Fr(1)p 2232 1068 105 4 v 2232 1144 a Fn(E)5 b(s)2347 1087 y(\037)p Fr(\()p Fm(j)p Fn(x)p Fr(\()p Fn(s)p Fr(\))20 b Fm(\000)e Fn(x)2754 1099 y Fk(j)2789 1087 y Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\024)f Fr(2\))p Fn(:)456 1311 y Fr(An)34 b(upp)r(er)f(b)r (ound)h(for)f(the)h(righ)n(t-hand)f(side)g(of)g(the)h(ab)r(o)n(v)n(e)f (inequalit)n(y)g(can)g(b)r(e)h(obtained)456 1410 y(in)e(the)h(follo)n (wing)e(w)n(a)n(y)-7 b(.)49 b(W)-7 b(e)33 b(\014rst)f(observ)n(e)e (that)j(b)n(y)g(\(4.5\))f(and)g(the)g(de\014nition)h(\(2.7\))f(only)456 1510 y(the)d(particles)f(whic)n(h)i(are)e(initially)h(in)g([)p Fm(\000)p Fn(Y)1856 1522 y Fk(t)1886 1510 y Fn(;)14 b Fm(\000)p Fn(L)p Fr(])k Fm([)i Fr([)p Fn(L;)14 b(Y)2326 1522 y Fk(t)2355 1510 y Fr(])29 b(ma)n(y)g(con)n(tribute)f(to)h(the)h (ab)r(o)n(v)n(e)456 1610 y(in)n(tegral.)35 b(W)-7 b(e)28 b(next)g(de\014ne:)1346 1749 y Fn(N)1413 1761 y Fk(q)1473 1749 y Fr(=)22 b Fm(f)p Fn(j)28 b Fm(2)23 b Fl(N)47 b Fr(:)37 b(2)1941 1715 y Fk(q)2000 1749 y Fn(<)23 b Fm(j)p Fn(x)2158 1761 y Fk(j)2193 1749 y Fm(j)g(\024)g Fr(2)2369 1715 y Fk(q)r Fo(+1)2489 1749 y Fm(g)p Fn(:)456 1888 y Fr(Since)k Fn(t)c(<)g(T)874 1858 y Fj(\003)939 1888 y Fr(w)n(e)k(ha)n(v)n(e:)1071 2014 y Fh(X)1116 2191 y Fk(j)1205 1980 y Fh(Z)1288 2001 y Fk(t)1251 2169 y(T)1290 2177 y Fg(L)1338 2093 y Fn(ds)1476 2037 y Fr(1)p 1444 2074 V 1444 2150 a Fn(E)5 b(s)1559 2093 y(\037)p Fr(\()p Fm(j)p Fn(x)1713 2105 y Fk(j)1749 2093 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)p Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\024)f Fr(2\))f Fm(\024)2431 1986 y Fk(q)2461 1994 y Fg(t)2489 1986 y Fo(+1)2442 2014 y Fh(X)2423 2189 y Fk(q)r Fo(=)p Fk(q)2536 2197 y Fg(L)2605 2037 y Fr(4)p Fm(j)p Fn(N)2737 2049 y Fk(q)2773 2037 y Fm(j)p 2605 2074 192 4 v 2615 2150 a Fn(E)2681 2126 y Fo(2)2718 2150 y Fn(t)2748 2126 y Fo(2)2748 2171 y Fk(q)2806 2093 y Fn(;)403 b Fr(\(4.18\))456 2327 y(where)23 b Fn(q)729 2339 y Fk(L)803 2327 y Fr([resp.)h Fn(q)1059 2339 y Fk(t)1088 2327 y Fr(])g(is)h(the)f(in)n(teger)f(suc)n(h)h(that)h(2)2029 2296 y Fk(q)2059 2304 y Fg(L)2129 2327 y Fn(<)e(L)g Fm(\024)f Fr(2)2426 2296 y Fk(q)2456 2304 y Fg(L)2500 2296 y Fo(+1)2613 2327 y Fr([resp.)h(2)2873 2296 y Fk(q)2903 2304 y Fg(t)2958 2327 y Fn(<)g(Y)3094 2339 y Fk(t)3146 2327 y Fm(\024)g Fr(2)3276 2296 y Fk(q)3306 2304 y Fg(t)3333 2296 y Fo(+1)3421 2327 y Fr(])456 2426 y(and)1089 2531 y Fn(t)1119 2543 y Fk(q)1199 2484 y Fn(:)1179 2531 y Fr(=)34 b(min)1266 2584 y Fk(j)s Fj(2)p Fk(N)1394 2592 y Fg(q)1441 2531 y Fr(inf)7 b Fm(f)p Fn(s)23 b Fm(2)g Fr([)p Fn(T)1796 2543 y Fk(L)1846 2531 y Fn(;)14 b(T)1944 2497 y Fj(\003)1981 2531 y Fr(\))37 b(:)g Fm(j)p Fn(x)p Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)2432 2543 y Fk(j)2468 2531 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fr(2)p Fm(g)p Fn(;)456 2697 y Fr(setting)33 b Fn(t)764 2709 y Fk(q)832 2697 y Fr(=)f(+)p Fm(1)h Fr(if)g(the)h(ab)r(o)n(v)n(e)d(set)i(is)g(empt)n(y) g(for)g(all)f Fn(j)38 b Fm(2)32 b Fn(N)2542 2709 y Fk(q)2578 2697 y Fr(.)54 b(W)-7 b(e)33 b(ma)n(y)f(assume)h Fn(L)f Fr(so)456 2797 y(large)27 b(\(i.e.)i Fn(a)871 2809 y Fo(0)937 2797 y Fr(small)f(enough\))g(that)h(2)1697 2767 y Fk(q)r Fj(\000)p Fo(1)1843 2797 y Fn(>)c Fr(log\()p Fn(e)19 b Fr(+)g(2)2256 2767 y Fk(q)r Fo(+1)2376 2797 y Fr(\))29 b(for)f(an)n(y)g Fn(q)g Fm(\025)d Fn(q)2915 2809 y Fk(L)2964 2797 y Fr(.)40 b(Then,)30 b(since)456 2897 y(2)498 2867 y Fk(q)r Fo(+1)640 2897 y Fm(\000)22 b Fr(2)769 2867 y Fk(q)837 2897 y Fr(=)32 b(2)976 2867 y Fk(q)1012 2897 y Fr(,)j(b)n(y)f(\(2.2\))f(and)g(\(2.3\))g(w)n(e)g(ha) n(v)n(e)e Fm(j)p Fn(N)2181 2909 y Fk(q)2218 2897 y Fm(j)h(\024)g Fr(2)p Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)2)2660 2867 y Fk(q)2728 2897 y Fm(\024)32 b Fn(Q)2891 2909 y Fk(E)2946 2897 y Fr(2)2988 2867 y Fk(q)r Fo(+1)3109 2897 y Fr(.)53 b(On)33 b(the)456 3009 y(other)c(hand,)i(from)f(\(4.6\))g(it)g(follo)n(ws)f (that)i Fn(t)1882 3021 y Fk(q)1946 3009 y Fn(>)2038 2935 y Fh(p)p 2121 2935 457 4 v 74 x Fr(2)2163 2985 y Fk(q)r Fj(\000)p Fo(1)2284 3009 y Fn(=)p Fr(\()2376 2988 y(\026)2358 3009 y Fn(C)6 b(Q)2489 3021 y Fk(E)2545 3009 y Fr(\).)45 b(Inserting)29 b(the)i(previous)456 3109 y(b)r(ounds)c(in)h(\(4.18\))f (w)n(e)g(obtain,)h(for)f(an)n(y)g Fn(t)c Fm(2)g Fr([)p Fn(T)1972 3121 y Fk(L)2021 3109 y Fn(;)14 b(T)2119 3079 y Fj(\003)2156 3109 y Fr(\),)802 3227 y Fh(X)846 3404 y Fk(j)935 3193 y Fh(Z)1018 3214 y Fk(t)982 3382 y(T)1021 3390 y Fg(L)1069 3306 y Fn(ds)1206 3250 y Fr(1)p 1175 3287 105 4 v 1175 3363 a Fn(E)5 b(s)1290 3306 y(\037)p Fr(\()p Fm(j)p Fn(x)1444 3318 y Fk(j)1479 3306 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(x)p Fr(\()p Fn(s)p Fr(\))p Fm(j)24 b(\024)f Fr(2\))g Fm(\024)g Fn(C)2229 3250 y(Q)2295 3220 y Fo(2)2295 3273 y Fk(E)p 2229 3287 122 4 v 2238 3363 a Fn(E)2304 3339 y Fo(2)2374 3306 y Fr(log)2505 3250 y Fn(Y)2553 3262 y Fk(t)p 2505 3287 78 4 v 2516 3363 a Fn(L)2616 3306 y Fm(\024)f Fn(C)2779 3250 y(Q)2845 3220 y Fo(2)2845 3273 y Fk(E)p 2779 3287 122 4 v 2788 3363 a Fn(E)2854 3339 y Fo(2)2924 3306 y Fr(log)14 b Fn(t;)456 3529 y Fr(so)27 b(that:)1058 3666 y Fm(j)p Fn(p)p Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(p)p Fr(\()p Fn(T)1442 3678 y Fk(L)1492 3666 y Fr(\))p Fm(j)23 b(\024)g Fn(C)1733 3610 y Fr(1)18 b(+)g Fn(Q)1942 3580 y Fo(3)1942 3633 y Fk(E)p 1733 3647 265 4 v 1814 3723 a Fn(E)1880 3699 y Fo(3)2021 3666 y Fr(log)c Fn(t)166 b Fm(8)14 b Fn(t)22 b Fm(2)i Fr([)p Fn(T)2602 3678 y Fk(L)2651 3666 y Fn(;)14 b(T)2749 3632 y Fj(\003)2786 3666 y Fr(\))p Fn(:)391 b Fr(\(4.19\))456 3816 y(Since)27 b Fn(v)s Fr(\()p Fn(T)796 3828 y Fk(L)846 3816 y Fr(\))19 b Fm(\000)f Fn(E)5 b(T)1095 3828 y Fk(L)1167 3816 y Fr(=)23 b(0)k(w)n(e)g(ha)n(v)n (e:)625 3956 y Fm(j)p Fn(v)s Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(E)5 b(t)p Fm(j)24 b(\024)e(j)p Fn(v)s Fr(\()p Fn(t)p Fr(\))e Fm(\000)e Fn(E)5 b(t)18 b Fm(\000)g Fn(p)p Fr(\()p Fn(t)p Fr(\))p Fm(j)h Fr(+)f Fm(j)p Fn(p)p Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(p)p Fr(\()p Fn(T)2222 3968 y Fk(L)2271 3956 y Fr(\))p Fm(j)h Fr(+)f Fm(j)p Fn(v)s Fr(\()p Fn(T)2575 3968 y Fk(L)2625 3956 y Fr(\))g Fm(\000)g Fn(E)5 b(T)2873 3968 y Fk(L)2941 3956 y Fm(\000)18 b Fn(p)p Fr(\()p Fn(T)3147 3968 y Fk(L)3196 3956 y Fr(\))p Fm(j)p Fn(:)456 4095 y Fr(By)k(the)h(de\014nition)f(of)h Fn(p)p Fr(\()p Fn(t)p Fr(\),)h(the)e(\014rst)h([resp.)f(third])g(term)h (on)f(the)h(righ)n(t-hand)e(side)h(is)g(smaller)456 4195 y(than)h(a)f(constan)n(t)g(m)n(ultiple)i(of)f Fn(N)9 b Fr(\()p Fn(t)p Fr(\))p Fn(=)p Fr(\()p Fn(E)c(t)p Fr(\))23 b([resp.)g Fn(N)9 b Fr(\()p Fn(T)2220 4207 y Fk(L)2269 4195 y Fr(\))p Fn(=)p Fr(\()p Fn(E)c(T)2490 4207 y Fk(L)2539 4195 y Fr(\)],)25 b(whic)n(h)e(w)n(e)f(ha)n(v)n(e)g(already)456 4294 y(sho)n(wn)k(to)h(b)r(e)g(b)r(ounded)h(b)n(y)f Fn(C)6 b Fr(\(1)17 b(+)g Fn(Q)1678 4306 y Fk(E)1734 4294 y Fr(\))p Fn(=E)32 b Fr(for)26 b(an)n(y)h Fn(t)c Fm(\025)f Fn(T)2373 4306 y Fk(L)2423 4294 y Fr(.)36 b(Finally)-7 b(,)28 b(the)f(second)g (term)g(is)456 4394 y(b)r(ounded)h(in)f(\(4.19\).)37 b(In)27 b(conclusion:)895 4587 y Fm(j)p Fn(v)s Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(E)5 b(t)p Fm(j)23 b(\024)g Fn(C)1462 4531 y Fr(1)18 b(+)g Fn(Q)1671 4543 y Fk(E)p 1462 4568 V 1561 4644 a Fn(E)1751 4470 y Fh(\022)1812 4587 y Fr(1)g(+)1965 4531 y Fn(Q)2031 4500 y Fo(2)2031 4553 y Fk(E)p 1965 4568 122 4 v 1974 4644 a Fn(E)2040 4620 y Fo(2)2110 4587 y Fr(log)c Fn(t)2261 4470 y Fh(\023)2502 4587 y Fm(8)g Fn(t)22 b Fm(2)i Fr([)p Fn(T)2766 4599 y Fk(L)2815 4587 y Fn(;)14 b(T)2913 4552 y Fj(\003)2950 4587 y Fr(\))p Fn(;)227 b Fr(\(4.20\))456 4771 y(whic)n(h)27 b(in)h(particular)e(implies,)i(if)g Fn(a)1598 4783 y Fo(0)1663 4771 y Fr(is)g(small)f(enough,)1311 4949 y Fm(j)p Fn(v)s Fr(\()p Fn(t)p Fr(\))20 b Fm(\000)e Fn(E)5 b(t)p Fm(j)23 b(\024)1813 4892 y Fn(E)5 b(t)p 1813 4929 97 4 v 1841 5006 a Fr(8)2085 4949 y Fm(8)14 b Fn(t)22 b Fm(2)i Fr([)p Fn(T)2349 4961 y Fk(L)2398 4949 y Fn(;)14 b(T)2496 4914 y Fj(\003)2533 4949 y Fr(\))p Fn(:)456 5116 y Fr(By)33 b(\(4.14\))g(and)f(the)i(last)e(b)r(ound)i(Eq.)e (\(4.11\))g(is)h(th)n(us)g(pro)n(v)n(ed,)g(whence)g Fn(T)2912 5086 y Fj(\003)2981 5116 y Fr(=)f(+)p Fm(1)p Fr(.)52 b(The)456 5216 y(limit)28 b(\(2.8\))f(th)n(us)h(follo)n(ws)f(from)g (\(4.20\))g(and)g(the)h(theorem)f(is)h(pro)n(v)n(ed.)658 b Ff(\003)p eop %%Page: 19 19 19 18 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(19)456 450 y Fp(Pr)l(o)l(of)36 b(of)h(The)l(or)l(em)g(2.2.)58 b Fr(W)-7 b(e)34 b(pro)n(v)n(e)f(there)g(exist)h Fn(a)2176 462 y Fo(0)2247 450 y Fm(2)g Fr(\(0)p Fn(;)14 b Fr(1\))34 b(and)g Fn(T)2772 462 y Fk(a)2845 450 y Fr(=)g Fn(T)2993 462 y Fk(a)3032 450 y Fr(\()p Fn(X)r(;)14 b(E)5 b Fr(\))34 b Fn(>)g Fr(0)456 550 y(suc)n(h)27 b(that,)h(for)f(an)n(y)g Fn(a)c Fm(2)g Fr(\(0)p Fn(;)14 b(a)1430 562 y Fo(0)1467 550 y Fr(],)1517 691 y Fn(v)s Fr(\()p Fn(t)p Fr(\))24 b Fm(\025)f(\000)p Fn(at)165 b Fm(8)14 b Fn(t)22 b Fm(\025)h Fn(T)2320 703 y Fk(a)2360 691 y Fn(:)849 b Fr(\(4.21\))555 833 y(This)38 b(cannot)f(b)r(e)h(ac)n(hiev)n(ed)e(b)n(y)h(the)h(same)f (strategy)f(used)h(in)h(the)g(pro)r(of)f(of)g(Theorem)456 932 y(2.3.)57 b(Let)34 b(us)h(\014rstly)f(review)g(the)h(pro)r(of)f(of) g(the)h(case)f(without)h(electric)f(\014eld.)58 b(W)-7 b(e)35 b(divide)456 1032 y(the)d(time)g(in)n(terv)-5 b(al)32 b([0)p Fn(;)14 b(T)e Fr(])30 b(in)i(man)n(y)g(subin)n(terv)-5 b(als)31 b([)p Fn(T)2183 1044 y Fk(k)q Fj(\000)p Fo(1)2308 1032 y Fn(;)14 b(T)2394 1044 y Fk(k)2435 1032 y Fr(])32 b(and)f(the)i(fast)f(particles)f(in)n(to)456 1131 y(man)n(y)25 b(disjoin)n(t)h(clusters.)36 b(By)26 b(using)f(the)i(equations)e(of)h (motion)g(and)g(some)f(tric)n(ks)g(w)n(e)h(pro)n(v)n(e)456 1231 y(that)h(the)g(v)n(elo)r(cit)n(y)f(of)h(the)h(cen)n(ter)e(of)h (mass)f(of)h(eac)n(h)f(cluster)h(remains)f(almost)h(constan)n(t.)35 b(Of)456 1331 y(course)21 b(a)g(cluster)h(ma)n(y)g(increase)f(its)h (size)g(due)h(to)f(the)g(in)n(ternal)g(forces,)g(th)n(us)h(approac)n (hing)d(an)456 1430 y(adjacen)n(t)k(one.)36 b(But)25 b(the)g(\014rst)g(time)g Fn(\034)1662 1442 y Fo(1)1725 1430 y Fr(\(in)g(the)h(bac)n(kw)n(ard)d(ev)n(olution\))h(when)h(this)g (happ)r(ens,)456 1530 y(w)n(e)31 b(decomp)r(ose)g(the)h(set)g(of)g (fast)g(particles)f(in)n(to)g(smaller)g(clusters)g(whic)n(h)h(remain)f (disjoin)n(t)456 1630 y(un)n(til)37 b(a)g(time)h Fn(\034)975 1642 y Fo(2)1050 1630 y Fr(and)f(so)g(on.)66 b(The)37 b(imp)r(ortan)n(t)g(p)r(oin)n(t)h(is)f(that)h(the)g(n)n(um)n(b)r(er)f (of)g(clusters)456 1729 y(increases)20 b(at)i(eac)n(h)g(step,)h(so)f (that)g(the)h(n)n(um)n(b)r(er)f(of)g(steps)g(is)g(not)g(bigger)f(than)h (the)h(cardinalit)n(y)456 1829 y(of)29 b(the)i(set)f(of)f(the)i(fast)e (particles.)43 b(This)30 b(pro)r(cedure)f(holds)g(in)h(eac)n(h)g(time)g (in)n(terv)-5 b(al)29 b(and)h(w)n(e)456 1928 y(go)25 b(bac)n(k)h(to)h(time)g(zero)e(with)i(some)g(fast)f(particles,)g(th)n (us)h(getting)f(a)h(con)n(tradiction)e(b)r(ecause)456 2028 y(of)i(the)h(initial)g(data)f(w)n(e)g(ha)n(v)n(e)g(c)n(hosen.)555 2128 y(The)j(ab)r(o)n(v)n(e)f(strategy)g(fails)h(in)g(the)h(presen)n(t) e(con)n(text.)44 b(In)30 b(fact,)h(due)g(to)f(the)g(presence)g(of)456 2227 y(the)35 b(electric)f(\014eld,)j(an)e(estimate)g(lik)n(e)f (\(3.46\))g(\(whic)n(h)h(implies)g(\(3.41\))f(and)h(hence)g(\(3.42\))o (\))456 2327 y(do)r(es)g(not)h(hold)f(for)g(the)h(cluster)g(con)n (taining)e(the)i(c)n(harged)e(particle:)53 b(the)36 b(v)n(elo)r(cit)n (y)f(of)g(its)456 2427 y(cen)n(ter)j(of)g(mass)h(decreases)e(during)h (the)h(bac)n(kw)n(ard)e(ev)n(olution.)69 b(Therefore)38 b(this)h(cluster)456 2526 y(could)e(approac)n(h)f(the)i(adjacen)n(t)f (one)g(without)i(mo)r(difying)e(its)h(size.)67 b(F)-7 b(or)37 b(this)h(reason,)h(a)456 2626 y(re\014nemen)n(t)34 b(in)n(to)h(smaller)f(clusters)g(do)r(es)h(not)f(an)n(ymore)g(guaran)n (tee)f(that)i(the)g(n)n(um)n(b)r(er)g(of)456 2725 y(clusters)i (increases.)68 b(W)-7 b(e)39 b(then)g(need)g(a)f(non)g(trivial)g(mo)r (di\014cation)g(of)h(that)g(part)f(of)g(the)456 2825 y(pro)r(of.)555 2925 y(Assume)27 b(that)g(the)g(c)n(harged)e(particle)h (v)n(elo)r(cit)n(y)g(is)h(negativ)n(e)f(and)g(v)n(ery)g(large)f(in)i (absolute)456 3024 y(v)-5 b(alue)25 b(at)f(time)i Fn(T)1002 3036 y Fk(k)1042 3024 y Fr(.)36 b(Consider)25 b(the)g(\014rst)g(time)g Fn(\033)1988 3036 y Fo(0)2049 3024 y Fm(2)f Fr([)p Fn(T)2200 3036 y Fk(k)q Fj(\000)p Fo(1)2325 3024 y Fn(;)14 b(T)2411 3036 y Fk(k)2451 3024 y Fr(\))26 b(in)f(the)g(bac)n(kw)n(ard)e(ev)n (olution)456 3124 y(when)33 b(the)g(particle)f(has)g(lost)h(some)f (amoun)n(t)g(of)h(its)g(\(negativ)n(e\))f(v)n(elo)r(cit)n(y)-7 b(.)52 b(W)-7 b(e)33 b(no)n(w)f(fo)r(cus)456 3224 y(our)24 b(atten)n(tion)h(on)g(the)g(cluster)g(whic)n(h)g(con)n(tains)f(the)h(c) n(harged)f(particle)g(\(for)h(the)h(other)e(ones)456 3323 y(the)29 b(pro)r(cedure)e(remains)h(unc)n(hanged\).)40 b(During)28 b(the)h(forw)n(ard)e(ev)n(olution)h(from)g(time)h Fn(\033)3304 3335 y Fo(0)3371 3323 y Fr(to)456 3423 y(the)h(time,)g(sa) n(y)f Fn(\033)1008 3435 y Fo(1)1072 3423 y Fm(2)e Fr(\()p Fn(\033)1233 3435 y Fo(0)1271 3423 y Fn(;)14 b(T)1357 3435 y Fk(k)1397 3423 y Fr(],)30 b(when)g(the)g(c)n(harged)e(particle)h (has)g(gained)g(some)g(amoun)n(t)h(of)456 3522 y(its)25 b(\(negativ)n(e\))g(v)n(elo)r(cit)n(y)-7 b(,)25 b(the)h(electric)f (\014eld)h(acts)f(in)h(the)g(go)r(o)r(d)e(direction)h(and)h(the)g(cen)n (ter)f(of)456 3622 y(mass)30 b(cannot)g(decrease)g(to)r(o)g(m)n(uc)n(h) h(its)g(v)n(elo)r(cit)n(y)-7 b(.)46 b(So)31 b(a)f(re\014nemen)n(t)h(at) g(time)g Fn(\033)3051 3634 y Fo(1)3120 3622 y Fr(allo)n(ws)f(to)456 3722 y(decrease)g(the)i(cardinalit)n(y)e(of)h(this)h(cluster.)49 b(W)-7 b(e)32 b(then)g(start)f(from)g(time)h Fn(\033)2914 3734 y Fo(1)2983 3722 y Fr(and)g(consider)456 3821 y(the)d(bac)n(kw)n (ard)f(ev)n(olution)h(up)g(to)h(the)g(time)g Fn(\033)1944 3833 y Fo(2)2007 3821 y Fm(2)d Fr([)p Fn(\033)2159 3833 y Fo(0)2197 3821 y Fn(;)14 b(\033)2281 3833 y Fo(1)2318 3821 y Fr(\))30 b(in)g(whic)n(h)f(the)h(c)n(harged)e(particle)456 3921 y(has)k(lost)g(a)g(smaller)f(amoun)n(t)h(of)h(its)f(\(negativ)n (e\))g(v)n(elo)r(cit)n(y)-7 b(.)51 b(Since)32 b(in)h(this)g(case)e(the) i(cen)n(ter)456 4021 y(of)c(mass)f(cannot)h(increase)e(its)j(v)n(elo)r (cit)n(y)-7 b(,)28 b(a)h(further)g(re\014nemen)n(t)g(decreases)e(the)j (cardinalit)n(y)456 4120 y(of)e(the)g(cluster.)39 b(So)28 b(on,)g(alternating)f(bac)n(kw)n(ard)f(and)i(forw)n(ard)e(ev)n (olutions,)i(w)n(e)g(are)f(able)h(to)456 4220 y(pro)n(v)n(e)f(that)i (if)h(the)f(c)n(harged)f(particle)g(has)h(a)f(large)g(negativ)n(e)g(v)n (elo)r(cit)n(y)g(at)h(time)g Fn(T)41 b Fr(then)29 b(the)456 4319 y(latter)23 b(cannot)g(increase)f(to)r(o)i(m)n(uc)n(h)f(during)g (the)h(time)g(in)n(terv)-5 b(als)23 b([)p Fn(T)2599 4331 y Fk(k)q Fj(\000)p Fo(1)2725 4319 y Fn(;)14 b(T)2811 4331 y Fk(k)2851 4319 y Fr(].)36 b(It)24 b(then)g(arriv)n(es)456 4419 y(at)j(time)h(zero)f(to)r(o)g(fast,)h(th)n(us)f(getting)h(a)f(con) n(tradiction.)555 4519 y(Let)h(us)g(no)n(w)e(giv)n(e)h(the)h(rigorous)d (demonstration.)36 b(Eq.)27 b(\(4.21\))g(will)h(b)r(e)g(pro)n(v)n(ed)e (with)1406 4676 y Fn(T)1455 4688 y Fk(a)1495 4676 y Fr(\()p Fn(X)r(;)14 b(E)5 b Fr(\))1777 4629 y Fn(:)1756 4676 y Fr(=)23 b Fn(a)1888 4641 y Fj(\000)p Fk(K)t Fo([1+)p Fk(Q)2155 4649 y Fg(E)2202 4641 y Fo(\()p