This is a multi-part message in MIME format. ---------------0002281139419 Content-Type: text/plain; name="00-90.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-90.keywords" Thermostatted Lorentz gas, steady state current, smoothness, regularity, symbolic dynamics ---------------0002281139419 Content-Type: application/postscript; name="unad.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="unad.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: unad.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips unad -o unad.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.01.21:0933 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 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/sfreccia { gsave % uso: x1 y1 x2 y2 n setlinewidth freccia grestore } def 115 15 moveto 185 15 lineto 115 15 moveto 115 85 lineto 115 85 moveto 185 85 lineto 185 15 moveto 185 85 lineto stroke /q1 { gsave 0 90 newpath arc stroke grestore} def /q2 {gsave 90 180 newpath arc stroke grestore} def /q3 { gsave 180 270 newpath arc stroke grestore} def /q4 { gsave 270 360 newpath arc stroke grestore} def /cerchio { gsave % uso: x1 y1 r cerchio 0 360 newpath arc stroke grestore} def 115 15 30 q1 115 85 30 q4 185 85 30 q3 185 15 30 q2 150 50 15 cerchio 145 10 115 10 1. sfreccia 155 10 185 10 1. sfreccia 115 15 136 36 1. sfreccia 150 50 160 60 1. sfreccia 150 30 160 35 1. sfreccia stroke %%EndDocument @endspecial -236 1048 a Fh(Fig.)33 b(1)p Fi(:)59 b(General)21 b(billiard)g(structure)g(with)h(scatterers)f(of)h(radius)g Fg(R)1579 1056 y Ff(1)1634 1048 y Fi(and)g Fg(R)1813 1056 y Ff(2)1867 1048 y Fi(in)g(a)h(p)r(erio)r(dic)d(b)r(o)n(x)i(with)g (side)f(length)h Fg(l)p Fi(.)1257 1328 y Fe(8)1257 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v 1981 2176 a Fi(2)2065 2147 y Fp(\014xed)f(so)f(that)i(the)f(motion)g(tak)n(es)e (place)i(on)g(the)g(three)-236 2247 y(dimensional)27 b(surface)h(of)g(constan)n(t)f(energy)g(whic)n(h)h(is)g(the)h(direct)f (pro)r(duct)g(of)g(a)g(torus)f(\(with)i(holes)f(due)g(to)g(the)g (obstacles\))-236 2347 y(and)e(the)h(circle)g(with)g Fk(j)p Fm(v)q Fk(j)d Fp(=)e Fl(v)719 2359 y Fi(0)757 2347 y Fp(.)37 b(W)-7 b(e)27 b(can)f(set)h Fl(v)f Fp(=)d Fl(v)1433 2359 y Fi(0)1473 2347 y Fp(^)-45 b Fl(v)30 b Fp(where)g(^)-45 b Fl(v)30 b Fp(is)c(a)h(unit)g(v)n(ector)i(^)-46 b Fl(v)27 b Fp(=)22 b(\()p Fl(cos#;)14 b Fp(sin)g Fl(#)p Fp(\).)38 b(The)26 b(micro)r(canonical)-236 2446 y(or)i(uniform)g (distribution)h(on)g(this)g(three)g(dimensional)f(energy)g(surface)g (is)g(in)n(v)-5 b(arian)n(t)28 b(under)h(the)g(dynamics)f(when)h Fl(E)h Fp(=)25 b(0.)-236 2546 y(F)-7 b(or)25 b Fl(E)k Fk(6)p Fp(=)22 b(0)k(the)h(dynamics)f(is)g(not)g(Hamiltonian.)36 b(There)26 b(is)g(in)h(fact)f(a)g(phase)g(space)g(con)n(traction)f Fl(\015)31 b Fp(on)26 b(the)g(energy)f(surface)-236 2645 y(equal)i(to)g(\()p Fm(E)19 b Fk(\001)f Fm(v)q Fp(\))p Fl(=v)407 2615 y Fi(2)404 2666 y(0)473 2645 y Fp([MH][GG][CELS].)