Fk(X)5 b Fo(\))2313 4616 y Fe(7)2346 4641 y Fo(])p Fk(=a)2435 4616 y Fe(6)2471 4676 y Fn(;)738 b Fr(\(4.22\))456 4817 y(where)20 b(the)i(parameter)e Fn(K)28 b Fm(\025)23 b Fr(1)d(will)i(b)r(e)f(\014xed)h(later.)34 b(In)21 b(the)h(sequel)f(w)n(e)f(mo)r(dify)i(the)g(previous)456 4917 y(notation)31 b(and)h(denote)g(b)n(y)g(\()p Fn(x)1429 4929 y Fo(0)1467 4917 y Fn(;)14 b(v)1544 4929 y Fo(0)1581 4917 y Fr(\))33 b(the)f(p)r(osition)g(and)g(v)n(elo)r(cit)n(y)f(of)h (the)h(c)n(harged)d(particle.)456 5016 y(W)-7 b(e)38 b(pro)r(ceed)e(b)n(y)i(con)n(tradiction.)65 b(W)-7 b(e)38 b(assume)f(there)g(exists)g Fn(T)51 b(>)39 b(T)2782 5028 y Fk(a)2822 5016 y Fr(\()p Fn(X)r(;)14 b(E)5 b Fr(\))38 b(for)f(whic)n(h)456 5116 y Fn(v)496 5128 y Fo(0)533 5116 y Fr(\()p Fn(T)12 b Fr(\))23 b Fn(<)f Fm(\000)p Fn(aT)37 b Fr(and)27 b(w)n(e)e(sho)n(w)h(that,)h(for)f Fn(a)1821 5128 y Fo(0)1884 5116 y Fr(small)g(enough)g(and)g Fn(K)j Fm(\025)22 b Fr(1)k(large)f(enough,)h(this)456 5216 y(implies)h Fn(v)777 5228 y Fo(0)815 5216 y Fr(\(0\))c Fn(<)g Fm(\000)p Fn(aT)9 b(=)p Fr(2)26 b(for)h(an)n(y)f Fn(a)d Fm(2)h Fr(\(0)p Fn(;)14 b(a)1893 5228 y Fo(0)1930 5216 y Fr(],)27 b(th)n(us)h(getting)f(an)h(absurd)f(since)g Fn(v)3099 5228 y Fo(0)3136 5216 y Fr(\(0\))d(=)e(0.)p eop %%Page: 20 20 20 19 bop 456 251 a Fo(20)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)555 450 y Fr(Using)33 b(\(4.2\),)g(\(4.3\),)g(and)f(\(4.4\))o(,)h(w) n(e)f(can)f(rep)r(eat)h(the)g(same)g(reasoning)e(from)h(Eq.)h(\(3.8\)) 456 550 y(to)h(Eq.)f(\(3.16\),)i(with)g(the)g(c)n(harged)d(particle)i (pla)n(ying)f(the)i(role)e(of)h(the)h Fn(i)p Fr(-th)f(particle.)53 b(W)-7 b(e)456 650 y(then)36 b(de\014ne)f(in)h(the)g(presen)n(t)e(con)n (text)h(the)h(sets)f Fm(P)7 b Fr(,)38 b Fm(P)2260 662 y Fo(1)2297 650 y Fr(,)f Fm(J)2413 662 y Fk(t)2443 650 y Fr(,)g(and)e(the)h(functions)g Fn(y)3233 615 y Fj(\006)3230 671 y Fk(t)3289 650 y Fr(,)h Fn(\015)3392 662 y Fk(t)3421 650 y Fr(,)456 750 y(b)n(y)e(replacing)f(ev)n(erywhere)g Fn(x)1429 762 y Fk(i)1493 750 y Fr(=)i Fn(x)1641 762 y Fk(i)1669 750 y Fr(\(0\))g([resp.)f Fn(Q)p Fr(\()p Fn(X)7 b Fr(\)])35 b(with)h Fn(x)2549 762 y Fo(0)2586 750 y Fr(\(0\))h(=)f(0)f([resp.)g Fn(Q)3203 762 y Fk(E)3258 750 y Fr(\()p Fn(X)7 b Fr(\)].)456 850 y(W)-7 b(e)30 b(analogously)d(in)n(tro)r(duce)i(the)h(times)g Fn(T)1842 862 y Fk(k)1930 802 y Fn(:)1909 850 y Fr(=)d Fn(k)s("T)12 b Fr(,)29 b Fn(k)g Fr(=)d(0)p Fn(;)14 b(:)g(:)g(:)f(;)h(")2627 819 y Fj(\000)p Fo(1)2716 850 y Fr(,)30 b(with)h Fn(")e Fr(as)g(in)h(\(3.19\))456 949 y(\(where)d Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))27 b(is)h(again)e(replaced)h(b)n(y)g Fn(Q)1775 961 y Fk(E)1831 949 y Fr(\()p Fn(X)7 b Fr(\)\).)555 1049 y(W)-7 b(e)37 b(shall)e(pro)n(v)n(e)g(that,)j(if)f Fn(a)1484 1061 y Fo(0)1557 1049 y Fr(is)f(small)f(enough)h(and)g Fn(K)41 b Fr(large)35 b(enough)g(then,)k(for)c(an)n(y)456 1148 y Fn(k)26 b Fr(=)c(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(")877 1118 y Fj(\000)p Fo(1)966 1148 y Fr(,)705 1339 y Fn(v)745 1351 y Fo(0)783 1339 y Fr(\()p Fn(T)864 1351 y Fk(k)905 1339 y Fr(\))23 b Fn(<)g Fm(\000)1123 1283 y Fn(a)p Fr(\()p Fn(T)29 b Fr(+)18 b Fn(T)1409 1295 y Fk(k)1449 1283 y Fr(\))p 1123 1320 360 4 v 1281 1396 a(2)1681 1339 y(=)-14 b Fm(\))188 b Fn(v)2043 1351 y Fo(0)2081 1339 y Fr(\()p Fn(T)2162 1351 y Fk(k)q Fj(\000)p Fo(1)2288 1339 y Fr(\))23 b Fn(<)f Fm(\000)2505 1283 y Fn(a)p Fr(\()p Fn(T)30 b Fr(+)18 b Fn(T)2792 1295 y Fk(k)q Fj(\000)p Fo(1)2917 1283 y Fr(\))p 2505 1320 445 4 v 2706 1396 a(2)2959 1339 y Fn(:)250 b Fr(\(4.23\))456 1514 y(By)24 b(\(4.23\))f(and)h(the)g (assumption)f Fn(v)1591 1526 y Fo(0)1629 1514 y Fr(\()p Fn(T)12 b Fr(\))22 b Fn(<)h Fm(\000)p Fn(aT)34 b Fr(it)25 b(follo)n(ws)d(that)i Fn(v)2620 1526 y Fo(0)2658 1514 y Fr(\(0\))f Fn(<)g Fm(\000)p Fn(aT)9 b(=)p Fr(2,)22 b(and)i(the)456 1614 y(theorem)j(is)g(pro)n(v)n(ed.)555 1713 y(Let)h Fn(k)e Fm(2)d(f)p Fr(1;)14 b Fn(:)g(:)g(:)f Fr(;)h Fn(")1158 1683 y Fj(\000)p Fo(1)1247 1713 y Fm(g)27 b Fr(and)g(assume)g Fn(v)1804 1725 y Fo(0)1842 1713 y Fr(\()p Fn(T)1923 1725 y Fk(k)1963 1713 y Fr(\))d Fn(<)e Fm(\000)p Fn(a)p Fr(\()p Fn(T)30 b Fr(+)18 b Fn(T)2458 1725 y Fk(k)2498 1713 y Fr(\))p Fn(=)p Fr(2.)36 b(W)-7 b(e)28 b(de\014ne:)680 1911 y Fn(\033)727 1923 y Fo(0)808 1864 y Fn(:)787 1911 y Fr(=)23 b(inf)990 1794 y Fh(\032)1052 1911 y Fn(t)g Fm(2)g Fr([)p Fn(T)1255 1923 y Fk(k)q Fj(\000)p Fo(1)1381 1911 y Fn(;)14 b(T)1467 1923 y Fk(k)1507 1911 y Fr(\))37 b(:)g Fn(v)1676 1923 y Fo(0)1714 1911 y Fr(\()p Fn(s)p Fr(\))23 b Fn(<)g(v)1968 1923 y Fo(0)2005 1911 y Fr(\()p Fn(T)2086 1923 y Fk(k)2127 1911 y Fr(\))c(+)2271 1854 y Fn(a"T)p 2271 1891 143 4 v 2321 1968 a Fr(2)2506 1911 y Fm(8)14 b Fn(s)22 b Fm(2)i Fr([)p Fn(t;)14 b(T)2846 1923 y Fk(k)2886 1911 y Fr(])2909 1794 y Fh(\033)2985 1911 y Fn(:)224 b Fr(\(4.24\))456 2108 y(Equation)28 b(\(4.23\))g(follo)n(ws)g(if)h Fn(\033)1455 2120 y Fo(0)1518 2108 y Fr(=)c Fn(T)1657 2120 y Fk(k)q Fj(\000)p Fo(1)1783 2108 y Fr(.)41 b(T)-7 b(o)28 b(pro)n(v)n(e)g(this)h(w)n(e)f(argue)g(b)n (y)h(con)n(tradiction,)f(b)n(y)456 2207 y(assuming)e Fn(\033)863 2219 y Fo(0)924 2207 y Fn(>)d(T)1061 2219 y Fk(k)q Fj(\000)p Fo(1)1186 2207 y Fr(.)37 b(Analogously)26 b(to)h(\(3.17\))g(and)h(\(3.18\))o(,)g(w)n(e)f(in)n(tro)r(duce)g(the)h (sets:)759 2355 y Fm(V)810 2367 y Fk(\033)848 2375 y Fe(0)989 2308 y Fn(:)968 2355 y Fr(=)1116 2287 y Fh(\010)1164 2355 y Fn(j)g Fm(2)c(P)43 b Fr(:)37 b Fm(j)p Fn(v)1529 2367 y Fk(j)1564 2355 y Fr(\()p Fn(\033)1643 2367 y Fo(0)1681 2355 y Fr(\))p Fm(j)24 b Fn(>)e(\015)1890 2367 y Fk(\033)1928 2375 y Fe(0)1965 2355 y Fn(aT)7 b(;)97 b(y)2229 2320 y Fj(\000)2226 2375 y Fk(\033)2264 2383 y Fe(0)2323 2355 y Fm(\024)23 b Fn(x)2458 2367 y Fk(j)2493 2355 y Fr(\()p Fn(\033)2572 2367 y Fo(0)2610 2355 y Fr(\))h Fm(\024)e Fn(y)2797 2320 y Fo(+)2794 2375 y Fk(\033)2832 2383 y Fe(0)2869 2287 y Fh(\011)2931 2355 y Fn(;)278 b Fr(\(4.25\))710 2481 y Fm(M)810 2493 y Fk(\033)848 2501 y Fe(0)989 2434 y Fn(:)968 2481 y Fr(=)83 b Fm(f)o Fn(j)28 b Fm(2)c(P)43 b Fr(:)37 b Fm(j)p Fn(v)1522 2493 y Fk(j)1557 2481 y Fr(\()p Fn(\033)1636 2493 y Fo(0)1674 2481 y Fr(\))p Fm(j)24 b(\024)e Fn(\015)1883 2493 y Fk(\033)1921 2501 y Fe(0)1958 2481 y Fn(aT)12 b Fm(g)h Fn(:)1091 b Fr(\(4.26\))456 2628 y(By)26 b(\(4.24\))e(the)i(c)n(harged)d(particle)i(b)r(elongs)g (to)g(the)g(set)h(of)f(fast)g(particles,)g(i.e.)36 b(0)23 b Fm(2)g(V)3167 2640 y Fk(\033)3205 2648 y Fe(0)3242 2628 y Fr(.)36 b(The)456 2728 y(same)e(argumen)n(ts)g(leading)g(from)g (Eq.)h(\(3.19\))f(to)h(Eq.)f(\(3.21\))g(apply)h(here)f(to)h(the)h(set)e Fm(V)3369 2740 y Fk(\033)3407 2748 y Fe(0)456 2828 y Fr(with)d Fn(\014)k Fr(as)30 b(in)h(\(3.21\))f(\(where)h Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))30 b(is)g(replaced)g(b)n(y)h Fn(Q)2292 2840 y Fk(E)2347 2828 y Fr(\()p Fn(X)7 b Fr(\)\).)47 b(The)31 b(particles)f(of)g(the)h(set)456 2927 y Fm(V)507 2939 y Fk(\033)545 2947 y Fe(0)604 2927 y Fr(can)22 b(th)n(us)h(in)n (teract)f(only)g(among)g(themselv)n(es)g(and)g(with)h(the)g(bac)n (kground)e Fm(M)3103 2939 y Fk(\033)3141 2947 y Fe(0)3200 2927 y Fr(of)h(slo)n(w)456 3027 y(particles)k(during)i(the)g(time)g(in) n(terv)-5 b(al)27 b([)p Fn(\033)1754 3039 y Fo(0)1791 3027 y Fn(;)14 b(T)1877 3039 y Fk(k)1918 3027 y Fr(].)37 b(Moreo)n(v)n(er,)25 b(b)n(y)i([the)h(analogous)d(of)6 b(])28 b(\(3.15\))o(,)1526 3227 y Fm(jV)1600 3239 y Fk(\033)1638 3247 y Fe(0)1675 3227 y Fm(j)23 b Fn(<)f(\032)1851 3239 y Fj(\003)1899 3171 y Fr(1)c(+)g Fn(Q)2108 3183 y Fk(E)2164 3171 y Fr(\()p Fn(X)7 b Fr(\))2304 3141 y Fo(2)p 1899 3208 442 4 v 2080 3284 a Fn(a)2124 3260 y Fo(2)2351 3227 y Fn(:)858 b Fr(\(4.27\))456 3403 y(Finally)-7 b(,)27 b(analogously)e(to)j(Lemma)f(3.1,)g(for)g Fn(a)1930 3415 y Fo(0)1995 3403 y Fr(small)g(enough)g(and)h Fn(")f Fr(as)g(in)h (\(3.22\))o(,)1302 3550 y(max)1267 3603 y Fk(j)s Fj(2M)1420 3611 y Fg(\033)1454 3623 y Fe(0)1582 3550 y Fr(sup)1505 3623 y Fk(s)p Fj(2)p Fo([)p Fk(\033)1638 3631 y Fe(0)1670 3623 y Fk(;T)1729 3632 y Fg(k)1766 3623 y Fo(])1798 3550 y Fm(j)p Fn(v)1861 3562 y Fk(j)1897 3550 y Fr(\()p Fn(s)p Fr(\))p Fm(j)23 b(\024)g Fr(\()p Fn(\015)2209 3562 y Fk(\033)2247 3570 y Fe(0)2303 3550 y Fr(+)18 b(2)p Fn(\014)t Fr(\))p Fn(aT)7 b(:)598 b Fr(\(4.28\))456 3765 y(Observ)n(e)33 b(in)i(fact)f(that)h(in)g(the)g(pro)r(of)f(of)h(the)g(lemma)g(w)n(e)f (ha)n(v)n(e)f(only)i(to)f(replace)g Fm(kr)p Fr(\010)p Fm(k)3375 3777 y Fj(1)456 3864 y Fr(b)n(y)i(max)o Fm(fkr)p Fr(\010)p Fm(k)989 3876 y Fj(1)1058 3864 y Fr(;)14 b Fm(kr)p Fr(\011)p Fm(k)1313 3876 y Fj(1)1382 3864 y Fm(g)36 b Fr(in)h(the)g(righ)n(t-hand)e(side)h(of)h(Eq.)f(\(3.24\))o(,)i(th)n (us)f(taking)f(in)n(to)456 3964 y(accoun)n(t)26 b(of)i(the)g(in)n (teraction)f(with)h(the)g(c)n(harged)e(particle.)555 4063 y(Let)f(\005)763 4075 y Fo(0)826 4063 y Fr(b)r(e)g(the)h (partition)e(of)h Fm(V)1563 4075 y Fk(\033)1601 4083 y Fe(0)1663 4063 y Fr(in)n(to)g(\()p Fn(\014)1912 4033 y Fk(`)1940 4041 y Fe(0)1977 4063 y Fn(;)14 b(\033)2061 4075 y Fo(0)2099 4063 y Fr(\)-clusters)24 b(with)h Fn(`)2678 4075 y Fo(0)2740 4063 y Fr(the)h(smallest)e(in)n(teger)456 4163 y(suc)n(h)j(that:)1368 4311 y(max)1377 4364 y Fk(i)p Fj(2C)1481 4372 y Fe(0)1537 4311 y Fm(j)p Fn(v)1600 4323 y Fk(i)1628 4311 y Fr(\()p Fn(\033)1707 4323 y Fo(0)1745 4311 y Fr(\))18 b Fm(\000)g Fn(v)1918 4323 y Fj(C)1954 4331 y Fe(0)1991 4311 y Fr(\()p Fn(\033)2070 4323 y Fo(0)2108 4311 y Fr(\))p Fm(j)24 b Fn(<)2284 4255 y(\014)2335 4225 y Fk(`)2363 4233 y Fe(0)2400 4255 y Fn("T)p 2284 4292 215 4 v 2350 4368 a Fr(32)2509 4311 y Fn(;)700 b Fr(\(4.29\))456 4477 y(where)25 b Fm(C)738 4489 y Fo(0)802 4477 y Fr(denotes)h(the)g (cluster)g(of)g(\005)1664 4489 y Fo(0)1728 4477 y Fr(whic)n(h)g(con)n (tains)f(the)i(c)n(harged)e(particle.)36 b(W)-7 b(e)26 b(de\014ne:)947 4624 y Fn(w)1006 4636 y Fo(0)1147 4577 y Fn(:)1126 4624 y Fr(=)83 b(min)1275 4678 y Fk(i)p Fj(2C)1379 4686 y Fe(0)1426 4624 y Fn(v)1466 4636 y Fk(i)1494 4624 y Fr(\()p Fn(\033)1573 4636 y Fo(0)1611 4624 y Fr(\))p Fn(;)97 b(W)1841 4636 y Fo(0)1950 4577 y Fn(:)1930 4624 y Fr(=)50 b(max)2054 4678 y Fk(i)p Fj(2C)2158 4686 y Fe(0)2213 4624 y Fn(v)2253 4636 y Fk(i)2281 4624 y Fr(\()p Fn(\033)2360 4636 y Fo(0)2398 4624 y Fr(\))p Fn(;)843 4792 y(D)912 4804 y Fo(0)949 4792 y Fr(\()p Fn(t)p Fr(\))1147 4745 y Fn(:)1126 4792 y Fr(=)147 b(inf)1274 4849 y Fk(s)p Fj(2)p Fo([)p Fk(\033)1407 4857 y Fe(0)1440 4849 y Fk(;t)p Fo(])1517 4792 y Fr(min)p Fm(fj)p Fn(v)1760 4804 y Fk(i)1788 4792 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)2033 4804 y Fk(j)2068 4792 y Fr(\()p Fn(s)p Fr(\))p Fm(j)37 b Fr(:)g Fn(i)23 b Fm(2)h(C)2466 4804 y Fo(0)2502 4792 y Fn(;)42 b(j)28 b Fm(2)23 b(V)2758 4804 y Fk(\033)2796 4812 y Fe(0)2852 4792 y Fm(n)18 b(C)2956 4804 y Fo(0)2993 4792 y Fm(g)p Fn(;)456 4986 y Fr(and)567 5169 y Fn(\033)614 5181 y Fo(1)695 5121 y Fn(:)675 5169 y Fr(=)k(sup)901 5051 y Fh(\032)963 5169 y Fn(t)h Fm(2)h Fr(\()p Fn(\033)1174 5181 y Fo(0)1212 5169 y Fn(;)14 b(T)1298 5181 y Fk(k)1338 5169 y Fr(])37 b(:)101 b(inf)1458 5226 y Fk(s)p Fj(2)p Fo([)p Fk(\033)1591 5234 y Fe(0)1624 5226 y Fk(;t)p Fo(])1701 5169 y Fn(v)1741 5181 y Fo(0)1779 5169 y Fr(\()p Fn(s)p Fr(\))23 b Fn(>)g(w)2052 5181 y Fo(0)2108 5169 y Fm(\000)2201 5112 y Fn(\014)2252 5082 y Fk(`)2280 5090 y Fe(0)2317 5112 y Fn("T)p 2201 5149 V 2287 5225 a Fr(8)2425 5169 y Fn(;)42 b(D)2559 5181 y Fo(0)2596 5169 y Fr(\()p Fn(t)p Fr(\))24 b Fn(>)2811 5112 y(\014)2862 5082 y Fk(`)2890 5090 y Fe(0)2927 5112 y Fn("T)p 2811 5149 V 2898 5225 a Fr(2)3036 5051 y Fh(\033)3098 5169 y Fn(:)111 b Fr(\(4.30\))p eop %%Page: 21 21 21 20 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(21)555 450 y Fr(W)-7 b(e)20 b(no)n(w)f(consider)f(the)i(bac)n(kw)n (ard)d(ev)n(olution)i(of)g(the)h(cluster)f Fm(C)2545 462 y Fo(0)2601 450 y Fr(during)g(the)h(time)g(in)n(terv)-5 b(al)456 550 y([)p Fn(\033)526 562 y Fo(0)563 550 y Fn(;)14 b(\033)647 562 y Fo(1)685 550 y Fr(].)52 b(Let)32 b(\005)998 562 y Fo(1)1068 550 y Fr(b)r(e)h(the)g(partition)f(of)h Fm(C)1829 562 y Fo(0)1898 550 y Fr(in)n(to)f(\()p Fn(\014)2154 520 y Fk(`)2182 528 y Fe(1)2219 550 y Fn(;)14 b(\033)2303 562 y Fo(1)2341 550 y Fr(\)-clusters)32 b(with)h Fn(`)2936 562 y Fo(1)3005 550 y Fr(the)g(smallest)456 649 y(in)n(teger)26 b(greater)g(or)h(equal)g(to)g Fn(`)1469 661 y Fo(0)1525 649 y Fr(+)18 b(1)27 b(suc)n(h)g(that:)1368 848 y(max)1377 901 y Fk(i)p Fj(2C)1481 909 y Fe(1)1537 848 y Fm(j)p Fn(v)1600 860 y Fk(i)1628 848 y Fr(\()p Fn(\033)1707 860 y Fo(1)1745 848 y Fr(\))18 b Fm(\000)g Fn(v)1918 860 y Fj(C)1954 868 y Fe(1)1991 848 y Fr(\()p Fn(\033)2070 860 y Fo(1)2108 848 y Fr(\))p Fm(j)24 b Fn(<)2284 791 y(\014)2335 761 y Fk(`)2363 769 y Fe(1)2400 791 y Fn("T)p 2284 828 215 4 v 2350 904 a Fr(32)2509 848 y Fn(;)456 1031 y Fr(where)j Fm(C)740 1043 y Fo(1)804 1031 y Fr(is)h(the)g (cluster)f(of)h(\005)1456 1043 y Fo(1)1521 1031 y Fr(whic)n(h)f(con)n (tains)g(the)h(c)n(harged)e(particle.)36 b(W)-7 b(e)28 b(next)g(de\014ne:)969 1177 y Fn(w)1028 1189 y Fo(1)1169 1130 y Fn(:)1149 1177 y Fr(=)82 b(min)1297 1231 y Fk(i)p Fj(2C)1401 1239 y Fe(1)1449 1177 y Fn(v)1489 1189 y Fk(i)1516 1177 y Fr(\()p Fn(\033)1595 1189 y Fo(1)1633 1177 y Fr(\))p Fn(;)98 b(W)1864 1189 y Fo(1)1973 1130 y Fn(:)1952 1177 y Fr(=)50 b(max)2076 1231 y Fk(i)p Fj(2C)2180 1239 y Fe(1)2236 1177 y Fn(v)2276 1189 y Fk(i)2304 1177 y Fr(\()p Fn(\033)2383 1189 y Fo(1)2420 1177 y Fr(\))p Fn(;)865 1345 y(D)934 1357 y Fo(1)971 1345 y Fr(\()p Fn(t)p Fr(\))1169 1298 y Fn(:)1149 1345 y Fr(=)146 b(inf)1296 1402 y Fk(s)p Fj(2)p Fo([)p Fk(t;\033)1474 1410 y Fe(1)1507 1402 y Fo(])1539 1345 y Fr(min)q Fm(fj)p Fn(v)1783 1357 y Fk(i)1810 1345 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)2055 1357 y Fk(j)2091 1345 y Fr(\()p Fn(s)p Fr(\))p Fm(j)37 b Fr(:)g Fn(i)23 b Fm(2)g(C)2488 1357 y Fo(1)2525 1345 y Fn(;)42 b(j)28 b Fm(2)23 b(C)2774 1357 y Fo(0)2829 1345 y Fm(n)18 b(C)2933 1357 y Fo(1)2970 1345 y Fm(g)p Fn(;)456 1538 y Fr(and)27 b(set:)572 1720 y Fn(\033)619 1732 y Fo(2)700 1673 y Fn(:)680 1720 y Fr(=)22 b(inf)882 1603 y Fh(\032)944 1720 y Fn(t)h Fm(2)h Fr([)p Fn(\033)1146 1732 y Fo(0)1183 1720 y Fn(;)14 b(\033)1267 1732 y Fo(1)1305 1720 y Fr(\))37 b(:)89 b(sup)1434 1794 y Fk(s)p Fj(2)p Fo([)p Fk(t;\033)1612 1802 y Fe(1)1645 1794 y Fo(])1677 1720 y Fn(v)1717 1732 y Fo(0)1755 1720 y Fr(\()p Fn(s)p Fr(\))23 b Fn(<)g(W)2047 1732 y Fo(1)2103 1720 y Fr(+)2196 1664 y Fn(\014)2247 1634 y Fk(`)2275 1642 y Fe(1)2312 1664 y Fn("T)p 2196 1701 V 2283 1777 a Fr(8)2421 1720 y Fn(;)41 b(D)2554 1732 y Fo(1)2591 1720 y Fr(\()p Fn(t)p Fr(\))24 b Fn(>)2806 1664 y(\014)2857 1634 y Fk(`)2885 1642 y Fe(1)2922 1664 y Fn("T)p 2806 1701 V 2893 1777 a Fr(2)3031 1603 y Fh(\033)3093 1720 y Fn(:)116 b Fr(\(4.31\))456 1930 y(W)-7 b(e)26 b(iterate)f(the)h(ab)r(o)n(v)n(e)f(pro)r(cedure)g (and)g(de\014ne)h(recursiv)n(ely)e(a)i(sequence)f(of)h(time)g(in)n (terv)-5 b(als,)651 2076 y([)p Fn(\033)721 2088 y Fo(0)759 2076 y Fn(;)14 b(T)845 2088 y Fk(k)886 2076 y Fr(])23 b Fm(\023)f Fr([)p Fn(\033)1089 2088 y Fo(0)1127 2076 y Fn(;)14 b(\033)1211 2088 y Fo(1)1249 2076 y Fr(])23 b Fm(\023)f Fr([)p Fn(\033)1452 2088 y Fo(2)1490 2076 y Fn(;)14 b(\033)1574 2088 y Fo(1)1612 2076 y Fr(])23 b Fm(\023)f Fr([)p Fn(\033)1815 2088 y Fo(2)1853 2076 y Fn(;)14 b(\033)1937 2088 