-203 2746 y(De\014ning)30 b(a)f(new)g(dimensionless)g(time)1094 2712 y Fg(v)1127 2720 y Ff(0)p 1094 2727 66 4 v 1116 2774 a Fg(l)1169 2746 y Fl(t)h Fp(where)f Fl(l)i Fp(is)e(a)g(c)n (haracteristic)e(length)j(of)f(the)h(system)f(\(sa)n(y)g(half)g(the)h (length)g(of)-236 2856 y(the)25 b(torus\))f(and)h(scaling)f Fm(q)h Fp(b)n(y)g Fl(l)r Fp(,)g Fm(v)i Fp(b)n(y)d Fl(v)1067 2868 y Fi(0)1130 2856 y Fp(and)g Fl(E)30 b Fp(b)n(y)1525 2823 y Fg(l)p 1502 2837 68 4 v 1502 2887 a(v)1537 2867 y Ff(2)1535 2907 y(0)1605 2856 y Fp(lea)n(v)n(e)23 b(the)i(dynamics)g (unc)n(hanged.)35 b(W)-7 b(e)25 b(shall)g(therefore)f(tak)n(e)g(from) -236 2977 y(no)n(w)g(on)i Fl(l)e Fp(=)f(2)i(and)g Fl(v)451 2989 y Fi(0)511 2977 y Fp(=)e(1)i(\(w)n(e)g(c)n(ho)r(ose)f Fl(l)h Fp(=)d(2)j(b)r(ecause)g(in)h(this)f(w)n(a)n(y)f(w)n(e)h(can)g (see)g(the)h(obstacle)e(as)h(ha)n(ving)f(cen)n(ters)h(at)g(\(0)p Fl(:)p Fp(0\))-236 3077 y(and)34 b(\(1)p Fl(;)14 b Fp(1\)\).)56 b(Then)35 b(giv)n(en)e(an)n(y)g(initial)i(absolutely)e(con)n(tin)n (uous)g(densit)n(y)i Fl( )2256 3089 y Fi(0)2293 3077 y Fp(\()p Fl(#;)14 b Fm(q)p Fp(\))35 b(w)n(e)f(shall)g(b)r(e)g(in)n (terested)g(in)h(the)f(time)-236 3177 y(ev)n(olv)n(ed)24 b(densit)n(y)i Fl( )396 3189 y Fg(t)426 3177 y Fp(\()p Fl(#;)14 b Fm(q)p Fp(\))27 b(in)f(the)g(limit)h(when)f Fl(t)d Fk(!)g(1)j Fp(whic)n(h)g(describ)r(es)g(the)g(stationary)f (nonequilibrium)h(state)f(\(SNS\))j(of)-236 3276 y(this)g(system.)-203 3377 y(Moran)20 b(and)i(Ho)r(o)n(v)n(er)e([MH])i(carried)e(out)h (extensiv)n(e)g(n)n(umerical)g(studies)h(of)f Fl( )2249 3389 y Fg(t)2278 3377 y Fp(,)i(or)e(more)g(precisely)f(of)i(the)g (induced)g(measure)-236 3476 y Fl(\027)-195 3488 y Fg(t)-139 3476 y Fp(on)27 b(the)h(P)n(oincar)n(\023)-39 b(e)24 b(cross)i(section)h(parametrized)f(b)n(y)h(the)h(p)r(oin)n(ts)f Fl(P)35 b Fp(=)23 b(\()p Fl(\013;)14 b(\014)t Fp(\))24 b Fk(2)f Fl(S)2489 3446 y Fi(1)2544 3476 y Fk(\002)18 b Fp([)p Fk(\000)p Fl(\031)s(=)p Fp(2)p Fl(;)c(\031)s(=)p Fp(2],)26 b Fl(\013)h Fp(corresp)r(onding)f(to)-236 3576 y(the)i(angle)g(lo)r(cating)f(the)i(p)r(oin)n(t)f(on)g(the)h(p)r (erimeter)f(of)g(the)h(obstacle)e(where)h(the)g(collision)g(to)r(ok)f (place)h(and)g Fl(\014)33 b Fp(to)28 b(the)g(angle)-236 3675 y(b)r(et)n(w)n(een)d(the)h(normal)e(v)n(ector)g(at)h Fl(\013)h Fp(and)f(the)h(unit)g(v)n(elo)r(cit)n(y)e(v)n(ector)j(^)-45 b Fl(v)29 b Fp(\(see)c(subsection)g(2.1)f(for)h(a)g(more)f(precise)h (de\014nition\).)-236 3775 y(F)-7 b(ollo)n(wing)21 b(the)h(discrete)g (time)h(tra)5 b(jectory)20 b Fl(P)1187 3787 y Fg(n)1255 3775 y Fp(\(in)j(their)f(case)f(there)h(w)n(as)f(only)h(one)g(obstacle) f(p)r(er)i(unit)f(cell)h(of)f(the)g(triangular)-236 3875 y(lattice,)j(in)g(the)g(case)f(of)h(the)g(billiard)f(in)h(Fig.)36 b(1)24 b(one)g(should)h(add)f(an)h(in)n(teger)e(comp)r(onen)n(t)i(to)f Fl(P)2823 3887 y Fg(n)2869 3875 y Fp(,)h Fc(i.e.)j Fl(P)3104 3887 y Fg(n)3173 3875 y Fp(=)22 b(\()p Fl(\013;)14 b(\014)t(;)g(s)p Fp(\))26 b(where)-236 3974 y Fl(s)42 b Fk(2)i(f)p Fp(0)p Fl(;)14 b Fp(1)p Fk(g)37 b Fp(indicates)i(whic)n(h)h(obstacle)e(is)i (hit\))g(they)f(found)h(that)g(the)g(densit)n(y)f(of)g(p)r(oin)n(ts)h (for)f(large)f Fl(n)h Fp(had)g(a)g(fractal)-236 4074 y(app)r(earance,)30 b Fc(i.e.)d Fp(the)k(stationary)e(measure)h(app)r (eared)f(to)i(b)r(e)g(singular)e(with)i(resp)r(ect)f(to)g(the)h (uniform)g(\(micro)r(canonical\))-236 4174 y(measure)26 b(\(4)p Fl(\031)s Fp(\))243 4143 y Fo(\000)p Fi(1)347 4174 y Fp(cos)13 b Fl(\013d\013d\036)29 b Fp(on)e Fl(P)40 b Fp(whic)n(h)27 b(is)h(stationary)e(and)h(approac)n(hed)f(\(in)i(a)f (w)n(eak)g(sense\))g(as)g Fl(t)c Fk(!)g(1)28 b Fp(when)g Fm(E)23 b Fp(=)f(0.)