y Fo(3)1975 2076 y Fr(])23 b Fm(\023)g Fr([)p Fn(\033)2179 2088 y Fo(4)2216 2076 y Fn(;)14 b(\033)2300 2088 y Fo(3)2338 2076 y Fr(])23 b Fm(\023)g Fr([)p Fn(\033)2542 2088 y Fo(4)2579 2076 y Fn(;)14 b(\033)2663 2088 y Fo(5)2701 2076 y Fr(])23 b Fm(\023)g Fr([)p Fn(\033)2905 2088 y Fo(6)2942 2076 y Fn(;)14 b(\033)3026 2088 y Fo(5)3064 2076 y Fr(])28 b Fn(:)14 b(:)g(:)f(;)456 2223 y Fr(according)25 b(to)j(the)g(follo)n (wing)e(rule.)37 b(F)-7 b(or)27 b(an)n(y)g(in)n(teger)f Fn(n)d Fm(\025)g Fr(1)k(w)n(e)g(set:)687 2370 y Fn(D)756 2382 y Fo(2)p Fk(n)834 2370 y Fr(\()p Fn(t)p Fr(\))1033 2323 y Fn(:)1012 2370 y Fr(=)165 b(inf)1160 2427 y Fk(s)p Fj(2)p Fo([)p Fk(\033)1293 2435 y Fe(2)p Fg(n)1362 2427 y Fk(;t)p Fo(])1439 2370 y Fr(min)q Fm(fj)p Fn(v)1683 2382 y Fk(i)1710 2370 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)1955 2382 y Fk(j)1990 2370 y Fr(\()p Fn(s)p Fr(\))p Fm(j)38 b Fr(:)f Fn(i)22 b Fm(2)i(C)2388 2382 y Fo(2)p Fk(n)2466 2370 y Fn(;)41 b(j)28 b Fm(2)c(C)2715 2382 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2896 2370 y Fm(n)18 b(C)3000 2382 y Fo(2)p Fk(n)3078 2370 y Fm(g)p Fn(;)603 2547 y(D)672 2559 y Fo(2)p Fk(n)p Fo(+1)834 2547 y Fr(\()p Fn(t)p Fr(\))1033 2500 y Fn(:)1012 2547 y Fr(=)200 b(inf)1160 2605 y Fk(s)p Fj(2)p Fo([)p Fk(t;\033)1338 2613 y Fe(2)p Fg(n)p Fe(+1)1478 2605 y Fo(])1510 2547 y Fr(min)q Fm(fj)p Fn(v)1754 2559 y Fk(i)1781 2547 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)2026 2559 y Fk(j)2061 2547 y Fr(\()p Fn(s)p Fr(\))p Fm(j)38 b Fr(:)f Fn(i)22 b Fm(2)i(C)2459 2559 y Fo(2)p Fk(n)p Fo(+1)2621 2547 y Fn(;)41 b(j)28 b Fm(2)c(C)2870 2559 y Fo(2)p Fk(n)2966 2547 y Fm(n)18 b(C)3070 2559 y Fo(2)p Fk(n)p Fo(+1)3232 2547 y Fm(g)p Fn(;)456 2741 y Fr(and)611 2923 y Fn(\033)658 2935 y Fo(2)p Fk(n)p Fo(+1)924 2876 y Fn(:)904 2923 y Fr(=)82 b(sup)1190 2806 y Fh(\032)1252 2923 y Fn(t)23 b Fm(2)h Fr(\()p Fn(\033)1463 2935 y Fo(2)p Fk(n)1542 2923 y Fn(;)14 b(\033)1626 2935 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)1789 2923 y Fr(])37 b(:)120 b(inf)1909 2981 y Fk(s)p Fj(2)p Fo([)p Fk(\033)2042 2989 y Fe(2)p Fg(n)2112 2981 y Fk(;t)p Fo(])2189 2923 y Fn(v)2229 2935 y Fo(0)2267 2923 y Fr(\()p Fn(s)p Fr(\))23 b Fn(>)g(w)2540 2935 y Fo(2)p Fk(n)2637 2923 y Fm(\000)2730 2867 y Fn(\014)2781 2837 y Fk(`)2809 2845 y Fe(2)p Fg(n)2882 2867 y Fn("T)p 2730 2904 252 4 v 2835 2980 a Fr(8)1383 3163 y(and)28 b Fn(D)1614 3175 y Fo(2)p Fk(n)1692 3163 y Fr(\()p Fn(t)p Fr(\))23 b Fn(>)1907 3107 y(\014)1958 3077 y Fk(`)1986 3085 y Fe(2)p Fg(n)2059 3107 y Fn("T)p 1907 3144 V 2012 3220 a Fr(2)2168 3046 y Fh(\033)2230 3163 y Fn(;)979 b Fr(\(4.32\))611 3402 y Fn(\033)658 3414 y Fo(2)p Fk(n)p Fo(+2)924 3355 y Fn(:)904 3402 y Fr(=)82 b(inf)1166 3285 y Fh(\032)1228 3402 y Fn(t)23 b Fm(2)h Fr([)p Fn(\033)1430 3414 y Fo(2)p Fk(n)1509 3402 y Fn(;)14 b(\033)1593 3414 y Fo(2)p Fk(n)p Fo(+1)1755 3402 y Fr(\))37 b(:)143 b(sup)1884 3476 y Fk(s)p Fj(2)p Fo([)p Fk(t;\033)2062 3484 y Fe(2)p Fg(n)p Fe(+1)2202 3476 y Fo(])2235 3402 y Fn(v)2275 3414 y Fo(0)2313 3402 y Fr(\()p Fn(s)p Fr(\))23 b Fn(<)g(W)2605 3414 y Fo(2)p Fk(n)p Fo(+1)2786 3402 y Fr(+)2879 3346 y Fn(\014)2930 3316 y Fk(`)2958 3324 y Fe(2)p Fg(n)p Fe(+1)3102 3346 y Fn("T)p 2879 3383 322 4 v 3019 3459 a Fr(8)3211 3402 y Fn(;)1383 3659 y Fr(and)28 b Fn(D)1614 3671 y Fo(2)p Fk(n)p Fo(+1)1776 3659 y Fr(\()p Fn(t)p Fr(\))23 b Fn(>)1991 3603 y(\014)2042 3573 y Fk(`)2070 3581 y Fe(2)p Fg(n)p Fe(+1)2214 3603 y Fn("T)p 1991 3640 V 2131 3716 a Fr(2)2323 3542 y Fh(\033)2385 3659 y Fn(;)824 b Fr(\(4.33\))456 3851 y(where,)27 b(for)g(an)n(y)g Fn(n)c Fm(\025)f Fr(2,)1224 3997 y Fn(w)1283 4009 y Fk(n)1372 3950 y Fn(:)1351 3997 y Fr(=)k(min)1439 4051 y Fk(i)p Fj(2C)1543 4059 y Fg(n)1598 3997 y Fn(v)1638 4009 y Fk(i)1666 3997 y Fr(\()p Fn(\033)1745 4009 y Fk(n)1790 3997 y Fr(\))p Fn(;)181 b(W)2104 4009 y Fk(n)2193 3950 y Fn(:)2172 3997 y Fr(=)23 b(max)2265 4051 y Fk(i)p Fj(2C)2369 4059 y Fg(n)2428 3997 y Fn(v)2468 4009 y Fk(i)2496 3997 y Fr(\()p Fn(\033)2575 4009 y Fk(n)2621 3997 y Fr(\))p Fn(;)456 4181 y Fr(and)f(the)h(set)g Fm(C)919 4193 y Fk(n)987 4181 y Fr(is)g(the)g(cluster)f(of)h(\005)1619 4193 y Fk(n)1687 4181 y Fr(whic)n(h)g(con)n(tains)f(the)h(c)n(harged)e (particle,)i(\005)3062 4193 y Fk(n)3131 4181 y Fr(denoting)456 4280 y(the)28 b(partition)f(of)g Fm(C)1083 4292 y Fk(n)p Fj(\000)p Fo(1)1241 4280 y Fr(in)n(to)g(\()p Fn(\014)1492 4250 y Fk(`)1520 4258 y Fg(n)1565 4280 y Fn(;)14 b(\033)1649 4292 y Fk(n)1695 4280 y Fr(\)-clusters)27 b(with)h Fn(`)2280 4292 y Fk(n)2352 4280 y Fr(the)g(smallest)f(in)n(teger)g(greater)e(or) 456 4380 y(equal)i(to)g Fn(`)811 4392 y Fk(n)p Fj(\000)p Fo(1)959 4380 y Fr(+)18 b(1)28 b(suc)n(h)f(that:)1352 4581 y(max)1357 4635 y Fk(i)p Fj(2C)1461 4643 y Fg(n)1520 4581 y Fm(j)p Fn(v)1583 4593 y Fk(i)1611 4581 y Fr(\()p Fn(\033)1690 4593 y Fk(n)1736 4581 y Fr(\))19 b Fm(\000)f Fn(v)1910 4593 y Fj(C)1946 4601 y Fg(n)1991 4581 y Fr(\()p Fn(\033)2070 4593 y Fk(n)2116 4581 y Fr(\))p Fm(j)24 b Fn(<)2292 4525 y(\014)2343 4495 y Fk(`)2371 4503 y Fg(n)2416 4525 y Fn("T)p 2292 4562 223 4 v 2362 4638 a Fr(32)2525 4581 y Fn(:)684 b Fr(\(4.34\))456 4769 y(W)-7 b(e)36 b(remark)e(that)i(the)g(function)h Fn(D)1641 4781 y Fo(2)p Fk(n)1719 4769 y Fr(\()p Fm(\001)p Fr(\))f([resp.)g Fn(D)2142 4781 y Fo(2)p Fk(n)p Fo(+1)2304 4769 y Fr(\()p Fm(\001)p Fr(\)])g(is)g(not)g(increasing)e([resp.)h(not)456 4869 y(decreasing].)555 4968 y(In)28 b(the)g(sequel)f(w)n(e)g(shall)h (assume)f Fn(a)1697 4980 y Fo(0)1757 4968 y Fn(<)22 b Fr(1)p Fn(=)p Fr(\(32)p Fn(\032)2087 4980 y Fj(\003)2124 4968 y Fr(\))28 b(so)f(that,)h(b)n(y)g(\(4.27\))o(,)1626 5115 y Fn(\014)1677 5085 y Fk(`)1705 5093 y Fg(n)p Fd(\000)p Fe(1)1823 5115 y Fn("T)p 1626 5152 296 4 v 1647 5228 a Fr(32)p Fm(jV)1805 5240 y Fk(\033)1843 5248 y Fe(0)1879 5228 y Fm(j)1955 5171 y Fn(>)23 b(\014)2094 5137 y Fk(`)2122 5145 y Fg(n)2167 5171 y Fn("T)7 b(:)p eop %%Page: 22 22 22 21 bop 456 251 a Fo(22)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(Since)h Fm(jC)735 462 y Fk(n)780 450 y Fm(j)g(\024)g(jC)981 462 y Fo(0)1018 450 y Fm(j)g(\024)g(jV)1226 462 y Fk(\033)1264 470 y Fe(0)1301 450 y Fm(j)g Fr(then)h Fn(`)1567 462 y Fo(0)1627 450 y Fm(\024)f(jV)1789 462 y Fk(\033)1827 470 y Fe(0)1864 450 y Fm(j)g Fr(and)g Fn(`)2102 462 y Fk(n)2170 450 y Fm(\024)g Fr(1)10 b(+)g Fn(`)2420 462 y Fk(n)p Fj(\000)p Fo(1)2559 450 y Fr(+)g Fm(jC)2701 462 y Fk(n)p Fj(\000)p Fo(1)2830 450 y Fm(j)g(\000)g(jC)3005 462 y Fk(n)3050 450 y Fm(j)23 b Fr(for)g Fn(n)g Fm(\025)g Fr(1,)456 550 y(whence:)1661 711 y Fn(`)1696 723 y Fk(n)1764 711 y Fm(\024)f Fr(2)p Fm(jV)1967 723 y Fk(\033)2005 731 y Fe(0)2042 711 y Fm(j)c Fr(+)g Fn(n:)993 b Fr(\(4.35\))456 896 y Fv(Lemma)26 b(4.1.)37 b Fp(F)-6 b(or)28 b Fn(a)1175 908 y Fo(0)1239 896 y Fp(smal)t(l)g(enough)g(and)f Fn(K)33 b Fp(lar)l(ge)28 b(enough,)g(if)g Fn(\033)2627 908 y Fo(0)2688 896 y Fn(>)23 b(T)2825 908 y Fk(k)q Fj(\000)p Fo(1)2977 896 y Fp(then,)28 b(for)g(any)456 996 y(inte)l(ger)h Fn(n)23 b Fr(=)g(0)p Fn(;)14 b(:)g(:)g(:)f(;)h Fr(2)p Fm(jV)1233 1008 y Fk(\033)1271 1016 y Fe(0)1307 996 y Fm(j)p Fp(,)1575 1213 y Fn(D)1644 1225 y Fk(n)1689 1213 y Fr(\()p Fn(\033)1768 1225 y Fk(n)p Fo(+1)1898 1213 y Fr(\))23 b Fm(\025)2051 1157 y Fr(5)p Fn(\014)2144 1127 y Fk(`)2172 1135 y Fg(n)2216 1157 y Fn("T)p 2051 1194 265 4 v 2162 1270 a Fr(8)3232 1213 y(\(4.36\))456 1416 y Fp(and)689 1775 y Fn(v)729 1787 y Fj(C)765 1795 y Fg(n)810 1775 y Fr(\()p Fn(t)p Fr(\))c Fm(\000)f Fn(v)1046 1787 y Fj(C)1082 1795 y Fg(n)1127 1775 y Fr(\()p Fn(\033)1206 1787 y Fk(n)1252 1775 y Fr(\))1298 1530 y Fh(8)1298 1605 y(>)1298 1630 y(>)1298 1655 y(>)1298 1679 y(<)1298 1829 y(>)1298 1854 y(>)1298 1879 y(>)1298 1904 y(:)1413 1647 y Fm(\025)23 b(\000)1576 1591 y Fn(\014)1627 1561 y Fk(`)1655 1569 y Fg(n)1699 1591 y Fn("T)p 1576 1628 223 4 v 1645 1704 a Fr(64)1893 1647 y Fm(8)14 b Fn(t)22 b Fm(2)h Fr([)p Fn(\033)2154 1659 y Fk(n)2200 1647 y Fn(;)14 b(\033)2284 1659 y Fk(n)p Fo(+1)2414 1647 y Fr(])83 b(if)34 b Fn(n)28 b Fr(is)f(ev)n(en)p Fn(;)1413 1931 y Fm(\024)c Fr(+)1576 1875 y Fn(\014)1627 1844 y Fk(`)1655 1852 y Fg(n)1699 1875 y Fn("T)p 1576 1912 V 1645 1988 a Fr(64)1893 1931 y Fm(8)14 b Fn(t)22 b Fm(2)h Fr([)p Fn(\033)2154 1943 y Fk(n)p Fo(+1)2284 1931 y Fn(;)14 b(\033)2368 1943 y Fk(n)2414 1931 y Fr(])83 b(if)34 b Fn(n)28 b Fr(is)f(o)r(dd)p Fn(:)3232 1775 y Fr(\(4.37\))555 2145 y(Before)34 b(pro)n(ving)e(the)j (lemma)f(let)h(us)f(conclude)g(the)h(pro)r(of)f(of)g(the)h(theorem.)56 b(Let)39 b(\026)-46 b Fn(n)3401 2098 y(:)3380 2145 y Fr(=)456 2245 y(2)p Fm(jV)572 2257 y Fk(\033)610 2265 y Fe(0)646 2245 y Fm(j)28 b Fr(and)f(set)h Fn(\033)1035 2257 y Fj(\000)p Fo(1)1168 2198 y Fn(:)1147 2245 y Fr(=)23 b Fn(T)1284 2257 y Fk(k)1324 2245 y Fr(.)37 b(By)28 b(con)n(tin)n(uit)n (y)f(and)g(Eq.)g(\(4.36\))g(w)n(e)g(ha)n(v)n(e:)560 2497 y Fn(\033)607 2509 y Fo(2)p Fk(n)p Fo(+1)793 2497 y Fn(<)22 b(\033)927 2509 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)1091 2497 y Fn(;)97 b Fr(1)22 b Fm(\024)h Fr(2)p Fn(n)18 b Fr(+)g(1)k Fn(<)27 b Fr(\026)-46 b Fn(n)97 b Fr(=)-14 b Fm(\))96 b Fn(v)2125 2509 y Fo(0)2163 2497 y Fr(\()p Fn(\033)2242 2509 y Fo(2)p Fk(n)p Fo(+1)2405 2497 y Fr(\))23 b(=)g Fn(w)2607 2509 y Fo(2)p Fk(n)2704 2497 y Fm(\000)2797 2441 y Fn(\014)2848 2411 y Fk(`)2876 2419 y Fe(2)p Fg(n)2949 2441 y Fn("T)p 2797 2478 252 4 v 2902 2554 a Fr(8)3058 2497 y Fn(;)151 b Fr(\(4.38\))787 2714 y Fn(\033)834 2726 y Fo(2)p Fk(n)936 2714 y Fn(>)22 b(\033)1070 2726 y Fo(2)p Fk(n)p Fj(\000)p Fo(2)1234 2714 y Fn(;)97 b Fr(2)22 b Fm(\024)h Fr(2)p Fn(n)f Fm(\024)27 b Fr(\026)-46 b Fn(n)97 b Fr(=)-14 b Fm(\))96 b Fn(v)2125 2726 y Fo(0)2163 2714 y Fr(\()p Fn(\033)2242 2726 y Fo(2)p Fk(n)2321 2714 y Fr(\))23 b(=)g Fn(W)2542 2726 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2724 2714 y Fr(+)2817 2658 y Fn(\014)2868 2628 y Fk(`)2896 2636 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)3042 2658 y Fn("T)p 2817 2695 325 4 v 2958 2771 a Fr(8)3151 2714 y Fn(:)58 b Fr(\(4.39\))456 2934 y(It)25 b(then)h(follo)n(ws)e(that)h (for)g Fn(a)1346 2946 y Fo(0)1408 2934 y Fr(small)g(enough)f(the)h (times)h Fn(T)2315 2946 y Fk(k)2355 2934 y Fn(;)14 b(\033)2439 2946 y Fo(0)2477 2934 y Fn(;)g(\033)2561 2946 y Fo(1)2598 2934 y Fn(;)g(:)g(:)g(:)g(;)g(\033)2834 2946 y Fo(\026)-37 b Fk(n)2901 2934 y Fr(are)24 b(distinct.)36 b(In)456 3034 y(fact,)27 b(if)i Fn(\033)766 3046 y Fo(1)826 3034 y Fr(=)23 b Fn(T)963 3046 y Fk(k)1031 3034 y Fr(w)n(e)k(get)h(an)f (absurd,)g(since:)488 3281 y Fn(v)528 3293 y Fo(0)565 3281 y Fr(\()p Fn(\033)644 3293 y Fo(1)682 3281 y Fr(\))84 b Fm(\025)e Fn(w)1004 3293 y Fo(0)1060 3281 y Fm(\000)1153 3225 y Fn(\014)1204 3195 y Fk(`)1232 3203 y Fe(0)1269 3225 y Fn("T)p 1153 3262 215 4 v 1240 3338 a Fr(8)1401 3281 y Fn(>)23 b(v)1529 3293 y Fo(0)1566 3281 y Fr(\()p Fn(\033)1645 3293 y Fo(0)1683 3281 y Fr(\))c Fm(\000)1827 3225 y Fr(2)p Fn(\014)1920 3195 y Fk(`)1948 3203 y Fe(0)1984 3225 y Fn("T)p 1827 3262 256 4 v 1913 3338 a Fr(32)2111 3281 y Fm(\000)2204 3225 y Fn(\014)2255 3195 y Fk(`)2283 3203 y Fe(0)2320 3225 y Fn("T)p 2204 3262 215 4 v 2291 3338 a Fr(8)2452 3281 y(=)j Fn(v)2579 3293 y Fo(0)2617 3281 y Fr(\()p Fn(T)2698 3293 y Fk(k)2738 3281 y Fr(\))d(+)2882 3225 y Fn(a"T)p 2882 3262 143 4 v 2933 3338 a Fr(2)3053 3281 y Fm(\000)3146 3225 y Fr(3)p Fn(\014)3239 3195 y Fk(`)3267 3203 y Fe(0)3303 3225 y Fn("T)p 3146 3262 256 4 v 3233 3338 a Fr(16)798 3434 y Fn(>)82 b(v)985 3446 y Fo(0)1023 3434 y Fr(\()p Fn(T)1104 3446 y Fk(k)1144 3434 y Fr(\))p Fn(:)456 3630 y Fr(where)30 b(w)n(e)h(used)h(\(4.29\))o (,)g(de\014nition)g(\(4.30\))o(,)h(and)e(that)h Fn(v)2316 3642 y Fo(0)2353 3630 y Fr(\()p Fn(\033)2432 3642 y Fo(0)2470 3630 y Fr(\))e(=)e Fn(v)2665 3642 y Fo(0)2703 3630 y Fr(\()p Fn(T)2784 3642 y Fk(k)2825 3630 y Fr(\))21 b(+)f Fn(a"T)9 b(=)p Fr(2)30 b(\(recall)456 3730 y(w)n(e)25 b(are)f(assuming)h Fn(\033)1118 3742 y Fo(0)1178 3730 y Fn(>)e(T)1315 3742 y Fk(k)q Fj(\000)p Fo(1)1440 3730 y Fr(\).)37 b(W)-7 b(e)26 b(then)g(pro)r(ceed)f(b)n(y)g(induction.)36 b(If)26 b Fn(n)d Fm(\025)g Fr(1)i(and)g Fn(\033)3193 3742 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)3380 3730 y Fn(<)456 3830 y(\033)503 3842 y Fo(2)p Fk(n)p Fj(\000)p Fo(3)694 3830 y Fr(then,)j(b)n(y)h(\(4.34\))d(and)i(\(4.38\))o(,)645 4082 y Fn(v)685 4094 y Fo(0)723 4082 y Fr(\()p Fn(\033)802 4094 y Fo(2)p Fk(n)881 4082 y Fr(\))83 b Fm(\024)g Fn(W)1222 4094 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)1404 4082 y Fr(+)1497 4026 y Fn(\014)1548 3996 y Fk(`)1576 4004 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)1722 4026 y Fn("T)p 1497 4063 325 4 v 1638 4139 a Fr(8)1868 4082 y Fm(\024)36 b Fn(v)2009 4094 y Fo(0)2047 4082 y Fr(\()p Fn(\033)2126 4094 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2290 4082 y Fr(\))18 b(+)2433 4026 y(2)p Fn(\014)2526 3996 y Fk(`)2554 4004 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2700 4026 y Fn("T)p 2433 4063 366 4 v 2575 4139 a Fr(32)2828 4082 y(+)2921 4026 y Fn(\014)2972 3996 y Fk(`)3000 4004 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)3146 4026 y Fn("T)p 2921 4063 325 4 v 3062 4139 a Fr(8)996 4299 y(=)83 b Fn(w)1203 4311 y Fo(2)p Fk(n)p Fj(\000)p Fo(2)1385 4299 y Fm(\000)1478 4243 y Fn(\014)1529 4213 y Fk(`)1557 4221 y Fe(2)p Fg(n)p Fd(\000)p Fe(2)1703 4243 y Fn("T)p 1478 4280 V 1619 4356 a Fr(8)1830 4299 y(+)1923 4243 y(3)p Fn(\014)2016 4213 y Fk(`)2044 4221 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2190 4243 y Fn("T)p 1923 4280 366 4 v 2065 4356 a Fr(16)2336 4299 y Fn(<)36 b(v)2477 4311 y Fo(0)2515 4299 y Fr(\()p Fn(\033)2594 4311 y Fo(2)p Fk(n)p Fj(\000)p Fo(2)2758 4299 y Fr(\))p Fn(;)456 4523 y Fr(whic)n(h)28 b(implies)h Fn(\033)1024 4535 y Fo(2)p Fk(n)1128 4523 y Fn(>)c(\033)1265 4535 y Fo(2)p Fk(n)p Fj(\000)p Fo(2)1457 4523 y Fr(\(in)k(the)h(last)e(inequalit)n(y)g(w)n (e)h(used)f(that)i Fn(`)2807 4535 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2994 4523 y Fn(>)25 b(`)3119 4535 y Fo(2)p Fk(n)p Fj(\000)p Fo(2)3311 4523 y Fr(and)456 4623 y Fn(\014)k(<)24 b Fr(2)p Fn(=)p Fr(3)j(for)h Fn(a)946 4635 y Fo(0)1012 4623 y Fr(small\).)40 b(Analogously)-7 b(,)28 b(if)h Fn(n)24 b Fm(\025)h Fr(1)j(and)g Fn(\033)2306 4635 y Fo(2)p Fk(n)2410 4623 y Fn(>)c(\033)2546 4635 y Fo(2)p Fk(n)p Fj(\000)p Fo(2)2739 4623 y Fr(then,)29 b(b)n(y)h(\(4.34\))e(and) 456 4722 y(\(4.39\))o(,)807 4975 y Fn(v)847 4987 y Fo(0)885 4975 y Fr(\()p Fn(\033)964 4987 y Fo(2)p Fk(n)p Fo(+1)1127 4975 y Fr(\))83 b Fm(\025)g Fn(w)1449 4987 y Fo(2)p Fk(n)1546 4975 y Fm(\000)1639 4919 y Fn(\014)1690 4889 y Fk(`)1718 4897 y Fe(2)p Fg(n)1791 4919 y Fn("T)p 1639 4956 252 4 v 1744 5032 a Fr(8)1937 4975 y Fm(\025)36 b Fn(v)2078 4987 y Fo(0)2116 4975 y Fr(\()p Fn(\033)2195 4987 y Fo(2)p Fk(n)2274 4975 y Fr(\))19 b Fm(\000)2418 4919 y Fr(2)p Fn(\014)2511 4889 y Fk(`)2539 4897 y Fe(2)p Fg(n)2611 4919 y Fn("T)p 2418 4956 293 4 v 2522 5032 a Fr(32)2739 4975 y Fm(\000)2832 4919 y Fn(\014)2883 4889 y Fk(`)2911 4897 y Fe(2)p Fg(n)2984 4919 y Fn("T)p 2832 4956 252 4 v 2936 5032 a Fr(8)1242 5192 y(=)83 b Fn(W)1468 5204 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)1650 5192 y Fr(+)1743 5136 y Fn(\014)1794 5106 y Fk(`)1822 5114 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)1969 5136 y Fn("T)p 