-203 4274 y(This)h(problem)g(w)n(as)f(later)g (studied)i(analytically)e(b)n(y)h(Cherno)n(v)f(et)h(al.)35 b([CELS])23 b(who)g(pro)n(v)n(ed)f(that)h(starting)f(with)i(an)n(y)e (initial)-236 4373 y(densit)n(y)30 b Fl( )107 4385 y Fi(0)144 4373 y Fp(\()p Fl(\022)r(;)14 b Fm(q)p Fp(\))32 b(there)e(do)r(es)g(indeed)h(exist)f(a)g(limiting)h(measure)e Fl(\027)1989 4385 y Fg(t)2046 4373 y Fk(!)f Fl(\027)2203 4338 y Fi(+)2198 4398 y Fg(E)2289 4373 y Fp(for)i(the)h(induced)g (measure)e(on)h(the)h(P)n(oincar)n(\023)-39 b(e)-236 4473 y(section)20 b(as)h Fl(t)i Fk(!)g(1)p Fp(,)f(whose)f(Hausdor\013)f (dimension)h(is,)h(for)e Fl(E)29 b Fk(6)p Fp(=)22 b(0,)g(strictly)f (less)f(than)i(t)n(w)n(o)e(\(the)h(corresp)r(onding)e(measure)h Fl(\026)3723 4438 y Fi(+)3723 4497 y Fg(E)-236 4573 y Fp(on)28 b(the)i(energy)d(surface)h(has)h(Hausdor\013)f(dimension)h (less)f(than)h(three\).)41 b(They)28 b(also)g(sho)n(w)n(ed)g(that)h (the)g(stationary)e(curren)n(t)-236 4672 y Fm(j)p Fp(\()p Fm(E)p Fp(\))d(=)g Fk(h)p Fm(v)q Fk(i)148 4684 y Fh(E)227 4672 y Fp(=)g Fl(\033)363 4684 y Fh(0)405 4672 y Fm(E)p Fp([)p Fm(1)18 b Fp(+)h Fm(o)p Fp(\()p Fk(j)p Fm(E)p Fk(j)p Fp(\)])28 b(for)g Fm(E)g Fp(small)g(enough)f Fk(j)p Fm(E)p Fk(j)d Fl(<)g(E)1919 4684 y Fi(0)1957 4672 y Fp(,)k Fl(E)2069 4684 y Fi(0)2131 4672 y Fl(<<)23 b Fp(1)28 b(\(p)r(ossibly)g(as)g(small)g(as)f(10)3213 4642 y Fo(\000)p Fi(20)3334 4672 y Fp(\))i(where)f Fk(h\001i)3723 4684 y Fg(E)-236 4772 y Fp(is)k(the)i(a)n(v)n(erage)c(with)j(resp)r(ect)g (to)f Fl(\026)944 4736 y Fi(+)944 4796 y Fg(E)1033 4772 y Fp(and)h Fl(\033)1247 4784 y Fi(0)1318 4772 y Fp(\(generally)e(a)i (tensor\))f(is)h(giv)n(en)f(b)n(y)g(the)h(Kub)r(o)g(form)n(ula.)51 b(\(It)34 b(also)e(follo)n(ws)-236 4872 y(from)27 b(their)h(analysis)e (that)i Fm(j)p Fp(\()p Fm(E)p Fp(\))g(is)f(con)n(tin)n(uous)g(for)g Fk(j)p Fm(E)p Fk(j)c Fl(<)g(E)1742 4884 y Fi(0)1779 4872 y Fp(.\))-203 4972 y(In)35 b(the)f(presen)n(t)g(w)n(ork)f(w)n(e)h (examine)g Fm(j)p Fp(\()p Fm(E)p Fp(\))h(and)f(other)g(prop)r(erties)f (of)i(the)f(stationary)f(measure)h Fl(\027)3034 4942 y Fi(+)3123 4972 y Fp(for)g Fl(E)40 b Fk(2)34 b Fp(\(0)p Fl(:)p Fp(025)p Fl(;)14 b Fp(2\))-236 5072 y(and)33 b(^)-45 b Fl(e)27 b Fp(=)33 b(^)-48 b Fl(x)q Fp(,)p Fc(i.e.)28 b Fp(the)i(\014eld)h(is)f(in)h(the)g Fl(x)g Fp(direction)e(and)i(so)e Fm(j)p Fp(\()p Fm(E)p Fp(\))f(=)g Fl(j)5 b Fp(\()p Fl(E)g Fp(\))g(^)-47 b Fl(x)q Fp(.)45 b(Carrying)28 b(out)j(a)f(n)n(umerical)f (ev)-5 b(aluation)30 b(of)g(the)-236 5171 y(Kub)r(o)c(form)n(ula)f(for) h Fl(\033)467 5183 y Fi(0)531 5171 y Fp(\()p Fc(i.e.)i Fp(an)e(in)n(tegral)f(of)h(the)g(auto)r(correlation)e(function)j(at)f (zero)f(\014eld\))i(w)n(e)f(\014nd)h(go)r(o)r(d)e(agreemen)n(t)g(with) -236 5271 y(the)30 b(results)f(of)h(the)h(sim)n(ulation)e(for)g Fl(j)5 b Fp(\()p Fl(E)g Fp(\))31 b(as)e Fl(E)35 b Fp(tends)30 b(to)n(w)n(ards)e(zero.)43 b(A)30 b(similar)f(in)n(v)n(estigation,)g (with)i(less)e(precision)g(for)-236 5370 y(small)g(v)-5 b(alues)29 b(of)g Fl(E)5 b Fp(,)30 b(w)n(as)e(carried)g(out)h(for)g(a)g (triangular)e(geometry)h(in)i([LNRM],)g(where)f(the)g(results,)h(whic)n (h)f(lo)r(ok)f(similar)-236 5470 y(to)j(ours)g(for)g Fl(E)37 b Fp(not)32 b(to)r(o)f(small,)i(w)n(ere)e(analyzed)f(in)j (terms)e(of)h(p)r(erio)r(dic)f(orbits.)49 b(Here)31 b(w)n(e)h(fo)r(cus) g(on)f(the)h(b)r(eha)n(vior)f(of)g(the)-236 5570 y(curren)n(t)23 b(for)h Fl(E)k Fk(2)23 b Fp(\(0)p Fl(:)p Fp(025)p Fl(;)14 b Fp(0)p Fl(:)p Fp(5\))22 b(where)i(rather)f(precise)h(n)n(umerical)f (sim)n(ulations)h(indicate)g(nonsmo)r(oth)g(b)r(eha)n(vior)f(of)h Fl(j)5 b Fp(\()p Fl(E)g Fp(\))25 b(vs.)-236 5669 y Fl(E)5 b Fp(.)46 b(In)31 b(particular)e(w)n(e)h(analyze)f(the)i(apparen)n(t)f (singularities)f(of)i Fl(\027)1924 5634 y Fi(+)1919 5694 y Fg(E)2010 5669 y Fp(as)f(a)g(function)h(of)g Fl(E)k Fp(in)c(terms)f(of)h(the)g(c)n(hange)e(in)i(the)1751 5869 y(2)p eop %%Page: 3 3 3 2 bop -236 83 a Fp(singularities)24 b(of)h(the)g(map)g(from)g Fl(P)885 95 y Fg(n)955 83 y Fp(to)g Fl(P)1107 95 y Fg(n)p Fi(+1)1262 83 y Fp(caused)f(b)n(y)h(grazing)e(collisions)h(as)h Fl(E)30 b Fp(c)n(hanges.)35 b(W)-7 b(e)25 b(also)f(analyze)g(the)h (formal)-236 183 y(expression)k(for)h(the)h(deriv)-5 b(ativ)n(es)30 b(of)g Fl(j)5 b Fp(\()p Fl(E)g Fp(\))32 b(obtained)e(from)g(the)h(Ka)n(w)n(asaki)d(form)n(ula)i([CELS],)g(and)h (the)g(equiv)-5 b(alen)n(t)30 b(Ruelle)-236 282 y(formalism)20 b([R].)h(W)-7 b(e)22 b(iden)n(tify)f(p)r(ossible)g(origins)e(of)i (singularities)f(and)h(argue)f(for)g(a)h(function)g Fl(j)5 b Fp(\()p Fl(E)g Fp(\))22 b(that)f(is)g(non)g(di\013eren)n(tiable)-236 382 y(or,)j(at)f(most,)i(admits)f(a)f(w)n(eak)g(deriv)-5 b(ativ)n(e)23 b(with)i(a)e(dense)h(set)g(of)g(discon)n(tin)n(uities)f (although)h(the)g(non)g(rigorous)d(nature)j(of)f(our)-236 482 y(argumen)n(t)29 b(do)r(es)i(not)g(p)r(ermit)g(a)f(de\014nitiv)n(e) h(statemen)n(t)g(\(a)g(similar)f(statemen)n(t)g(should)h(hold)g(for)f (the)h(dep)r(endence)g(of)g Fm(j)p Fp(\()p Fm(E)p Fp(\))-236 581 y(on)f(^)-45 b Fl(e)p Fp(\).)-203 684 y(It)32 b(should)f(b)r(e)h (noted)g(that,)h(as)e(sho)n(wn)g(in)g([ER])h(the)g(stationary)e(curren) 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Fp(\003)1809 1131 y Fo(\000)p Fi(1)1898 1165 y Fp(\()p Fl(t)1960 1131 y Fo(0)1984 1165 y Fl(;)14 b(X)7 b Fp(\))e(^)-47 b Fl(xJ)8 b Fp(\()p Fl(S)2313 1177 y Fg(E)2369 1165 y Fp(\()p Fl(t;)14 b(X)7 b Fp(\)\)\))p Fk(i)2672 1177 y Fi(0)2710 1165 y Fp(+)1195 1289 y Fe(Z)1278 1310 y Fo(1)1241 1478 y Fi(0)1362 1402 y Fl(dt)1449 1289 y Fe(Z)1532 1310 y Fg(t)1495 1478 y Fi(0)1575 1402 y Fl(dt)1648 1368 y Fo(0)1672 1402 y Fk(h)e Fp(^)-47 b Fl(x)p Fp(\003)1809 1368 y Fo(\000)p Fi(1)1898 1402 y Fp(\()p Fl(t)1960 1368 y Fo(0)1984 1402 y Fl(;)14 b(X)7 b Fp(\))18 b Fk(\001)g Fm(v)q Fl(J)8 b Fp(\()p Fl(S)2376 1414 y Fg(E)2433 1402 y Fp(\()p Fl(t)2495 1368 y Fo(0)2519 1402 y Fl(;)14 b(X)7 b Fp(\)\))p Fl(J)h Fp(\()p Fl(S)2833 1414 y Fg(E)2889 1402 y Fp(\()p Fl(t;)14 b(X)7 b Fp(\)\))p Fk(i)3160 1414 y Fi(0)3609 1160 y Fp(\(2)p Fl(:)p Fp(5\))-236 1676 y(where)40 b(\003\()p Fl(t;)14 b(X)7 b Fp(\))41 b(is)f(the)i(tangen)n(t)e(\015o)n(w)g(generated)g(b)n (y)h Fl(S)5 b Fp(\()p Fl(t;)14 b(X)7 b Fp(\),)43 b Fc(i.e.)28 b Fp(\003\()p Fl(t;)14 b(X)7 b Fp(\))45 b(=)2490 1636 y Fg(@)t(S)s Fi(\()p Fg(t;X)5 b Fi(\))p 2490 1657 239 4 v 2561 1705 a Fg(@)t(X)2739 1676 y Fp(.)77 b(Eq.)f(\(2.4\),)44 b(together)c(with)-236 1776 y(analogous)27 b(expressions)h(for)h(the)h (higher)f(order)f(deriv)-5 b(ativ)n(es)29 b(can)g(b)r(e)h(obtained)f (easily)g(from)g(the)h(formalism)f(dev)n(elop)r(ed)g(in)-236 1876 y([R].)j(Note)g(that)g(since)g(the)g(system)g(has)f(Ly)n(apuno)n (v)g(exp)r(onen)n(ts)g Fl(\025)1933 1846 y Fg(u)1977 1876 y Fp(\()p Fl(E)5 b Fp(\))31 b Fl(>)f Fp(0)h(and)h(one)g Fl(\025)2677 1846 y Fg(s)2713 1876 y Fp(\()p Fl(E)5 b Fp(\))31 b Fl(<)e Fp(0)j(\(at)g(least)g(for)f(not)h(to)r(o)-236 1975 y(large)26 b Fl(E)5 b Fp(\))29 b(one)f(should)f(exp)r(ect)i(that)k (^)-47 b Fl(x)q Fp(\003)1060 1945 y Fo(\000)p Fi(1)1148 1975 y Fp(\()p Fl(t)1210 1945 y Fo(0)1234 1975 y Fl(;)14 b(X)7 b Fp(\))e(^)-47 b