1743 5173 325 4 v 1884 5249 a Fr(8)2096 5192 y Fm(\000)2189 5136 y Fr(3)p Fn(\014)2282 5106 y Fk(`)2310 5114 y Fe(2)p Fg(n)2382 5136 y Fn("T)p 2189 5173 293 4 v 2293 5249 a Fr(16)2528 5192 y Fn(>)37 b(v)2670 5204 y Fo(0)2707 5192 y Fr(\()p Fn(\033)2786 5204 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2950 5192 y Fr(\))p Fn(;)p eop %%Page: 23 23 23 22 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(23)456 450 y Fr(whic)n(h)29 b(implies)g Fn(\033)1025 462 y Fo(2)p Fk(n)p Fo(+1)1213 450 y Fn(<)c(\033)1350 462 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)1514 450 y Fr(.)41 b(W)-7 b(e)30 b(no)n(w)e(observ)n(e)f(that,)j(b)n(y)g(\(4.34\))o(,)g (\(4.38\))o(,)f(and)g(\(4.37\))o(,)456 550 y(for)e(an)n(y)f(1)d Fm(\024)g Fr(2)p Fn(n)17 b Fr(+)i(1)j Fn(<)27 b Fr(\026)-46 b Fn(n)p Fr(,)868 741 y Fn(v)908 753 y Fo(0)945 741 y Fr(\()p Fn(\033)1024 753 y Fo(2)p Fk(n)p Fo(+1)1187 741 y Fr(\))19 b Fm(\000)f Fn(v)1361 753 y Fj(C)1397 761 y Fe(2)p Fg(n)1470 741 y Fr(\()p Fn(\033)1549 753 y Fo(2)p Fk(n)p Fo(+1)1712 741 y Fr(\))84 b(=)e([)p Fn(w)2057 753 y Fo(2)p Fk(n)2154 741 y Fm(\000)18 b Fn(v)2277 753 y Fj(C)2313 761 y Fe(2)p Fg(n)2387 741 y Fr(\()p Fn(\033)2466 753 y Fo(2)p Fk(n)2545 741 y Fr(\)])h Fm(\000)2712 685 y Fn(\014)2763 655 y Fk(`)2791 663 y Fe(2)p Fg(n)2864 685 y Fn("T)p 2712 722 252 4 v 2816 798 a Fr(8)1975 893 y(+)14 b([)p Fn(v)2117 905 y Fj(C)2153 913 y Fe(2)p Fg(n)2226 893 y Fr(\()p Fn(\033)2305 905 y Fo(2)p Fk(n)2384 893 y Fr(\))19 b Fm(\000)f Fn(v)2558 905 y Fj(C)2594 913 y Fe(2)p Fg(n)2667 893 y Fr(\()p Fn(\033)2746 905 y Fo(2)p Fk(n)p Fo(+1)2909 893 y Fr(\)])1828 1074 y Fm(\024)1985 1018 y Fr(2)p Fn(\014)2078 988 y Fk(`)2106 996 y Fe(2)p Fg(n)2179 1018 y Fn("T)p 1985 1055 293 4 v 2090 1131 a Fr(32)2306 1074 y Fm(\000)2399 1018 y Fn(\014)2450 988 y Fk(`)2478 996 y Fe(2)p Fg(n)2551 1018 y Fn("T)p 2399 1055 252 4 v 2504 1131 a Fr(8)2679 1074 y(+)2772 1018 y Fn(\014)2823 988 y Fk(`)2851 996 y Fe(2)p Fg(n)2924 1018 y Fn("T)p 2772 1055 V 2855 1131 a Fr(64)1828 1291 y(=)82 b Fm(\000)2050 1234 y Fr(3)p Fn(\014)2143 1204 y Fk(`)2171 1212 y Fe(2)p Fg(n)2244 1234 y Fn("T)p 2050 1272 293 4 v 2155 1348 a Fr(64)2352 1291 y Fn(;)456 1458 y Fr(and,)27 b(b)n(y)i(\(4.34\))o(,)e(\(4.39\),)g(and)h(\(4.37\))o(,)g (for)f(an)n(y)g(2)22 b Fm(\024)h Fr(2)p Fn(n)f Fm(\024)27 b Fr(\026)-46 b Fn(n)p Fr(,)766 1654 y Fn(v)806 1666 y Fo(0)843 1654 y Fr(\()p Fn(\033)922 1666 y Fo(2)p Fk(n)1001 1654 y Fr(\))19 b Fm(\000)f Fn(v)1175 1666 y Fj(C)1211 1674 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)1358 1654 y Fr(\()p Fn(\033)1437 1666 y Fo(2)p Fk(n)1516 1654 y Fr(\))83 b(=)g([)p Fn(W)1880 1666 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2062 1654 y Fm(\000)18 b Fn(v)2185 1666 y Fj(C)2221 1674 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2368 1654 y Fr(\()p Fn(\033)2447 1666 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2611 1654 y Fr(\)])g(+)2777 1598 y Fn(\014)2828 1568 y Fk(`)2856 1576 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)3003 1598 y Fn("T)p 2777 1635 325 4 v 2919 1711 a Fr(8)1779 1815 y(+)1871 1747 y Fh(\002)1906 1815 y Fn(v)1946 1827 y Fj(C)1982 1835 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2128 1815 y Fr(\()p Fn(\033)2207 1827 y Fo(2)p Fk(n)p Fj(\000)p Fo(1)2371 1815 y Fr(\))h Fm(\000)f Fn(v)2545 1827 y Fj(C)2581 1835 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2728 1815 y Fr(\()p Fn(\033)2807 1827 y Fo(2)p Fk(n)2886 1815 y Fr(\))2918 1747 y Fh(\003)1631 2003 y Fm(\025)83 b(\000)1867 1947 y Fr(2)p Fn(\014)1960 1917 y Fk(`)1988 1925 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2134 1947 y Fn("T)p 1867 1984 366 4 v 2008 2060 a Fr(32)2261 2003 y(+)2354 1947 y Fn(\014)2405 1917 y Fk(`)2433 1925 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2580 1947 y Fn("T)p 2354 1984 325 4 v 2496 2060 a Fr(8)2707 2003 y Fm(\000)2800 1947 y Fn(\014)2851 1917 y Fk(`)2879 1925 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)3025 1947 y Fn("T)p 2800 1984 V 2920 2060 a Fr(64)1631 2220 y(=)1789 2164 y(3)p Fn(\014)1882 2134 y Fk(`)1910 2142 y Fe(2)p Fg(n)p Fd(\000)p Fe(1)2056 2164 y Fn("T)p 1789 2201 366 4 v 1930 2277 a Fr(64)2164 2220 y Fn(:)456 2383 y Fr(In)27 b(conclusion:)1006 2558 y Fm(j)p Fn(v)1069 2570 y Fo(0)1107 2558 y Fr(\()p Fn(\033)1186 2570 y Fk(n)1232 2558 y Fr(\))19 b Fm(\000)f Fn(v)1406 2570 y Fj(C)1442 2578 y Fg(n)p Fd(\000)p Fe(1)1560 2558 y Fr(\()p Fn(\033)1639 2570 y Fk(n)1685 2558 y Fr(\))p Fm(j)23 b(\025)1861 2502 y Fr(3)p Fn(\014)1954 2472 y Fk(`)1982 2480 y Fg(n)p Fd(\000)p Fe(1)2100 2502 y Fn("T)p 1861 2539 338 4 v 1988 2615 a Fr(64)2374 2558 y Fm(8)14 b Fn(n)22 b Fr(=)h(1)p Fn(;)14 b(:)g(:)g(:)f(;)18 b Fr(\026)-46 b Fn(n;)456 2726 y Fr(whic)n(h)35 b(in)g(particular)f(implies)h Fm(jC)1552 2738 y Fk(n)p Fj(\000)p Fo(1)1682 2726 y Fm(j)h Fn(>)g Fr(1)e(for)h(all)g Fn(n)g Fr(=)h(1)p Fn(;)14 b(:)g(:)g(:)f(;)18 b Fr(\026)-46 b Fn(n)p Fr(.)59 b(On)35 b(the)h(other)e(hand,)456 2825 y(recalling)i(\(4.27\))o(,)i(if)f Fn(a)1200 2837 y Fo(0)1275 2825 y Fn(<)g Fr(3)p Fn(=)p Fr(\(64)p Fn(\032)1620 2837 y Fj(\003)1656 2825 y Fr(\))g(then)f Fm(j)p Fr(\005)2007 2837 y Fk(n)2053 2825 y Fm(j)h Fn(>)g Fr(1,)h(i.e.)f Fm(jC)2528 2837 y Fk(n)2573 2825 y Fm(j)g Fn(<)g Fm(jC)2802 2837 y Fk(n)p Fj(\000)p Fo(1)2932 2825 y Fm(j)f Fr(for)g(an)n(y)g Fn(n)h Fr(=)456 2925 y(1)p Fn(;)14 b(:)g(:)g(:)f(;)18 b Fr(\026)-46 b Fn(n)o Fr(.)71 b(But)39 b Fm(jC)1068 2937 y Fo(0)1105 2925 y Fm(j)i(\024)h(jV)1350 2937 y Fk(\033)1388 2945 y Fe(0)1425 2925 y Fm(j)f(\024)46 b Fr(\026)-47 b Fn(n=)p Fr(2,)41 b(whence)e Fm(jC)2162 2937 y Fk(n)2203 2945 y Fd(\003)2241 2925 y Fm(j)j Fr(=)f(1)d(for)h (some)f Fn(n)2900 2937 y Fj(\003)2979 2925 y Fm(\024)46 b Fr(\026)-47 b Fn(n=)p Fr(2,)41 b(then)456 3024 y(getting)30 b(an)f(absurd.)44 b(The)30 b(assumption)g Fn(\033)1832 3036 y Fo(0)1897 3024 y Fn(>)d(T)2038 3036 y Fk(k)q Fj(\000)p Fo(1)2194 3024 y Fr(is)j(th)n(us)g(dispro)n(v)n(ed)e(and)i(the)h (theorem)456 3124 y(follo)n(ws.)2655 b Ff(\003)456 3264 y Fp(Pr)l(o)l(of)31 b(of)h(L)l(emma)e(4.1.)42 b Fr(W)-7 b(e)28 b(pro)n(v)n(e)f(the)i(result)f(b)n(y)g(induction.)40 b(W)-7 b(e)29 b(\014rst)f(consider)g(the)g(case)456 3363 y Fn(n)33 b Fr(=)h(0.)56 b(The)34 b(forw)n(ard)f(motion)h(of)g(the)g (particles)g(in)g Fm(V)2280 3375 y Fk(\033)2318 3383 y Fe(0)2378 3363 y Fm(n)22 b(C)2486 3375 y Fo(0)2557 3363 y Fr(during)34 b(the)g(time)h(in)n(terv)-5 b(al)456 3463 y([)p Fn(\033)526 3475 y Fo(0)563 3463 y Fn(;)14 b(\033)647 3475 y Fo(1)685 3463 y Fr(])26 b(can)g(b)r(e)g(studied)h(as) e(the)h(bac)n(kw)n(ard)e(motion)i(of)g(the)g(fast)g(particles)g Fm(V)2952 3475 y Fk(k)3018 3463 y Fr(in)h(the)f(pro)r(of)456 3563 y(of)d(Theorem)f(3.1.)34 b(More)23 b(precisely)-7 b(,)23 b(w)n(e)g(in)n(tro)r(duce)f(a)h(sequence)f(of)h(partitions)2980 3542 y(\026)2970 3563 y(\005)3032 3575 y Fk(`)3087 3563 y Fr(of)g Fm(V)3228 3575 y Fk(\033)3266 3583 y Fe(0)3313 3563 y Fm(n)9 b(C)3408 3575 y Fo(0)456 3662 y Fr(in)n(to)28 b(\()669 3640 y(\026)657 3662 y Fn(\014)708 3632 y Fk(`)741 3662 y Fn(;)18 b Fr(\026)-46 b Fn(\034)814 3674 y Fk(`)846 3662 y Fr(\)-clusters,)1243 3640 y(\026)1231 3662 y Fn(\014)29 b Fr(=)c Fn(\014)1448 3632 y Fk(`)1476 3640 y Fe(0)1512 3662 y Fn(=)p Fr(4,)j Fn(`)d Fr(=)f(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(`) 2057 3632 y Fj(\003)2095 3662 y Fr(,)29 b(where)f(the)h(times)k(\026) -46 b Fn(\034)2791 3674 y Fo(1)2853 3662 y Fm(\024)29 b Fr(\026)-46 b Fn(\034)2979 3674 y Fo(2)3041 3662 y Fm(\024)24 b Fn(:)14 b(:)g(:)25 b Fm(\024)k Fr(\026)-47 b Fn(\034)3377 3674 y Fk(`)3405 3658 y Fd(\003)456 3762 y Fr(and)27 b(the)h(in)n(teger)f Fn(`)1070 3732 y Fj(\003)1135 3762 y Fr(are)g(de\014ned,)h(analogously)d(to)i(Eq.)g(\(3.37\),)g(b)n (y:)667 3834 y Fh(8)667 3909 y(>)667 3934 y(>)667 3959 y(<)667 4108 y(>)667 4133 y(>)667 4158 y(:)786 3901 y Fr(\026)-46 b Fn(\034)818 3913 y Fo(1)899 3854 y Fn(:)879 3901 y Fr(=)22 b Fn(\033)1013 3913 y Fo(0)1051 3901 y Fn(;)786 4057 y Fr(\026)-46 b Fn(\034)818 4069 y Fk(`)p Fo(+1)978 4010 y Fn(:)957 4057 y Fr(=)23 b(inf)1160 3940 y Fh(\032)1222 4057 y Fn(t)g Fm(2)h Fr([)t(\026)-46 b Fn(\034)1413 4069 y Fk(`)1445 4057 y Fn(;)14 b(\033)1529 4069 y Fo(1)1566 4057 y Fr(])37 b(:)1695 4035 y(\026)1686 4057 y Fn(\030)1722 4069 y Fk(`)1755 4057 y Fr(\()p Fn(t)p Fr(\))23 b Fn(<)1982 3979 y Fr(\026)1970 4001 y Fn(\014)2021 3971 y Fk(`)2053 4001 y Fn("T)p 1970 4038 183 4 v 2040 4114 a Fr(8)2162 4057 y Fn(;)47 b Fr(\026)-48 b Fn(\021)2267 4069 y Fk(`)2300 4057 y Fr(\()p Fn(t)p Fr(\))23 b Fn(<)2527 3979 y Fr(\026)2515 4001 y Fn(\014)2566 3971 y Fk(`)2598 4001 y Fn("T)p 2515 4038 V 2585 4114 a Fr(8)2707 3940 y Fh(\033)2769 4057 y Fn(;)180 b(`)23 b Fm(\025)f Fr(1)p Fn(;)782 4206 y(`)817 4176 y Fj(\003)878 4206 y Fr(=)g(min)q Fm(f)p Fn(`)g Fm(2)h Fl(N)47 b Fr(:)42 b(\026)-47 b Fn(\034)1474 4218 y Fk(`)p Fo(+1)1614 4206 y Fr(=)22 b Fn(\033)1748 4218 y Fo(1)1786 4206 y Fm(g)p Fn(;)456 4342 y Fr(where:)943 4459 y(\026)935 4481 y Fn(\030)971 4493 y Fk(`)1003 4481 y Fr(\()p Fn(t)p Fr(\))1201 4434 y Fn(:)1181 4481 y Fr(=)129 b(sup)1328 4555 y Fk(s)p Fj(2)p Fo([)t(\026)-37 b Fk(\034)1454 4564 y Fg(`)1483 4555 y Fk(;t)p Fo(])1560 4481 y Fr(max)1729 4414 y Fh(\010)1777 4481 y Fm(j)p Fn(v)1840 4493 y Fj(C)1883 4481 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)2128 4493 y Fj(C)2172 4481 y Fr(\()p Fn(\034)2240 4493 y Fk(`)2272 4481 y Fr(\))p Fm(j)37 b Fr(:)g Fm(C)28 b(2)2585 4460 y Fr(\026)2574 4481 y(\005)2636 4493 y Fk(`)2669 4414 y Fh(\011)2731 4481 y Fn(;)936 4681 y Fr(\026)-48 b Fn(\021)971 4693 y Fk(`)1003 4681 y Fr(\()p Fn(t)p Fr(\))1201 4634 y Fn(:)1181 4681 y Fr(=)129 b(sup)1328 4755 y Fk(s)p Fj(2)p Fo([)t(\026)-37 b Fk(\034)1454 4764 y Fg(`)1483 4755 y Fk(;t)p Fo(])1560 4681 y Fr(max)1729 4614 y Fh(\010)1777 4681 y Fm(j)p Fn(v)1840 4693 y Fk(j)1875 4681 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)2120 4693 y Fj(C)2164 4681 y Fr(\()p Fn(s)p Fr(\))p Fm(j)37 b Fr(:)g Fn(j)28 b Fm(2)23 b(C)5 b Fn(;)41 b Fm(C)28 b(2)2801 4660 y Fr(\026)2790 4681 y(\005)2852 4693 y Fk(`)2885 4614 y Fh(\011)2947 4681 y Fn(;)456 4897 y Fr(and)h(setting)34 b(\026)-46 b Fn(\034)930 4909 y Fk(`)p Fo(+1)1073 4897 y Fr(=)31 b(\026)-47 b Fn(\034)1200 4909 y Fk(`)1263 4897 y Fr(if)36 b(\026)-48 b Fn(\021)1382 4909 y Fk(`)1414 4897 y Fr(\()t(\026)i Fn(\034)1482 4909 y Fk(`)1515 4897 y Fr(\))27 b Fm(\025)1677 4875 y Fr(\026)1665 4897 y Fn(\014)1716 4866 y Fk(`)1748 4897 y Fn("T)9 b(=)p Fr(8.)42 b(By)31 b(\(4.28\))e(and)h(since)f Fn(D)2808 4909 y Fo(0)2845 4897 y Fr(\()p Fn(t)p Fr(\))f Fm(\025)e Fn(\014)3109 4866 y Fk(`)3137 4874 y Fe(0)3174 4897 y Fn("T)9 b(=)p Fr(2)25 b(=)456 4996 y(2)510 4974 y(\026)498 4996 y Fn(\014)s("T)12 b Fr(,)27 b(analogously)e(to)j (\(3.39\))e(and)i(\(3.40\))o(,)g(for)f(an)n(y)g Fn(t)c Fm(2)g Fr([)t(\026)-46 b Fn(\034)2388 5008 y Fk(`)2420 4996 y Fn(;)19 b Fr(\026)-47 b Fn(\034)2493 5008 y Fk(`)p Fo(+1)2610 4996 y Fr(],)898 5192 y(min)1050 5125 y Fh(\010)1099 5192 y Fm(j)p Fn(v)1162 5204 y Fk(j)1197 5192 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1433 5204 y Fk(q)1470 5192 y Fr(\()p Fn(t)p Fr(\))p Fm(j)37 b Fr(:)g Fm(C)28 b(2)1845 5171 y Fr(\026)1834 5192 y(\005)1896 5204 y Fk(`)1928 5192 y Fn(;)42 b(j)28 b Fm(2)23 b(C)5 b Fn(;)42 b(q)26 b Fm(2)d(V)2439 5204 y Fk(\033)2477 5212 y Fe(0)2532 5192 y Fm(n)18 b(C)2641 5125 y Fh(\011)2712 5192 y Fm(\025)2822 5114 y Fr(\026)2810 5136 y Fn(\014)2861 5106 y Fk(`)2893 5136 y Fn("T)p 2810 5173 V 2880 5249 a Fr(2)p eop %%Page: 24 24 24 23 bop 456 251 a Fo(24)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(and)939 577 y(min)1091 509 y Fh(\010)1139 577 y Fm(j)p Fn(v)1202 589 y Fk(j)1238 577 y Fr(\()p Fn(t)p Fr(\))d Fm(\000)f Fn(v)1474 589 y Fk(q)1511 577 y Fr(\()p Fn(t)p Fr(\))p Fm(j)37 b Fr(:)g Fm(C)28 b(2)1886 556 y Fr(\026)1875 577 y(\005)1937 589 y Fk(`)1969 577 y Fn(;)42 b(j)28 b Fm(2)23 b(C)5 b Fn(;)42 b(q)26 b Fm(2)d(M)2529 589 y Fk(\033)2567 597 y Fe(0)2604 509 y Fh(\011)2675 577 y Fm(\025)2773 520 y Fn(\014)t(aT)p 2773 558 156 4 v 2830 634 a Fr(2)2938 577 y Fn(:)456 726 y Fr(Moreo)n(v)n(er,)30 b(b)r(oth)i Fn(`)1084 738 y Fo(0)1153 726 y Fr(and)g Fn(`)1354 696 y Fj(\003)1424 726 y Fr(are)f(not)h(bigger)f(than)h Fm(jV)2245 738 y Fk(\033)2283 746 y Fe(0)2320 726 y Fm(j)p Fr(.)50 b(Then,)33 b(recalling)e(Remark)g(3.2,)456 826 y(b)n(y)d(\(4.27\))f(and)g(the)h(c)n(hoice)f(\(4.22\))o(,)h(w)n(e)f (get:)1100 1021 y(sup)1028 1094 y Fk(t)p Fj(2)p Fo([)p Fk(\033)1155 1102 y Fe(0)1187 1094 y Fk(;\033)1245 1102 y Fe(1)1278 1094 y Fo(])1311 1021 y Fm(j)p Fn(v)1374 1033 y Fk(j)1409 1021 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1645 1033 y Fk(j)1680 1021 y Fr(\()p Fn(\033)1759 1033 y Fo(0)1797 1021 y Fr(\))p Fm(j)23 b(\024)1973 964 y Fn(\014)2024 934 y Fk(`)2052 942 y Fe(0)2089 964 y Fn("T)p 1973 1001 215 4 v 2060 1077 a Fr(8)2363 1021 y Fm(8)14 b Fn(j)27 b Fm(2)d(V)2615 1033 y Fk(\033)2653 1041 y Fe(0)2708 1021 y Fm(n)18 b(C)2812 1033 y Fo(0)2849 1021 y Fn(:)360 b Fr(\(4.40\))456 1222 y(W)-7 b(e)28 b(omit)f(the)h(details.)37 b(W)-7 b(e)28 b(shall)f(pro)n(v)n(e)f(b)r (elo)n(w)h(that,)h(for)f Fn(a)2401 1234 y Fo(0)2466 1222 y Fr(small)g(enough,)675 1412 y Fn(w)734 1424 y Fo(0)790 1412 y Fm(\000)883 1356 y Fn(\014)934 1326 y Fk(`)962 1334 y Fe(0)999 1356 y Fn("T)p 883 1393 V 970 1469 a Fr(4)1131 1412 y Fm(\024)22 b Fn(v)1258 1424 y Fk(i)1286 1412 y Fr(\()p Fn(t)p Fr(\))i Fm(\024)e Fn(W)1569 1424 y Fo(0)1625 1412 y Fr(+)1718 1356 y Fn(\014)1769 1326 y Fk(`)1797 1334 y Fe(0)1834 1356 y Fn("T)p 1718 1393 V 1805 1469 a Fr(4)2109 1412 y Fm(8)14 b Fn(t)22 b Fm(2)h Fr([)p Fn(\033)2370 1424 y Fo(0)2408 1412 y Fn(;)14 b(\033)2492 1424 y Fo(1)2530 1412 y Fr(])166 b Fm(8)14 b Fn(i)21 b Fm(2)j(C)2953 1424 y Fo(0)2990 1412 y Fn(:)219 b Fr(\(4.41\))456 1587 y(Since)27 b Fn(D)741 1599 y Fo(0)778 1587 y Fr(\()p Fn(\033)857 1599 y Fo(0)895 1587 y Fr(\))d Fm(\025)e Fn(\014)1089 1557 y Fk(`)1117 1565 y Fe(0)1154 1587 y Fn("T)12 b Fr(,)26 b(b)n(y)j(\(4.40\))e(and)g(\(4.41\))o(,)h(w)n(e)f (ha)n(v)n(e:)1099 1782 y Fn(D)1168 1794 y Fo(0)1205 1782 y Fr(\()p Fn(\033)1284 1794 y Fo(1)1322 1782 y Fr(\))c Fm(\025)g Fn(D)1534 1794 y Fo(0)1571 1782 y Fr(\()p Fn(\033)1650 1794 y Fo(0)1688 1782 y Fr(\))18 b Fm(\000)1831 1726 y Fn(\014)1882 1696 y Fk(`)1910 1704 y Fe(0)1947 1726 y Fn("T)p 1831 1763 V 1918 1839 a Fr(4)2074 1782 y Fm(\000)2167 1726 y Fn(\014)2218 1696 y Fk(`)2246 1704 y Fe(0)2283 1726 y Fn("T)p 2167 1763 V 2254 1839 a Fr(8)2415 1782 y Fm(\025)2512 1726 y Fr(5)p Fn(\014)2605 1696 y Fk(`)2633 1704 y Fe(0)2669 1726 y Fn("T)p 2512 1763 256 4 v 2619 1839 a Fr(8)2778 1782 y Fn(;)456 1949 y Fr(whic)n(h)27 b(pro)n(v)n(es)f(\(4.36\))h(for)g Fn(n)c Fr(=)f(0.)555 2049 y(T)-7 b(o)27 b(obtain)h(\(4.41\))f(w)n(e)g(ha)n(v)n(e)f(to)i(sho) n(w)e(that)i Fn(\033)2023 2061 y Fj(\003)2085 2049 y Fr(=)23 b Fn(\033)2220 2061 y Fo(1)2257 2049 y Fr(,)28 b(where:)482 2244 y Fn(\033)529 2256 y Fj(\003)611 2197 y Fn(:)590 2244 y Fr(=)23 b(sup)817 2127 y Fh(\032)879 2244 y Fn(t)g Fm(2)g Fr(\()p Fn(\033)1089 2256 y Fo(0)1127 2244 y Fn(;)14 b(\033)1211 2256 y Fo(1)1249 2244 y Fr(])37 b(:)g Fn(w)1428 2256 y Fo(0)1484 2244 y Fm(\000)1577 2188 y Fn(\014)1628 2157 y Fk(`)1656 2165 y Fe(0)1693 2188 y Fn("T)p 1577 2225 215 4 v 1663 2301 a Fr(4)1824 2244 