Fl(x)24 b Fk(')f Fl(e)1577 1945 y Fo(\000)p Fg(\025)1668 1920 y Fa(s)1700 1945 y Fi(\()p Fg(E)s Fi(\))p 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Fp(=)22 b(\()p Fl(\013;)14 b(\014)t(;)g(s)p Fp(\).)37 b(With)24 b(a)f(sligh)n(t)g(abuse)g(of)h(notation)e(w)n(e)i(will)f(still)h (indicate)f(our)g(dynamical)g(system)g(in)h(the)g(new)f(co)r(ordinates) -236 3372 y(b)n(y)k Fk(S)-71 3384 y Fg(E)-15 3372 y Fp(\()p Fl(P)12 b Fp(\))29 b(\(note)e(that)h(for)f Fl(E)h Fp(=)23 b(0)k(the)h(in)n(v)-5 b(arian)n(t)27 b(measure)f(on)i Fk(T)49 b Fp(is)27 b(prop)r(ortional)f(to)h Fl(sin\014)t(d\014)t(d\013) p Fp(\).)-203 3474 y(As)g(w)n(e)g(already)f(men)n(tioned)h(in)h(the)g (in)n(tro)r(duction)f(the)g(in)n(v)-5 b(arian)n(t)26 b(measure)h Fl(\026)2316 3439 y Fi(+)2316 3498 y Fg(E)2399 3474 y Fp(for)g Fl(S)2577 3486 y Fg(E)2633 3474 y Fp(\()p Fl(t;)14 b(X)7 b Fp(\))27 b(will)g(induce)h(a)f(measure)f Fl(\027)3724 3439 y Fi(+)3719 3498 y Fg(E)-236 3574 y Fp(for)h Fk(S)6 b Fp(\()p Fl(P)12 b Fp(\))28 b(on)g Fk(T)21 b Fp(.)37 b(The)28 b(a)n(v)n(erage)c(of)k(an)n(y)f(function)h Fl(G)p Fp(\()p Fl(X)7 b Fp(\))28 b(with)g(resp)r(ect)f(to)h Fl(\026)2249 3538 y 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b(v)n(ery)f(useful)h(to)r(ol)f(for)g(the)i(discussion)e(of)g(regularit) n(y)f(prop)r(erties)h(of)h(dynamical)f(system)g(in)h(terms)g(of)f (external)g(parameters)-236 4255 y(consists)28 b(in)h(the)g (construction)f(of)h(a)g(conjugating)f(map,)h Fc(i.e.)e Fp(w)n(e)i(can)f(lo)r(ok)h(for)f(a)g(family)h(of)g(functions)g Fl(h)3165 4267 y Fg(E)s(;E)3289 4250 y Fb(0)3315 4255 y Fp(\()p Fl(P)12 b Fp(\))30 b(with)f(the)-236 4354 y(prop)r(ert)n(y)d (that:)1195 4562 y Fk(S)1245 4574 y Fg(E)1302 4562 y Fp(\()p Fl(h)1382 4574 y Fg(E)s(;E)1506 4558 y Fb(0)1532 4562 y Fp(\()p Fl(P)12 b Fp(\)\))24 b(=)e Fl(h)1852 4574 y Fg(E)s(;E)1976 4558 y Fb(0)2002 4562 y Fp(\()p Fk(S)2084 4574 y Fg(E)2136 4558 y Fb(0)2163 4562 y Fp(\()p Fl(P)12 b Fp(\)\))p Fl(:)1262 b Fp(\(2)p Fl(:)p Fp(7\))-236 4726 y(see)36 b([BKL])g(for)g(an)g(example)h(of)f(suc)n(h)g(a)g (construction)g(for)g(an)h(Anoso)n(v)e(systems.)63 b(It)37 b(is)g(easy)e(to)i(see)f(that)h(for)f(smo)r(oth)-236 4826 y(dynamical)24 b(systems)h(\()p Fc(e.g.)j 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690 y Fp(.)37 b(The)28 b(set)g Fk(E)2861 650 y Fi(1)p Fg(;)p Fi(1)2854 712 y(0)2979 690 y Fp(is)f(sho)n(wn)g(in)h(Fig.)37 b(5.)3128 3444 y Fg(\013)427 1018 y(\014)2160 1076 y(E)s Fi(=0)304 3432 y Fo(\000)p Fi(3)p Fg(:)p Fi(14)485 b Fo(\000)p Fi(1)p Fg(:)p Fi(57)511 b(0)p Fg(:)p Fi(00)536 b(1)p Fg(:)p Fi(57)546 b(3)p Fg(:)p Fi(14)156 3249 y Fo(\000)p Fi(1)p Fg(:)p Fi(570)156 2742 y Fo(\000)p Fi(0)p Fg(:)p Fi(785)208 2227 y(0)p Fg(:)p Fi(000)208 1713 y(0)p Fg(:)p Fi(785)208 1198 y(1)p Fg(:)p Fi(570)95 3419 y @beginspecial @setspecial %%BeginDocument: disc0.ps gsave 0.3 setlinewidth 36 10 translate /dash0 [] def /dash1 [3 3] def /dash2 [5 1] def /dash3 [3 2 1 2] def /m {moveto } def /l {lineto } def /rm {rmoveto } def /rl {rlineto } def /punto {1.000000 0 360 arc fill } def /croce {m 1.000000 dup dup rm 2 mul neg dup dup rl dup neg 0 rm dup neg rl} def /barretta0 {/wi 1.000000 def wi neg 0 rm wi 2 mul 0 rl wi neg 0 rm} def /barretta1p {/wi 1.000000 def wi neg wi rl wi 2 mul 0 rm wi neg dup rl} def /barretta1m {/wi 1.000000 def wi neg dup rl wi 2 mul 0 rm wi neg wi rl} def /dato_err0 { /p_tmp gsave [] 0 setdash exch def p_tmp