y Fn(<)23 b(v)1952 2256 y Fk(i)1980 2244 y Fr(\()p Fn(s)p Fr(\))h Fn(<)e(W)2272 2256 y Fo(0)2328 2244 y Fr(+)2421 2188 y Fn(\014)2472 2157 y Fk(`)2500 2165 y Fe(0)2537 2188 y Fn("T)p 2421 2225 V 2508 2301 a Fr(4)2673 2244 y Fm(8)14 b Fn(i)22 b Fm(2)h(C)2907 2256 y Fo(0)2944 2244 y Fn(;)28 b(s)23 b Fm(2)g Fr([)p Fn(\033)3205 2256 y Fo(0)3243 2244 y Fn(;)14 b(t)p Fr(])3333 2127 y Fh(\033)3395 2244 y Fn(:)456 2432 y Fr(W)-7 b(e)39 b(\014rst)h(observ)n(e)d(that,)43 b(b)n(y)c(the)h(assumption)f Fn(\033)2099 2444 y Fo(0)2179 2432 y Fn(>)k(T)2336 2444 y Fk(k)q Fj(\000)p Fo(1)2500 2432 y Fr(and)d(the)f(de\014nition)h(\(4.24\))o(,)456 2532 y Fn(v)496 2544 y Fo(0)533 2532 y Fr(\()p Fn(t)p Fr(\))24 b Fn(<)e(v)778 2544 y Fo(0)816 2532 y Fr(\()p Fn(\033)895 2544 y Fo(0)933 2532 y Fr(\))h Fm(\024)g Fn(W)1154 2544 y Fo(0)1219 2532 y Fr(for)k(all)g Fn(t)d Fm(2)f Fr(\()p Fn(\033)1672 2544 y Fo(0)1710 2532 y Fn(;)14 b(T)1796 2544 y Fk(k)1836 2532 y Fr(],)28 b(so)f(that,)h(b)n(y)g (\(4.30\))o(,)1163 2727 y Fn(w)1222 2739 y Fo(0)1278 2727 y Fm(\000)1371 2671 y Fn(\014)1422 2641 y Fk(`)1450 2649 y Fe(0)1487 2671 y Fn("T)p 1371 2708 V 1457 2784 a Fr(8)1618 2727 y Fm(\024)23 b Fn(v)1746 2739 y Fo(0)1783 2727 y Fr(\()p Fn(t)p Fr(\))h Fm(\024)f Fn(W)2067 2739 y Fo(0)2270 2727 y Fm(8)14 b Fn(t)22 b Fm(2)i Fr([)p Fn(\033)2532 2739 y Fo(0)2570 2727 y Fn(;)14 b(\033)2654 2739 y Fo(1)2691 2727 y Fr(])p Fn(:)495 b Fr(\(4.42\))456 2894 y(Giv)n(en)34 b Fn(\016)k(>)c Fr(0,)i(let)f(\005\()p Fn(\016)o(;)14 b(t)p Fr(\))34 b(b)r(e)h(the)g(partition)f(of)h Fm(C)2137 2906 y Fo(0)2208 2894 y Fr(in)n(to)g(\()p Fn(\016)o(;)14 b(t)p Fr(\)-clusters.)56 b(W)-7 b(e)35 b(denote)g(b)n(y)456 2993 y Fm(K)520 2963 y Fo(ext)519 3014 y Fj(\000)615 2993 y Fr(\()p Fn(\016)o(;)14 b(t)p Fr(\))27 b(the)h(set)g(of)f(all)h (the)g(particles)e(b)r(elonging)h(to)h(clusters)f Fm(C)g(2)d Fr(\005\()p Fn(\016)o(;)14 b(t)p Fr(\))28 b(suc)n(h)f(that:)1361 3195 y(max)1381 3249 y Fk(j)s Fj(2C)1530 3195 y Fn(v)1570 3207 y Fk(j)1605 3195 y Fr(\()p Fn(t)p Fr(\))d Fn(<)e(w)1869 3207 y Fo(0)1925 3195 y Fm(\000)2008 3078 y Fh(\022)2079 3139 y Fn(\014)2130 3109 y Fk(`)2158 3117 y Fe(0)p 2079 3176 116 4 v 2116 3252 a Fr(8)2223 3195 y(+)c Fn(\016)2346 3078 y Fh(\023)2421 3195 y Fn("T)7 b(;)456 3387 y Fr(and)30 b(b)n(y)h Fm(K)803 3357 y Fo(ext)802 3408 y(+)898 3387 y Fr(\()p Fn(\016)o(;)14 b(t)p Fr(\))31 b(the)g(set)g(of)g(all)g(the)h (particles)e(b)r(elonging)g(to)h(clusters)f Fm(C)k(2)29 b Fr(\005\()p Fn(\016)o(;)14 b(t)p Fr(\))31 b(suc)n(h)456 3487 y(that:)1546 3591 y(min)1558 3645 y Fk(j)s Fj(2C)1699 3591 y Fn(v)1739 3603 y Fk(j)1774 3591 y Fr(\()p Fn(t)p Fr(\))24 b Fn(>)e(W)2057 3603 y Fo(0)2113 3591 y Fr(+)c Fn(\016)s("T)7 b(:)456 3764 y Fr(Finally)-7 b(,)31 b Fm(K)822 3733 y Fo(ext)917 3764 y Fr(\()p Fn(\016)o(;)14 b(t)p Fr(\))1132 3716 y Fn(:)1111 3764 y Fr(=)27 b Fm(K)1267 3733 y Fo(ext)1266 3784 y Fj(\000)1362 3764 y Fr(\()p Fn(\016)o(;)14 b(t)p Fr(\))20 b Fm([)h(K)1689 3733 y Fo(ext)1688 3784 y(+)1784 3764 y Fr(\()p Fn(\016)o(;)14 b(t)p Fr(\))30 b(and)g Fm(K)2209 3733 y Fo(in)n(t)2294 3764 y Fr(\()p Fn(\016)o(;)14 b(t)p Fr(\))2509 3716 y Fn(:)2488 3764 y Fr(=)27 b Fm(C)2624 3776 y Fo(0)2682 3764 y Fm(n)19 b(K)2807 3733 y Fo(ext)2902 3764 y Fr(\()p Fn(\016)o(;)14 b(t)p Fr(\).)45 b(W)-7 b(e)31 b(then)456 3870 y(in)n(tro)r(duce)h(a)g(sequence)f(of)i(partitions)e(of)i Fm(C)1879 3882 y Fo(0)1948 3870 y Fr(in)n(to)f(\()p Fn(\016)2193 3840 y Fk(\024)2237 3870 y Fn(;)14 b(t)2304 3882 y Fk(\024)2347 3870 y Fr(\)-clusters,)33 b(with)f Fn(\016)i Fr(=)d Fn(\014)3148 3840 y Fk(`)3176 3848 y Fe(0)3209 3840 y Fo(+1)3297 3870 y Fn(=)p Fr(16,)456 3969 y Fn(\024)d Fr(=)g(1)p Fn(;)14 b(:)g(:)g(:)f(;)h(\024)899 3939 y Fj(\003)937 3969 y Fr(,)31 b(where)g(the)g(times)g Fn(t)1636 3981 y Fo(1)1701 3969 y Fm(\025)d Fn(t)1824 3981 y Fo(2)1890 3969 y Fm(\025)f Fn(:)14 b(:)g(:)29 b Fm(\025)e Fn(t)2230 3981 y Fk(\024)2269 3965 y Fd(\003)2339 3969 y Fr(and)k(the)g(in)n(teger)f Fn(\024)2976 3939 y Fj(\003)3044 3969 y Fr(are)g(de\014ned)456 4069 y(b)n(y:)903 4098 y Fh(8)903 4173 y(>)903 4197 y(>)903 4222 y(<)903 4372 y(>)903 4397 y(>)903 4422 y(:)1018 4168 y Fn(t)1048 4180 y Fo(1)1129 4121 y Fn(:)1108 4168 y Fr(=)23 b Fn(\033)1243 4180 y Fj(\003)1281 4168 y Fn(;)1018 4318 y(t)1048 4330 y Fk(\024)p Fo(+1)1219 4271 y Fn(:)1198 4318 y Fr(=)g(inf)1400 4201 y Fh(\032)1463 4318 y Fn(t)g Fm(2)g Fr([)p Fn(\033)1664 4330 y Fo(0)1702 4318 y Fn(;)14 b(t)1769 4330 y Fk(\024)1812 4318 y Fr(])37 b(:)g Fn(\022)1971 4330 y Fk(\024)2014 4318 y Fr(\()p Fn(t)p Fr(\))23 b Fn(>)2229 4262 y(\016)2269 4232 y Fk(\024)2312 4262 y Fn("T)p 2229 4299 182 4 v 2299 4375 a Fr(4)2421 4318 y Fn(;)2458 4201 y Fh(\033)2520 4318 y Fn(;)180 b(\024)23 b Fm(\025)f Fr(1)p Fn(;)1018 4466 y(\024)1066 4436 y Fj(\003)1127 4466 y Fr(=)h(min)p Fm(f)p Fn(\024)f Fm(2)i Fl(N)47 b Fr(:)37 b Fn(t)1731 4478 y Fk(\024)p Fo(+1)1881 4466 y Fr(=)22 b Fn(\033)2015 4478 y Fo(0)2053 4466 y Fm(g)p Fn(;)456 4584 y Fr(where:)697 4723 y Fn(\022)736 4735 y Fk(\024)779 4723 y Fr(\()p Fn(t)p Fr(\))918 4676 y Fn(:)897 4723 y Fr(=)83 b(inf)984 4780 y Fk(s)p Fj(2)p Fo([)p Fk(t;t)1149 4788 y Fg(\024)1188 4780 y Fo(])1220 4723 y Fr(min)q Fm(fj)p Fn(v)1464 4735 y Fk(i)1491 4723 y Fr(\()p Fn(s)p Fr(\))19 b Fm(\000)f Fn(v)1736 4735 y Fk(j)1771 4723 y Fr(\()p Fn(s)p Fr(\))p Fm(j)38 b Fr(:)f Fn(i)22 b Fm(2)i(K)2189 4689 y Fo(in)n(t)2273 4723 y Fr(\()p Fn(\016)2345 4689 y Fk(\024)2388 4723 y Fn(;)14 b(t)2455 4735 y Fk(\024)2498 4723 y Fr(\))p Fn(;)28 b(j)g Fm(2)c(K)2786 4689 y Fo(ext)2881 4723 y Fr(\()p Fn(\016)2953 4689 y Fk(\024)2996 4723 y Fn(;)14 b(t)3063 4735 y Fk(\024)3106 4723 y Fr(\))p Fm(g)p Fn(;)456 4917 y Fr(setting)30 b Fn(\022)770 4929 y Fk(\024)813 4917 y Fr(\()p Fn(t)p Fr(\))e(=)g(+)p Fm(1)i Fr(if)h Fm(K)1349 4887 y Fo(ext)1444 4917 y Fr(\()p Fn(\016)1516 4887 y Fk(\024)1559 4917 y Fn(;)14 b(t)1626 4929 y Fk(\024)1669 4917 y Fr(\))31 b(is)f(empt)n(y)-7 b(.)45 b(Observ)n(e)29 b(that,)j(b)n(y)f(\(4.42\))o (,)g(the)g(c)n(harged)456 5016 y(particle)c(b)r(elongs)g(to)g Fm(K)1219 4986 y Fo(in)n(t)1304 5016 y Fr(\()p Fn(\016)1376 4986 y Fk(\024)1419 5016 y Fn(;)14 b(t)1486 5028 y Fk(\024)1529 5016 y Fr(\))28 b(for)f(an)n(y)g Fn(\024)p Fr(.)37 b(The)28 b(bac)n(kw)n(ard)d(motion)j(of)f(the)h(particles)f(in)456 5116 y Fm(K)520 5086 y Fo(ext)615 5116 y Fr(\()p Fn(\016)687 5086 y Fk(\024)730 5116 y Fn(;)14 b(t)797 5128 y Fk(\024)840 5116 y Fr(\))31 b(during)f(the)h(time)g(in)n(terv)-5 b(al)30 b([)p Fn(t)1863 5128 y Fk(\024)p Fo(+1)1990 5116 y Fn(;)14 b(t)2057 5128 y Fk(\024)2100 5116 y Fr(])31 b(can)f(b)r(e)h(analyzed)e(analogously)g(to)h(the)456 5216 y(pro)r(of)c(of)34 b(\(4.40\).)i(In)28 b(fact,)g(b)n(y)g(\(4.27\)) f(and)g(the)h(c)n(hoice)f(\(4.22\))o(,)h(w)n(e)f(can)g(apply)g(the)h (strategy)p eop %%Page: 25 25 25 24 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(25)456 450 y Fr(of)31 b(Theorem)f(3.1)g(to)h(the)g(particles)f(in)h Fm(K)1796 420 y Fo(ext)1891 450 y Fr(\()p Fn(\016)1963 420 y Fk(\024)2007 450 y Fn(;)14 b(t)2074 462 y Fk(\024)2117 450 y Fr(\),)32 b(with)f(the)h(parameter)d Fn(\014)36 b Fr(replaced)30 b(b)n(y)456 553 y Fn(\016)496 523 y Fk(\024)539 553 y Fn(=)p Fr(2)41 b(=)g([)p Fn(\014)844 523 y Fo(\()p Fk(`)898 531 y Fe(0)931 523 y Fo(+1\))1045 553 y Fn(=)p Fr(16])1194 523 y Fk(\024)1236 553 y Fn(=)p Fr(2,)g(for)d(an)n(y)g Fn(\024)k Fm(\024)g Fr(min)p Fm(f)p Fn(\024)2115 523 y Fj(\003)2153 553 y Fr(;)14 b(2)p Fm(jC)2299 565 y Fo(0)2335 553 y Fm(jg)p Fr(,)42 b(see)d(Remark)f(3.2.)70 b(W)-7 b(e)39 b(th)n(us)456 653 y(obtain,)27 b(for)g Fn(a)908 665 y Fo(0)973 653 y Fr(small)g(enough)g(and)g Fn(\024)c Fm(\024)g Fr(min)p Fm(f)p Fn(\024)2024 622 y Fj(\003)2062 653 y Fr(;)14 b(2)p Fm(jC)2208 665 y Fo(0)2244 653 y Fm(jg)p Fr(,)1056 847 y(sup)953 920 y Fk(t)p Fj(2)p Fo(\()p Fk(t)1074 928 y Fg(\024)p Fe(+1)1183 920 y Fk(;t)1228 928 y Fg(\024)1266 920 y Fo(])1299 847 y Fm(j)p Fn(v)1362 859 y Fk(j)1397 847 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1633 859 y Fk(j)1668 847 y Fr(\()p Fn(t)1730 859 y Fk(\024)1774 847 y Fr(\))p Fm(j)23 b(\024)1950 791 y Fn(\016)1990 761 y Fk(\024)2033 791 y Fn("T)p 1950 828 182 4 v 2020 904 a Fr(4)2307 847 y Fm(8)14 b Fn(j)27 b Fm(2)d(K)2572 813 y Fo(ext)2667 847 y Fr(\()p Fn(\016)2739 813 y Fk(\024)2782 847 y Fn(;)14 b(t)2849 859 y Fk(\024)2892 847 y Fr(\))p Fn(:)285 b Fr(\(4.43\))456 1066 y(F)-7 b(or)35 b Fn(t)643 1078 y Fk(\024)p Fo(+1)807 1066 y Fn(>)h(\033)955 1078 y Fo(0)1029 1066 y Fr(and)f Fn(\024)i Fm(\024)g Fr(min)p Fm(f)p Fn(\024)1613 1036 y Fj(\003)1650 1066 y Fr(;)14 b(2)p Fm(jC)1796 1078 y Fo(0)1833 1066 y Fm(jg)p Fr(,)38 b(since)d Fn(\022)2209 1078 y Fk(\024)2252 1066 y Fr(\()p Fn(t)2314 1078 y Fk(\024)p Fo(+1)2442 1066 y Fr(\))i(=)f Fn(\016)2652 1036 y Fk(\024)2695 1066 y Fn("T)9 b(=)p Fr(4,)37 b(the)f(inequalit)n(y)456 1166 y(\(4.43\))26 b(implies)i(that)g(there)g(exists)f Fn(i)22 b Fm(2)i(K)1793 1135 y Fo(in)n(t)1877 1166 y Fr(\()p Fn(\016)1949 1135 y Fk(\024)1992 1166 y Fn(;)14 b(t)2059 1178 y Fk(\024)2102 1166 y Fr(\))28 b(suc)n(h)g(that:)1222 1373 y Fn(v)1262 1385 y Fk(i)1290 1373 y Fr(\()p Fn(t)1352 1385 y Fk(\024)p Fo(+1)1479 1373 y Fr(\))23 b Fm(\024)g Fn(w)1681 1385 y Fo(0)1737 1373 y Fm(\000)1820 1256 y Fh(\022)1891 1316 y Fn(\014)1942 1286 y Fk(`)1970 1294 y Fe(0)p 1891 1354 116 4 v 1928 1430 a Fr(8)2035 1373 y(+)18 b Fn(\016)2158 1338 y Fk(\024)2201 1256 y Fh(\023)2276 1373 y Fn("T)30 b Fr(+)2486 1316 y Fn(\016)2526 1286 y Fk(\024)2570 1316 y Fn("T)p 2486 1354 182 4 v 2557 1430 a Fr(2)456 1547 y(or)1387 1686 y Fn(v)1427 1698 y Fk(i)1455 1686 y Fr(\()p Fn(t)1517 1698 y Fk(\024)p Fo(+1)1645 1686 y Fr(\))23 b Fm(\025)f Fn(W)1865 1698 y Fo(0)1922 1686 y Fr(+)c Fn(\016)2045 1651 y Fk(\024)2088 1686 y Fn("T)29 b Fm(\000)2298 1630 y Fn(\016)2338 1599 y Fk(\024)2381 1630 y Fn("T)p 2298 1667 V 2368 1743 a Fr(2)2490 1686 y Fn(:)456 1847 y Fr(In)35 b(b)r(oth)g(cases,)h(b)n(y)f (\(4.27\))f(and)h(assuming)f Fn(a)1962 1859 y Fo(0)2034 1847 y Fr(so)g(small)h(that)g Fn(\016)j(<)d Fr(1)p Fn(=)p Fr(\(2)p Fm(jV)2962 1859 y Fk(\033)3000 1867 y Fe(0)3036 1847 y Fm(j)p Fr(\),)i(w)n(e)e(ha)n(v)n(e)456 1947 y Fn(i)h Fm(2)h(K)677 1917 y Fo(ext)772 1947 y Fr(\()p Fn(\016)844 1917 y Fk(\024)p Fo(+1)971 1947 y Fn(;)14 b(t)1038 1959 y Fk(\024)p Fo(+1)1165 1947 y Fr(\).)62 b(In)36 b(conclusion,)h(for)e Fn(a)2008 1959 y Fo(0)2081 1947 y Fr(small)h(enough,)h(if)f Fn(t)2740 1959 y Fk(\024)p Fo(+1)2904 1947 y Fn(>)g(\033)3052 1959 y Fo(0)3126 1947 y Fr(and)f Fn(\024)i Fm(\024)456 2046 y Fr(min)p Fm(f)p Fn(\024)684 2016 y Fj(\003)721 2046 y Fr(;)14 b(2)p Fm(jC)867 2058 y Fo(0)904 2046 y Fm(jg)27 b Fr(then:)1265 2199 y Fm(jK)1352 2164 y Fo(ext)1447 2199 y Fr(\()p Fn(\016)1519 2164 y Fk(\024)p Fo(+1)1646 2199 y Fn(;)14 b(t)1713 2211 y Fk(\024)p Fo(+1)1840 2199 y Fr(\))p Fm(j)24 b(\025)e(jK)2093 2164 y Fo(ext)2188 2199 y Fr(\()p Fn(\016)2260 2164 y Fk(\024)2304 2199 y Fn(;)14 b(t)2371 2211 y Fk(\024)2414 2199 y Fr(\))p Fm(j)k Fr(+)g(1)p Fn(;)456 2350 y Fr(so)27 b(that)g Fn(t)767 2362 y Fk(\024)p Fo(+1)918 2350 y Fr(=)22 b Fn(\033)1052 2362 y Fo(0)1118 2350 y Fr(if)28 b Fn(\024)23 b Fm(\025)f(jC)1419 2362 y Fo(0)1456 2350 y Fm(j)p Fr(,)28 b(i.e.)g Fn(\024)1712 2320 y Fj(\003)1773 2350 y Fm(\024)22 b(jC)1927 2362 y Fo(0)1964 2350 y Fm(j)p Fr(.)555 2449 y(The)f(pro)r(of)g(of)27 b(\(4.41\))21 b(no)n(w)f(follo)n(ws)g(b)n(y)h (con)n(tradiction.)33 b(F)-7 b(or)21 b Fn(a)2521 2461 y Fo(0)2579 2449 y Fr(small)g(enough,)h(if)f Fn(\033)3210 2461 y Fj(\003)3272 2449 y Fn(<)i(\033)3407 2461 y Fo(1)456 2549 y Fr(and)c Fn(\016)26 b Fr(=)d Fn(\014)811 2519 y Fk(`)839 2527 y Fe(0)872 2519 y Fo(+1)960 2549 y Fn(=)p Fr(16,)d(the)g(set)g Fm(K)1450 2519 y Fo(ext)1545 2549 y Fr(\()p Fn(\016)o(;)14 b(\033)1697 2561 y Fj(\003)1735 2549 y Fr(\))21 b(is)f(non)f(empt)n(y)h(whence,)i(b)n(y)e(the)g (previous)f(analysis,)456 2653 y(also)26 b Fm(K)686 2622 y Fo(ext)781 2653 y Fr(\()p Fn(\016)853 2622 y Fk(\024)892 2597 y Fd(\003)931 2653 y Fn(;)14 b(\033)1015 2665 y Fo(0)1053 2653 y Fr(\))28 b(is)f(non)h(empt)n(y)-7 b(,)27 b(whic)n(h)h(con)n(tradicts)e(the)i(de\014nitions)g(of)g Fn(w)2991 2665 y Fo(0)3056 2653 y Fr(and)f Fn(W)3295 2665 y Fo(0)3333 2653 y Fr(.)555 2752 y(W)-7 b(e)27 b(no)n(w)f (conclude)h(the)g(pro)r(of)f(of)33 b(\(4.36\))26 b(b)n(y)g(induction.) 37 b(W)-7 b(e)27 b(assume)f(it)h(is)g(true)f(for)g(an)n(y)456 2852 y(in)n(teger)h Fn(m)d Fr(=)g(0)p Fn(;)14 b(:)g(:)g(:)f(;)h(n)28 b Fr(and)g(w)n(e)g(pro)n(v)n(e)e(it)j(also)e(holds)h(for)g Fn(m)c Fr(=)g Fn(n)18 b Fr(+)h(1.)38 b(W)-7 b(e)29 b(\014rst)f(note)g (that,)456 2951 y(b)n(y)k(the)i(argumen)n(t)e(after)h(Eqs.)f(\(4.38\))o ({\(4.39\))o(,)j(this)e(assumption)g(implies)g(that)h(the)f(times)456 3051 y Fn(\033)503 3063 y Fo(0)540 3051 y Fn(;)14 b(:)g(:)g(:)g(;)g (\033)772 3063 y Fk(n)p Fo(+1)924 3051 y Fr(are)22 b(all)g(distinct.)36 b(On)22 b(the)h(other)f(hand,)h(\(setting)g Fn(\033)2555 3063 y Fj(\000)p Fo(1)2688 3004 y Fn(:)2668 3051 y Fr(=)f Fn(T)2804 3063 y Fk(k)2845 3051 y Fr(\),)i(for)e(an)n(y)f(in)n(teger) 456 3151 y Fn(p)h Fm(\025)h Fr(1,)584 3353 y Fn(\033)631 3365 y Fo(2)p Fk(p)p Fj(\000)p Fo(1)811 3353 y Fn(>)g(\033)946 3365 y Fo(2)p Fk(p)p Fj(\000)p Fo(3)1153 3353 y Fr(=)-14 b Fm(\))111 b Fn(w)1457 3365 y Fo(2)p Fk(p)p Fj(\000)p Fo(1)1637 3353 y Fm(\024)22 b Fn(v)1764 3365 y Fo(0)1802 3353 y Fr(\()p Fn(t)p Fr(\))h Fm(\024)g Fn(W)2085 3365 y Fo(2)p Fk(p)p Fj(\000)p Fo(1)2260 3353 y Fr(+)2353 3297 y Fn(\014)2404 3267 y Fk(`)2432 3275 y Fe(2)p Fg(p)p Fd(\000)p Fe(1)2573 3297 y Fn("T)p 2353 3334 319 4 v 2492 3410 a Fr(8)2765 3353 y Fm(8)14 b Fn(t)22 b Fm(2)h Fr([)p Fn(\033)3026 3365 y Fo(2)p Fk(p)3098 3353 y Fn(;)14 b(\033)3182 3365 y Fo(2)p Fk(p)p Fj(\000)p Fo(1)3339 3353 y Fr(])p Fn(;)585 3570 y(\033)632 3582 y Fo(2)p Fk(p)727 3570 y Fn(>)23 b(\033)862 3582 y Fo(2)p Fk(p)p Fj(\000)p Fo(2)1152 3570 y Fr(=)-14 b Fm(\))112 b Fn(w)1457 3582 y Fo(2)p Fk(p)1547 3570 y Fm(\000)1640 3514 y Fn(\014)1691 3484 y Fk(`)1719 3492 y Fe(2)p Fg(p)1787 3514 y Fn("T)p 1640 3551 246 4 v 1742 3627 a Fr(8)1918 3570 y Fm(\024)23 b Fn(v)2046 3582 y Fo(0)2083 3570 y Fr(\()p Fn(t)p Fr(\))h Fm(\024)f Fn(W)2367 3582 y Fo(2)p Fk(p)2771 3570 y Fm(8)14 b Fn(t)22 b Fm(2)h Fr([)p Fn(\033)3032 3582 y Fo(2)p Fk(p)3104 3570 y Fn(;)14 b(\033)3188 3582 y Fo(2)p Fk(p)p Fo(+1)3344 3570 y Fr(])p Fn(;)456 3749 y Fr(whic)n(h)28 b(giv)n(e)f(the)i Fc(apriori)e Fr(b)r(ounds)h(\(analogous)f(to)h (\(4.42\))o(\))h(on)f(the)h(v)n(elo)r(cit)n(y)e(of)h(the)h(c)n(harged) 456 3849 y(particle.)58 b(Th)n(us,)37 b(in)e(b)r(oth)h(cases)d Fn(n)24 b Fr(+)f(1)35 b(=)f(2)p Fn(p)23 b Fm(\000)g Fr(1)35 b(or)f Fn(n)23 b Fr(+)g(1)35 b(=)g(2)p Fn(p)p Fr(,)h(w)n(e)f(can)f(rep) r(eat)h(the)456 3948 y(same)h(argumen)n(t)g(used)h(in)g(the)g(case)f Fn(n)j Fr(=)f(0.)64 b(Recalling)38 b(\(4.35\))o(,)h(if)f Fn(a)2793 3960 y Fo(0)2867 3948 y Fr(is)f(small)f(enough)456 4048 y(then)26 b Fn(`)678 4060 y Fk(n)746 4048 y Fm(\024)c Fr(4)p Fm(jV)949 4060 y Fk(\033)987 4068 y Fe(0)1024 4048 y Fm(j)k Fr(for)f(an)n(y)g Fn(n)e Fm(\024)f Fr(2)p Fm(jV)1629 4060 y Fk(\033)1667 4068 y Fe(0)1704 4048 y Fm(j)p Fr(.)36 b(This)26 b(guaran)n(tees)d(that)j(the)g(c)n(hoice)f (\(4.22\))g(is)h(large)456 4147 y(enough)20 b(for)h(the)h(argumen)n(t)f (of)g(Theorem)g(3.1)f(to)i(w)n(ork)e(with)i(gaps)e(of)h(order)g