exch punto p_tmp exch m barretta0 p_tmp exch l barretta0 stroke grestore} def /dato_err1 { /p_tmp gsave [] 0 setdash exch def p_tmp exch croce p_tmp exch m barretta1p p_tmp exch l barretta1m stroke grestore} def /assey 158.400000 def /assex 134.640000 def /taccax {dup /t_l exch def sub m t_l 2 mul 0 exch rlineto stroke} def /taccay {/t_l exch def exch t_l sub exch m t_l 2 mul 0 rlineto stroke} def /ft_size 0.000000 def /ltx 316.800000 def /lty 269.280000 def 0 0 m 0 lty l ltx lty l ltx 0 l 0 0 l ltx lty m 0 10 rl 0 ltx 2 div sub 0 rl 0 -10 rl 0 11.205509 0.000000 taccay 0 72.922754 0.000000 taccay 0 134.640000 0.000000 taccay 0 196.357246 0.000000 taccay 0 258.074491 0.000000 taccay 0.072710 0 1.224000 taccax 79.236355 0 1.224000 taccax 158.400000 0 1.224000 taccax 237.563645 0 1.224000 taccax 316.727290 0 1.224000 taccax [5 5] 0 setdash 0 assex 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156.86 l 79.95 156.62 l 79.83 156.38 l 79.70 156.14 l 79.58 155.90 l 79.46 155.66 l 79.33 155.41 l 79.21 155.17 l 79.09 154.93 l 78.97 154.69 l 78.84 154.45 l 78.72 154.21 l 78.60 153.97 l 78.48 153.73 l 78.35 153.49 l 78.23 153.25 l 78.11 153.01 l 212.707269 252.485660 m 211.13 249.98 l 209.94 248.08 l 208.95 246.48 l 208.08 245.07 l 207.30 243.80 l 206.58 242.64 l 205.92 241.55 l 205.29 240.53 l 204.70 239.56 l 204.15 238.64 l 203.61 237.76 l 203.11 236.92 l 202.62 236.11 l 202.14 235.32 l 201.69 234.57 l 201.25 233.83 l 200.82 233.11 l 200.41 232.42 l 200.00 231.74 l 199.61 231.08 l 199.23 230.43 l 198.85 229.79 l 198.48 229.17 l 198.13 228.56 l 197.78 227.97 l 197.43 227.38 l 197.09 226.81 l 196.76 226.24 l 196.44 225.68 l 196.12 225.14 l 195.80 224.60 l 195.49 224.06 l 195.19 223.54 l 194.89 223.02 l 194.59 222.51 l 194.30 222.01 l 194.01 221.51 l 193.73 221.02 l 193.45 220.53 l 193.17 220.05 l 192.90 219.58 l 192.63 219.11 l 192.37 218.65 l 192.10 218.19 l 191.84 217.73 l 191.59 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176.60 l 169.32 176.33 l 169.18 176.07 l 169.04 175.80 l 168.90 175.54 l 168.76 175.27 l 168.63 175.01 l 168.49 174.75 l 168.35 174.48 l 168.22 174.22 l 168.08 173.96 l 167.94 173.70 l 167.81 173.44 l 167.67 173.18 l 167.54 172.92 l 167.40 172.66 l 167.27 172.40 l 167.13 172.14 l 167.00 171.88 l 166.87 171.62 l 166.73 171.36 l 166.60 171.11 l 166.46 170.85 l 166.33 170.59 l 166.20 170.34 l 166.07 170.08 l 165.93 169.83 l 165.80 169.57 l 165.67 169.31 l 165.54 169.06 l 165.41 168.81 l 165.28 168.55 l 165.15 168.30 l 165.01 168.04 l 164.88 167.79 l 164.75 167.54 l 164.62 167.29 l 164.49 167.04 l 164.36 166.78 l 164.23 166.53 l 164.10 166.28 l 163.98 166.03 l 163.85 165.78 l 163.72 165.53 l 163.59 165.28 l 163.46 165.03 l 163.33 164.78 l 163.20 164.53 l 163.08 164.28 l 162.95 164.03 l 162.82 163.78 l 162.69 163.54 l 162.57 163.29 l 162.44 163.04 l 162.31 162.79 l 162.19 162.55 l 162.06 162.30 l 161.93 162.05 l 161.81 161.81 l 161.68 161.56 l 161.55 161.32 l 161.43 161.07 l 161.30 160.82 l 161.18 160.58 l 161.05 160.33 l 160.93 160.09 l 160.80 159.84 l 160.68 159.60 l 160.55 159.36 l 160.43 159.11 l 160.30 158.87 l 160.18 158.62 l 160.05 158.38 l 159.93 158.14 l 159.80 157.90 l 159.68 157.65 l 159.56 157.41 l 159.43 157.17 l 159.31 156.92 l 159.18 156.68 l 159.06 156.44 l 158.94 156.20 l 158.81 155.96 l 158.69 155.72 l 158.57 155.47 l 158.45 155.23 l 158.32 154.99 l 158.20 154.75 l 158.08 154.51 l 157.95 154.27 l 157.83 