Fn(\014)2929 4117 y Fk(`)2957 4125 y Fg(n)2997 4117 y Fk(\024)3040 4147 y Fr(,)i Fn(\024)g Fm(\024)g Fr(2)p Fm(jC)3354 4159 y Fk(n)3398 4147 y Fm(j)p Fr(,)456 4247 y(for)k(an)n(y)f Fn(n)d Fm(\024)g Fr(2)p Fm(jV)1016 4259 y Fk(\033)1054 4267 y Fe(0)1090 4247 y Fm(j)p Fr(.)38 b(W)-7 b(e)27 b(omit)h(the)g(details.)555 4347 y(W)-7 b(e)35 b(are)e(left)j(with)f(the)f(pro)r(of)g(of)41 b(\(4.37\))o(.)58 b(W)-7 b(e)35 b(apply)f(the)h(p)r(erturbativ)n(e)e(argumen)n(t)h(to)456 4446 y(the)e(cluster)f Fm(C)919 4458 y Fk(n)964 4446 y Fr(.)50 b(In)32 b(fact,)h(for)f Fn(n)f Fr(ev)n(en)h(\(resp.)f(o)r (dd\))i(during)e(the)h(time)h(in)n(terv)-5 b(al)31 b([)p Fn(\033)3162 4458 y Fk(n)3208 4446 y Fn(;)14 b(\033)3292 4458 y Fk(n)p Fo(+1)3421 4446 y Fr(])456 4546 y(\(resp.)30 b([)p Fn(\033)760 4558 y Fk(n)p Fo(+1)890 4546 y Fn(;)14 b(\033)974 4558 y Fk(n)1019 4546 y Fr(]\))32 b(there)e(is)h(a)f(v)n (elo)r(cit)n(y)g(gap)g Fn(\014)2001 4516 y Fk(`)2029 4524 y Fg(n)2074 4546 y Fn("T)9 b(=)p Fr(2)30 b(b)r(et)n(w)n(een)h(the) g(particles)f(in)h Fm(C)3235 4558 y Fk(n)3311 4546 y Fr(and)456 4646 y(the)d(other)f(particles)f(whic)n(h)i(ma)n(y)f(in)n (teract)g(with)h(the)g(\014rst)f(ones.)555 4745 y(Since)36 b(the)g(c)n(harged)d(particle)i(b)r(elongs)g(to)g Fm(C)2012 4757 y Fk(n)2057 4745 y Fr(,)j(w)n(e)d(ha)n(v)n(e)f(to)i(mo)r(dify)f (the)h(p)r(erturbativ)n(e)456 4845 y(analysis)j(b)n(y)i(taking)g(in)n (to)g(accoun)n(t)f(of)h(the)g(electric)g(\014eld.)78 b(The)41 b(mass)g(of)g(the)g(c)n(harged)456 4944 y(particle)27 b(is)g Fn(M)9 b Fr(,)27 b(so)g(that)h(the)g(v)n(elo)r(cit)n(y)f(of)h (the)g(cen)n(ter)e(of)i(mass)f(of)g Fm(C)2632 4956 y Fk(n)2705 4944 y Fr(is:)1052 5130 y Fn(v)1092 5142 y Fj(C)1128 5150 y Fg(n)1173 5130 y Fr(\()p Fn(t)p Fr(\))d(=)1602 5074 y(1)p 1389 5111 470 4 v 1389 5187 a Fn(M)j Fr(+)18 b Fm(jC)1647 5199 y Fk(n)1692 5187 y Fm(j)g(\000)g Fr(1)1898 5051 y Fh(X)1881 5229 y Fk(j)s Fj(2C)1992 5237 y Fg(n)2034 5130 y Fr([\()p Fn(M)27 b Fm(\000)18 b Fr(1\))p Fn(\016)2391 5142 y Fk(j;)p Fo(0)2494 5130 y Fr(+)g(1])c Fn(v)2696 5142 y Fk(j)2730 5130 y Fr(\()p Fn(t)p Fr(\))p Fn(;)p eop %%Page: 26 26 26 25 bop 456 251 a Fo(26)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(and)27 b(the)h(asso)r(ciate)e(slo)n(wly)h(v)-5 b(arying)26 b(function:)608 688 y Fn(p)650 700 y Fj(C)686 708 y Fg(n)731 688 y Fr(\()p Fn(t)p Fr(\))929 641 y Fn(:)908 688 y Fr(=)83 b Fn(v)1096 700 y Fj(C)1132 708 y Fg(n)1177 688 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)1597 632 y Fr(1)p 1383 669 470 4 v 1383 745 a Fn(M)27 b Fr(+)18 b Fm(jC)1641 757 y Fk(n)1686 745 y Fm(j)g(\000)g Fr(1)1876 522 y Fh(2)1876 671 y(4)1931 688 y Fn(M)9 b(E)c(t)18 b Fm(\000)2308 610 y Fh(X)2218 792 y Fk(q)r Fj(2V)2337 800 y Fg(\033)2371 812 y Fe(0)2408 792 y Fj(nC)2478 800 y Fg(n)2548 610 y Fh(X)2532 788 y Fk(j)s Fj(2C)2643 796 y Fg(n)2698 575 y Fh(Z)2781 596 y Fk(t)2745 764 y Fo(0)2811 688 y Fn(ds)c Fm(r)p Fr(\010)3036 700 y Fk(j;q)3119 688 y Fr(\()p Fn(s)p Fr(\))3222 522 y Fh(3)3222 671 y(5)1056 972 y Fr(+)1358 916 y(1)p 1144 953 V 1144 1029 a Fn(M)27 b Fr(+)18 b Fm(jC)1402 1041 y Fk(n)1447 1029 y Fm(j)h(\000)f Fr(1)1690 893 y Fh(X)1637 1071 y Fk(q)r Fj(2M)1792 1079 y Fg(\033)1826 1091 y Fe(0)1893 893 y Fh(X)1876 1071 y Fk(j)s Fj(2C)1987 1079 y Fg(n)2155 916 y Fr(\010)2215 928 y Fk(j;q)2298 916 y Fr(\()p Fn(t)p Fr(\))p 2053 953 443 4 v 2053 1029 a Fn(v)2093 1041 y Fk(j)2128 1029 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(v)2364 1041 y Fk(q)2401 1029 y Fr(\()p Fn(t)p Fr(\))2505 972 y Fn(;)456 1215 y Fr(where)30 b(\010)759 1227 y Fk(j;q)842 1215 y Fr(\()p Fn(t)p Fr(\))f(=)e(\010\()p Fn(x)1196 1227 y Fk(j)1232 1215 y Fr(\()p Fn(t)p Fr(\))21 b Fm(\000)f Fn(x)1479 1227 y Fk(q)1516 1215 y Fr(\()p Fn(t)p Fr(\)\))31 b(if)h Fn(j;)14 b(q)31 b Fm(\025)c Fr(1,)k(\010)2140 1227 y Fk(j;q)2224 1215 y Fr(\()p Fn(t)p Fr(\))d(=)g(\011\()p Fn(x)2583 1227 y Fk(j)2618 1215 y Fr(\()p Fn(t)p Fr(\))21 b Fm(\000)f Fn(x)2865 1227 y Fk(q)2902 1215 y Fr(\()p Fn(t)p Fr(\)\))32 b(if)f Fn(j)i Fr(=)27 b(0)j(or)456 1314 y Fn(q)c Fr(=)c(0,)27 b(and)f(analogously)e (for)i Fm(r)p Fr(\010)1562 1326 y Fk(j;q)1646 1314 y Fr(\()p Fn(t)p Fr(\).)37 b(F)-7 b(rom)26 b(the)h(equations)e(of)i (motion)f(w)n(e)g(ha)n(v)n(e)f(in)i(this)456 1414 y(case:)762 1584 y(_)-40 b Fn(p)787 1596 y Fj(C)823 1604 y Fg(n)868 1584 y Fr(\()p Fn(t)p Fr(\))84 b(=)1417 1528 y(1)p 1203 1565 470 4 v 1203 1641 a Fn(M)27 b Fr(+)18 b Fm(jC)1461 1653 y Fk(n)1506 1641 y Fm(j)h(\000)f Fr(1)1749 1506 y Fh(X)1696 1684 y Fk(q)r Fj(2M)1851 1692 y Fg(\033)1885 1704 y Fe(0)1952 1506 y Fh(X)1936 1684 y Fk(j)s Fj(2C)2047 1692 y Fg(n)2265 1528 y Fr(\010)2325 1540 y Fk(j;q)2408 1528 y Fr(\()p Fn(t)p Fr(\))p 2112 1565 545 4 v 2112 1641 a(\()p Fn(v)2184 1653 y Fk(j)2219 1641 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(v)2455 1653 y Fk(q)2492 1641 y Fr(\()p Fn(t)p Fr(\)\))2618 1617 y Fo(2)1193 1891 y Fm(\002)1258 1749 y Fh(\()1360 1812 y(X)1338 1991 y Fk(s)p Fo(:)p Fk(s)p Fj(6)p Fo(=)p Fk(j)1516 1774 y Fh(\024)1697 1835 y Fm(r)p Fr(\010)1826 1847 y Fk(j;s)1909 1835 y Fr(\()p Fn(t)p Fr(\))p 1569 1872 562 4 v 1569 1948 a(\()p Fn(M)28 b Fm(\000)18 b Fr(1\))p Fn(\016)1904 1960 y Fk(j;)p Fo(0)2006 1948 y Fr(+)g(1)2159 1891 y Fm(\000)g Fn(E)5 b(\016)2345 1903 y Fk(j;)p Fo(0)2429 1774 y Fh(\025)2491 1891 y Fm(\000)2597 1812 y Fh(X)2575 1991 y Fk(s)p Fo(:)p Fk(s)p Fj(6)p Fo(=)p Fk(q)2753 1891 y Fm(r)p Fr(\010)2882 1903 y Fk(q)r(;s)2970 1891 y Fr(\()p Fn(t)p Fr(\))3064 1749 y Fh(\))3132 1891 y Fn(:)456 2125 y Fr(Then,)32 b(b)n(y)f(pro)r(ceeding)f(as)g(done)h(to)g(get)g(\(3.46\))o(,)h(w)n(e)f (obtain,)g(for)g Fn(n)g Fr(ev)n(en)f(\(resp.)h(o)r(dd\))h(and)456 2225 y(an)n(y)26 b Fn(t)d Fm(2)h Fr([)p Fn(\033)814 2237 y Fk(n)860 2225 y Fn(;)14 b(\033)944 2237 y Fk(n)p Fo(+1)1073 2225 y Fr(])28 b(\(resp.)f Fn(t)c Fm(2)h Fr([)p Fn(\033)1557 2237 y Fk(n)p Fo(+1)1686 2225 y Fn(;)14 b(\033)1770 2237 y Fk(n)1816 2225 y Fr(]\),)904 2297 y Fh(\014)904 2347 y(\014)904 2397 y(\014)904 2447 y(\014)931 2418 y Fn(v)971 2430 y Fj(C)1007 2438 y Fg(n)1053 2418 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1289 2430 y Fj(C)1325 2438 y Fg(n)1370 2418 y Fr(\()p Fn(\033)1449 2430 y Fk(n)1495 2418 y Fr(\))g Fm(\000)1651 2362 y Fn(M)9 b(E)c Fr(\()p Fn(t)18 b Fm(\000)g Fn(\033)2017 2374 y Fk(n)2063 2362 y Fr(\))p 1638 2399 470 4 v 1638 2475 a Fn(M)27 b Fr(+)18 b Fm(jC)1896 2487 y Fk(n)1941 2475 y Fm(j)h(\000)f Fr(1)2117 2297 y Fh(\014)2117 2347 y(\014)2117 2397 y(\014)2117 2447 y(\014)2168 2418 y Fm(\024)23 b Fn(C)2335 2301 y Fh(\024)2521 2362 y Fr(1)p 2389 2399 307 4 v 2389 2475 a Fn(\014)2440 2451 y Fk(`)2468 2459 y Fg(n)2509 2451 y Fo(+1)2597 2475 y Fn("T)2724 2418 y Fr(+)2840 2362 y(1)p 2817 2399 89 4 v 2817 2475 a Fn(\014)2868 2451 y Fo(2)2915 2301 y Fh(\025)2973 2418 y Fn(;)456 2611 y Fr(whic)n(h)k(implies)h(\(4.37\))f(b)n(y)h(\(4.22\))o(,)g(\(4.35\))o (,)g(and)f(since)h Fn(E)g Fm(\025)23 b Fr(0.)1757 2805 y Fq(Appendix)555 2955 y Fr(In)28 b(this)g(app)r(endix)f(w)n(e)g (brie\015y)g(discuss)g(the)h(deriv)-5 b(ation)27 b(of)g(Eqs.)g (\(4.2\),)g(\(4.3\),)g(and)h(\(4.4\))o(.)456 3054 y(Eqs.)33 b(\(3.2\))h(and)g(\(3.3\))g(are)f(reco)n(v)n(ered)e(b)n(y)j(putting)h (\011)f(=)f(0,)j Fn(E)j Fr(=)33 b(0,)j(and)d(neglecting)h(the)456 3154 y(c)n(harged)26 b(particle.)555 3254 y(In)31 b(the)g(sequel)g(w)n (e)f(adopt)g(the)h(last)g(notation,)g(denoting)f(b)n(y)h(\()p Fn(x)2630 3266 y Fo(0)2668 3254 y Fn(;)14 b(v)2745 3266 y Fo(0)2782 3254 y Fr(\))31 b(the)g(p)r(osition)g(and)456 3353 y(v)n(elo)r(cit)n(y)22 b(of)h(the)h(c)n(harged)e(particle)h(and)g (calling)2025 3332 y(\026)2002 3353 y Fn(X)29 b Fr(=)23 b Fm(f)p Fr(\()p Fn(x)2309 3365 y Fo(0)2346 3353 y Fn(;)14 b(v)2423 3365 y Fo(0)2460 3353 y Fr(\);)g Fn(X)7 b Fm(g)23 b Fr(the)h(state)f(of)g(the)h(whole)456 3469 y(system.)37 b(W)-7 b(e)28 b(also)e(denote)i(b)n(y)f Fn(t)c Fm(7!)1640 3448 y Fr(\026)1616 3469 y Fn(X)1692 3439 y Fo(\()p Fk(n)p Fo(\))1788 3469 y Fr(\()p Fn(t)p Fr(\))h(=)f Fm(f)p Fr(\()p Fn(x)2115 3426 y Fo(\()p Fk(n)p Fo(\))2115 3491 y(0)2212 3469 y Fr(\()p Fn(t)p Fr(\))p Fn(;)14 b(v)2386 3426 y Fo(\()p Fk(n)p Fo(\))2383 3491 y(0)2484 3469 y Fr(\()p Fn(t)p Fr(\)\);)g Fn(X)2723 3439 y Fo(\()p Fk(n)p Fo(\))2820 3469 y Fr(\()p Fn(t)p Fr(\))p Fm(g)28 b Fr(the)g Fn(n)p Fr(-partial)456 3569 y(dynamics)f(relativ)n(e)f(to)i(Eqs.)e(\(2.5\).) 555 3668 y(The)g(\014rst)g(step)g(is)f(the)h(con)n(trol)f(on)h(the)g (gro)n(wth)e(of)i(the)g(lo)r(cal)f(energy)g(and)h(densit)n(y)f(of)h (the)456 3768 y(whole)h(system)g(de\014ned)h(b)n(y:)684 3936 y(\026)665 3957 y Fn(Q)p Fr(\()787 3936 y(\026)763 3957 y Fn(X)6 b Fr(;)14 b Fn(\026;)g(R)q Fr(\))1102 3909 y Fn(:)1081 3957 y Fr(=)1169 3878 y Fh(X)1217 4054 y Fk(i)1303 3957 y Fn(\037)1355 3969 y Fk(i)1382 3957 y Fr(\()p Fn(\026;)g(R)q Fr(\))1611 3839 y Fh(\032)1684 3900 y Fr(\()p Fn(M)27 b Fm(\000)18 b Fr(1\))p Fn(\016)2018 3912 y Fk(i;)p Fo(0)2116 3900 y Fr(+)g(1)p 1684 3937 558 4 v 1941 4013 a(2)2251 3957 y Fn(v)2294 3922 y Fo(2)2291 3977 y Fk(i)2350 3957 y Fr(+)2443 3900 y(1)p 2443 3937 42 4 v 2443 4013 a(2)2526 3878 y Fh(X)2508 4056 y Fk(j)s Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)2677 3957 y Fr(\010)2737 3969 y Fk(i;j)2834 3957 y Fr(+)g(1)2959 3839 y Fh(\033)3020 3957 y Fn(;)210 b Fr(\(A.1\))456 4190 y(where)30 b(\010)759 4202 y Fk(i;j)865 4190 y Fr(=)d(\010\()p Fn(x)1096 4202 y Fk(i)1145 4190 y Fm(\000)20 b Fn(x)1277 4202 y Fk(j)1312 4190 y Fr(\))31 b(if)g Fn(i;)14 b(j)33 b Fm(\025)27 b Fr(1)j(and)h(\010)1976 4202 y Fk(i;j)2082 4190 y Fr(=)c(\011\()p Fn(x)2318 4202 y Fk(i)2367 4190 y Fm(\000)20 b Fn(x)2499 4202 y Fk(j)2534 4190 y Fr(\))31 b(if)g Fn(i)d Fr(=)f(0)j(or)g Fn(j)j Fr(=)27 b(0.)45 b(Note)456 4290 y(that)30 b(\010)698 4302 y Fk(i;j)806 4290 y Fr(is)h(sup)r(erstable)e(and)h(non-negativ)n (e.)43 b(Then,)31 b(recalling)e(the)i(de\014nition)f(\(4.1\),)h(for)456 4389 y(an)n(y)26 b Fn(X)k Fm(2)23 b(X)40 b Fr(and)28 b Fn(n)23 b Fm(2)g Fl(N)t Fr(,)983 4536 y(sup)1025 4601 y Fk(\026)1141 4515 y Fr(\026)1122 4536 y Fn(Q)o Fr(\()1243 4515 y(\026)1219 4536 y Fn(X)1295 4501 y Fo(\()p Fk(n)p Fo(\))1392 4536 y Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g(R)1673 4548 y Fk(n)1719 4536 y Fr(\()p Fn(t)p Fr(\)\))24 b Fm(\024)e Fn(C)6 b(Q)2087 4548 y Fk(E)2143 4536 y Fr(\()p Fn(X)h Fr(\))p Fn(R)2346 4548 y Fk(n)2391 4536 y Fr(\()p Fn(t)p Fr(\))167 b Fm(8)14 b Fn(t)22 b Fm(\025)h Fr(0)p Fn(;)335 b Fr(\(A.2\))456 4731 y(where)1359 4876 y Fn(R)1422 4888 y Fk(n)1467 4876 y Fr(\()p Fn(t)p Fr(\))1606 4829 y Fn(:)1585 4876 y Fr(=)22 b(log)q(\()p Fn(e)c Fr(+)g Fn(n)p Fr(\))h(+)2136 4763 y Fh(Z)2219 4784 y Fk(t)2182 4952 y Fo(0)2248 4876 y Fn(ds)14 b(V)2392 4888 y Fk(n)2438 4876 y Fr(\()p Fn(s)p Fr(\))712 b(\(A.3\))456 5044 y(and)1457 5152 y Fn(V)1505 5164 y Fk(n)1551 5152 y Fr(\()p Fn(t)p Fr(\))1689 5105 y Fn(:)1668 5152 y Fr(=)23 b(max)1764 5214 y Fk(i)p Fj(2)1840 5199 y Fo(\026)1832 5214 y Fk(I)1861 5222 y Fg(n)1958 5152 y Fr(sup)1924 5226 y Fk(s)p Fj(2)p Fo([0)p Fk(;t)p Fo(])2130 5152 y Fm(j)p Fn(v)2196 5109 y Fo(\()p Fk(n)p Fo(\))2193 5176 y Fk(i)2293 5152 y Fr(\()p Fn(s)p Fr(\))p Fm(j)p Fn(;)811 b Fr(\(A.4\))p eop %%Page: 27 27 27 26 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(27)456 450 y Fr(with)656 429 y(\026)646 450 y Fn(I)682 462 y Fk(n)774 403 y Fn(:)753 450 y Fr(=)26 b Fn(I)880 462 y Fk(n)945 450 y Fm([)20 b(f)p Fr(0)p Fm(g)p Fr(.)40 b(As)29 b(a)g(consequence)f(of)h(this)g(estimate)g(one)g(can)g(sho)n(w) f(that,)i(for)e(an)n(y)456 550 y Fn(X)h Fm(2)23 b(X)12 b Fr(,)28 b Fn(n)23 b Fm(2)h Fl(N)t Fr(,)33 b Fn(i)23 b Fm(2)1157 529 y Fr(\026)1147 550 y Fn(I)1183 562 y Fk(n)1228 550 y Fr(,)28 b(and)f Fn(t)c Fm(\025)g Fr(0,)1100 729 y Fm(j)p Fn(v)1166 686 y Fo(\()p Fk(n)p Fo(\))1163 752 y Fk(i)1264 729 y Fr(\()p Fn(t)p Fr(\))p Fm(j)g(\024)g Fn(C)1571 637 y Fh(h)1610 654 y(p)p 1693 654 638 4 v 75 x Fn(Q)1759 741 y Fk(E)1815 729 y Fr(\()p Fn(X)7 b Fr(\))14 b(log\()p Fn(e)k Fr(+)g Fn(n)p Fr(\))h(+)f Fn(Q)2498 741 y Fk(E)2554 729 y Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2724 637 y Fh(i)2777 729 y Fn(:)453 b Fr(\(A.5\))555 913 y(By)30 b(applying)f(an)g(iterativ)n(e)g(pro)r(cedure)g(one)h(obtains)f(from)g (\(A.5\))i(the)f(existence)f(of)h(the)456 1013 y(in\014nite)e (dynamics.)36 b(More)27 b(precisely)-7 b(,)27 b(de\014ning)1003 1184 y Fn(\016)1040 1196 y Fk(i)1067 1184 y Fr(\()p Fn(n;)14 b(t)p Fr(\))1293 1136 y Fn(:)1272 1184 y Fr(=)22 b Fm(j)p Fn(x)1429 1140 y Fo(\()p Fk(n)p Fo(\))1429 1207 y Fk(i)1527 1184 y Fr(\()p Fn(t)p Fr(\))d Fm(\000)f Fn(x)1770 1140 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1770 1207 y Fk(i)1953 1184 y Fr(\()p Fn(t)p Fr(\))p Fm(j)h Fr(+)f Fm(j)p Fn(v)2238 1140 y Fo(\()p Fk(n)p Fo(\))2235 1207 y Fk(i)2335 1184 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(v)2574 1140 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2571 1207 y Fk(i)2756 1184 y Fr(\()p Fn(t)p Fr(\))p Fm(j)p Fn(;)357 b Fr(\(A.6\))456 1343 y(there)33 b(is)h Fn(\013)g(>)g Fr(1)f(suc)n(h)h(that,)i(for)d(an) n(y)g Fn(i)g Fm(2)h Fl(N)f Fm([)23 b(f)p Fr(0)p Fm(g)p Fr(,)34 b Fn(X)40 b Fm(2)34 b(X)12 b Fr(,)36 b(and)e Fn(t)g Fm(\025)f Fr(0,)i(setting)f Fn(n)3308 1313 y Fj(\003)3380 1343 y Fr(=)456 1442 y Fn(n)506 1412 y Fj(\003)544 1442 y Fr(\()p Fn(i;)14 b(t)p Fr(\))23 b(=)f(In)n(t)934 1375 y Fh(\002)969 1442 y Fn(\013)p Fr(\(1)d(+)f Fn(x)1245 1412 y Fo(2)1245 1464 y Fk(i)1301 1442 y Fr(+)g Fn(Q)1450 1454 y Fk(E)1505 1442 y Fr(\()p Fn(X)7 b Fr(\))1645 1412 y Fo(2)1683 1442 y Fr(\))p Fn(e)1754 1412 y Fk(t)1783 1375 y Fh(\003)1817 1442 y Fr(,)28 b(w)n(e)f(ha)n(v)n(e:)629 1660 y Fn(\016)666 1672 y Fk(i)694 1660 y Fr(\()p Fn(n;)14 b(t)p Fr(\))23 b Fm(\024)g Fn(C)6 b(Q)1117 1672 y Fk(E)1173 1660 y Fr(\()p Fn(X)h Fr(\))14 b(exp)1467 1542 y Fh(\024)1511 1660 y Fm(\000)2025 1603 y Fn(n)p 1586 1640 929 4 v 1586 1725 a(C)6 b Fr([1)18 b(+)g Fn(Q)1883 1737 y Fk(E)1938 1725 y Fr(\()p Fn(X)7 b Fr(\)])14 b(log)2222 1688 y Fo(2)2260 1725 y Fr(\()p Fn(e)k Fr(+)g Fn(n)p Fr(\))2524 1542 y Fh(\025)2748 1660 y Fm(8)c Fn(n)22 b(>)g(n)3018 1625 y Fj(\003)3056 1660 y Fn(:)174 b Fr(\(A.7\))456 1876 y(F)-7 b(rom)27 b(\(A.7\))h(the)g(existence)f(of)h(the)g(limits)g (\(2.6\))f(follo)n(w.)36 b(Moreo)n(v)n(er:)1287 2052 y Fm(j)p Fn(v)1353 2009 y Fo(\()p Fk(n)p Fo(\))1350 2075 y Fk(i)1451 2052 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(v)1690 2009 y Fo(\()p Fk(n)1757 1984 y Fd(\003)1792 2009 y Fo(\))1687 2075 y Fk(i)1822 2052 y Fr(\()p Fn(t)p Fr(\))p Fm(j)23 b(\024)g Fn(C)172 b Fm(8)14 b Fn(n)22 b(>)h(n)2552 2018 y Fj(\003)2590 2052 y Fn(:)456 2211 y Fr(so)k(that,)g(b)n(y)i(\(A.5\),)f(for)f(an)n(y)g Fn(n)c Fm(2)g Fl(N)t Fr(,)550 2395 y Fm(j)p Fn(v)616 2352 y Fo(\()p Fk(n)p Fo(\))613 2418 y Fk(i)713 2395 y Fr(\()p Fn(t)p Fr(\))p Fm(j)h(\024)f Fn(C)1021 2303 y Fh(h)1060 2320 y(p)p 1143 2320 1072 4 v 75 x Fn(Q)1209 2407 y Fk(E)1265 2395 y Fr(\()p Fn(X)7 b Fr(\))14 b(log\()p Fn(e)k Fr(+)g Fm(j)p Fn(x)1768 2407 y Fk(i)1796 2395 y Fm(j)h Fr(+)f Fn(Q)1987 2407 y Fk(E)2043 2395 y Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g Fn(Q)2382 2407 y Fk(E)2438 2395 y Fr(\()p Fn(X)7 b Fr(\))p Fn(t)2608 2303 y Fh(i)2661 2395 y Fn(;)180 b Fm(8)14 b Fn(i)21 b Fm(2)3064 2374 y Fr(\026)3054 2395 y Fn(I)3090 2407 y Fk(n)3135 2395 y Fn(:)95 b Fr(\(A.8\))456 