154.03 l 157.71 153.79 l 157.59 153.55 l 157.47 153.31 l 157.34 153.07 l 292.432472 253.310942 m 290.68 250.53 l 289.42 248.52 l 288.39 246.86 l 287.49 245.41 l 286.69 244.11 l 285.96 242.92 l 285.28 241.81 l 284.65 240.78 l 284.05 239.80 l 283.49 238.87 l 282.95 237.98 l 282.43 237.13 l 281.94 236.31 l 281.46 235.52 l 281.01 234.75 l 280.56 234.01 l 280.13 233.29 l 279.71 232.59 l 279.31 231.91 l 278.91 231.24 l 278.52 230.59 l 278.15 229.95 l 277.78 229.33 l 277.42 228.72 l 277.07 228.12 l 276.72 227.53 l 276.38 226.95 l 276.05 226.38 l 275.72 225.82 l 275.40 225.27 l 275.08 224.73 l 274.77 224.20 l 274.47 223.67 l 274.17 223.15 l 273.87 222.64 l 273.58 222.13 l 273.29 221.63 l 273.00 221.14 l 272.72 220.65 l 272.45 220.17 l 272.17 219.70 l 271.90 219.23 l 271.64 218.76 l 271.37 218.30 l 271.11 217.84 l 270.85 217.39 l 270.60 216.95 l 270.35 216.50 l 270.10 216.06 l 269.85 215.63 l 269.60 215.20 l 269.36 214.77 l 269.12 214.35 l 268.88 213.93 l 268.65 213.51 l 268.42 213.10 l 268.18 212.69 l 267.96 212.28 l 267.73 211.88 l 267.50 211.48 l 267.28 211.08 l 267.06 210.68 l 266.84 210.29 l 266.62 209.90 l 266.40 209.51 l 266.19 209.13 l 265.97 208.75 l 265.76 208.37 l 265.55 207.99 l 265.34 207.62 l 265.14 207.24 l 264.93 206.87 l 264.73 206.50 l 264.52 206.14 l 264.32 205.78 l 264.12 205.41 l 263.92 205.05 l 263.72 204.70 l 263.53 204.34 l 263.33 203.99 l 263.14 203.63 l 262.94 203.28 l 262.75 202.93 l 262.56 202.59 l 262.37 202.24 l 262.18 201.90 l 262.00 201.56 l 261.81 201.22 l 261.62 200.88 l 261.44 200.54 l 261.26 200.20 l 261.07 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164.84 l 242.44 164.59 l 242.31 164.34 l 242.18 164.10 l 242.06 163.85 l 241.93 163.60 l 241.80 163.35 l 241.67 163.10 l 241.55 162.86 l 241.42 162.61 l 241.29 162.36 l 241.17 162.12 l 241.04 161.87 l 240.91 161.62 l 240.79 161.38 l 240.66 161.13 l 240.54 160.89 l 240.41 160.64 l 240.29 160.40 l 240.16 160.15 l 240.04 159.91 l 239.91 159.66 l 239.79 159.42 l 239.66 159.17 l 239.54 158.93 l 239.41 158.69 l 239.29 158.44 l 239.16 158.20 l 239.04 157.96 l 238.91 157.71 l 238.79 157.47 l 238.67 157.23 l 238.54 156.99 l 238.42 156.74 l 238.30 156.50 l 238.17 156.26 l 238.05 156.02 l 237.93 155.78 l 237.80 155.54 l 237.68 155.29 l 237.56 155.05 l 237.43 154.81 l 237.31 154.57 l 237.19 154.33 l 237.07 154.09 l 236.94 153.85 l 236.82 153.61 l 236.70 153.37 l 236.58 153.13 l 55.453831 254.309975 m 53.44 251.12 l 52.10 248.98 l 51.02 247.25 l 50.09 245.76 l 49.27 244.42 l 48.52 243.21 l 47.83 242.08 l 47.19 241.03 l 46.59 240.04 l 46.01 239.10 l 45.47 238.20 l 44.95 237.34 l 44.45 236.51 l 43.97 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b(this)h(w)n(ork)e(w)n(as)h(\014nished.)-236 3772 y Fm(References.)-236 3990 y Fp([A)n(CG])103 b(R.)31 b(Artuso,)f(G.)h(Casati,)f(I.)g (Guarneri:)41 b(Geometric)30 b(scaling)f(of)h(correlation)e(deca)n(y)h (in)i(c)n(haotic)e(billiards,)h Fn(Phys.)98 4090 y(R)l(ev.)39 b(E)28 b Fm(51)f Fp(\(1995\))f(R3807-R3810)-236 4264 y([BDGL])49 b(F.)34 b(Bonetto,)g(D.)f(Daems,)i(J.L.)d(Leb)r(o)n(witz,)j (V.)e(Ricci:)48 b(Prop)r(erties)32 b(of)h(Stationary)e(Nonequilibrium)j (States)f(in)98 4364 y(the)28 b(Thermostatted)f(P)n(erio)r(dic)f(Loren) n(tz)h(Gas)g(I)r(I:)h(The)g(man)n(y)f(p)r(oin)n(t)g(particles)g (system,)h(in)f(preparation)-236 4534 y([BDLR])53 b(F.)29 b(Bonetto,)g(D.)g(Daems,)g(P)-7 b(.L.)28 b(Garrido,)f(J.L.)i(Leb)r(o)n (witz:)38 b(Prop)r(erties)27 b(of)i(Stationary)e(Nonequilibrium)i (States)98 4633 y(in)f(the)g(Thermostatted)f(P)n(erio)r(dic)f(Loren)n (tz)h(Gas)g(I)r(I)r(I:)h(The)g(man)n(y)f(colliding)g(particles)g (system,)g(in)h(preparation)-236 4803 y([BGG])99 b(F.)33 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Fp(\(1993\),)g(569{601.)-236 357 y([EM])155 b(D.J.)27 b(Ev)-5 b(ans,)26 b(G.P)-7 b(.)27 b(Morris:)35 b Fn(Statistic)l(al)29 b(Me)l(chanics)h(of)g(none)l (quilibrium)g(\015uids)p Fp(,)d(Accademic)f(Press,)g(San)g(Diego)98 457 y(\(1990\))-236 631 y([EPR1])71 b(J.)33 b(P)-7 b(.)34 b(Ec)n(kmann,)g(C.-A.)f(Pillet)h(and)f(L.)g(Rey-Bellet:)49 b(En)n(trop)n(y)32 b(pro)r(duction)h(in)g(nonlinear,)i(thermally)e (driv)n(en)98 731 y(Hamiltonian)28 b(systems.)36 b Fn(J.)30 b(Statist.)38 b(Phys.)g Fm(95)27 b Fp(\(1999\))f(305{331.)-236 905 y([EPR2])71 b(J.)28 b(P)-7 b(.)28 b(Ec)n(kmann,)f(C.-A.)h(Pillet)g (and)g(L.)g(Rey-Bellet:)37 b(Non-equilibrium)28 b(statistical)f(mec)n (hanics)g(of)h(anharmonic)98 1005 y(c)n(hains)21 b(coupled)f(to)h(t)n (w)n(o)g(heat)g(baths)g(at)g(di\013eren)n(t)g(temp)r(eratures.)34 b Fn(Comm.)j(Math.)h(Phys.)e Fm(201)20 b Fp(\(1999\))g(657{697.)-236 1179 y([ER])170 b(J.P)-7 b(.)33 b(Ec)n(kmann,)g(D.)h(Ruelle:)48 b(Ergo)r(dic)32 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b(Statistic)l(al)f(Me)l(chanics)p Fp(,)f(Elsevier)e(\(1991\))-236 2420 y([JLl])170 b(M.)25 b(Jiang,)e(R.)i(de)f(la)g(LLa)n(v)n(e:)34 b(Smo)r(oth)24 b(dep)r(endence)g(of)h(thermo)r(dynamic)e(limits)i(of)f (SRB-measures)f(mp)p 3457 2433 28 4 v 28 w(arc/99-)98 2519 y(122)-236 2673 y([KD])160 b(R.)36 b(Klages,)f(J.R.)h(Dorfman:)52 b(Simple)35 b(Maps)g(with)h(F)-7 b(ractal)35 b(Di\013usion)g(Co)r (e\016cien)n(ts,)i Fn(Phys.)61 b(R)l(ew.)f(L)l(ett.)f Fm(74)98 2772 y Fp(\(1995\))26 b(387-390)-236 2947 y([LNRM])37 b(J.)30 b(Llo)n(yd,)g(M.)h(Niemey)n(er,)f(L.)g(Rondoni,)h(G.P)-7 b(.)30 b(Morris:)40 b(The)31 b(non-equilibrium)e(Loren)n(tz)g(gas,)h Fn(Chaos)i Fm(5)e Fp(\(1995\))98 3046 y(536-551)-236 3200 y([LS])190 b(J.L.)34 b(Leb)r(o)n(witz,)i(H.)e(Sp)r(ohn:)50 b(A)35 b(Galla)n(v)n(otti-Cohen)d(t)n(yp)r(e)i(symmetry)g(in)g(the)h (large)e(deviation)g(functional)h(for)98 3300 y(Sto)r(c)n(hastic)27 b(Dynamics,)h Fn(J.)i(Stat.)37 b(Phys.)i Fm(95)27 b Fp(\(1999\))f(333) -236 3474 y([MH])150 b(B.)28 b(Moran,)e(W.G.)i(Ho)r(o)n(v)n(er:)36 b(Di\013usion)28 b(in)g(the)g(p)r(erio)r(dic)f(Loren)n(tz)g(billiard)g Fn(J.)j(Stat.)38 b(Phys.)g Fm(48)p Fp(\(1987\))26 b(709)-236 3648 y([NR])165 b(W.H.)40 b(Press,)g(B.P)-7 b(.)39 b(Flannery)-7 b(,)41 b(S.A.)f(T)-7 b(euk)n(olsky)g(,)40 b(W.T.)g(W)-7 b(etterling)39 b Fn(Numeric)l(al)h(r)l(e)l(cip)l(es)h(in)g(C)p Fp(,)e(Cam)n(bridge)98 3748 y(Univ)n(ersit)n(y)27 b(Press)f(\(1993\)) -236 3922 y([R])227 b(D.)32 b(Ruelle:)44 b(Di\013eren)n(tiation)31 b(of)h(SRB)f(states)g Fn(Commun.)48 b(Math.)i(Phys.)f Fm(187)30 b Fp(\(1997\))g(227-241)e(and)j(Nonequi-)98 4022 y(librium)26 b(statistical)g(mec)n(hanics)f(near)g(equilibrium:)36 b(computing)26 b(higher-order)d(terms)j Fn(Nonline)l(arity)h Fm(11)e Fp(\(1998\))98 4122 y(5-18)-236 4275 y([Y])226 b(L.-S.)38 b(Y)-7 b(oung:)57 b(Statistical)38 b(prop)r(erties)e(of)i (systems)g(with)g(some)f(h)n(yp)r(erb)r(olicit)n(y)g(including)h (certain)f(billiards,)98 4375 y Fn(A)n(nnals)29 b(of)i(Math.)38 b Fp(to)28 b(app)r(ear)1730 5869 y(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0002281139419--