2579 y(By)27 b(taking)g(the)h(limit)g Fn(n)23 b Fm(!)g Fr(+)p Fm(1)k Fr(in)h(\(A.8\))g(w)n(e)g(get)f(Eqs.)g(\(4.2\))g(and)h (\(4.3\))o(.)555 2679 y(The)33 b(b)r(ounds)f(\(A.2\))q(,)h(\(A.5\))q(,) g(and)g(the)f(iterativ)n(e)g(argumen)n(t)f(leading)h(to)g(Eq.)g (\(A.7\))h(can)456 2778 y(b)r(e)27 b(found)g(in)g([10)o(])g(in)h(the)f (case)f(without)h(electric)g(\014eld)g Fn(E)5 b Fr(,)27 b(and)g(in)g([9,)g(Theorem)f(2.2])g(in)h(the)456 2878 y(case)33 b(of)h(a)g(\(more)g(complicated\))g(quasi)g(one-dimensional)f (mo)r(del.)57 b(W)-7 b(e)35 b(omit)g(the)f(pro)r(ofs.)456 2978 y(W)-7 b(e)32 b(instead)g(discuss)g(in)h(more)e(details)h(Eq.)g (\(4.4\))o(,)i(whic)n(h)e(is)g(a)g(corollary)e(of)i(the)h(previous)456 3077 y(estimates)g(but)h(it)g(has)g(b)r(een)g(nev)n(er)e(explicitly)i (stated)g(in)g(the)g(quoted)f(pap)r(ers.)55 b(W)-7 b(e)34 b(shall)456 3177 y(pro)n(v)n(e)20 b(the)i(analogous)e(result)h(for)g (the)i(function)f(in)g(\(A.1\))q(:)34 b(for)21 b(an)n(y)g Fn(\026)i Fm(2)h Fl(R)p Fr(,)29 b Fn(R)24 b(>)e Fr(log\()p Fn(e)7 b Fr(+)g Fm(j)p Fn(\026)p Fm(j)p Fr(\),)456 3277 y(and)27 b Fn(t)c Fm(\025)g Fr(0,)662 3415 y(\026)644 3436 y Fn(Q)o Fr(\()765 3415 y(\026)741 3436 y Fn(X)7 b Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g(R)q Fr(\))23 b Fm(\024)g Fn(C)6 b(Q)1373 3448 y Fk(E)1429 3436 y Fr(\()p Fn(X)h Fr(\))1583 3368 y Fh(\002)1617 3436 y Fn(R)19 b Fr(+)f(log\()p Fn(e)h Fr(+)f Fn(Q)2128 3448 y Fk(E)2184 3436 y Fr(\()p Fn(X)7 b Fr(\)\))18 b(+)g(\(1)g(+)g Fn(Q)2698 3448 y Fk(E)2754 3436 y Fr(\()p Fn(X)7 b Fr(\)\))p Fn(t)2956 3401 y Fo(2)2994 3368 y Fh(\003)3042 3436 y Fn(;)188 b Fr(\(A.9\))456 3595 y(whic)n(h)30 b(clearly)g(implies)g(\(4.4\))h(b)n (y)f(the)h(p)r(ositivit)n(y)f(of)h(the)g(in)n(teraction)e(\011.)46 b(W)-7 b(e)31 b(in)n(tro)r(duce)f(a)456 3694 y(molli\014ed)d(v)n (ersion)f(of)1188 3673 y(\026)1169 3694 y Fn(Q)p Fr(\()1291 3673 y(\026)1267 3694 y Fn(X)6 b Fr(;)14 b Fn(\026;)g(R)q Fr(\))28 b(b)n(y)f(de\014ning:)705 3903 y Fn(W)12 b Fr(\()851 3882 y(\026)827 3903 y Fn(X)6 b Fr(;)14 b Fn(\026;)g(R)q Fr(\))1166 3856 y Fn(:)1145 3903 y Fr(=)1233 3825 y Fh(X)1281 4001 y Fk(i)1366 3903 y Fn(f)1416 3864 y Fk(\026;R)1407 3927 y(i)1530 3786 y Fh(\032)1603 3847 y Fr(\()p Fn(M)27 b Fm(\000)18 b Fr(1\))p Fn(\016)1937 3859 y Fk(i;)p Fo(0)2036 3847 y Fr(+)g(1)p 1603 3884 558 4 v 1861 3960 a(2)2170 3903 y Fn(v)2213 3869 y Fo(2)2210 3924 y Fk(i)2269 3903 y Fr(+)2362 3847 y(1)p 2362 3884 42 4 v 2362 3960 a(2)2445 3825 y Fh(X)2427 4003 y Fk(j)s Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)2596 3903 y Fr(\010)2656 3915 y Fk(i;j)2753 3903 y Fr(+)g(1)2878 3786 y Fh(\033)2939 3903 y Fn(;)249 b Fr(\(A.10\))456 4148 y(where)1573 4302 y Fn(f)1623 4262 y Fk(\026;R)1614 4325 y(i)1761 4302 y Fr(=)22 b Fn(f)1912 4185 y Fh(\022)1983 4246 y Fm(j)p Fn(x)2053 4258 y Fk(i)2099 4246 y Fm(\000)c Fn(\026)p Fm(j)p 1983 4283 273 4 v 2087 4359 a Fn(R)2265 4185 y Fh(\023)3211 4302 y Fr(\(A.11\))456 4493 y(and)30 b Fn(f)36 b Fm(2)28 b Fn(C)845 4463 y Fj(1)915 4493 y Fr(\()p Fl(R)1002 4505 y Fo(+)1063 4493 y Fr(\))i(is)h(not)f (increasing)f(and)h(satis\014es:)41 b Fn(f)9 b Fr(\()p Fn(x)p Fr(\))29 b(=)e(1)j(for)f Fn(x)f Fm(2)g Fr([0)p Fn(;)14 b Fr(1],)31 b Fn(f)9 b Fr(\()p Fn(x)p Fr(\))28 b(=)f(0)456 4593 y(for)g Fn(x)c Fm(\025)g Fr(2,)k(and)g Fm(j)p Fn(f)1067 4563 y Fj(0)1090 4593 y Fr(\()p Fn(x)p Fr(\))p Fm(j)e(\024)d Fr(2.)37 b(Ob)n(viously:)1224 4731 y(\026)1205 4752 y Fn(Q)p Fr(\()1327 4731 y(\026)1303 4752 y Fn(X)7 b Fr(;)14 b Fn(\026;)g(R)q Fr(\))22 b Fm(\024)h Fn(W)12 b Fr(\()1855 4731 y(\026)1831 4752 y Fn(X)7 b Fr(;)14 b Fn(\026;)g(R)q Fr(\))22 b Fm(\024)2256 4731 y Fr(\026)2237 4752 y Fn(Q)p Fr(\()2359 4731 y(\026)2335 4752 y Fn(X)6 b Fr(;)14 b Fn(\026;)g Fr(2)p Fn(R)q Fr(\))p Fn(;)516 b Fr(\(A.12\))456 4911 y(whic)n(h)18 b(in)h(particular)e(sho)n (ws)g(it)i(is)f(enough)g(to)g(pro)n(v)n(e)f(Eq.)h(\(A.9\))g(with)2657 4890 y(\026)2638 4911 y Fn(Q)p Fr(\()2760 4890 y(\026)2736 4911 y Fn(X)7 b Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g(R)q Fr(\))k(replaced)456 5011 y(b)n(y)27 b Fn(W)12 b Fr(\()717 4990 y(\026)693 5011 y Fn(X)6 b Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g(R)q Fr(\).)38 b(Giv)n(en)27 b Fn(\013)1436 5023 y Fo(0)1497 5011 y Fm(\025)22 b Fr(1,)27 b Fn(\026)d Fm(2)f Fl(R)p Fr(,)34 b(and)27 b Fn(R)d(>)e Fr(log)q(\()p Fn(e)c Fr(+)g Fm(j)p Fn(\026)p Fm(j)p Fr(\))28 b(let:)1005 5195 y Fn(n)1055 5207 y Fo(0)1115 5195 y Fr(=)23 b Fn(n)1253 5207 y Fo(0)1290 5195 y Fr(\()p Fn(\026;)14 b(R)q(;)g(t)p Fr(\))23 b(=)g(In)n(t)1803 5103 y Fh(h)1842 5195 y Fn(\013)1895 5207 y Fo(0)1932 5195 y Fr(\()p Fn(e)18 b Fr(+)g Fn(Q)2170 5207 y Fk(E)2226 5195 y Fr(\()p Fn(X)7 b Fr(\))2366 5161 y Fo(2)2403 5195 y Fr(\))p Fn(e)2474 5161 y Fo(2\()p Fk(R)p Fo(+)p Fk(t)p Fo(\))2690 5103 y Fh(i)2747 5195 y Fr(+)18 b(1)p Fn(:)p eop %%Page: 28 28 28 27 bop 456 251 a Fo(28)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fr(Since)27 b(log)q(\()p Fn(e)18 b Fr(+)g Fn(n)1002 462 y Fo(0)1039 450 y Fr(\))23 b Fn(>)g(R)q Fr(,)28 b(b)n(y)g(\(A.2\),)g(\(A.5\))q(,)f(and)h(\(A.12\),)545 596 y Fn(W)12 b Fr(\()691 575 y(\026)667 596 y Fn(X)743 562 y Fo(\()p Fk(n)810 570 y Fe(0)841 562 y Fo(\))872 596 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))83 b Fm(\024)1436 575 y Fr(\026)1417 596 y Fn(Q)o Fr(\()1538 575 y(\026)1514 596 y Fn(X)1590 562 y Fo(\()p Fk(n)1657 570 y Fe(0)1689 562 y Fo(\))1719 596 y Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g Fr(2)p Fn(R)2042 608 y Fk(n)2083 616 y Fe(0)2120 596 y Fr(\()p Fn(t)p Fr(\)\))24 b Fm(\024)e Fn(C)e Fr(sup)2480 662 y Fk(\027)2594 575 y Fr(\026)2575 596 y Fn(Q)p Fr(\()2697 575 y(\026)2673 596 y Fn(X)2749 562 y Fo(\()p Fk(n)2816 570 y Fe(0)2848 562 y Fo(\))2878 596 y Fr(\()p Fn(t)p Fr(\);)14 b Fn(\027)q(;)g(R)3151 608 y Fk(n)3192 616 y Fe(0)3229 596 y Fr(\()p Fn(t)p Fr(\)\))1269 775 y Fm(\024)83 b Fn(C)6 b(Q)1548 787 y Fk(E)1604 775 y Fr(\()p Fn(X)h Fr(\))p Fn(R)1807 787 y Fk(n)1848 795 y Fe(0)1884 775 y Fr(\()p Fn(t)p Fr(\))24 b Fm(\024)e Fn(C)6 b(Q)2220 787 y Fk(E)2276 775 y Fr(\()p Fn(X)h Fr(\))2430 708 y Fh(\002)2465 775 y Fr(log\()p Fn(e)18 b Fr(+)g Fn(n)2794 787 y Fo(0)2831 775 y Fr(\))h(+)f Fn(Q)3031 787 y Fk(E)3087 775 y Fr(\()p Fn(X)7 b Fr(\))p Fn(t)3257 741 y Fo(2)3294 708 y Fh(\003)1269 909 y Fm(\024)83 b Fn(C)6 b(Q)1548 921 y Fk(E)1604 909 y Fr(\()p Fn(X)h Fr(\))1758 842 y Fh(\002)1792 909 y Fn(R)19 b Fr(+)f(log\()p Fn(e)h Fr(+)f Fn(Q)2303 921 y Fk(E)2358 909 y Fr(\()p Fn(X)7 b Fr(\)\))19 b(+)f(\(1)g(+)g Fn(Q)2873 921 y Fk(E)2929 909 y Fr(\()p Fn(X)7 b Fr(\)\))p Fn(t)3131 875 y Fo(2)3168 842 y Fh(\003)3217 909 y Fn(;)456 1048 y Fr(where)27 b(in)h(the)h(second)e(inequalit)n(y)h(w)n(e)f(used)h(the)h(p)r (ositivit)n(y)f(of)g(the)g(p)r(oten)n(tial,)g(see)g([10)o(,)g(Eq.)456 1148 y(\(A.12\)].)37 b(On)27 b(the)h(other)f(hand:)671 1294 y Fn(W)12 b Fr(\()817 1273 y(\026)793 1294 y Fn(X)7 b Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g(R)q Fr(\))83 b Fm(\024)g Fn(W)12 b Fr(\()1560 1273 y(\026)1536 1294 y Fn(X)1612 1259 y Fo(\()p Fk(n)1679 1267 y Fe(0)1711 1259 y Fo(\))1741 1294 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))1414 1435 y(+)1515 1356 y Fh(X)1493 1530 y Fk(n>n)1627 1538 y Fe(0)1672 1435 y Fm(j)p Fn(W)e Fr(\()1841 1414 y(\026)1817 1435 y Fn(X)1893 1400 y Fo(\()p Fk(n)p Fo(\))1990 1435 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))19 b Fm(\000)f Fn(W)12 b Fr(\()2552 1414 y(\026)2528 1435 y Fn(X)2604 1400 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2785 1435 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))p Fm(j)p Fn(:)66 b Fr(\(A.13\))456 1656 y(Let)27 b(us)h(estimate)f(the)h (sum)g(on)f(the)h(righ)n(t-hand)f(side)g(of)34 b(\(A.13\).)j(W)-7 b(e)28 b(ha)n(v)n(e:)735 1802 y Fm(j)p Fn(W)12 b Fr(\()904 1781 y(\026)880 1802 y Fn(X)956 1768 y Fo(\()p Fk(n)p Fo(\))1053 1802 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))19 b Fm(\000)f Fn(W)12 b Fr(\()1615 1781 y(\026)1591 1802 y Fn(X)1667 1768 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1848 1802 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))p Fm(j)901 2002 y(\024)1003 1923 y Fh(X)1051 2100 y Fk(i)1136 2002 y Fn(f)1200 1860 y Fh( )1276 1945 y Fm(j)p Fn(x)1346 1902 y Fo(\()p Fk(n)p Fo(\))1346 1969 y Fk(i)1443 1945 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(\026)p Fm(j)p 1276 1983 437 4 v 1462 2059 a Fn(R)1722 1860 y Fh(!)1802 1931 y(\014)1802 1981 y(\014)1829 2002 y Fn(")1868 1959 y Fo(\()p Fk(n)p Fo(\))1868 2025 y Fk(i)1984 2002 y Fm(\000)g Fn(")2106 1959 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2106 2025 y Fk(i)2287 1931 y Fh(\014)2287 1981 y(\014)1067 2286 y Fr(+)1146 2208 y Fh(X)1194 2384 y Fk(i)1279 2141 y Fh(\014)1279 2191 y(\014)1279 2241 y(\014)1279 2291 y(\014)1279 2340 y(\014)1307 2286 y Fn(f)1370 2144 y Fh( )1446 2230 y Fm(j)p Fn(x)1516 2187 y Fo(\()p Fk(n)p Fo(\))1516 2253 y Fk(i)1614 2230 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(\026)p Fm(j)p 1446 2267 V 1633 2343 a Fn(R)1893 2144 y Fh(!)1977 2286 y Fm(\000)g Fn(f)2123 2144 y Fh( )2199 2230 y Fm(j)p Fn(x)2269 2187 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2269 2253 y Fk(i)2452 2230 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(\026)p Fm(j)p 2199 2267 522 4 v 2428 2343 a Fn(R)2731 2144 y Fh(!)2797 2141 y(\014)2797 2191 y(\014)2797 2241 y(\014)2797 2291 y(\014)2797 2340 y(\014)2838 2286 y Fn(")2877 2243 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2877 2310 y Fk(i)3059 2286 y Fn(;)129 b Fr(\(A.14\))456 2498 y(where)1103 2629 y Fn(")1142 2585 y Fo(\()p Fk(n)p Fo(\))1142 2652 y Fk(i)1262 2629 y Fr(=)1360 2572 y(\()p Fn(M)27 b Fm(\000)18 b Fr(1\))p Fn(\016)1694 2584 y Fk(i;)p Fo(0)1793 2572 y Fr(+)g(1)p 1360 2610 558 4 v 1618 2686 a(2)1927 2629 y Fm(j)p Fn(v)1993 2585 y Fo(\()p Fk(n)p Fo(\))1990 2652 y Fk(i)2091 2629 y Fm(j)2114 2594 y Fo(2)2169 2629 y Fr(+)2262 2572 y(1)p 2262 2610 42 4 v 2262 2686 a(2)2345 2550 y Fh(X)2328 2728 y Fk(j)s Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)2497 2629 y Fr(\010)2557 2585 y Fo(\()p Fk(n)p Fo(\))2557 2652 y Fk(i;j)2672 2629 y Fr(+)g(1)456 2865 y(and)h(\010)669 2822 y Fo(\()p Fk(n)p Fo(\))669 2888 y Fk(i;j)785 2865 y Fr(is)g(de\014ned)g(as)g(\010)1291 2877 y Fk(i;j)1388 2865 y Fr(in)g(\(A.1\))h(with)g Fn(x)1915 2877 y Fk(i)1943 2865 y Fr(,)h Fn(x)2034 2877 y Fk(j)2088 2865 y Fr(replaced)e(b)n(y)f Fn(x)2561 2822 y Fo(\()p Fk(n)p Fo(\))2561 2888 y Fk(i)2659 2865 y Fr(,)j Fn(x)2750 2822 y Fo(\()p Fk(n)p Fo(\))2750 2888 y Fk(j)2867 2865 y Fr(resp)r(ectiv)n(ely)-7 b(.)33 b(By)456 2994 y(\(A.8\),)22 b(if)g Fm(j)p Fn(x)832 2951 y Fo(\()p Fk(n)p Fo(\))832 3017 y Fk(i)929 2994 y Fr(\()p Fn(t)p Fr(\))5 b Fm(\000)g Fn(\026)p Fm(j)23 b(\024)g Fr(2)p Fn(R)e Fr(then)h(all)e(the)h(particles)f Fn(j)28 b Fm(2)c Fn(I)2338 3006 y Fk(n)2404 2994 y Fr(suc)n(h)d(that)g Fm(j)p Fn(x)2828 2951 y Fo(\()p Fk(n)p Fo(\))2828 3017 y Fk(i)2926 2994 y Fr(\()p Fn(t)p Fr(\))5 b Fm(\000)g Fn(x)3142 2951 y Fo(\()p Fk(n)p Fo(\))3142 3017 y Fk(j)3239 2994 y Fr(\()p Fn(t)p Fr(\))p Fm(j)24 b(\024)456 3124 y Fr(1)i(or)h Fm(j)p Fn(x)696 3080 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))696 3147 y Fk(i)878 3124 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)e Fn(x)1120 3080 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1120 3147 y Fk(j)1303 3124 y Fr(\()p Fn(t)p Fr(\))p Fm(j)23 b(\024)g Fr(1)k(are)f(initially)h(con)n(tained)g(in)h(the)f(in) n(terv)-5 b(al)27 b(with)h(cen)n(ter)e Fn(\026)456 3229 y Fr(and)h(radius)g Fn(R)q Fr(\()p Fn(t)p Fr(\),)h(where:)1398 3368 y Fn(R)q Fr(\()p Fn(t)p Fr(\))23 b(=)f Fn(C)6 b Fr([)p Fn(R)20 b Fr(+)e Fn(Q)1986 3380 y Fk(E)2042 3368 y Fr(\()p Fn(X)7 b Fr(\)\(1)18 b(+)g Fn(t)2387 3334 y Fo(2)2424 3368 y Fr(\)])p Fn(:)456 3507 y Fr(In)24 b(particular,)g(b)n (y)g(c)n(ho)r(osing)e Fn(\013)1458 3519 y Fo(0)1520 3507 y Fr(large)h(enough,)h(for)g(an)n(y)f Fn(n)g Fm(\025)g Fn(n)2514 3519 y Fo(0)2575 3507 y Fr(eac)n(h)h(particle)f Fn(i)h Fr(suc)n(h)g(that)456 3618 y Fm(j)p Fn(x)526 3575 y Fo(\()p Fk(n)p Fo(\))526 3641 y Fk(i)623 3618 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(\026)p Fm(j)23 b(\024)g Fr(2)p Fn(R)28 b Fr(do)r(es)f(not)h(in)n(teract)f(with)h(the)g (particles)f Fn(j)h Fm(2)23 b Fn(I)2619 3630 y Fk(n)2683 3618 y Fm(n)18 b Fn(I)2779 3630 y Fk(n)p Fj(\000)p Fo(1)2910 3618 y Fr(,)27 b(so)g(that:)540 3788 y Fh(\014)540 3838 y(\014)568 3859 y Fn(")607 3816 y Fo(\()p Fk(n)p Fo(\))607 3882 y Fk(i)722 3859 y Fm(\000)18 b Fn(")844 3816 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))844 3882 y Fk(i)1026 3788 y Fh(\014)1026 3838 y(\014)1076 3859 y Fm(\024)23 b Fn(C)1243 3692 y Fh(2)1243 3842 y(4)1309 3803 y Fm(j)p Fn(v)1375 3760 y Fo(\()p Fk(n)p Fo(\))1372 3826 y Fk(i)1472 3803 y Fr(\()p Fn(t)p Fr(\))p Fm(j)c Fr(+)f Fm(j)p Fn(v)1757 3760 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1754 3826 y Fk(i)1939 3803 y Fr(\()p Fn(t)p Fr(\))p Fm(j)p 1309 3840 749 4 v 1662 3916 a Fr(2)2067 3859 y Fn(\016)2104 3871 y Fk(i)2131 3859 y Fr(\()p Fn(n;)c(t)p Fr(\))19 b(+)2432 3780 y Fh(X)2414 3959 y Fk(j)s Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)2569 3792 y Fj(\003)2621 3859 y Fr([)p Fn(\016)2681 3871 y Fk(i)2709 3859 y Fr(\()p Fn(n;)14 b(t)p Fr(\))k(+)g Fn(\016)3028 3871 y Fk(j)3063 3859 y Fr(\()p Fn(n;)c(t)p Fr(\)])3268 3692 y Fh(3)3268 3842 y(5)3337 3859 y Fn(;)456 4104 y Fr(\(recall)38 b(the)h(de\014nition)h(\(A.6\)\))f(where)1770 4042 y Fh(P)1857 4062 y Fj(\003)1857 4129 y Fk(j)s Fo(:)p Fk(j)s Fj(6)p Fo(=)p Fk(i)2055 4104 y Fr(denotes)g(the)g(sum)g (restricted)g(to)f(all)h(the)456 4235 y(particles)30 b Fn(j)35 b Fm(2)30 b Fn(I)982 4247 y Fk(n)p Fj(\000)p Fo(1)1145 4235 y Fr(suc)n(h)h(that)h Fm(j)p Fn(x)1590 4192 y Fo(\()p Fk(n)p Fo(\))1590 4258 y Fk(i)1688 4235 y Fr(\()p Fn(t)p Fr(\))21 b Fm(\000)g Fn(x)1936 4192 y Fo(\()p Fk(n)p Fo(\))1936 4258 y Fk(j)2034 4235 y Fr(\()p Fn(t)p Fr(\))p Fm(j)30 b(\024)f Fr(1)j(or)e Fm(j)p Fn(x)2524 4192 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2524 4258 y Fk(i)2707 4235 y Fr(\()p Fn(t)p Fr(\))22 b Fm(\000)e Fn(x)2955 4192 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2955 4258 y Fk(j)3138 4235 y Fr(\()p Fn(t)p Fr(\))p Fm(j)30 b(\024)g Fr(1.)456 4335 y(The)d(n)n(um)n(b)r(er)h(of)f(these)h(particles)f(is)g(therefore) g(b)r(ounded)h(b)n(y:)1432 4474 y Fn(N)9 b Fr(\()p Fn(X)e Fr(;)14 b Fn(\026;)g(R)q Fr(\()p Fn(t)p Fr(\)\))22 b Fm(\024)h Fr(2)p Fn(Q)p Fr(\()p Fn(X)7 b Fr(\))p Fn(R)q Fr(\()p Fn(t)p Fr(\))p Fn(;)456 4613 y Fr(where)27 b(w)n(e)g(used)g (\(2.2\),)h(\(2.3\),)f(and)h(that)g Fn(R)23 b(>)g Fr(log\()p Fn(e)c Fr(+)f Fm(j)p Fn(\026)p Fm(j)p Fr(\).)37 b(Then,)28 b(setting)1229 4752 y(\001)1298 4764 y Fk(n)1344 4752 y Fr(\()p Fn(t)p Fr(\))1482 4704 y Fn(:)1461 4752 y Fr(=)23 b(max)o Fm(f)p Fn(\016)1782 4764 y Fk(i)1809 4752 y Fr(\()p Fn(n;)14 b(t)p Fr(\))38 b(:)f Fm(j)p Fn(x)2158 4764 y Fk(i)2204 4752 y Fm(\000)18 b Fn(\026)p Fm(j)23 b(\024)g Fn(R)q Fr(\()p Fn(t)p Fr(\))p Fm(g)456 4890 y Fr(and)k(using)h(\(A.8\)) q(,)g(if)g Fn(i)f Fr(is)g(suc)n(h)h(that)g Fm(j)p Fn(x)1731 4902 y Fk(i)1777 4890 y Fm(\000)18 b Fn(\026)p Fm(j)23 b(\024)g Fn(R)q Fr(\()p Fn(t)p Fr(\),)28 b(for)f(an)n(y)g Fn(n)c(>)f(n)2747 4902 y Fo(0)2784 4890 y Fr(,)509 4983 y Fh(\014)509 5033 y(\014)537 5054 y Fn(")576 5011 y Fo(\()p Fk(n)p Fo(\))576 5077 y Fk(i)691 5054 y Fm(\000)c Fn(")813 5011 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))813 5077 y Fk(i)995 4983 y Fh(\014)995 5033 y(\014)1106 5054 y Fm(\024)82 b Fn(C)1332 4962 y Fh(h)1371 4979 y(p)p 1455 4979 944 4 v 1455 5054 a Fn(Q)1521 5066 y Fk(E)1576 5054 y Fr(\()p Fn(X)7 b Fr(\))14 b(log\()p Fn(e)19 b Fr(+)f Fm(j)p Fn(\026)p Fm(j)g Fr(+)g Fn(R)q Fr(\()p Fn(t)p Fr(\)\))h(+)f Fn(Q)2565 5066 y Fk(E)2621 5054 y Fr(\()p Fn(X)7 b Fr(\)\()p Fn(t)18 b Fr(+)h Fn(R)q Fr(\()p Fn(t)p Fr(\)\))3115 4962 y Fh(i)3182 5054 y Fr(\001)3251 5066 y Fk(n)3296 5054 y Fr(\()p Fn(t)p Fr(\))1106 5216 y Fm(\024)82 b Fn(C)6 b Fr([1)18 b(+)g Fn(Q)1550 5228 y Fk(E)1606 5216 y Fr(\()p Fn(X)7 b Fr(\))1746 5181 y Fo(2)1783 5216 y Fr(])14 b(log)1927 5178 y Fo(2)1965 5216 y Fr(\()p Fn(e)k Fr(+)g Fn(n)p Fr(\))c(\001)2302 5228 y Fk(n)2347 5216 y Fr(\()p Fn(t)p Fr(\))p Fn(:)747 b Fr(\(A.15\))p eop %%Page: 29 29 29 28 bop 796 251 a Fo(ON)29 b(THE)g(VIOLA)-5 b(TION)29 b(OF)g(OHM'S)f(LA)-7 b(W)29 b(F)n(OR)g(BOUNDED)g(INTERA)n(CTIONS)275 b(29)456 450 y Fr(On)27 b(the)h(other)f(hand:)831 512 y Fh(\014)831 561 y(\014)831 611 y(\014)831 661 y(\014)831 711 y(\014)858 657 y Fn(f)922 515 y Fh( )997 601 y Fm(j)p Fn(x)1067 558 y Fo(\()p Fk(n)p Fo(\))1067 624 y Fk(i)1165 601 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(\026)p Fm(j)p 997 638 437 4 v 1184 714 a Fn(R)1444 515 y Fh(!)1528 657 y Fm(\000)g Fn(f)1675 515 y Fh( )1750 601 y Fm(j)p Fn(x)1820 558 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1820 624 y Fk(i)2003 601 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(\026)p Fm(j)p 1750 638 522 4 v 1979 714 a Fn(R)2282 515 y Fh(!)2348 512 y(\014)2348 561 y(\014)2348 611 y(\014)2348 661 y(\014)2348 711 y(\014)1163 937 y Fm(\024)k Fr(2)1302 881 y Fm(j)p Fn(x)1372 838 y Fo(\()p Fk(n)p Fo(\))1372 904 y Fk(i)1469 881 y Fr(\()p Fn(t)p Fr(\))d Fm(\000)f Fn(x)1712 838 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1712 904 y Fk(i)1895 881 y Fr(\()p Fn(t)p Fr(\))p Fm(j)p 1302 918 711 4 v 1625 994 a Fn(R)2023 937 y(\037)2075 870 y Fh(\000)2113 937 y Fm(j)p Fn(x)2183 894 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2183 960 y Fk(i)2365 937 y Fr(\()p Fn(t)p Fr(\))h Fm(\000)f Fn(\026)p Fm(j)23 b(\024)g Fn(\016)2782 949 y Fk(i)2809 937 y Fr(\()p Fn(n;)14 b(t)p Fr(\))19 b(+)f(2)p Fn(R)3198 870 y Fh(\001)1163 1114 y Fm(\024)k Fn(C)6 b(\037)1367 1046 y Fh(\000)1406 1114 y Fm(j)p Fn(x)1476 1071 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1476 1137 y Fk(i)1658 1114 y Fr(\()p Fn(t)p Fr(\))19 b Fm(\000)f Fn(\026)p Fm(j)23 b(\024)g Fr(\001)2107 1126 y Fk(n)2152 1114 y Fr(\()p Fn(t)p Fr(\))c(+)f(2)p Fn(R)2454 1046 y Fh(\001)2505 1114 y Fr(\001)2574 1126 y Fk(n)2620 1114 y Fr(\()p Fn(t)p Fr(\))p Fn(:)474 b Fr(\(A.16\))456 1267 y(By)28 b(\(A.7\))g(and)g(the)g(de\014nition)g(of)f Fn(n)1624 1279 y Fo(0)1661 1267 y Fr(,)h(if)g Fn(\013)1841 1279 y Fo(0)1906 1267 y Fr(is)g(large)e(enough,)627 1469 y(\001)696 1481 y Fk(n)742 1469 y Fr(\()p Fn(t)p Fr(\))d Fm(\024)g Fn(C)6 b(Q)1078 1481 y Fk(E)1134 1469 y Fr(\()p Fn(X)h Fr(\))14 b(exp)1428 1352 y Fh(\024)1472 1469 y Fm(\000)1986 1413 y Fn(n)p 1547 1450 929 4 v 1547 1535 a(C)6 b Fr([1)18 b(+)g Fn(Q)1844 1547 y Fk(E)1900 1535 y Fr(\()p Fn(X)7 b Fr(\)])14 b(log)2184 1498 y Fo(2)2221 1535 y Fr(\()p Fn(e)k Fr(+)g Fn(n)p Fr(\))2485 1352 y Fh(\025)2709 1469 y Fm(8)c Fn(n)22 b(>)h(n)2980 1481 y Fo(0)3017 1469 y Fn(:)171 b Fr(\(A.17\))456 1679 y(In)27 b(particular)g(\001)1012 1691 y Fk(n)1057 1679 y Fr(\()p Fn(t)p Fr(\))d Fm(\024)e Fn(C)6 b Fr(.)38 b(Then,)27 b(inserting)g(the)h(b)r(ounds)g(\(A.15\))g (and)f(\(A.16\))h(in)g(\(A.14\),)751 1840 y Fm(j)p Fn(W)12 b Fr(\()920 1819 y(\026)896 1840 y Fn(X)972 1805 y Fo(\()p Fk(n)p Fo(\))1069 1840 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))k Fm(\000)g Fn(W)12 b Fr(\()1630 1819 y(\026)1606 1840 y Fn(X)1682 1805 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))1864 1840 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(R)q Fr(\))p Fm(j)1249 1971 y(\024)23 b Fn(C)6 b Fr([1)18 b(+)g Fn(Q)1634 1983 y Fk(E)1690 1971 y Fr(\()p Fn(X)7 b Fr(\))1830 1937 y Fo(2)1867 1971 y Fr(])14 b(log)2011 1934 y Fo(2)2048 1971 y Fr(\()p Fn(e)19 b Fr(+)f Fn(n)p Fr(\))p Fn(N)9 b Fr(\()p Fn(X)2487 1937 y Fo(\()p Fk(n)p Fo(\))2583 1971 y Fr(\()p Fn(t)p Fr(\);)14 b Fn(\026;)g Fr(2)p Fn(R)q Fr(\))g(\001)3022 1983 y Fk(n)3067 1971 y Fr(\()p Fn(t)p Fr(\))1581 2103 y(+)g Fn(W)e Fr(\()1806 2082 y(\026)1782 2103 y Fn(X)1858 2069 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2039 2103 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g Fr(\001)2326 2115 y Fk(n)2372 2103 y Fr(\()p Fn(t)p Fr(\))19 b(+)f(2)p Fn(R)q Fr(\))c(\001)2789 2115 y Fk(n)2834 2103 y Fr(\()p Fn(t)p Fr(\))1249 2235 y Fm(\024)23 b Fn(C)6 b Fr([1)18 b(+)g Fn(Q)1634 2247 y Fk(E)1690 2235 y Fr(\()p Fn(X)7 b Fr(\))1830 2201 y Fo(2)1867 2235 y Fr(])14 b(log)2011 2198 y Fo(2)2048 2235 y Fr(\()p Fn(e)19 b Fr(+)f Fn(n)p Fr(\))p Fn(W)12 b Fr(\()2449 2214 y(\026)2425 2235 y Fn(X)2501 2201 y Fo(\()p Fk(n)p Fo(\))2597 2235 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g Fr(2)p Fn(R)2920 2247 y Fk(n)2965 2235 y Fr(\()p Fn(t)p Fr(\)\))g(\001)3174 2247 y Fk(n)3221 2235 y Fr(\()p Fn(t)p Fr(\))1581 2367 y(+)g Fn(W)e Fr(\()1806 2346 y(\026)1782 2367 y Fn(X)1858 2333 y Fo(\()p Fk(n)p Fj(\000)p Fo(1\))2039 2367 y Fr(\()p Fn(t)p Fr(\);)i Fn(\026;)g(C)25 b Fr(+)18 b(2)p Fn(R)2529 2379 y Fk(n)p Fj(\000)p Fo(1)2659 2367 y Fr(\()p Fn(t)p Fr(\)\))c(\001)2868 2379 y Fk(n)2914 2367 y Fr(\()p Fn(t)p Fr(\))1249 2495 y Fm(\024)23 b Fn(C)6 b Fr([1)18 b(+)g Fn(Q)1634 2507 y Fk(E)1690 2495 y Fr(\()p Fn(X)7 b Fr(\))1830 2461 y Fo(2)1867 2495 y Fr(])14 b(log)2011 2458 y Fo(2)2048 2495 y Fr(\()p Fn(e)19 b Fr(+)f Fn(n)p Fr(\))c([)p Fn(R)2403 2507 y Fk(n)p Fj(\000)p Fo(1)2533 2495 y Fr(\()p Fn(t)p Fr(\))19 b(+)f Fn(R)2792 2507 y Fk(n)2837 2495 y Fr(\()p Fn(t)p Fr(\)])28 b(\001)3051 2507 y Fk(n)3097 2495 y Fr(\()p Fn(t)p Fr(\))p Fn(:)456 2648 y Fr(where)35 b(in)h(the)h(last)e (inequalit)n(y)h(w)n(e)g(used)g(the)g(p)r(ositivit)n(y)g(of)g(the)g(p)r (oten)n(tial,)i(\(A.12\),)h(and)456 2752 y(\(A.2\).)52 b(Again)32 b(b)n(y)i(\(A.8\))f(w)n(e)f(ha)n(v)n(e)g(that)h Fn(R)1887 2764 y Fk(n)1932 2752 y Fr(\()p Fn(t)p Fr(\))f Fm(\024)f Fn(C)6 b Fr([1)22 b(+)f Fn(Q)2458 2764 y Fk(E)2514 2752 y Fr(\()p Fn(X)7 b Fr(\))2654 2722 y Fo(2)2691 2752 y Fr(])14 b(log)2835 2715 y Fo(2)2872 2752 y Fr(\()p Fn(e)22 b Fr(+)g Fn(n)p Fr(\))32 b(for)g Fn(n)g Fm(\025)456 2852 y Fn(n)506 2864 y Fo(0)543 2852 y Fr(.)45 b(By)31 b(\(A.17\))g(w)n(e)f(then)h(conclude)f(that)h(the)f(sum)h(on)f(the)h (righ)n(t-hand)e(side)h(of)37 b(\(A.13\))31 b(is)456 2951 y(b)r(ounded)d(b)n(y)f(a)g(constan)n(t.)1708 3173 y Fq(References)491 3306 y Fu([1])35 b(Bahn,)24 b(C.,)f(P)n(ark,)h (Y.M.,)e(Y)-6 b(o)r(o,)24 b(H.J.:)31 b(Nonequilibrium)22 b(dynamics)h(of)h(in\014nite)h(particle)f(systems)f(with)601 3389 y(in\014nite)h(range)g(in)n(teractions.)g(J.)g(Mathematical)f(Ph)n (ys.)h Fb(40)p Fu(,)e(4337{4358)k(\(1999\).)491 3472 y([2])35 b(Boldrighini,)24 b(C.:)35 b(Bernoulli)25 b(prop)r(ert)n(y)h (for)f(a)h(one-dimensional)f(system)g(with)h(lo)r(calized)g(in)n (teraction.)601 3555 y(Comm)n(un.)21 b(Math.)j(Ph)n(ys.)f Fb(103)p Fu(,)f(499{514)k(\(1986\).)491 3638 y([3])35 b(Boldrighini,)30 b(C.,)g(Cosimi,)g(G.C.,)g(F)-6 b(rigio,)31 b(S.,)g(Nogueira,)g(A.:)43 b(Con)n(v)n(ergence)32 b(to)f(a)f (stationary)h(state)601 3721 y(and)23 b(di\013usion)g(for)g(a)g(c)n (harged)h(particle)f(in)g(a)h(standing)g(medium.)c(Probab.)j(Theory)h (Relat.)f(Fields)f Fb(80)p Fu(,)601 3804 y(481{500)j(\(1989\).)491 3887 y([4])35 b(Boldrighini,)g(C.,)h(P)n(ellegrinotti,)h(A.,)f (Presutti,)h(E.,)f(Sinai,)g(Y)-6 b(a.G.,)37 b(Solo)n(v)n(eyc)n(hik,)h (M.R.:)51 b(Ergo)r(dic)601 3970 y(prop)r(erties)19 b(of)f(a)i (semi-in\014nite)d(one-dimensional)h(system)h(of)f(statistical)i(mec)n (hanics.)e(Comm)n(un.)f(Math.)601 4053 y(Ph)n(ys.)23 b Fb(101)p Fu(,)f(363{382)k(\(1985\).)491 4136 y([5])35 b(Boldrighini,)17 b(C.,)g(Solo)n(v)n(eitc)n(hik,)j(M.:)27 b(Drift)16 b(and)i(di\013usion)f(for)g(a)g(mec)n(hanical)g(system.)g (Probab.)g(Theory)601 4219 y(Relat.)23 b(Fields)h Fb(103)p Fu(,)e(349{379)j(\(1995\).)491 4302 y([6])35 b(Boldrighini,)22 b(C.,)g(Solo)n(v)n(eitc)n(hik,)j(M.:)30 b(On)23 b(the)h(Einstein)g (relation)f(for)g(a)h(mec)n(hanical)f(system.)f(Probab.)601 4385 y(Theory)i(Relat.)f(Fields)h Fb(107)p Fu(,)e(493{515)j(\(1997\).) 491 4468 y([7])35 b(Bruneau,)k(L.,)f(De)f(Bi)n(\023)-33 b(evre,)38 b(S.:)56 b(A)36 b(Hamiltonian)f(mo)r(del)g(for)g(linear)h (friction)f(in)h(a)g(homogeneous)601 4551 y(medium.)21 b(Comm)n(un.)g(Math.)j(Ph)n(ys.)f Fb(229)p Fu(,)g(511{542)i(\(2002\).) 491 4634 y([8])35 b(Butt\022)-35 b(a,)36 b(P)-6 b(.,)35 b(Caglioti,)g(E.,)g(Marc)n(hioro,)g(C.:)50 b(On)33 b(the)h(long)g(time) e(b)r(eha)n(vior)i(of)f(in\014nitely)h(extended)601 4717 y(systems)23 b(of)g(particles)h(in)n(teracting)h(via)e(Kac)h(p)r(oten)n (tials.)h(J.)e(Stat.)h(Ph)n(ys.)g Fb(108)p Fu(,)e(317{339)k(\(2002\).) 491 4800 y([9])35 b(Butt\022)-35 b(a,)25 b(P)-6 b(.,)24 b(Caglioti,)h(E.,)f(Marc)n(hioro,)g(C.:)32 b(On)25 b(the)g(motion)g(of) f(a)h(c)n(harged)g(particle)g(in)n(teracting)h(with)601 4883 y(an)e(in\014nitely)g(extended)i(system.)c(Comm)n(un.)g(Math.)h (Ph)n(ys.)h Fb(233)p Fu(,)e(545{569)k(\(2003\).)456 4967 y([10])35 b(Caglioti,)22 b(E.,)f(Marc)n(hioro,)h(C.:)30 b(On)22 b(the)h(long)g(time)e(b)r(eha)n(vior)i(of)f(a)g(particle)h(in)f (an)g(in\014nitely)h(extended)601 5050 y(system)g(in)g(one)h (dimension.)f(J.)g(Stat.)i(Ph)n(ys.)e Fb(106)p Fu(,)f(663{680)k (\(2002\).)456 5133 y([11])35 b(Caglioti,)26 b(E.,)g(Marc)n(hioro,)g (C.,)g(Pulviren)n(ti,)g(M.:)35 b(Non-equilibrium)24 b(dynamics)h(of)h (three-dimensional)601 5216 y(in\014nite)e(particle)g(systems.)e(Comm)n (un.)g(Math.)h(Ph)n(ys.)h Fb(215)p Fu(,)e(25{43)j(\(2000\).)p eop %%Page: 30 30 30 29 bop 456 251 a Fo(30)676 b(P)-5 b(.)21 b(BUTT)1475 236 y(\022)1467 251 y(A,)j(E.)e(CA)n(GLIOTI,)h(AND)g(C.)f(MAR)n(CHIOR)n (O)456 450 y Fu([12])35 b(Calderoni,)20 b(P)-6 b(.,)20 b(Caprino,)g(S.:)28 b(Time)19 b(ev)n(olution)i(of)e(in\014nitely)i(man) n(y)e(particles:)29 b(an)20 b(existence)i(theorem.)601 533 y(J.)h(Stat.)i(Ph)n(ys.)e Fb(28)p Fu(,)f(815{833)k(\(1982\).)456 616 y([13])35 b(Calderoni,)d(P)-6 b(.,)32 b(D)r(\177)-37 b(urr,)31 b(D.:)46 b(The)31 b(Smoluc)n(ho)n(wski)f(limit)f(for)i(a)g (simple)e(mec)n(hanical)i(mo)r(del.)f(J.)g(Stat.)601 699 y(Ph)n(ys.)23 b Fb(55)p Fu(,)g(695{738)i(\(1989\).)456 782 y([14])35 b(Dobrushin,)f(R.L.,)g(F)-6 b(ritz,)35 b(J.:)49 b(Non-equilibrium)31 b(dynamics)h(of)h(one-dimensional)f (in\014nite)h(particle)601 865 y(systems)23 b(with)g(a)h(hard-core)g (in)n(teraction.)g(Comm)n(un.)e(Math.)h(Ph)n(ys.)h Fb(55)p Fu(,)e(275{292)k(\(1977\).)456 948 y([15])35 b(F)-6 b(ritz,)34 b(J.,)g(Dobrushin,)g(R.L.:)48 b(Non-equilibrium)30 b(dynamics)i(of)g(t) n(w)n(o-dimensional)g(in\014nite)h(particle)601 1031 y(systems)23 b(with)g(a)h(singular)f(in)n(teraction.)i(Comm)n(un.)c (Math.)j(Ph)n(ys.)f Fb(57)p Fu(,)g(67{81)i(\(1977\).)456 1114 y([16])35 b(Landau,)i(L.D.,)e(Lifshitz,)h(E.M.:)50 b Fs(Physic)l(al)36 b(Kinetics)p Fu(,)f(Course)f(of)g(Theoretical)h(Ph) n(ysics,)h(V)-6 b(ol.)33 b(10.)601 1197 y(Oxford,)22 b(New)i(Y)-6 b(ork,)23 b(F)-6 b(rankfurt:)31 b(P)n(ergamon)24 b(Press,)e(1981.)456 1280 y([17])35 b(Lanford,)18 b(O.E.:)27 b(Classical)17 b(Mec)n(hanics)i(of)e(one-dimensional)g(systems)g(with)h (in\014nitely)g(man)n(y)g(particles.)601 1363 y(I)24 b(An)f(existence)j(theorem.)d(Comm)n(un.)e(Math.)j(Ph)n(ys.)f Fb(9)p Fu(,)g(176{191)i(\(1968\).)456 1446 y([18])35 b(Lanford,)18 b(O.E.:)27 b(Classical)17 b(Mec)n(hanics)i(of)e (one-dimensional)g(systems)g(with)h(in\014nitely)g(man)n(y)g (particles.)601 1529 y(I)r(I)24 b(Kinetic)g(Theory)-6 b(.)24 b(Comm)n(un.)d(Math.)j(Ph)n(ys.)f Fb(11)p Fu(,)g(257{292)i (\(1969\).)456 1612 y([19])35 b(Marc)n(hioro,)24 b(C.,)h(P)n (ellegrinotti,)g(A.,)g(Pulviren)n(ti,)g(M:)f(Remarks)g(on)i(the)g (existence)h(of)e(nonequilibrium)601 1695 y(dynamics.)i(Random)i (\014elds,)g(V)-6 b(ol.)28 b(I,)h(I)r(I)g(\(Esztergom,)h(1979\),)h (733{746,)h(Collo)r(q.)c(Math.)h(So)r(c.)g(J\023)-35 b(anos)601 1778 y(Boly)n(ai,)23 b(27,)g(North-Holland,)g(Amsterdam-New) f(Y)-6 b(ork,)23 b(1981.)456 1861 y([20])35 b(Marc)n(hioro,)26 b(C.,)g(Pulviren)n(ti,)g(M.:)35 b(Time)25 b(ev)n(olution)j(of)d (in\014nite)i(one-dimensional)f(Coulom)n(b)f(system,)601 1944 y(J.)e(Stat.)i(Ph)n(ys.)e Fb(27)p Fu(,)f(809{822)k(\(1982\).)456 2028 y([21])35 b(P)n(ellegrinotti,)25 b(A.,)f(Sidora)n(vicius,)g(V.,)h (V)-6 b(ares,)24 b(M.E.:)32 b(Stationary)27 b(state)f(and)f (di\013usion)g(for)f(a)h(c)n(harged)601 2111 y(particle)20 b(in)g(a)g(one-dimensional)g(medium)e(with)i(lifetimes.)e(T)-6 b(eor.)20 b(V)-6 b(ero)n(y)n(atnost.)21 b(i)f(Primenen.)f Fb(44)p Fu(,)g(796{)601 2194 y(825)24 b(\(1999\);)h(translation)f(in)g (Theory)g(Probab.)f(Appl.)g Fb(44)p Fu(,)g(697{721)i(\(2000\).)456 2277 y([22])35 b(Piasec)n(ki,)f(J.,)f(W)-6 b(a)t(jnryb,)34 b(E.:)48 b(Long-time)31 b(b)r(eha)n(vior)i(of)f(the)h(Loren)n(tz)g (electron)g(gas)g(in)e(a)i(constan)n(t,)601 2360 y(uniform)21 b(electric)j(\014eld.)g(J.)f(Stat.)i(Ph)n(ys.)e Fb(21)p Fu(,)g(549{559)i(\(1979\).)456 2443 y([23])35 b(Pulviren)n(ti,)21 b(M.:)30 b(On)22 b(the)h(time)f(ev)n(olution)h(of)f(the)h(states)h(of)e (in\014nitely)g(extended)j(particle)d(systems.)g(J.)601 2526 y(Stat.)i(Ph)n(ys.)g Fb(27)p Fu(,)e(693{709)k(\(1982\).)456 2609 y([24])35 b(Ruelle,)21 b(D.:)29 b Fs(Statistic)l(al)24 b(Me)l(chanics.)f(R)n(igor)l(ous)i(R)l(esults)p Fu(.)c(New)h(Y)-6 b(ork-Amsterdam:)27 b(W.A.)21 b(Benjamin,)601 2692 y(Inc,)j(1969.)456 2775 y([25])35 b(Sidora)n(vicius,)17 b(V.,)f(T)-6 b(riolo,)16 b(L.,)h(V)-6 b(ares,)17 b(M.E.:)26 b(Mixing)15 b(prop)r(erties)i(for)e (mec)n(hanical)h(motion)f(of)h(a)g(c)n(harged)601 2858 y(particle)24 b(in)f(a)h(random)f(medium.)e(Comm)n(un.)g(Math.)j(Ph)n (ys.)f Fb(219)p Fu(,)f(323{355)k(\(2001\).)456 2941 y([26])35 b(Sinai,)23 b(Y)-6 b(a.G.,)24 b(Solo)n(v)n(eyc)n(hik,)i(M.R.:)31 b(One-dimensional)23 b(classical)h(massiv)n(e)f(particle)h(in)g(the)h (ideal)f(gas.)601 3024 y(Comm)n(un.)d(Math.)j(Ph)n(ys.)f Fb(104)p Fu(,)f(423{443)k(\(1986\).)555 3179 y Ft(P)-6 b(a)o(olo)18 b(Butt)946 3173 y(\022)944 3179 y(a,)i(Dip)l(ar)l(timento) e(di)g(Ma)l(tema)l(tica,)g(Universit)2344 3173 y(\022)2342 3179 y(a)g(di)h(R)n(oma)f(`La)g(Sapienza)l(',)h(Piazzale)456 3262 y(Aldo)26 b(Mor)o(o)e(2,)h(00185)f(R)n(oma,)h(It)l(al)l(y)555 3345 y Fs(E-mail)h(addr)l(ess)5 b Fu(:)33 b Fa(butta@mat.uniroma1.it) 555 3487 y Ft(Emanuele)c(Ca)o(glioti,)h(Dip)l(ar)l(timento)f(di)h(Ma)l (tema)l(tica,)f(Universit)2607 3481 y(\022)2605 3487 y(a)g(di)h(R)n(oma)f(`La)h(Sapienza)l(',)456 3570 y(Piazzale)25 b(Aldo)h(Mor)o(o)f(2,)g(00185)f(R)n(oma,)h(It)l(al)l(y)555 3653 y Fs(E-mail)h(addr)l(ess)5 b Fu(:)33 b Fa(caglioti@mat.uniroma1.)q (it)555 3794 y Ft(Carlo)h(Mar)o(chior)o(o,)h(Dip)l(ar)l(timento)e(di)i (Ma)l(tema)l(tica,)f(Universit)2589 3788 y(\022)2587 3794 y(a)g(di)g(R)n(oma)g(`La)g(Sapienza)l(',)456 3877 y(Piazzale)25 b(Aldo)h(Mor)o(o)f(2,)g(00185)f(R)n(oma,)h(It)l(al)l(y) 555 3960 y Fs(E-mail)h(addr)l(ess)5 b Fu(:)33 b Fa (marchior@mat.uniroma1.)q(it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF