Content-Type: multipart/mixed; boundary="-------------0002281123905" This is a multi-part message in MIME format. ---------------0002281123905 Content-Type: text/plain; name="00-89.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-89.keywords" Fourier's Law, heat conduction, steady states, Geen-Kubo formula ---------------0002281123905 Content-Type: application/postscript; name="luc10.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="luc10.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: luc10.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips luc10 -o luc10.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.02.15:1500 %%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 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b(a)n(v)n(oid)e(con)n(v)n(ection)g(one)h(has)g(to)g(mak)n(e)664 2375 y Fq(T)713 2387 y Fn(h)779 2375 y Fq(>)22 b(T)915 2387 y Fn(b)972 2375 y Fr(or)h(k)n(eep)g Fm(j)p Fq(T)1327 2387 y Fn(h)1380 2375 y Fm(\000)10 b Fq(T)1504 2387 y Fn(b)1536 2375 y Fm(j)24 b Fr(small\).)36 b(Assuming)23 b(uniformit)n(y)g(in)h(the)g(direction)f(parallel)664 2475 y(to)f(the)h(v)n(ertical)e Fq(x)p Fr(-axis)h(one)g(has)g(in)g(the) h(stationary)e(state)h(a)g(temp)r(erature)g(pro\014le)3331 2454 y(~)3315 2475 y Fq(T)11 b Fr(\()p Fq(x)p Fr(\))664 2590 y(with)869 2569 y(~)853 2590 y Fq(T)g Fr(\(0\))23 b(=)g Fq(T)1179 2602 y Fn(b)1212 2590 y Fr(,)1279 2569 y(~)1263 2590 y Fq(T)11 b Fr(\()p Fq(h)p Fr(\))23 b(=)g Fq(T)1595 2602 y Fn(h)1665 2590 y Fr(and)k Fq(\024)p Fr(\()1923 2569 y(~)1906 2590 y Fq(T)12 b Fr(\))2009 2558 y Fn(d)2056 2543 y Fp(~)2044 2558 y Fn(T)p 2009 2572 83 4 v 2014 2619 a(dx)2125 2590 y Fr(=Const.)36 b(for)27 b Fq(x)c Fm(2)h Fr(\(0)p Fq(;)14 b(h)p Fr(\).)623 2748 y(F)-7 b(rom)29 b(a)g(ph)n(ysical)g(p)r(oin)n(t)g(of)h(view,)g (whic)n(h)f(is)h(ho)n(w)f(w)n(e)g(presen)n(ted)g(them,)h(the)g(t)n(w)n (o)f(cases)498 2848 y(are)f(conceptually)g(v)n(ery)g(similar)g(\(some)h (ph)n(ysicists)f(w)n(ould)h(ev)n(en)f(sa)n(y)g(iden)n(tical\).)41 b(W)-7 b(e)29 b(ha)n(v)n(e)498 2947 y(implicitly)e(assumed)f(that)g (the)h(system)e(is)h(describ)r(ed)g(fully)h(b)n(y)f(sp)r(ecifying)g (its)g(temp)r(erature)498 3047 y Fq(T)12 b Fr(\()p Fz(r)p Fq(;)i(t)p Fr(\))40 b(ev)n(erywhere)e(in)i(\003.)73 b(What)40 b(this)g(means)f(on)g(the)h(microscopic)f(lev)n(el)g(is)g(that)h(w)n(e) 498 3146 y(imagine)31 b(the)h(system)f(to)h(b)r(e)g(in)g(lo)r(cal)f (thermal)g(equilibrium)h(\(L)-7 b(TE\).)31 b(T)-7 b(o)32 b(mak)n(e)e(this)i(a)f(bit)498 3246 y(more)20 b(precise)g(w)n(e)h(migh) n(t)f(think)i(of)f(the)g(system)f(as)h(b)r(eing)g(divided)g(up)g(\(men) n(tally\))g(in)n(to)g(man)n(y)498 3346 y(little)29 b(cub)r(es,)f(eac)n (h)g(big)g(enough)f(to)h(con)n(tain)g(v)n(ery)f(man)n(y)g(atoms)h(y)n (et)f(small)h(enough)g(on)g(the)498 3445 y(macroscopic)g(scale)g(to)i (b)r(e)g(accurately)e(describ)r(ed,)i(at)f(a)g(sp)r(eci\014ed)h(time)g Fq(t)p Fr(,)g(as)f(a)g(system)h(in)498 3545 y(equilibrium)35 b(at)f(temp)r(erature)g Fq(T)12 b Fr(\()p Fz(r)1666 3557 y Fn(i)1694 3545 y Fq(;)i(t)p Fr(\),)36 b(where)e Fz(r)2138 3557 y Fn(i)2201 3545 y Fr(is)h(the)g(cen)n(ter)f(of)g(the)h Fq(i)p Fr(-th)f(cub)r(e.)59 b(F)-7 b(or)498 3645 y(slo)n(w)27 b(v)-5 b(ariation)26 b(in)i(space)f(and)g(time)h(w)n(e)g(can)f(then)h (use)f(a)h(con)n(tin)n(uous)e(description)h Fq(T)12 b Fr(\()p Fz(r)p Fq(;)i(t)p Fr(\).)623 3744 y(This)29 b(notion)g(is)g (made)h(precise)e(in)i(the)g(so)e(called)h(h)n(ydro)r(dynamic)g (scaling)f(limit)i(\(HSL\))498 3844 y(where)41 b(the)g(ratio)f(of)h (micro)g(to)f(macro)g(scale)h(go)r(es)f(to)h(zero)2523 3814 y Fp(47)p Fn(;)p Fp(79)p Fn(;)p Fp(57)2764 3844 y Fr(.)77 b(The)41 b(macroscopic)498 3943 y(co)r(ordinates)33 b Fz(r)h Fr(and)g Fq(t)h Fr(are)e(related)g(to)h(the)h(microscopic)d (ones)i Fz(q)h Fr(and)f Fq(\034)9 b Fr(,)36 b(b)n(y)e Fz(r)g Fr(=)g Fq(\017)p Fz(q)g Fr(and)498 4043 y Fq(t)27 b Fr(=)g Fq(\017)681 4013 y Fn(\013)728 4043 y Fq(\034)9 b Fr(,)32 b Fo(i.e.)e Fr(if)h(\003)e(is)h(a)g(cub)r(e)h(of)f (macroscopic)e(sides)h Fq(l)r Fr(,)i(then)f(its)h(sides,)f(no)n(w)g (measured)f(in)498 4143 y(microscopic)h(length)i(units,)h(are)e(of)g (length)h Fq(L)d Fr(=)h Fq(\017)2158 4113 y Fk(\000)p Fp(1)2247 4143 y Fq(l)r Fr(.)48 b(W)-7 b(e)32 b(then)g(supp)r(ose)g (that)g(at)f Fq(t)f Fr(=)f(0)498 4242 y(our)e(system)g(of)h Fq(N)k Fr(=)22 b Fq(\032L)1302 4212 y Fn(d)1368 4242 y Fr(particles)27 b(with)h(Hamiltonian)812 4483 y Fq(H)7 b Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))23 b(=)1251 4379 y Fn(N)1221 4404 y Fj(X)1227 4581 y Fn(i)p Fp(=1)1355 4316 y Fj(2)1355 4465 y(4)1432 4426 y Fz(p)1485 4396 y Fp(2)1485 4448 y Fn(i)p 1420 4463 115 4 v 1420 4540 a Fr(2)p Fq(m)1563 4483 y Fr(+)1646 4404 y Fj(X)1653 4582 y Fn(j)s Fk(6)p Fp(=)p Fn(i)1779 4483 y Fq(\036)p Fr(\()p Fz(q)1910 4495 y Fn(j)1965 4483 y Fm(\000)18 b Fz(q)2098 4495 y Fn(i)2126 4483 y Fr(\))h(+)f Fq(u)p Fr(\()p Fz(q)2390 4495 y Fn(i)2418 4483 y Fr(\))2450 4316 y Fj(3)2450 4465 y(5)2528 4483 y Fr(=)2647 4379 y Fn(N)2616 4404 y Fj(X)2622 4581 y Fn(i)p Fp(=1)2772 4426 y Fz(p)2825 4396 y Fp(2)2825 4448 y Fn(i)p 2760 4463 V 2760 4540 a Fr(2)p Fq(m)2903 4483 y Fr(+)g Fm(V)7 b Fr(\()p Fq(Q)p Fr(\))207 b(\(3\))498 4727 y(is)32 b(describ)r(ed)h(b)n(y)f(an)g(equilibrium)g(Gibbs)h (measure)e(with)i(a)f(temp)r(erature)g Fq(T)12 b Fr(\()p Fz(r)p Fr(\))31 b(=)g Fq(T)12 b Fr(\()p Fq(\017)p Fz(q)p Fr(\):)498 4827 y(roughly)26 b(sp)r(eaking)h(the)h(phase)f(space)g (ensem)n(ble)g(densit)n(y)h(has)f(the)h(form,)792 5067 y Fq(\026)842 5079 y Fp(0)880 5067 y Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))23 b Fm(\030)g Fr(exp)1354 4897 y Fj(8)1354 4972 y(<)1354 5121 y(:)1427 5067 y Fm(\000)1536 4963 y Fn(N)1506 4988 y Fj(X)1512 5165 y Fn(i)p Fp(=1)1640 5067 y Fq(\014)1687 5079 y Fp(0)1724 5067 y Fr(\()p Fq(\017)p Fz(q)1840 5079 y Fn(i)1868 5067 y Fr(\))1914 4900 y Fj(2)1914 5050 y(4)1979 5011 y Fz(p)2032 5023 y Fn(i)2060 4981 y Fp(2)p 1979 5048 118 4 v 1981 5124 a Fr(2)p Fq(m)2126 5067 y Fr(+)2209 4988 y Fj(X)2216 5167 y Fn(j)s Fk(6)p Fp(=)p Fn(i)2342 5067 y Fq(\036)p Fr(\()p Fz(q)2473 5079 y Fn(j)2528 5067 y Fm(\000)18 b Fz(q)2661 5079 y Fn(i)2689 5067 y Fr(\))h(+)f Fq(u)p Fr(\()p Fz(q)2953 5079 y Fn(i)2981 5067 y Fr(\))3013 4900 y Fj(3)3013 5050 y(5)3068 4897 y(9)3068 4972 y(=)3068 5121 y(;)3170 5067 y Fq(;)188 b Fr(\(4\))p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(2)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 3 3 3 2 bop 498 658 a Fr(where)22 b Fq(Q)h Fr(=)g(\()p Fz(q)992 670 y Fp(1)1030 658 y Fq(;)14 b(:)g(:)g(:)f(;)h Fz(q)1264 670 y Fn(N)1327 658 y Fr(\))24 b Fm(2)f Fr(\003)1519 628 y Fn(dN)1616 658 y Fr(,)h Fq(P)35 b Fr(=)23 b(\()p Fz(p)1924 670 y Fp(1)1961 658 y Fq(;)14 b(:)g(:)g(:)g(;)g Fz(p)2199 670 y Fn(N)2262 658 y Fr(\))23 b Fm(2)g Fi(R)2467 610 y Fn(dN)2565 658 y Fr(,)h Fq(\036)p Fr(\()p Fz(q)p Fr(\))g(is)e(some)g(short)g(range)498 758 y(in)n(ter)27 b(particle)g(p)r(oten)n(tial,)h Fq(u)p Fr(\()p Fz(q)1498 770 y Fn(i)1526 758 y Fr(\))g(an)f(external)g(p)r(oten)n(tial)g(and)h Fq(\014)2585 722 y Fk(\000)p Fp(1)2581 780 y(0)2674 758 y Fr(\()p Fz(r)p Fr(\))c(=)f Fq(T)2938 770 y Fp(0)2975 758 y Fr(\()p Fz(r)p Fr(\))3078 728 y Fp(79)3149 758 y Fr(.)623 858 y(In)28 b(the)h(limit)g Fq(\017)24 b Fm(!)g Fr(0,)k Fq(\032)g Fr(\014xed,)h(the)f(system)g(at)g Fq(t)d Fr(=)e(0)28 b(will)h(b)r(e)f(macroscopically)e(in)i(L)-7 b(TE)498 957 y(with)35 b(a)g(lo)r(cal)f(temp)r(erature)g Fq(T)1503 969 y Fp(0)1540 957 y Fr(\()p Fz(r)p Fr(\))i(\(as)e(already)g (noted)g(w)n(e)h(suppress)f(here)g(the)h(v)-5 b(ariation)498 1057 y(in)33 b(the)g(particle)e(densit)n(y)i Fq(n)p Fr(\()p Fz(r)p Fr(\)\).)52 b(W)-7 b(e)33 b(are)e(in)n(terested)h(in)h(the)g(b)r (eha)n(vior)e(of)h(a)g(macroscopic)498 1156 y(system,)23 b(for)e(whic)n(h)h Fq(\017)h(<<)f Fr(1,)h(at)f(macroscopic)d(times)j Fq(t)i Fm(\025)e Fr(0,)h(corresp)r(onding)d(to)i(microscopic)498 1256 y(times)i Fq(\034)33 b Fr(=)23 b Fq(\017)907 1226 y Fk(\000)p Fn(\013)1006 1256 y Fq(t)p Fr(,)i Fq(\013)e Fr(=)g(2)g(for)h(heat)g(conduction)g(or)f(other)h(di\013usiv)n(e)g(b)r (eha)n(vior.)34 b(The)24 b(implicit)498 1356 y(assumption)f(then)i (made)e(in)i(the)f(macroscopic)e(description)h(giv)n(en)g(earlier)f(is) i(that)g(since)g(the)498 1455 y(v)-5 b(ariations)21 b(in)h Fq(T)1014 1467 y Fp(0)1051 1455 y Fr(\()p Fz(r)p Fr(\))h(are)e(of)h (order)f Fq(\017)h Fr(on)f(a)h(microscopic)e(scale,)j(then)f(for)g Fq(\017)h(<<)f Fr(1,)h(the)f(system)498 1555 y(will,)27 b(also)e(at)h(time)g Fq(t)p Fr(,)h(b)r(e)f(in)h(a)e(state)h(v)n(ery)f (close)g(to)h(L)-7 b(TE)26 b(with)g(a)g(temp)r(erature)g Fq(T)12 b Fr(\()p Fz(r)p Fq(;)i(t)p Fr(\))26 b(that)498 1655 y(ev)n(olv)n(es)g(in)i(time)g(according)e(to)h(F)-7 b(ourier's)27 b(la)n(w,)g(Eq.\(1\).)623 1754 y(>F)-7 b(rom)32 b(a)h(mathematical)g(p)r(oin)n(t)g(of)g(view)f(the)i (di\016cult)g(problem)e(is)h(pro)n(ving)e(that)j(the)498 1854 y(system)26 b(sta)n(ys)f(in)h(L)-7 b(TE)25 b(for)h Fq(t)d(>)f Fr(0)k(when)g(the)g(dynamics)g(are)f(giv)n(en)g(b)n(y)g(a)h (Hamiltonian)f(time)498 1954 y(ev)n(olution.)35 b(This)24 b(requires)f(pro)n(ving)g(that)h(the)h(macroscopic)d(system)i(has)g (some)f(v)n(ery)g(strong)498 2053 y(ergo)r(dic)37 b(prop)r(erties,)i Fo(e.g.)f Fr(that)g(the)g(only)g(time)g(in)n(v)-5 b(arian)n(t)37 b(measures)g(lo)r(cally)g(absolutely)498 2153 y(con)n(tin)n(uous)f (w.r.t)g(Leb)r(esgue)g(measure)g(are,)i(for)e(in\014nitely)h(extended)g (spatially)f(uniform)498 2252 y(systems,)d(of)e(the)h(Gibbs)g(t)n(yp)r (e)1516 2222 y Fp(30)p Fn(;)p Fp(70)p Fn(;)p Fp(63)1759 2252 y Fr(.)49 b(This)32 b(has)f(only)g(b)r(een)i(pro)n(v)n(en)d(so)h (far)g(for)g(systems)498 2352 y(ev)n(olving)17 b(via)h(sto)r(c)n (hastic)g(dynamics,)i Fo(e.g.)e Fr(in)n(teracting)g(Bro)n(wnian)f (particles)h(or)g(lattice)g(gases.)498 2452 y(In)39 b(these)h(systems)e (the)i(relev)-5 b(an)n(t)38 b(conserv)n(ed)g(quan)n(tit)n(y)g(is)h (usually)g(the)h(particle)e(densit)n(y)498 2551 y(rather)e(than)i(the)g (energy)e(densit)n(y)-7 b(.)66 b(W)-7 b(e)37 b(shall)g(not)h(discuss)f (suc)n(h)g(sto)r(c)n(hastic)f(ev)n(olutions)498 2651 y(here)27 b(but)h(refer)f(the)h(reader)e(to)i([79,47])e(for)h(a)g (mathematical)g(exp)r(osition.)623 2751 y(The)j(only)f(Hamiltonian)h (system)f(for)g(whic)n(h)h(a)g(macroscopic)d(transp)r(ort)i(la)n(w)g (has)h(b)r(een)498 2850 y(deriv)n(ed)36 b(is)g(a)g(gas)f(of)i(nonin)n (teracting)e(particles)h(mo)n(ving)f(among)g(a)h(\014xed)h(arra)n(y)d (of)i(p)r(eri-)498 2950 y(o)r(dic)27 b(con)n(v)n(ex)f(scatterers)g(\(p) r(erio)r(dic)h(Loren)n(tz)f(gas)g(or)h(Sinai)g(billiard\).)36 b(F)-7 b(or)27 b(this)g(system)g(one)498 3049 y(can)38 b(pro)n(v)n(e)f(a)h(di\013usion)g(equation)g(lik)n(e)g(Eq.\(1\))g(for)g (the)g(densit)n(y)g(of)h(the)f(particles,)j(b)r(oth)498 3149 y(for)36 b(the)h(initial)g(v)-5 b(alue)36 b(and)g(the)h(suitably)f (de\014ned)h(stationary)e(state)h(problem,)j(with)e(the)498 3249 y(\(self)6 b(\)di\013usion)39 b(constan)n(t)e(giv)n(en)g(b)n(y)g (the)h(Einstein-Green-Kub)r(o)e(form)n(ula)3005 3219 y Fp(10)p Fn(;)p Fp(56)3161 3249 y Fr(.)67 b(Unfor-)498 3348 y(tunately)-7 b(,)36 b(the)f(absence)e(of)i(in)n(teractions)e(b)r (et)n(w)n(een)h(particles)f(mak)n(es)g(this)i(system)f(a)g(p)r(o)r(or) 498 3448 y(mo)r(del)24 b(for)g(heat)g(conduction)g(in)h(realistic)e (systems.)35 b(In)25 b(particular)e(there)h(is)g(no)f(mec)n(hanism)498 3548 y(for)28 b(ac)n(hieving)f(L)-7 b(TE.)27 b(The)i(sp)r(eed)f(of)g (eac)n(h)f(particle)h Fm(j)p Fz(v)q Fm(j)h Fr(do)r(es)f(not)g(c)n (hange)f(in)i(the)f(course)f(of)498 3647 y(time)j(and)g(the)f (di\013usion)h(constan)n(t)f(for)g(eac)n(h)g(particle)g(is)g(prop)r (ortional)f(to)h(its)h(sp)r(eed.)43 b(The)498 3747 y(di\013usion)34 b(equation)g(for)f(the)i(densit)n(y)f(men)n(tioned)g(ab)r(o)n(v)n(e)f (are)g(therefore)g(in)h(fact)g(separate)498 3846 y(uncoupled)24 b(equations)f(for)h(particles)f(with)i(sp)r(eci\014ed)f(sp)r(eeds.)36 b(It)24 b(corresp)r(onds)e(to)i(the)g(usual)498 3946 y(di\013usion)k(equation)f(only)g(when)h(all)f(the)h(particles)f(ha)n (v)n(e)f(the)i(same)f(sp)r(eed.)623 4046 y(T)-7 b(o)28 b(remedy)f(this)i(problem)f(it)h(w)n(ould)f(b)r(e)g(necessary)f(to)h (add)g(in)n(teractions)f(b)r(et)n(w)n(een)i(the)498 4145 y(mo)n(ving)39 b(particles,)i Fo(e.g.)e Fr(instead)g(of)h(p)r(oin)n(ts) f(mak)n(e)g(them)h(little)g(balls,)i(and)d(then)h(deriv)n(e)498 4245 y(coupled)33 b(equations)f(for)g(the)i(di\013usion)f(of)g(b)r(oth) g(particle)g(and)f(energy)g(densities.)53 b(This)33 b(is)498 4345 y(what)f(w)n(e)f(w)n(ould)h(consider)f(a)g(satisfactory)f(answ)n (er)g(to)i(the)g(c)n(hallenge)f(in)h(the)g(title)h(of)f(this)498 4444 y(article)37 b(and)g(w)n(e)g(o\013er)f(a)h(b)r(ottle)h(of)f(v)n (ery)g(go)r(o)r(d)f(wine)i(to)f(an)n(y)n(one)e(who)i(pro)n(vides)f(it.) 67 b(W)-7 b(e)498 4544 y(b)r(eliev)n(e)33 b(that)f(this)h(system,)h (with)f(only)f(t)n(w)n(o)g(conserv)-5 b(ation)31 b(la)n(ws)h(and)g(an)g (external)g(source)498 4643 y(\(the)24 b(\014xed)e(con)n(v)n(ex)g (scatterers\))f(for)h(c)n(haotic)g(dynamics)g(ma)n(y)g(b)r(e)h(the)g (simplest)g(Hamiltonian)498 4743 y(system)k(for)h(whic)n(h)f(suc)n(h)g (results)g(could)h(b)r(e)g(pro)n(v)n(en)e(rigorously)-7 b(.)623 4843 y(Just)26 b(ho)n(w)g(far)g(w)n(e)g(are)g(from)g(suc)n(h)h (results)f(will)g(b)r(ecome)h(clear)e(as)h(w)n(e)h(describ)r(e)f(our)g (cur-)498 4942 y(ren)n(t)k(mathematical)g(understanding)g(of)g(the)h (stationary)e(nonequilibrium)h(state)g(\(SNS\))i(of)498 5042 y(macroscopic)i(systems)i(whose)f(ends)i(are,)g(as)f(in)g(the)h (example)e(of)i(the)f(Benard)f(problem,)498 5142 y(k)n(ept)29 b(at)f(\014xed)g(temp)r(eratures)g Fq(T)1543 5154 y Fp(1)1609 5142 y Fr(and)g Fq(T)1820 5154 y Fp(2)1857 5142 y Fr(.)39 b(The)29 b(heat)f(conductivit)n(y)h(in)f(this)h(situation)f(can)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(3)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 4 4 4 3 bop 498 646 a Fr(b)r(e)36 b(de\014ned)g(precisely)f(without)h(in)n (v)n(oking)e(L)-7 b(TE.)36 b(T)-7 b(o)35 b(do)g(this)h(w)n(e)g(let)2841 625 y(~)2821 646 y Fq(J)43 b Fr(b)r(e)36 b(the)h(exp)r(ecta-)498 746 y(tion)31 b(v)-5 b(alue)31 b(in)g(the)g(SNS,)h Fo(i.e.)f Fr(w)n(e)g(assume)f(that)h(the)g(SNS)h(is)f(describ)r(ed)g(b)n(y)f(a)h (phase-space)498 846 y(measure)g(\(whose)g(existence)h(w)n(e)f(discuss) g(later\),)i(of)f(the)g(energy)e(or)h(heat)h(curren)n(t)f(\015o)n(wing) 498 945 y(from)d(reserv)n(oir)d(1)j(to)g(reserv)n(oir)d(2.)38 b(W)-7 b(e)28 b(then)h(de\014ne)f(the)h(conductivit)n(y)f Fq(\024)2889 957 y Fn(L)2966 945 y Fr(as)3089 924 y(~)3069 945 y Fq(J)8 b(=)p Fr(\()p Fq(A\016)s(T)h(=L)p Fr(\))498 1045 y(where)31 b Fq(\016)s(T)9 b(=L)28 b Fr(=)h(\()p Fq(T)1142 1057 y Fp(1)1200 1045 y Fm(\000)20 b Fq(T)1334 1057 y Fp(2)1371 1045 y Fr(\))p Fq(=L)30 b Fr(is)i(the)f(e\013ectiv)n (e)h(temp)r(erature)e(gradien)n(t)h(for)f(a)h(cylinder)g(of)498 1144 y(microscopic)c(length)h Fq(L)f Fr(and)h(uniform)g(crossection)e Fq(A)j Fr(and)e Fq(\024)p Fr(\()p Fq(T)12 b Fr(\))28 b(as)f(the)i(limit)f(of)g Fq(\024)3220 1156 y Fn(L)3298 1144 y Fr(when)498 1244 y Fq(\016)s(T)34 b Fm(!)23 b Fr(0)i(\()p Fq(T)875 1256 y Fp(1)935 1244 y Fr(=)d Fq(T)1071 1256 y Fp(2)1131 1244 y Fr(=)h Fq(T)12 b Fr(\))24 b(and)h Fq(L)d Fm(!)h(1)1763 1214 y Fp(54)1834 1244 y Fr(.)36 b(The)24 b(existence)h(of)f(suc)n(h)h(a)f(limit)i(with)f Fq(\024)f Fr(p)r(ositiv)n(e)498 1344 y(and)k(\014nite)g(is)f(what)h (one)f(w)n(ould)g(lik)n(e)g(to)h(pro)n(v)n(e.)498 1583 y Fz(2)95 b(Heat)32 b(Conduction)g(in)f(Gases)498 1767 y Fr(Before)37 b(going)g(on)g(to)h(a)g(mathematical)f(discussion)g(of)h (heat)g(conducting)g(SNS,)g(w)n(e)g(turn)498 1867 y(brie\015y)26 b(to)g(the)h(\\kinetic)f(theory")f(analysis)g(of)i(heat)f(conduction)g (in)g(gases.)35 b(This)27 b(is)f(histori-)498 1967 y(cally)19 b(the)h(\014rst)g(example)f(of)h(a)f(microscopic)g(description)g(of)g (this)i(macroscopic)c(phenomenon.)498 2066 y(It)32 b(go)r(es)e(bac)n(k) h(to)g(the)h(w)n(orks)e(of)h(Clausius,)h(Maxw)n(ell)e(and)i(Boltzmann) 2853 2036 y Fp(9)2921 2066 y Fr(who)f(obtained)g(a)498 2166 y(theoretical)f(expression)f(for)h(the)g(heat)h(conductivit)n(y)f (of)g(gases,)g Fq(\024)e Fm(\030)2763 2096 y(p)p 2832 2096 61 4 v 70 x Fq(T)11 b Fr(,)32 b(indep)r(enden)n(t)f(of)498 2266 y(the)j(gas)e(densit)n(y)-7 b(.)54 b(This)34 b(agrees)d(with)j (exp)r(erimen)n(t)f(\(when)h(the)g(densit)n(y)f(is)g(not)h(to)r(o)f (high\))498 2365 y(and)28 b(w)n(as)e(a)h(ma)5 b(jor)27 b(early)f(ac)n(hiev)n(emen)n(t)h(of)g(the)h(atomic)f(theory)g(of)h (matter)2917 2335 y Fp(9)2954 2365 y Fr(.)623 2465 y(Clausius)j(and)h (Maxw)n(ell)f(used)h(the)g(concept)g(of)g(a)f(\\mean)g(free)h(path")g Fq(\025)p Fr(:)46 b(the)32 b(a)n(v)n(erage)498 2564 y(distance)27 b(a)f(particle)h(\(atom)g(or)f(molecule\))h(tra)n(v)n(els)e(b)r(et)n(w) n(een)i(collisions)f(in)h(a)g(gas)f(with)h(par-)498 2664 y(ticle)f(densit)n(y)f Fq(\032)p Fr(.)37 b(Straigh)n(tforw)n(ard)22 b(analysis)j(giv)n(es)f Fq(\025)g Fm(\030)e Fr(1)p Fq(=\032\031)s(\033) 2556 2634 y Fp(2)2593 2664 y Fr(,)k Fq(\033)h Fm(\030)e Fr(an)g(\\e\013ectiv)n(e")g(hard)498 2764 y(core)32 b(diameter)h(of)h (a)f(particle.)53 b(They)33 b(considered)g(a)g(gas)f(with)i(temp)r (erature)f(gradien)n(t)f(in)498 2863 y(the)i Fq(x)p Fr(-direction)f (and)g(assumed)g(that)g(the)h(gas)e(is)h(\(appro)n(ximately\))g(in)g (lo)r(cal)g(equilibrium)498 2963 y(with)h(densit)n(y)f Fq(\032)g Fr(and)g(temp)r(erature)g Fq(T)12 b Fr(\()p Fq(x)p Fr(\).)54 b(Bet)n(w)n(een)32 b(collisions)g(a)h(particle)g(mo)n (v)n(es)f(a)g(dis-)498 3063 y(tance)d Fq(\025)g Fr(carrying)e(a)i (kinetic)g(energy)f(prop)r(ortional)f(to)i Fq(T)12 b Fr(\()p Fq(x)p Fr(\))29 b(from)g Fq(x)g Fr(to)g Fq(x)20 b Fr(+)f Fq(\025=)3135 2994 y Fm(p)p 3204 2994 42 4 v 69 x Fr(3)o(,)30 b(while)498 3162 y(in)24 b(the)g(opp)r(osite)g (direction)f(the)h(amoun)n(t)g(carried)e(is)i(prop)r(ortional)e(to)i Fq(T)12 b Fr(\()p Fq(x)f Fr(+)g Fq(\025)3037 3093 y Fm(p)p 3105 3093 V 3105 3162 a Fr(3\).)36 b(T)-7 b(aking)498 3262 y(in)n(to)32 b(accoun)n(t)f(the)h(fact)g(that)g(the)g(sp)r(eed)g (is)f(prop)r(ortional)f(to)2536 3191 y Fm(p)p 2605 3191 61 4 v 71 x Fq(T)43 b Fr(the)32 b(amoun)n(t)f(of)h(energy)498 3361 y(transp)r(orted)27 b(p)r(er)g(unit)h(area)e(and)i(time)g(across)d (a)i(plane)g(p)r(erp)r(endicular)g(to)h(the)g Fq(x)p Fr(-axis)f Fq(J)35 b Fr(is)498 3461 y(appro)n(ximately)-7 b(,)1123 3662 y Fq(J)31 b Fm(\030)23 b Fq(\032)1331 3588 y Fm(p)p 1400 3588 V 74 x Fq(T)1474 3570 y Fj(h)1513 3662 y Fq(T)12 b Fr(\()p Fq(x)p Fr(\))18 b Fm(\000)g Fq(T)12 b Fr(\()p Fq(x)18 b Fr(+)h Fq(\025)2076 3590 y Fm(p)p 2145 3590 42 4 v 72 x Fr(3\))2219 3570 y Fj(i)2281 3662 y Fm(')k(\000)p Fq(\033)2484 3628 y Fk(\000)p Fp(2)2573 3588 y Fm(p)p 2642 3588 61 4 v 74 x Fq(T)2712 3606 y(dT)p 2712 3643 104 4 v 2718 3719 a(dx)2839 3662 y(;)519 b Fr(\(5\))498 3868 y(and)28 b(so)e Fq(\024)d Fm(\030)920 3797 y(p)p 989 3797 61 4 v 71 x Fq(T)39 b Fr(indep)r(enden)n(t)28 b(of)g Fq(\032)p Fr(,)g(in)f(agreemen)n(t)g(with)h(exp)r(erimen)n(t.) 623 3967 y(It)g(w)n(as)f(clear)g(to)h(the)h(founding)f(fathers)f(that)i (starting)e(with)i(a)e(lo)r(cal)h(equilibrium)g(situ-)498 4067 y(ation)f(\(corresp)r(onding)f(to)h(a)g(Maxw)n(ellian)f (distribution)i(of)f(v)n(elo)r(cities\))g(there)g(will)h(dev)n(elop,) 498 4166 y(as)e(time)h(go)r(es)f(on,)g(a)g(deviation)g(from)g(L)-7 b(TE.)26 b(They)h(reasoned)e(ho)n(w)n(ev)n(er)f(that)j(this)g (deviation)498 4266 y(from)i(lo)r(cal)f(equilibrium)h(will)g(b)r(e)h (small)e(when)h(\()p Fq(\025=T)12 b Fr(\))p Fq(dT)d(=dx)25 b(<<)g Fr(1,)k(the)g(regime)f(in)h(whic)n(h)498 4366 y(F)-7 b(ourier's)20 b(la)n(w)g(is)h(exp)r(ected)g(to)g(hold,)h(and)f (the)g(ab)r(o)n(v)n(e)f(calculation)g(should)g(yield,)j(up)e(to)g(some) 498 4465 y(factor)28 b(of)g(order)f(unit)n(y)-7 b(,)28 b(the)h(righ)n(t)e(heat)h(conductivit)n(y)-7 b(.)39 b(In)29 b(fact)f(if)h(one)e(computes)h(the)h(heat)498 4565 y(\015ux)h(at)g(a)g (p)r(oin)n(t)g Fq(x)g Fr(b)n(y)g(a)n(v)n(eraging)d(the)j(microscopic)f (energy)f(curren)n(t)h(at)h Fq(x)h(j)h Fr(=)26 b Fq(\032)p Fz(v)q Fr(\()3257 4532 y Fp(1)p 3258 4546 34 4 v 3258 4594 a(2)3301 4565 y Fq(mv)3417 4535 y Fp(2)3455 4565 y Fr(\))498 4665 y(o)n(v)n(er)33 b(the)i(one)g(particle)f(distribution) h(function)g Fq(f)9 b Fr(\()p Fz(r)p Fq(;)14 b Fz(v)q Fq(;)g(t)p Fr(\))36 b(then)f(it)g(is)g(only)f(the)i(deviation)498 4764 y(from)d(lo)r(cal)g(equilibrium)h(whic)n(h)f(mak)n(es)g(a)g(con)n (tribution.)54 b(The)34 b(result)f(ho)n(w)n(ev)n(er)e(is)j(essen-)498 4864 y(tially)28 b(the)g(same)g(as)f(Eq.\(5\).)38 b(This)28 b(w)n(as)f(sho)n(wn)g(b)n(y)h(Boltzmann)g(who)f(deriv)n(ed)h(an)f (accurate)498 4963 y(form)n(ula)e(for)f Fq(\024)i Fr(in)f(gases)f(b)n (y)h(using)g(the)h(Boltzmann)f(equation)g(to)g(compute)g Fq(\024)p Fr(.)36 b(If)26 b(one)f(tak)n(es)498 5063 y Fq(\024)31 b Fr(from)f(exp)r(erimen)n(t)h(the)g(ab)r(o)n(v)n(e)e (analysis)g(yields)i(a)f(v)-5 b(alue)31 b(for)f Fq(\033)s Fr(,)i(the)f(e\013ectiv)n(e)f(size)h(of)f(an)498 5163 y(atom)c(or)g(molecule,)h(whic)n(h)g(turns)f(out)h(to)g(b)r(e)g(close)f (to)g(other)h(determinations)f(of)g(the)i(c)n(har-)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(4)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 5 5 5 4 bop 498 637 a Fr(acteristic)35 b(size)g(of)h(an)f(atom)1438 607 y Fp(9)1475 637 y Fr(.)61 b(This)36 b(ga)n(v)n(e)d(evidence)j(for)f (the)h(realit)n(y)f(of)g(atoms)g(and)h(the)498 737 y(molecular)27 b(theory)f(of)i(heat.)623 837 y(Using)22 b(ideas)f(of)i(h)n(ydro)r (dynamical)d(space)i(and)g(time)h(scaling)e(describ)r(ed)h(earlier)f (it)h(is)h(p)r(os-)498 936 y(sible)28 b(to)f(deriv)n(e)g(a)g(con)n (trolled)g(expansion)f(for)h(the)i(solution)e(of)g(the)h(stationary)e (Boltzmann)498 1036 y(equation)38 b(describing)g(the)h(steady)f(state)g (of)h(a)f(gas)f(coupled)h(to)h(temp)r(erature)f(reserv)n(oirs)498 1136 y(at)32 b(the)g(top)f(and)h(b)r(ottom)1333 1105 y Fp(23)p Fn(;)p Fp(24)p Fn(;)p Fp(25)1575 1136 y Fr(.)49 b(The)32 b(coupling)f(is)h(implemen)n(ted)g(b)n(y)f(the)h(imp)r (osition)g(of)498 1235 y(\\Maxw)n(ell)i(b)r(oundary)h(conditions":)52 b(when)36 b(a)f(particle)g(hits)h(the)g(left)h(\(righ)n(t\))e(w)n(all)g (it)h(get)498 1335 y(re\015ected)27 b(with)i(a)e(distribution)g(of)h(v) n(elo)r(cities)1037 1532 y Fq(f)1078 1544 y Fn(\013)1125 1532 y Fr(\()p Fq(d)p Fz(v)q Fr(\))d(=)1518 1476 y Fq(m)1591 1446 y Fp(2)p 1406 1513 336 4 v 1406 1589 a Fr(2)p Fq(\031)s Fr(\()p Fq(k)s(T)1625 1601 y Fn(\013)1671 1589 y Fr(\))1703 1565 y Fp(2)1751 1532 y Fm(j)p Fq(v)1814 1544 y Fn(x)1856 1532 y Fm(j)14 b Fr(exp)2033 1415 y Fj(\024)2077 1532 y Fm(\000)2162 1476 y Fq(m)p Fz(v)2286 1446 y Fp(2)p 2152 1513 184 4 v 2152 1589 a Fr(2)p Fq(k)s(T)2289 1601 y Fn(\013)2345 1415 y Fj(\025)2403 1532 y Fq(d)p Fz(v)168 b Fq(\013)23 b Fr(=)g(1)p Fq(;)14 b Fr(2)432 b(\(6\))498 1747 y(corresp)r(onding)49 b(to)h(a)h(temp)r(erature)f Fq(T)1817 1759 y Fp(1)1904 1747 y Fr(\()p Fq(T)1985 1759 y Fp(2)2022 1747 y Fr(\))h(at)g(the)g(left)g(\(righ)n(t\))g(w)n(all.) 105 b(One)50 b(then)498 1847 y(sho)n(ws)710 1817 y Fp(23)p Fn(;)p Fp(24)p Fn(;)p Fp(25)988 1847 y Fr(that)36 b(for)g Fq(\017)i(<<)f Fr(1,)h Fq(\017)f Fr(b)r(eing)f(no)n(w)g(the)h(ratio)f Fq(\025=L)p Fr(,)i(the)f(Boltzmann)f(equa-)498 1946 y(tion)h(for)g Fq(f)46 b Fr(in)37 b(the)h(slab)e(has)h(a)g(time)h(indep)r(enden)n(t)g (solution)e(whic)n(h)h(is)g(close)g(to)g(a)g(lo)r(cal)498 2046 y(Maxw)n(ellian,)24 b(corresp)r(onding)e(to)i(L)-7 b(TE,)24 b(\(apart)f(from)h(b)r(oundary)g(la)n(y)n(er)e(terms\))i(with) h(a)f(lo)r(cal)498 2146 y(temp)r(erature)d(and)h(densit)n(y)g(giv)n(en) e(b)n(y)i(the)g(solution)f(of)h(the)g(Na)n(vier-Stok)n(es)d(equations)i (whic)n(h)498 2245 y(incorp)r(orates)h(F)-7 b(ourier's)23 b(la)n(w)g(as)h(expressed)e(in)j(Eq.\(2\).)35 b(The)24 b(main)g(mathematical)g(problem)498 2345 y(is)i(in)g(con)n(trolling)e (the)i(remainder)f(in)h(an)f(asymptotic)h(expansion)e(of)i Fq(f)35 b Fr(in)26 b(p)r(o)n(w)n(er)e(of)i Fq(\017)p Fr(.)36 b(This)498 2445 y(requires)25 b(that)i(the)f(macroscopic)f (temp)r(erature)h(gradien)n(t,)f Fo(i.e.)h Fm(j)p Fq(T)2618 2457 y Fp(1)2671 2445 y Fm(\000)16 b Fq(T)2801 2457 y Fp(2)2837 2445 y Fm(j)p Fq(=h)p Fr(,)26 b(where)g Fq(h)d Fr(=)g Fq(\017L)498 2544 y Fr(is)28 b(the)g(thic)n(kness)f(of)g(the)h (slab)f(on)h(the)g(macroscopic)d(scale,)i(b)r(e)h(small.)623 2644 y(Ev)n(en)g(if)i(this)g(apparen)n(tly)e(tec)n(hnical)h(problem)g (could)h(b)r(e)f(o)n(v)n(ercome)f(w)n(e)h(w)n(ould)g(still)h(b)r(e)498 2743 y(left)h(with)g(the)f(question)g(of)g(justifying)h(the)f (Boltzmann)g(equation)f(for)h(suc)n(h)g(steady)f(states)498 2843 y(and)h(of)h(course)e(it)i(w)n(ould)f(not)g(tell)h(us)f(an)n (ything)g(ab)r(out)g(dense)g(\015uids)h(or)e(crystals.)44 b(In)31 b(fact)498 2943 y(the)22 b(Boltzmann)f(equation)g(itself)h(is)g (really)e(closer)h(to)g(a)g(macroscopic)f(then)i(to)f(a)h(microscopic) 498 3042 y(description.)35 b(It)22 b(is)g(obtained)f(in)i(a)e(w)n(ell)h (de\014ned)g(kinetic)g(scaling)f(limit)i(in)f(whic)n(h)g(in)g(addition) 498 3142 y(to)28 b(rescaling)e(space)g(and)i(time)g(the)g(particle)f (densit)n(y)g(go)r(es)g(to)h(zero)2677 3112 y Fp(79)2746 3142 y Fr(,)f Fo(i.e.)h Fq(\025)c(>>)e(\033)s Fr(.)498 3383 y Fz(3)95 b(Heat)32 b(conduction)g(in)g(insulating)e(crystals)498 3569 y Fr(Excellen)n(t)g(accoun)n(ts)g(of)g(the)i(historical)d(dev)n (elopmen)n(t)h(of)h(the)g(theory)f(of)h(heat)f(conduction)498 3668 y(in)i(solids)f(exist)1030 3638 y Fp(72)p Fn(;)p Fp(51)1218 3668 y Fr(so)g(w)n(e)g(will)h(con)n(ten)n(t)f(ourselv)n(es)e (here)i(with)h(some)f(brief)h(remarks.)47 b(In)498 3768 y(\(electrically\))37 b(insulating)f(solids,)j(heat)e(is)g(transmitted) g(through)f(the)h(vibrations)f(of)h(the)498 3868 y(lattice)h(\(in)g (conductors)f(the)h(electronic)e(con)n(tribution)i(is)f(in)h(general)e (m)n(uc)n(h)i(larger)d(then)498 3967 y(the)30 b(con)n(tribution)e(due)i (to)f(the)g(lattice)g(vibrations\).)41 b(In)29 b(order)f(to)h(use)g (concepts)g(of)g(kinetic)498 4067 y(theory)-7 b(,)42 b(it)e(is)g(useful)g(to)g(picture)f(a)h(solid)f(as)g(a)g(gas)g(of)g (phonons)g(whic)n(h)h(can)f(store)g(and)498 4166 y(transmit)33 b(heat.)52 b(In)33 b(a)g(p)r(erfectly)g(harmonic)f(crystal,)h(the)g (phonons)f(b)r(eha)n(v)n(e)g(lik)n(e)h(a)f(gas)g(of)498 4266 y(nonin)n(teracting)g(particles)h(and)h(therefore)e(the)i(thermal) f(curren)n(t)g(will)h(not)f(decrease)f(with)498 4366 y(the)f(length)g(of)f(the)h(crystal)e(placed)i(b)r(et)n(w)n(een)f(t)n (w)n(o)g(thermal)g(reserv)n(oirs.)42 b(Th)n(us)31 b(a)f(p)r(erfectly) 498 4465 y(harmonic)i(crystal)h(has)f(an)h(in\014nite)h(thermal)f (conductivit)n(y:)48 b(in)33 b(the)h(language)e(of)h(kinetic)498 4565 y(theory)d Fq(\033)h Fr(=)c(0)j(and)g(the)h(mean)f(free)g(path)g Fq(\025)h Fr(is)f(in\014nite.)46 b(A)31 b(real)e(crystal)g(is)i(not)f (harmonic)498 4665 y(and,)21 b(in)f(the)h(phonon)e(picture,)i(an)n(y)e (thermal)h(curren)n(t)f(will)h(b)r(e)g(degraded)f(b)n(y)g(the)h (anharmonic)498 4764 y(forces)37 b(in)i(the)f(lattice.)69 b(Another)38 b(source)f(of)h(\014nite)h(thermal)f(conductivit)n(y)g(ma) n(y)f(b)r(e)i(the)498 4864 y(lattice)e(imp)r(erfections)f(and)g (impurities)h(whic)n(h)f(will)g(scatter)g(the)h(phonons)e(and)h (degrade)498 4963 y(the)28 b(thermal)f(curren)n(t)g(to)r(o.)623 5063 y(Deb)n(y)n(e)887 5033 y Fp(18)998 5063 y Fr(devised)41 b(a)g(kind)g(of)h(kinetic)f(theory)g(for)f(phonons)h(in)g(order)f(to)h (describ)r(e)498 5163 y(thermal)22 b(conductivit)n(y)-7 b(.)35 b(One)21 b(assumes)g(that)h(a)g(small)f(gradien)n(t)g(of)h(temp) r(erature)f(is)h(imp)r(osed)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(5)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 6 6 6 5 bop 498 628 a Fr(and)23 b(that)g(the)g(collisions)e(b)r(et)n(w)n (een)i(phonons)f(main)n(tain)g(lo)r(cal)g(equilibrium.)36 b(An)23 b(elemen)n(tary)498 728 y(argumen)n(t)868 698 y Fp(2)931 728 y Fr(giv)n(es)i(a)h(thermal)f(conductivit)n(y)h (analogous)e(to)i(Eq.\(5\))f(obtained)h(in)g(Section)g(2)498 827 y(for)k(gases,)f(\(remem)n(b)r(ering)g(ho)n(w)n(ev)n(er)f(that)i (the)g(densit)n(y)g(of)g(phonon)g(is)f(itself)i(a)f(function)g(of)498 927 y Fq(T)12 b Fr(\))1798 1080 y Fq(\024)23 b Fm(\030)f Fq(c)1992 1092 y Fn(v)2032 1080 y Fq(c)2068 1046 y Fp(2)2105 1080 y Fq(\034)h(:)1194 b Fr(\(7\))498 1233 y(Comparing)32 b(Eq.\(7\))h(and)h(Eq.\(5\))f(w)n(e)g(see)g(that)h Fq(\032)f Fr(has)g(b)r(een)h(replaced)f(b)n(y)g Fq(c)3004 1245 y Fn(v)3077 1233 y Fr(the)h(sp)r(eci\014c)498 1332 y(heat)g(of)f(the)h (phonons,)1296 1262 y Fm(p)p 1366 1262 61 4 v 1366 1332 a Fq(T)44 b Fr(b)n(y)34 b Fq(c)f Fr(the)h(\(mean\))g(v)n(elo)r(cit)n(y) f(of)g(the)h(phonons,)h(and)e Fq(\025)h Fr(b)n(y)f Fq(c\034)9 b Fr(,)498 1432 y(where)35 b Fq(\034)45 b Fr(is)35 b(the)h(e\013ectiv)n (e)f(mean)g(free)g(time)g(b)r(et)n(w)n(een)h(phonon)e(collisions.)59 b(The)35 b(thermal)498 1532 y(conductivit)n(y)23 b(dep)r(ends)h(on)f (the)h(temp)r(erature)f(via)g Fq(\034)33 b Fr(and)23 b(a)g(more)g(re\014ned)g(theory)g(is)g(needed)498 1631 y(to)38 b(accoun)n(t)f(for)g(this)h(dep)r(endence.)68 b(P)n(eierls)2021 1601 y Fp(71)2129 1631 y Fr(used)38 b(a)f(Boltzmann)g(t)n(yp)r(e)h(equation)g(for)498 1731 y(phonons)18 b(to)h(in)n(v)n(estigate)f(this)h(problem.)33 b(The)19 b(P)n(eierls)f(theory)g(singles)g(out)h(one)f(phenomenon)498 1831 y(whic)n(h)29 b(giv)n(es)g(rise)f(to)h(a)g(\014nite)h(thermal)f (conductivit)n(y)2246 1800 y Fp(72)2316 1831 y Fr(.)43 b(The)29 b(momen)n(tum)h(of)f(phonons)g(in)498 1930 y(collisions)39 b(is)h(conserv)n(ed)e(only)h(mo)r(dulo)h(a)g(v)n(ector)e(of)i(the)h (recipro)r(cal)d(lattice.)74 b(One)39 b(can)498 2030 y(therefore)25 b(classify)g(the)h(collisions)f(of)h(phonons)f(in)n(to)g (t)n(w)n(o)g(classes:)35 b(the)26 b(ones)f(where)g(phonon)498 2129 y(momen)n(tum)j(is)f(conserv)n(ed)e(\(the)j(normal)f(pro)r (cesses\))f(and)h(the)h(ones)e(where)h(the)h(initial)f(and)498 2229 y(\014nal)d(momen)n(ta)g(di\013er)g(b)n(y)g(a)g(non-zero)e (recipro)r(cal)h(lattice)h(v)n(ector)f(\(the)i(umklap)f(pro)r (cesses\).)498 2329 y(P)n(eierls)37 b(theory)g(ma)n(y)h(b)r(e)g (summarized)g(\(v)n(ery)f(roughly\))h(as)f(follo)n(ws:)57 b(in)39 b(the)g(absence)e(of)498 2428 y(umklap)k(pro)r(cesses)e(the)i (mean)g(free)g(path)g(and)f(th)n(us)h(the)g(thermal)g(conductivit)n(y)f (of)h(an)498 2528 y(insulating)27 b(solid)h(is)f(in\014nite.)623 2628 y(A)f(success)g(of)g(P)n(eierls)f(theory)g(is)h(to)g(describ)r(e)g (correctly)f(the)i(temp)r(erature)f(dep)r(endence)498 2727 y(of)31 b(the)h(thermal)f(conductivit)n(y)1537 2697 y Fp(2)1574 2727 y Fr(.)48 b(F)-7 b(urthermore,)31 b(on)g(the)g(basis)g (of)g(this)g(theory)-7 b(,)32 b(one)e(do)r(es)498 2827 y(not)22 b(exp)r(ect)g(a)f(\014nite)h(thermal)g(conductivit)n(y)f(in)h (1-dimensional)f(mono-atomic)f(lattices)h(with)498 2926 y(pair)27 b(in)n(teractions:)36 b(this)28 b(seems)f(so)g(far)g(a)g (correct)f(prediction,)h(see)h(Section)f(10.)623 3026 y(The)21 b(justi\014cation)h(of)f(the)h(Boltzmann)g(equation)e(for)h (phonons)g(has)g(b)r(een)h(questioned)3394 2996 y Fp(46)3464 3026 y Fr(.)498 3126 y(V)-7 b(arious)19 b(alternativ)n(e)h(mec)n (hanisms)f(ha)n(v)n(e)g(b)r(een)i(prop)r(osed)f(whic)n(h)g(w)n(ould)g (giv)n(e)f(rise)h(to)g(a)g(\014nite)498 3225 y(thermal)27 b(conductivit)n(y)-7 b(,)27 b(but)g(it)h(seems)e(fair)g(to)h(sa)n(y)f (that,)h(so)g(far,)f(no)h(b)r(etter)g(theory)f(of)h(heat)498 3325 y(conduction)40 b(in)g(insulators)e(has)h(b)r(een)i(prop)r(osed.) 72 b(As)40 b(P)n(eierls)e(himself)i(puts)g(it)3229 3295 y Fp(72)3299 3325 y Fr(:)61 b(\\It)498 3425 y(seems)24 b(there)g(is)g(no)f(problem)h(in)g(mo)r(dern)g(ph)n(ysics)f(for)h(whic) n(h)g(there)g(are)f(on)h(record)e(as)i(man)n(y)498 3524 y(false)29 b(starts,)g(and)h(as)e(man)n(y)h(theories)g(whic)n(h)g(o)n (v)n(erlo)r(ok)e(some)i(essen)n(tial)f(feature,)i(as)f(in)h(the)498 3624 y(problem)d(of)h(the)g(thermal)f(conductivit)n(y)g(of)h ([electrically])f(non-conducting)f(crystals".)623 3723 y(T)-7 b(o)31 b(\014nd)h(a)f(mathematical)g(description)g(of)h(thermal) f(conduction)h(in)f(crystals)g(w)n(e)g(need)498 3823 y(to)g(sp)r(ecify)h(the)g(Hamiltonian)f(of)g(the)h(system)f(or)g(at)g (least)g(some)g(appropriately)e(idealized)498 3923 y(v)n(ersion)g(of)i (it.)46 b(A)31 b(mo)r(del)g(crystal)e(is)h(c)n(haracterized)f(b)n(y)h (the)h(fact)g(that)g(all)f(atoms)g(oscillate)498 4022 y(around)23 b(giv)n(en)g(equilibrium)h(p)r(ositions.)36 b(The)24 b(equilibrium)g(p)r(ositions)f(can)h(b)r(e)g(though)n(t)g(of)g (as)498 4135 y(the)j(p)r(oin)n(ts)g(of)f(a)h(regular)e(lattice)h(in)h Fi(R)1752 4087 y Fn(d)1791 4135 y Fr(.)37 b(F)-7 b(or)26 b(simplicit)n(y)h(w)n(e)f(will)h(assume)f(that)h(the)g(lattice)498 4248 y(is)e(simply)f Fi(Z)908 4200 y Fn(d)946 4248 y Fr(.)36 b(Although)25 b Fq(d)e Fr(=)g(3)h(is)g(the)h(ph)n(ysical)f (situation)g(one)g(can)g(b)r(e)h(in)n(terested)f(also)g(in)498 4348 y(the)h(case)f Fq(d)f Fr(=)g(1)p Fq(;)14 b Fr(2.)35 b(\(The)25 b Fq(d)e Fr(=)g(1)h(system)g(ma)n(y)g(sho)n(w)g(\014nite)h (thermal)f(conductivit)n(y)h(without)498 4447 y(violating)32 b(the)h(P)n(eierls)e(criteria)g(if)j(w)n(e)e(admit)h(one)f(particle,)h (non)g(momen)n(tum)g(conserving)498 4547 y(in)n(teractions.\))623 4665 y(Let)38 b(\003)j Fm(\032)g Fi(Z)1054 4616 y Fn(d)1131 4665 y Fr(b)r(e)e(a)f(\014nite)h(set)g(and)f(denote)h(b)n(y)f Fq(N)47 b Fr(its)39 b(cardinalit)n(y)-7 b(.)69 b(Eac)n(h)37 b(atom)h(is)498 4764 y(iden)n(ti\014ed)33 b(b)n(y)f(its)h(p)r(osition)f Fz(x)1477 4776 y Fh(i)1533 4764 y Fr(=)f Fz(i)21 b Fr(+)g Fz(q)1813 4776 y Fh(i)1872 4764 y Fr(where)31 b Fz(i)g Fm(2)g Fr(\003)h(is)h(the)f(equilibrium)h(p)r(osition)f(and)498 4864 y Fz(q)548 4876 y Fh(i)605 4864 y Fm(2)g Fz(R)764 4834 y Fn(d)835 4864 y Fr(is)g(the)h(displacemen)n(t)f(of)g(the)h (particle)f(at)h(lattice)f(site)h Fz(i)f Fr(from)g(this)h(equilibrium) 498 4963 y(p)r(osition,)f(and)e(w)n(e)h(denote)g(b)n(y)f Fz(p)1574 4975 y Fh(i)1631 4963 y Fr(its)h(momen)n(tum)g(and)g Fq(m)g Fr(its)g(mass.)46 b(Since)31 b(in)n(ter)f(atomic)498 5063 y(forces)35 b(in)h(real)e(solids)h(ha)n(v)n(e)g(short)g(range,)h (it)g(is)g(reasonable)d(to)j(assume)f(that)h(the)g(atoms)498 5163 y(in)n(teract)26 b(only)f(with)i(their)f(nearest)f(neigh)n(b)r (ors)g(via)h(a)g(p)r(oten)n(tial)g(that)g(dep)r(ends)h(only)f(on)g(the) p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(6)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 7 7 7 6 bop 498 628 a Fr(relativ)n(e)27 b(distance)g(with)h(resp)r(ect)f (to)h(the)g(equilibrium)f(distance.)623 728 y(As)36 b(already)f(noted)h (it)h(is)f(useful)h(to)f(allo)n(w)f(an)h(external)f(con\014ning)h(1-b)r (o)r(dy)g(p)r(oten)n(tial)498 827 y(whic)n(h)i(breaks)f(the)h (translation)e(in)n(v)-5 b(ariance.)67 b(Accordingly)37 b(the)h(Hamiltonians)f(that)h(w)n(e)498 927 y(consider)27 b(ha)n(v)n(e)f(the)i(general)e(form)638 1151 y Fq(H)7 b Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))38 b(=)1075 1072 y Fj(X)1079 1251 y Fh(i)p Fk(2)p Fp(\003)1231 1095 y Fz(p)1284 1064 y Fp(2)1284 1118 y Fh(i)p 1219 1132 115 4 v 1219 1208 a Fr(2)p Fq(m)1362 1151 y Fr(+)1495 1072 y Fj(X)1445 1254 y Fk(j)p Fh(i)p Fk(\000)p Fh(j)p Fk(j)p Fp(=1)1678 1151 y Fq(V)19 b Fr(\()p Fz(q)1827 1163 y Fh(i)1871 1151 y Fm(\000)f Fz(q)2004 1163 y Fh(j)2032 1151 y Fr(\))h(+)2166 1072 y Fj(X)2215 1251 y Fh(i)2300 1151 y Fq(U)2357 1163 y Fh(i)2382 1151 y Fr(\()p Fz(q)2464 1163 y Fh(i)2490 1151 y Fr(\))k(=)2646 1072 y Fj(X)2651 1251 y Fh(i)p Fk(2)p Fp(\003)2802 1095 y Fz(p)2855 1064 y Fp(2)2855 1118 y Fh(i)p 2790 1132 V 2790 1208 a Fr(2)p Fq(m)2933 1151 y Fr(+)18 b Fm(V)7 b Fr(\()p Fq(Q)p Fr(\))14 b Fq(;)140 b Fr(\(8\))498 1423 y(where)21 b Fq(P)35 b Fr(=)22 b(\()p Fz(p)992 1435 y Fh(i)1018 1423 y Fr(\))1050 1435 y Fh(i)p Fk(2)p Fp(\003)1187 1423 y Fr(and)f(analogously)e(for)i Fq(Q)p Fr(.)35 b(W)-7 b(e)21 b(shall)g(further)g(assume)g(that)h(as)e Fm(j)p Fz(q)p Fm(j)k(!)f(1)498 1523 y Fr(so)33 b(do)g Fq(U)784 1535 y Fh(i)809 1523 y Fr(\()p Fz(q)p Fr(\))h(and)g Fq(V)18 b Fr(\()p Fz(q)p Fr(\).)56 b(The)33 b(addition)g(of)h Fq(U)2051 1535 y Fh(i)2076 1523 y Fr(\()p Fz(q)p Fr(\))g(pins)g(do)n (wn)f(the)g(crystal)g(and)g(ensures)498 1622 y(that)19 b(exp)13 b([)q Fm(\000)p Fq(\014)t(H)7 b Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\)])k(is)h(in)n(tegrable)e(with)i(resp)r(ect)f(to)g Fq(dP)12 b(dQ)19 b Fr(and)f(th)n(us)g(the)h(corresp)r(onding)498 1722 y(Gibbs)29 b(measure)e(is)h(w)n(ell)g(de\014ned.)40 b(Observ)n(e)27 b(that)h(for)g(man)n(y)g(purp)r(ose)f(it)i(is)f(enough) g(to)g(put)498 1821 y(the)c(p)r(oten)n(tial)f Fq(U)1040 1833 y Fh(i)1088 1821 y Fr(on)g(only)g(some)g(of)g(the)g(atoms,)h Fo(e.g.)f Fr(the)g(ones)g(on)g(the)g(b)r(oundary)g(of)g(\003)3308 1791 y Fp(75)p Fn(;)p Fp(67)3464 1821 y Fr(.)498 1934 y(W)-7 b(e)36 b(note)g(\014nally)g(that)g(when)g(\003)g Fm(\032)g Fi(Z)1777 1886 y Fn(d)1851 1934 y Fr(one)g(can)f(still)h (consider)f(that)h Fz(p)2915 1946 y Fh(i)2976 1934 y Fr(and)g Fz(q)3196 1946 y Fh(i)3259 1934 y Fm(2)h Fi(R)3423 1886 y Fn(\027)3464 1934 y Fr(,)498 2047 y Fq(\027)28 b Fm(6)p Fr(=)23 b Fq(d)p Fr(,)28 b(but)g(w)n(e)g(will)f(generally)f (assume)h(that)h Fz(q)2050 2059 y Fh(i)2099 2047 y Fm(2)23 b Fi(R)2249 1999 y Fn(d)2288 2047 y Fr(.)498 2147 y Fz(Remark)p Fr(:)58 b(While)40 b(the)f(Hamiltonian)f(in)i(Eq.\(8\))e(lo)r(oks)g (similar)g(to)g(that)h(in)g(Eq.\(3\))g(the)498 2246 y(meaning)g(and)g (domain)g(of)h(the)f Fq(Q)g Fr(v)-5 b(ariables)39 b(is)g(en)n(tirely)g (di\013eren)n(t.)72 b(In)40 b(a)f(\015uid)h(all)f(the)498 2346 y(particle)26 b(are)g(iden)n(tical)h(and)g(the)g(particle)f(with)h (lab)r(el)g Fq(i)g Fr(in)n(teracts)f(with)h(an)n(y)f(other)h(particle) 498 2459 y(whose)j(p)r(osition)g Fz(q)1115 2471 y Fn(j)1181 2459 y Fr(is)g(close)f(to)i Fz(q)1623 2471 y Fn(i)1651 2459 y Fr(,)g Fz(q)1755 2471 y Fn(l)1808 2459 y Fm(2)d Fr(\003)g Fm(\032)f Fi(R)2141 2411 y Fn(d)2180 2459 y Fr(,)k Fq(l)e Fr(=)e(1)p Fq(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fr(.)45 b(The)30 b(pair)g(in)n(teraction)498 2559 y(p)r(oten)n(tial) 22 b Fq(\036)p Fr(\()p Fz(q)p Fr(\))i(is)e(of)g(\014nite)g(range,)g Fo(e.g.)g Fr(hard)f(balls,)i(or)e(deca)n(ys)g(rapidly)g(with)i (distance.)35 b(F)-7 b(or)498 2676 y(the)30 b(crystal)f(in)h(Eq.\(8\))f Fz(q)1324 2688 y Fn(i)1382 2676 y Fr(is)g(the)i Fl(deviation)g Fr(from)f(an)f(equilibrium)h(p)r(osition)f Fz(i)d Fm(2)h Fr(\003)g Fm(\032)f Fi(Z)3425 2628 y Fn(d)3464 2676 y Fr(,)498 2776 y(etc.)498 3028 y Fz(4)95 b(Microscopic)32 b(mo)s(dels)c(of)k(heat)g(reserv)m(oirs)498 3216 y Fr(T)-7 b(o)27 b(pro)r(duce)h(a)f(stationary)f(heat)h(\015o)n(w)g(in)h(a)f (system,)h(b)r(e)g(it)g(a)f(gas)g(or)f(a)i(crystal,)e(the)i(system)498 3315 y(m)n(ust)20 b(b)r(e)h(coupled)f(to)g(at)g(least)g(t)n(w)n(o)f (heat)h(reserv)n(oirs)d(at)j(di\013eren)n(t)h(temp)r(eratures.)33 b(A)21 b(ph)n(ysical)498 3415 y(coupling)i(is)f(one)h(whic)n(h)g(acts)f (only)h(at)g(the)g(b)r(oundary)f(of)h(the)h(system)e(lea)n(ving)g(the)h (dynamics)498 3515 y(in)28 b(the)g(bulk)g(purely)f(Hamiltonian.)623 3614 y(Since)k(a)f(realistic)h(description)f(of)h(heat)g(reserv)n(oirs) d(and)j(coupling)f(is)h(out)g(of)g(the)h(ques-)498 3714 y(tion)g(v)-5 b(arious)31 b(mo)r(del)h(reserv)n(oirs)d(ha)n(v)n(e)h(b)r (een)i(used)g(in)g(analytical)f(and)h(n)n(umerical)f(studies.)498 3814 y(W)-7 b(e)32 b(giv)n(e)e(here)h(some)f(examples)h(whic)n(h)g (will)g(b)r(e)h(used)f(later)f(\(other)h(c)n(hoices)f(are)g(of)h (course)498 3913 y(p)r(ossible\).)36 b(The)27 b(exp)r(ectation)e(is)h (of)g(course)f(that)h(the)h(di\013eren)n(t)f(mo)r(dels)g(will)g(giv)n (e)f(the)i(same)498 4013 y(b)r(eha)n(vior)j(a)n(w)n(a)n(y)g(from)h(the) h(b)r(oundary)e(when)i(the)g(system)f(is)g(macroscopic.)47 b(This)31 b(has)g(not)498 4112 y(b)r(een)d(pro)n(v)n(en)e(in)i(an)n(y)f (example,)g(see)g([32,33])498 4356 y Fl(4.1)86 b(Sto)l(chastic)31 b(r)l(eservoirs.)498 4515 y Fr(W)-7 b(e)31 b(ha)n(v)n(e)f(already)f (discussed)i(one)f(suc)n(h)h(mo)r(del)g(of)g(reserv)n(oirs)c(commonly)k (used)f(for)h(\015uids)498 4614 y(in)d(Section)f(2.)37 b(This)27 b(corresp)r(onds)e(to)j(Maxw)n(ell)e(b)r(oundary)h(condition) g(discussed)g(in)g(Eq.\(4\))498 4732 y(for)j(a)g(gas)f(in)i(a)f (rectangular)f(slab.)45 b(More)29 b(generally)g(giv)n(en)h(a)g(\015uid) h(in)f(a)g(domain)g(\003)e Fm(\032)f Fi(R)3448 4684 y Fn(d)498 4832 y Fr(a)i(particle)f(hitting)i(the)f(w)n(all)f(of)h(the)g (con)n(tainer)f(con\014ning)h(the)g(system)g(at)g(a)f(p)r(oin)n(t)h Fz(r)d Fm(2)g Fq(@)5 b Fr(\003)498 4931 y(will)28 b(b)r(ounce)g(bac)n (k)e(in)n(to)i(\003)f(with)h(a)f(Maxw)n(ellian)g(distribution)g(of)h (momen)n(ta)1349 5139 y Fq(f)1390 5151 y Fh(r)1425 5139 y Fr(\()p Fq(d)p Fz(p)p Fr(\))c(=)1711 5083 y Fq(\014)t Fr(\()p Fz(r)p Fr(\))1865 5053 y Fp(2)p 1707 5120 202 4 v 1707 5196 a Fr(2)p Fq(\031)s(m)1872 5172 y Fp(2)1919 5139 y Fz(p)18 b Fm(\001)23 b Fr(^)-46 b Fq(n)o Fr(\()p Fz(r)p Fr(\))p Fq(e)2223 5105 y Fk(\000)p Fn(\014)s Fp(\()p Fh(r)p Fp(\))2415 5077 y Fg(p)2450 5060 y Ff(2)p 2409 5092 80 4 v 2409 5125 a(2)p Fe(m)2503 5139 y Fq(d)p Fz(p)14 b Fq(;)745 b Fr(\(9\))p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(7)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 8 8 8 7 bop 498 637 a Fr(where)32 b(^)-46 b Fq(n)p Fr(\()p Fz(r)p Fr(\))29 b(is)f(the)g(in)n(w)n(ard)f(directed)h(unit)h(v)n (ector)d(normal)h(to)h Fq(@)5 b Fr(\003)28 b(at)g Fz(r)g Fr(and)g Fq(\014)3066 607 y Fk(\000)p Fp(1)3156 637 y Fr(\()p Fz(r)p Fr(\))h(is)f(the)498 737 y(preassigned)e(temp)r(erature) h(at)h Fz(r)p Fr(.)623 837 y(F)-7 b(or)22 b(solids,)i(whic)n(h)f(are)g (usually)f(not)i(con\014ned)f(to)g(an)n(y)g(\014xed)g(spatial)g(region) f(b)n(y)h(external)498 936 y(w)n(alls,)34 b(it)g(is)f(sometimes)g (mathematically)g(con)n(v)n(enien)n(t)f(to)i(use)f(Langevin)f(t)n(yp)r (e)i(reserv)n(oirs)498 1036 y(whic)n(h)24 b(act)f(on)h(the)g(atoms)f (at)h(the)g(\\edge")e(of)i(the)g(crystal.)35 b(F)-7 b(or)23 b(de\014niteness)h(w)n(e)f(will)h(c)n(ho)r(ose)498 1136 y(\003)29 b(to)g(b)r(e)h(a)f(c)n(hain)f(of)i(particles)e(or)g(a)h (parallelepip)r(ed)g(in)g(higher)g(dimension)g(\(with)h(suitable)498 1253 y(b)r(oundary)22 b(conditions\),)h(\003)f(=)h Fm(f)p Fz(i)f Fm(2)h Fi(Z)1719 1205 y Fn(d)1771 1253 y Fr(;)28 b(1)23 b Fm(\024)f Fq(i)2003 1265 y Fn(k)2067 1253 y Fm(\024)h Fq(N)2222 1265 y Fn(k)2276 1253 y Fq(;)28 b Fr(1)22 b Fm(\024)h Fq(k)j Fm(\024)c Fq(d)p Fm(g)p Fr(.)35 b(W)-7 b(e)23 b(assume)e(that)i(the)498 1353 y(particles)28 b(at)h(the)g(\\left")f(b)r(oundary)g Fm(f)p Fz(i)c Fm(2)h Fr(\003)14 b(;)27 b Fq(i)2007 1365 y Fp(1)2069 1353 y Fr(=)e(1)p Fm(g)i Fr(are)h(coupled)h(to)f(a)h(heat)f(reserv)n(oir)e(at) 498 1452 y(temp)r(erature)j Fq(T)1022 1464 y Fn(L)1101 1452 y Fr(and)g(that)g(the)h(particles)e(at)h(the)h(\\righ)n(t)e(")h(b) r(oundary)f Fm(f)p Fz(i)d Fm(2)h Fr(\003)14 b(;)27 b Fq(i)3188 1464 y Fp(1)3251 1452 y Fr(=)e Fq(N)3408 1464 y Fp(1)3445 1452 y Fm(g)498 1552 y Fr(are)31 b(coupled)g(to)h(a)f(heat) h(reserv)n(oir)d(at)i(temp)r(erature)h Fq(T)2291 1564 y Fn(R)2345 1552 y Fr(.)49 b(W)-7 b(e)32 b(set)g Fm(L)e(\021)f Fq(N)2946 1564 y Fp(1)3015 1552 y Fr(the)j(length)g(of)498 1652 y(the)c(crystal)f(and)g Fq(A)c Fm(\021)g Fq(N)1313 1664 y Fp(2)1364 1652 y Fm(\001)14 b(\001)g(\001)g Fq(N)1542 1664 y Fn(\027)1610 1652 y Fr(its)28 b(cross)e(section.)623 1751 y(F)-7 b(or)23 b(the)h(particles)f(at)h(the)g(b)r(oundary)f(of)h (the)g(crystal)f(in)h(con)n(tact)f(with)i(a)e(heat)h(reserv)n(oir,)498 1851 y(the)29 b(Hamiltonian)e(equations)h(of)g(motion)f(are)h(mo)r (di\014ed)g(b)n(y)g(the)g(addition)g(of)g(an)g(Ornstein-)498 1951 y(Uehlen)n(b)r(ec)n(k)g(pro)r(cess:)1080 2103 y Fq(m)1153 2115 y Fh(i)1193 2102 y Fr(_)1178 2103 y Fz(p)1231 2115 y Fh(i)1295 2103 y Fr(=)39 b Fm(\000r)1533 2115 y Fh(q)1573 2124 y Fg(i)1599 2103 y Fm(V)7 b Fr(\()p Fz(q)p Fr(\))19 b Fm(\000)f Fq(\025)1921 2115 y Fn(\013)1968 2103 y Fz(p)2021 2115 y Fh(i)2047 2103 y Fq(=m)2162 2115 y Fh(i)2205 2103 y Fr(+)g(\(2)p Fq(\025)2410 2115 y Fn(\013)2457 2103 y Fq(T)2506 2115 y Fn(\013)2553 2103 y Fr(\))2585 2069 y Fp(1)p Fn(=)p Fp(2)2690 2103 y Fq(\030)2726 2115 y Fn(\013)2773 2103 y Fr(\()p Fq(t)p Fr(\))c Fq(:)435 b Fr(\(10\))498 2255 y(In)23 b(Eq.\(10\),)g Fq(\013)h Fm(2)f(f)p Fq(L;)14 b(R)q Fm(g)21 b Fr(are)h(indices)h(of)f(the)i (reserv)n(oirs,)c Fq(\025)2400 2267 y Fn(\013)2471 2255 y Fr(describ)r(es)i(the)h(strength)g(of)g(the)498 2355 y(coupling)i(to)h(the)g(reserv)n(oir)c(with)k(temp)r(erature)g Fq(T)2112 2367 y Fn(\013)2184 2355 y Fr(of)g(the)g(reserv)n(oir,)d(and) i Fq(\030)2972 2367 y Fn(\013)3020 2355 y Fr(\()p Fq(t)p Fr(\))h(is)f(a)h(white)498 2455 y(noise,)d Fo(i.e.)f Fr(,)i(a)d(Gaussian)h(random)f(pro)r(cesses)g(with)h(co)n(v)-5 b(ariance)21 b Fm(h)p Fq(\030)2616 2467 y Fn(\013)2663 2455 y Fr(\()p Fq(t)p Fr(\))p Fq(\030)2793 2467 y Fn(\014)2839 2455 y Fr(\()p Fq(s)p Fr(\))p Fm(i)j Fr(=)f Fq(\016)3123 2467 y Fn(\013\014)3210 2455 y Fq(\016)s Fr(\()p Fq(t)8 b Fm(\000)g Fq(s)p Fr(\).)498 2554 y(The)24 b(form)g(of)g(the)h(co)r (e\016cien)n(ts)e(is)h(c)n(hosen)g(so)f(that)i(the)f(dynamics)g (satisfy)g(detailed)g(balance.)498 2654 y(This)39 b(implies)h(in)f (particular)f(that)h(if)h(the)f(system)g(is)g(coupled)g(to)g(a)g (single)g(reserv)n(oir)d(at)498 2753 y(temp)r(erature)26 b Fq(T)12 b Fr(,)25 b(then)i(the)g(Gibbs)f(measure)f(with)i(densit)n(y) f Fq(Z)2506 2723 y Fk(\000)p Fp(1)2608 2753 y Fr(exp)2749 2686 y Fj(\000)2787 2753 y Fm(\000)p Fq(T)2913 2723 y Fk(\000)p Fp(1)3001 2753 y Fq(H)7 b Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))3299 2686 y Fj(\001)3363 2753 y Fr(is)26 b(a)498 2853 y(stationary)g(state)i(of)f(the)h(system.)623 2953 y(With)c(an)n(y)e(suc)n(h)h(a)g(c)n(hoice)f(of)h(sto)r(c)n(hastic) g(pro)r(cess)f(to)h(mo)r(del)g(the)h(reserv)n(oirs)c(the)j(dynam-)498 3052 y(ics)32 b(is)h(describ)r(ed)f(b)n(y)g(a)g(stationary)f(Mark)n(o)n (v)f(pro)r(cess)i(in)g(the)h(phase)f(space)g(of)g(the)h(system)498 3152 y(\012.)498 3371 y Fl(4.2)86 b(Hamiltonian)31 b(r)l(eservoirs.)498 3525 y Fr(In)c(this)h(case)e(the)i(reserv)n(oirs)c(themselv)n(es)i(are) h(mo)r(deled)g(b)n(y)g(in\014nite)h(Hamiltonian)e(systems)498 3624 y(and)c(the)g(full)g(system)g(consisting)f(of)g(reserv)n (oirs+system)d(is)k(Hamiltonian)2871 3594 y Fn(a)2911 3624 y Fr(.)35 b(An)22 b(alternativ)n(e)498 3724 y(but)33 b(equiv)-5 b(alen)n(t)32 b(p)r(oin)n(t)g(of)g(view)g(is)f(to)h(start)g (with)g(an)g(in\014nite)h(system)f(and)g(to)f(consider)g(a)498 3824 y(\014nite)j(subsystem)g(of)f(it)h(as)f(the)h(system)g(and)f(the)h (remaining)f(part)g(as)g(the)h(reserv)n(oirs.)52 b(A)498 3923 y(non-equilibrium)25 b(situation)f(is)h(obtained)g(b)n(y)g(c)n(ho) r(osing)e(suitable)i(initial)h(conditions)e(for)h(the)498 4023 y(part)33 b(of)f(the)i(total)e(system)h(whic)n(h)g(describ)r(es)f (the)h(reserv)n(oirs,)f Fo(e.g.)g Fr(the)h(initial)g(conditions)498 4123 y(of)28 b(the)g(reserv)n(oirs)c(are)j(assumed)g(to)g(b)r(e)h (distributed)g(according)e(to)h(a)g(Gibbs)h(measure)e(with)498 4222 y(corresp)r(onding)g(temp)r(eratures)1511 4192 y Fn(b)1543 4222 y Fr(.)623 4322 y(Studied)31 b(in)f([77,80,69])d(the)k (simplest)f(v)n(ersion)f(of)h(suc)n(h)g(a)f(total)h(system)g(consists)f (of)h(an)498 4421 y(in\014nite)24 b(c)n(hain.)34 b(The)23 b(\\left")f(reserv)n(oir)e(consists)i(of)h(the)g(particles)e(with)i (lab)r(els)g(in)g(\()p Fm(\0001)p Fq(;)14 b Fm(\000)p Fq(N)9 b Fr(])498 4521 y(and)24 b(the)h(righ)n(t)f(reserv)n(oir)e (consists)h(of)i(the)f(particles)g(with)h(lab)r(els)f(in)h([)p Fq(N)t(;)14 b Fr(+)p Fm(1)p Fr(\).)36 b(The)24 b(system)p 498 4615 1196 4 v 498 4663 a Fd(a)536 4687 y Fw(A)32 b(series)f(of)h(results)g(has)h(b)r(een)g(obtained)g(for)f(systems)f (coupled)j(to)e(a)h(single)f(reserv)n(oir,)h(suc)n(h)g(as)f(non-)498 4765 y(relativistic)25 b(atoms)f(coupled)i(to)g(the)g(quan)n(tized)h (electromagnetic)f(\014eld)f(at)g(temp)r(erature)h(zero)3152 4742 y Fc(3)p Fd(;)p Fc(4)p Fd(;)p Fc(5)3285 4765 y Fw(,)f(\014nite)498 4844 y(lev)n(el)d(atoms)e(coupled)j(to)f(a)f(b)r(oson)h(\014eld)g(at)g (p)r(ositiv)n(e)g(temp)r(erature)2352 4821 y Fc(43)p Fd(;)p Fc(44)p Fd(;)p Fc(19)2577 4844 y Fw(,)f(classical)g(particles)h (coupled)g(to)498 4923 y(a)i(scalar)f(\014eld)h(at)g(temp)r(erature)g (zero)1547 4900 y Fc(49)p Fd(;)p Fc(50)1716 4923 y Fw(or)f(at)h(p)r (ositiv)n(e)h(temp)r(erature)2551 4900 y Fc(45)2617 4923 y Fw(.)498 4981 y Fd(b)530 5005 y Fw(F)-6 b(or)35 b(quan)n(tum)g(spin)g (systems)g(axioms)f(are)i(form)n(ulated)e(in)h([78])g(whic)n(h)h (establish)f(the)i(existence)g(of)e(a)498 5084 y(stationary)21 b(state)f(and)g(its)f(mixing)f(prop)r(ert)n(y)i(for)f(\014nite)h (systems)f(coupled)h(to)g(sev)n(eral)g(reserv)n(oirs)e(at)i(di\013eren) n(t)498 5163 y(temp)r(eratures)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(8)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 9 9 9 8 bop 498 628 a Fr(consists)31 b(of)g(the)h(particles)e(in)i(the)f (middle.)49 b(A)n(t)32 b(time)f Fq(t)f Fr(=)e(0,)k(the)g(reserv)n(oirs) c(are)j(assumed)498 728 y(to)i(b)r(e)h(in)f(thermal)g(equilibrium)h(at) f(temp)r(eratures)f Fq(T)2255 740 y Fn(L)2338 728 y Fr(and)h Fq(T)2554 740 y Fn(R)2608 728 y Fr(.)54 b(This)33 b(is)g(clearly)f (readily)498 827 y(generalized)26 b(to)i(higher)f(dimensions.)498 1069 y Fl(4.3)86 b(Hamiltonian/Sto)l(chastic)32 b(r)l(eservoir.)498 1228 y Fr(While)27 b(Hamiltonian)f(reserv)n(oirs)e(are)h(in)i (principle)f(the)h(righ)n(t)f(ones)g(to)g(use)g(they)h(are)e(totally) 498 1328 y(in)n(tractable)d(without)i(further)e(simpli\014cations.)35 b(When)24 b(this)f(is)g(done)g(it)g(is)g(actually)f(p)r(ossible)498 1427 y(to)e(\014nd)g(examples)g(in)g(whic)n(h)g(one)f(starts)g(with)i (a)e(Hamiltonian)h(reserv)n(oir)d(and)j(b)n(y)f(in)n(tegrating)498 1527 y(out)g(o)n(v)n(er)e(the)j(degrees)d(of)i(freedom)g(of)f(the)i (reserv)n(oirs,)e(ends)g(up)i(with)f(a)g(sto)r(c)n(hastic)f(ev)n (olution.)498 1626 y(Man)n(y)26 b(mo)r(dels)h(of)g(this)g(t)n(yp)r(e)g (ha)n(v)n(e)e(b)r(een)i(constructed)2256 1596 y Fp(38)p Fn(;)p Fp(39)p Fn(;)p Fp(28)2498 1626 y Fr(.)37 b(W)-7 b(e)27 b(describ)r(e)f(here)g(a)h(mo)r(del)498 1726 y(considered)j(in)h ([45,21,22,20].)42 b(The)31 b(system)g(is)f(a)h(\014nite)g(c)n(hain)f (of)h(anharmonic)e(oscillators)498 1826 y(coupled)21 b(at)h(eac)n(h)e(end)i(to)f(a)h(reserv)n(oir)c(mo)r(deled)k(b)n(y)f(a)g (linear)g Fq(d)p Fr(-dimensional)g(w)n(a)n(v)n(e)f(equation,)498 1925 y(whic)n(h)j(is)g(the)g(con)n(tin)n(uum)g(limit)h(of)f(a)f Fq(d)p Fr(-dimensional)h(lattice)g(of)g(harmonic)f(oscillators.)33 b(The)498 2025 y(dynamics)24 b(of)g(the)h(in\014nite)g(system,)f (crystal+reserv)n(oirs,)d(is)j(Hamiltonian.)36 b(One)23 b(mak)n(es)h(the)498 2125 y(statistical)h(assumption)f(that,)i(at)f (time)h Fq(t)d Fr(=)f(0,)j(the)h(reserv)n(oirs)c(are)i(in)h(thermal)g (equilibrium)498 2224 y(at)39 b(temp)r(eratures)e Fq(T)1176 2236 y Fn(L)1264 2224 y Fr(and)i Fq(T)1486 2236 y Fn(R)1539 2224 y Fr(.)70 b(Since)39 b(the)g(reserv)n(oirs)c(are)j(linear,)i(one)e (ma)n(y)g(in)n(tegrate)498 2324 y(them)32 b(out,)f(and,)h(b)n(y)e(our)g (assumptions)h(on)f(the)h(initial)g(conditions)g(of)g(the)g(reserv)n (oirs,)e(the)498 2423 y(resulting)h(dynamics)f(for)h(the)g(crystal)f (is)h(sto)r(c)n(hastic,)g(though)g(in)h(general)d(not)j(Mark)n(o)n (vian.)498 2523 y(Nev)n(ertheless,)22 b(the)g(fact)g(that)g(the)g (reserv)n(oirs)d(are)h(describ)r(ed)i(b)n(y)f(a)h(w)n(a)n(v)n(e)e (equation,)i(together)498 2623 y(with)g(sp)r(ecial)g(c)n(hoices)f(of)h (the)g(coupling)f(b)r(et)n(w)n(een)h(the)g(reserv)n(oirs)d(and)j(the)g (c)n(hain,)h(p)r(ermits)3417 2593 y Fp(21)498 2722 y Fr(enlarging)j(the)h(phase)g(space)f(of)i(the)f(crystal)f(with)i(a)f (\014nite)g(n)n(um)n(b)r(er)g(of)h(auxiliary)d(v)-5 b(ariables)498 2822 y(so)27 b(that)h(the)g(dynamics)f(is)h(Mark)n(o)n(vian)d(on)i(the) h(enlarged)e(phase)h(space.)623 2922 y(In)35 b(the)h(simplest)f(case)f (of)h(coupling)g(one)g(v)-5 b(ariable)34 b(p)r(er)h(reserv)n(oir)d(is)j (enough)g(and)g(the)498 3021 y(resulting)27 b(equations)g(for)g(the)h Fq(N)36 b Fr(oscillators)26 b(are)1150 3194 y(\177)1145 3195 y Fz(q)1195 3207 y Fp(1)1272 3195 y Fr(=)38 b Fm(\000r)1509 3207 y Fh(q)1549 3215 y Ff(1)1585 3195 y Fm(V)7 b Fr(\()p Fq(Q)p Fr(\))18 b(+)g Fz(r)1913 3207 y Fn(L)1977 3195 y Fq(;)1152 3319 y Fr(\177)1147 3320 y Fz(q)1197 3332 y Fn(j)1272 3320 y Fr(=)38 b Fm(\000r)1509 3332 y Fh(q)1549 3340 y Fe(j)1583 3320 y Fm(V)7 b Fr(\()p Fq(Q)p Fr(\))14 b Fq(;)97 b(j)28 b Fr(=)23 b(2)p Fq(;)14 b(:)g(:)g(:)f(;)h(N)27 b Fm(\000)18 b Fr(1)c Fq(;)1142 3443 y Fr(\177)1137 3444 y Fz(q)1187 3456 y Fn(n)1272 3444 y Fr(=)38 b Fm(\000r)1509 3456 y Fh(q)1549 3464 y Fe(n)1593 3444 y Fm(V)7 b Fr(\()p Fq(Q)p Fr(\))19 b(+)f Fz(r)1922 3456 y Fn(R)1990 3444 y Fq(;)1152 3578 y Fr(_)1144 3579 y Fz(r)1183 3591 y Fn(L)1272 3579 y Fr(=)38 b Fm(\000)p Fq(\015)1483 3591 y Fn(L)1532 3579 y Fr(\()p Fz(r)1603 3591 y Fn(L)1672 3579 y Fm(\000)18 b Fq(\025)1803 3545 y Fp(2)1803 3599 y Fn(L)1853 3579 y Fz(q)1903 3591 y Fp(1)1941 3579 y Fr(\))g(+)g(\(2)p Fq(\015)2191 3591 y Fn(L)2241 3579 y Fq(\025)2289 3545 y Fp(2)2289 3599 y Fn(L)2339 3579 y Fq(T)2388 3591 y Fn(L)2437 3579 y Fr(\))2469 3545 y Fp(1)p Fn(=)p Fp(2)2600 3579 y Fr(_)-49 b Fq(w)2633 3591 y Fn(L)2683 3579 y Fr(\()p Fq(t)p Fr(\))14 b Fq(;)1147 3682 y Fr(_)1145 3703 y Fz(_)-33 b(r)1178 3715 y Fn(R)1272 3703 y Fr(=)38 b Fm(\000)p Fq(\015)1483 3715 y Fn(R)1537 3703 y Fr(\()p Fz(r)1608 3715 y Fn(R)1681 3703 y Fm(\000)18 b Fq(\025)1812 3669 y Fp(2)1812 3724 y Fn(R)1867 3703 y Fz(q)1917 3715 y Fp(1)1955 3703 y Fr(\))h(+)f(\(2)p Fq(\015)2206 3715 y Fn(R)2260 3703 y Fq(\025)2308 3669 y Fp(2)2308 3724 y Fn(R)2363 3703 y Fq(T)2412 3715 y Fn(R)2466 3703 y Fr(\))2498 3669 y Fp(1)p Fn(=)p Fp(2)2629 3703 y Fr(_)-50 b Fq(w)2661 3715 y Fn(R)2716 3703 y Fr(\()p Fq(t)p Fr(\))14 b Fq(;)492 b Fr(\(11\))498 3877 y(In)27 b(Eqs.\(11\))f Fq(\025)980 3889 y Fn(L)1058 3877 y Fr(and)h Fq(\025)1267 3889 y Fn(R)1349 3877 y Fr(describ)r(e)g(the)g(coupling)g (strength)f(to)h(the)h(reserv)n(oirs,)c Fq(\015)3151 3889 y Fn(L)3201 3877 y Fr(,)j Fq(\015)3294 3889 y Fn(R)3376 3877 y Fr(are)498 3977 y(parameters)f(describing)h(the)h(coupling)f (and)53 b(_)-49 b Fq(w)2016 3989 y Fn(L)2066 3977 y Fr(,)54 b(_)-49 b Fq(w)2176 3989 y Fn(R)2259 3977 y Fr(are)26 b(white)i(noises.)623 4077 y(If)34 b(the)h(temp)r(eratures)f(of)g(b)r (oth)g(reserv)n(oirs)e(are)h(the)i(same,)g Fq(T)2639 4089 y Fn(L)2722 4077 y Fr(=)f Fq(T)2870 4089 y Fn(R)2958 4077 y Fr(=)g Fq(T)12 b Fr(,)35 b(then)g(the)498 4176 y(stationary)26 b(state)i(is)f(giv)n(en)g(b)n(y)g(the)h(generalized)e (Gibbs)i(measure)f(with)h(densit)n(y)1512 4395 y Fq(Z)1575 4361 y Fk(\000)p Fp(1)1677 4395 y Fr(exp)1818 4278 y Fj(\022)1879 4395 y Fm(\000)1963 4339 y Fr(1)p 1954 4376 61 4 v 1954 4452 a Fq(T)2024 4395 y(G)p Fr(\()p Fq(P)r(;)14 b(Q;)g(R)q Fr(\))2412 4278 y Fj(\023)3339 4395 y Fr(\(12\))498 4614 y(where)26 b Fq(Z)32 b Fr(is)26 b(a)g(normalization)f(constan)n(t) h(and)g(the)h(generalized)e(\\Hamiltonian")g Fq(G)i Fr(is)f(giv)n(en) 498 4714 y(b)n(y)883 4937 y Fq(G)p Fr(\()p Fq(P)r(;)14 b(Q;)g(R)q Fr(\))38 b(=)1410 4819 y Fj(\022)1507 4880 y Fz(r)1546 4850 y Fp(2)1546 4903 y Fn(L)p 1481 4917 140 4 v 1481 4993 a Fr(2)p Fq(\025)1571 4965 y Fp(2)1571 5018 y Fn(L)1649 4937 y Fm(\000)18 b Fz(q)1782 4949 y Fp(1)1820 4937 y Fz(r)1859 4949 y Fn(L)1909 4819 y Fj(\023)1989 4937 y Fr(+)2072 4819 y Fj(\022)2168 4880 y Fz(r)2207 4850 y Fp(2)2207 4903 y Fn(R)p 2143 4917 145 4 v 2143 4993 a Fr(2)p Fq(\025)2233 4965 y Fp(2)2233 5018 y Fn(R)2316 4937 y Fm(\000)g Fz(q)2449 4949 y Fn(N)2512 4937 y Fq(r)2549 4949 y Fn(R)2604 4819 y Fj(\023)2683 4937 y Fr(+)g Fq(H)7 b Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))g Fq(:)238 b Fr(\(13\))498 5163 y(If)31 b(one)e(in)n(tegrates)g(the)h(generalized)f(Gibbs)h (state,)g(Eq.\(12\),)g(o)n(v)n(er)e(the)j(auxiliary)d(v)-5 b(ariables)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)745 b Fz(9)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 10 10 10 9 bop 498 628 a Fz(r)537 640 y Fn(L)615 628 y Fr(and)27 b Fz(r)815 640 y Fn(R)898 628 y Fr(one)g(\014nds)689 707 y Fj(Z)786 820 y Fq(d)p Fz(r)868 832 y Fn(L)918 820 y Fq(d)p Fz(r)1000 832 y Fn(R)1069 820 y Fq(Z)1132 786 y Fk(\000)p Fp(1)1235 820 y Fr(exp)1375 703 y Fj(\022)1436 820 y Fm(\000)1520 764 y Fr(1)p 1511 801 61 4 v 1511 877 a Fq(T)1581 820 y(G)p Fr(\()p Fq(P)r(;)14 b(Q;)g(R)q Fr(\))1969 703 y Fj(\023)2068 820 y Fr(=)2186 799 y(~)2169 820 y Fq(Z)2232 786 y Fk(\000)p Fp(1)2335 820 y Fr(exp)2475 703 y Fj(\022)2536 820 y Fm(\000)2620 764 y Fr(1)p 2611 801 V 2611 877 a Fq(T)2681 820 y(H)2750 832 y Fp(e\013)2828 820 y Fr(\()p Fq(P)r(;)g(Q)p Fr(\))3050 703 y Fj(\023)3125 820 y Fq(;)191 b Fr(\(14\))498 1035 y(where)28 b Fq(H)808 1047 y Fp(e\013)886 1035 y Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))25 b(=)g Fq(H)7 b Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))19 b(+)g Fq(\025)1672 1005 y Fp(2)1672 1058 y Fn(L)1722 1035 y Fz(q)1772 1047 y Fp(1)1810 1035 y Fq(=)p Fr(2)f(+)h Fq(\025)2044 1005 y Fp(2)2044 1058 y Fn(R)2098 1035 y Fz(q)2148 1047 y Fn(n)2194 1035 y Fq(=)p Fr(2)28 b(and)2486 1014 y(~)2468 1035 y Fq(Z)34 b Fr(a)29 b(normalization)e(constan)n(t.) 498 1135 y(In)h(view)f(of)h(this)g(it)g(is)f(natural)g(to)h(consider)e Fq(H)2001 1147 y Fp(e\013)2079 1135 y Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))28 b(as)f(the)h(energy)f(of)g(the)h(c)n(hain.)498 1352 y Fl(4.4)86 b(Thermostats.)498 1505 y Fr(A)42 b(fourth)g(w)n(a)n (y)e(of)i(mo)r(deling)f(the)h(reserv)n(oirs)d(is)i(b)n(y)g (deterministic)h(\(non-Hamiltonian\))498 1605 y(forces)705 1575 y Fp(27)p Fn(;)p Fp(31)860 1605 y Fr(.)63 b(Suc)n(h)36 b(mo)r(dels)g(of)g(reserv)n(oirs)d(are)i(usually)g(called)h (thermostats.)61 b(An)37 b(exam-)498 1705 y(ple)d(of)f(suc)n(h)g (reserv)n(oirs)d(whic)n(h)j(are)g(widely)g(used)g(in)h(n)n(umerical)e (w)n(ork,)i(are)e(the)i(so)e(called)498 1804 y(Nos)n(\023)-39 b(e-Ho)r(o)n(v)n(er)959 1774 y Fp(68)p Fn(;)p Fp(41)1146 1804 y Fr(thermostats.)55 b(Imp)r(osing)34 b(these)g(thermostats)g(on)f (small)h(parts)g(of)g(the)498 1904 y(system)746 1874 y Fn(c)819 1904 y Fr(\(on)39 b(the)g(left)h(and)f(on)f(the)i(righ)n (t\))e(\003)2045 1916 y Fn(L)2134 1904 y Fr(and)h(\003)2365 1916 y Fn(R)2419 1904 y Fr(,)j(the)d(equations)f(of)h(motion)g(of)498 2004 y(particles)27 b(in)h(those)f(region)f(of)i(the)g(b)r(o)n(x)f(are) g(resp)r(ectiv)n(ely)1508 2154 y Fq(m)1585 2153 y Fr(\177)1581 2154 y Fz(q)1631 2166 y Fh(i)1693 2154 y Fr(=)41 b Fm(\000r)1933 2166 y Fh(q)1973 2175 y Fg(i)1999 2154 y Fm(V)7 b Fr(\()p Fq(Q)p Fr(\))18 b Fm(\000)g Fq(\020)2324 2166 y Fn(L)2388 2153 y Fr(_)2374 2154 y Fz(q)2424 2166 y Fh(i)1508 2254 y Fq(m)1585 2253 y Fr(\177)1581 2254 y Fz(q)1631 2266 y Fh(i)1693 2254 y Fr(=)39 b Fm(\000r)1931 2266 y Fh(q)1971 2275 y Fg(i)1996 2254 y Fm(V)7 b Fr(\()p Fq(Q)p Fr(\))19 b Fm(\000)f Fq(\020)2322 2266 y Fn(R)2390 2253 y Fr(_)2377 2254 y Fz(q)2427 2266 y Fh(i)3339 2254 y Fr(\(15\))498 2405 y(where)32 b Fz(q)793 2417 y Fn(i)852 2405 y Fm(2)f Fr(\003)996 2417 y Fn(\013)1075 2405 y Fr(for)h(a)g(\015uid)h(and)f Fz(i)e Fm(2)i Fr(\003)1843 2417 y Fn(\013)1922 2405 y Fr(for)g(a)g(crystal,)g Fq(\013)f Fm(2)h(f)p Fq(L;)14 b(R)q Fm(g)p Fr(.)49 b(The)32 b(v)-5 b(ariable)31 b Fq(\020)3439 2417 y Fn(\013)498 2504 y Fr(mo)r(del)d(the)g(action)f(of)g(the)h (thermostat)g(and)f(satisfy)g(the)h(equations)1452 2703 y(_)1436 2725 y Fq(\020)1472 2737 y Fn(\013)1556 2725 y Fr(=)1698 2669 y(1)p 1668 2706 102 4 v 1668 2782 a(\002)1733 2758 y Fp(2)1793 2583 y Fj( )1896 2669 y Fr(1)p 1869 2706 96 4 v 1869 2782 a Fq(T)1918 2794 y Fn(\013)2005 2646 y Fj(X)1989 2825 y Fh(i)p Fk(2)p Fp(\003)2100 2833 y Fe(\013)2178 2669 y Fz(p)2231 2639 y Fp(2)2231 2693 y Fh(i)p 2166 2706 115 4 v 2166 2782 a Fr(2)p Fq(m)2308 2725 y Fm(\000)18 b Fr(1)2433 2583 y Fj(!)2526 2725 y Fq(:)790 b Fr(\(16\))498 2963 y(In)37 b(Eq.\(16\),)g(\002)f(is)g(in)n (terpreted)g(as)f(the)i(resp)r(onse)e(time)i(of)f(the)h(reserv)n(oir)c (and)j Fq(T)3196 2975 y Fn(\013)3279 2963 y Fr(is)h(the)498 3063 y(temp)r(erature)27 b(of)h(the)g Fq(\013)p Fr(-th)g(reserv)n(oir.) 623 3163 y(A)22 b(limiting)g(case)g(of)f(Eqs.\(15\)\(16\))g(is)h(when)g (w)n(e)g(let)g(\002)h Fm(!)g Fr(0.)34 b(This)22 b(limit)h(can)e(b)r(e)i (formally)498 3262 y(tak)n(en)e(and)g(the)h(mo)r(del)f(b)r(ecomes)g (equiv)-5 b(alen)n(t)22 b(to)f(the)h(so)e(called)h(Gaussian)g (thermostat.)34 b(This)498 3362 y(means)22 b(that)g(one)f(computes)h Fq(\020)1467 3374 y Fn(\013)1537 3362 y Fr(as)f(a)g(function)i(of)e Fq(P)34 b Fr(and)22 b Fq(Q)g Fr(in)g(suc)n(h)f(a)h(w)n(a)n(y)e(that)j (the)f(kinetic)498 3461 y(energy)33 b(of)i(the)g(particles)e(in)i(\003) 1523 3473 y Fn(L)1607 3461 y Fr(or)f(\003)1774 3473 y Fn(R)1862 3461 y Fr(is)h(a)f(constan)n(t)f(of)i(the)g(motion.)57 b(After)35 b(a)f(simple)498 3561 y(calculation)27 b(one)g(gets)g(for)g (the)h(c)n(hain:)896 3765 y Fq(\020)932 3777 y Fn(L)982 3765 y Fr(\()p Fq(Q;)14 b(P)e Fr(\))23 b(=)1335 3637 y Fj(P)1422 3724 y Fn(i)f Fr(0)g(will)h(not)f (b)r(e)h(the)g(in)n(v)-5 b(arian)n(t)21 b(state)h(at)g(a)498 3768 y(giv)n(en)f(temp)r(erature)h(whic)n(h)f(is)h(stationary)e(for)i (the)g(isolated)f(reserv)n(oir.)32 b(It)22 b(is)g(this)g(fact)g(whic)n (h)498 3868 y(mak)n(es)h(the)i(problem)f(of)h(general)e(Hamiltonian)h (reserv)n(oirs)d(m)n(uc)n(h)j(more)g(di\016cult)h(that)g(than)498 3967 y(of)32 b(sto)r(c)n(hastic)f(reserv)n(oirs.)46 b(It)32 b(is)g(only)g(in)g(v)n(ery)e(sp)r(ecial)i(cases)e(\(essen)n(tially)h (no)h(in)n(teraction)498 4067 y(inside)23 b(the)g(reserv)n(oir\))d(suc) n(h)i(as)g(that)h(discussed)f(in)h(subsection)f(4.3)g(where)g(this)h (can)f(b)r(e)h(dealt)498 4166 y(with.)623 4266 y(F)-7 b(or)26 b(thermostated)h(systems)f(the)i(temp)r(erature)e(of)h(the)h (reserv)n(oirs)c(is)j(already)e(giv)n(en)h(b)n(y)498 4366 y(the)f(equations)f(of)g(motion)h(so)f(w)n(e)g(exp)r(ect)h(again)e (to)i(ha)n(v)n(e)e(a)i(unique)f(in)n(v)-5 b(arian)n(t)24 b(distribution.)498 4465 y(Moreo)n(v)n(er)g(these)i(systems)g(ha)n(v)n (e)f(the)i(prop)r(ert)n(y)e(that)i(the)g(phase)f(space)f(v)n(olume)h (is)g(not)h(con-)498 4565 y(serv)n(ed)d(b)n(y)i(the)g(dynamics)f(so)f (that,)j(in)e(general,)g(no)g(in)n(v)-5 b(arian)n(t)25 b(measure)f(will)i(b)r(e)g(absolutely)498 4665 y(con)n(tin)n(uous)k (with)i(resp)r(ect)e(to)h(Leb)r(esgue)g(measure.)45 b(In)32 b(this)f(case)f(w)n(e)h(need)g(a)f(criterion)g(to)498 4764 y(c)n(hose)25 b(the)h(\\ph)n(ysical")e(in)n(v)-5 b(arian)n(t)24 b(distribution.)37 b(A)26 b(natural)f(c)n(hoice)g(are)f (the)i(so)g(called)f(SRB)498 4864 y(states.)36 b(These)24 b(can)g(b)r(e)h(c)n(haracterized)d(b)n(y)j(assuming)e(that)i(the)g (system)g(w)n(as)e(in)i(equilibrium)498 4963 y(in)i(the)g(v)n(ery)f (distan)n(t)g(past)h(and)f(that)h(at)g(some)f(p)r(oin)n(t)h(a)f (forcing)g(w)n(as)f(switc)n(hed)i(on)f(and)h(this)498 5063 y(dro)n(v)n(e)i(the)h(system)h(to)f(a)g(steady)f(state)i (distribution.)45 b(More)29 b(mathematically)h(this)g(means)498 5163 y(that)d(w)n(e)g(consider)e(the)i(w)n(eak)f(limit)i(of)e(a)h (probabilit)n(y)e(distribution)i(absolutely)f(con)n(tin)n(uous)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(11)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 12 12 12 11 bop 498 628 a Fr(with)28 b(resp)r(ect)g(to)f(Leb)r(esgue,)g Fo(e.g.)h Fr(the)g(canonical)e(distribution)i(that)g(c)n(haracterize)d (the)j(sys-)498 728 y(tem)23 b(when)f(all)g(thermostats)g(ha)n(v)n(e)f (the)h(same)g(temp)r(erature,)h(under)f(the)h(time)f(ev)n(olution,)h Fo(i.e.)498 827 y Fr(if)j(\010)632 797 y Fn(t)687 827 y Fr(is)f(the)h(\015o)n(w)f(describing)g(the)g(ev)n(olution)g(of)h(the) f(thermostated)g(system)h(then)g(a)f(state)g Fq(\026)498 927 y Fr(is)j(called)f(SRB)h(if)g(it)g(has)f(the)h(prop)r(ert)n(y)e (that)1186 1070 y Fq(\026)1236 1082 y Fn(S)s(RB)1388 1070 y Fr(\()p Fq(dX)7 b Fr(\))23 b(=)43 b(lim)1681 1120 y Fn(t)p Fk(!1)1862 1049 y Fr(\026)1853 1070 y(\010)1913 1036 y Fn(t)1942 1070 y Fq(\025)p Fr(\()p Fq(dX)7 b Fr(\))14 b Fq(;)180 b(X)30 b Fr(=)22 b(\()p Fq(P)r(;)14 b(Q)p Fr(\))541 b(\(18\))498 1249 y(where)759 1228 y(\026)749 1249 y(\010)809 1218 y Fn(t)878 1249 y Fr(indicates)38 b(the)h(adjoin)n(t)g(and)g Fq(\025)p Fr(\()p Fq(dX)7 b Fr(\))39 b(is)g(the)g(giv)n(en)f(initial)h(distribution.)71 b(W)-7 b(e)498 1348 y(observ)n(e)37 b(that)i(w)n(e)f(often)h(cannot)f (c)n(ho)r(ose)f(directly)h(the)h(Leb)r(esgue)f(distribution)g(b)r (ecause)498 1448 y(the)28 b(phase)e(space)h(of)g(our)f(system)h(in)g (not)h(compact)e(lik)n(e)h(for)g(the)g(Nos)n(\023)-39 b(e-Ho)r(o)n(v)n(er)24 b(thermostat)498 1547 y(Eq.\(16\).)623 1647 y(Although)j(the)g(de\014nition)g(Eq.\(18\))f(is)h(in)n(teresting) f(for)g(its)h(similarit)n(y)f(to)h(the)g(ones)f(used)498 1747 y(in)35 b(the)g(previous)e(system+reserv)n(oirs)e(mo)r(dels)j (another)g(c)n(haracterization)e(of)i(these)h(mea-)498 1846 y(sure)c(is)h(obtained)g(b)n(y)g(sa)n(ying)e(that)i(they)g (represen)n(t)f(the)h(statistics)g(of)g(the)g(motion.)50 b(More)498 1946 y(precisely)-7 b(,)31 b(giv)n(en)f(an)n(y)g(observ)-5 b(able,)30 b(the)i(a)n(v)n(erage)c(of)i(this)i(observ)-5 b(able)29 b(with)i(resp)r(ect)g(to)g(the)498 2046 y(SRB)21 b(distribution)g(is)f(equal)h(to)f(its)h(time)h(a)n(v)n(erage)17 b(along)j(a)g(tra)5 b(jectory)19 b(starting)h(from)h(almost)498 2145 y(ev)n(ery)28 b(p)r(oin)n(t)i(\(with)g(resp)r(ect)f(to)h(Leb)r (esgue)e(measure\).)42 b(In)30 b(form)n(ulae)e(w)n(e)h(can)g(sa)n(y)f (that)i(if)g Fq(\026)498 2245 y Fr(is)e(the)g(SRB)f(distribution)h (then)1416 2425 y(lim)1395 2475 y Fn(t)p Fk(!1)1576 2369 y Fr(1)p 1576 2406 42 4 v 1582 2482 a Fq(t)1641 2312 y Fj(Z)1724 2333 y Fn(t)1687 2501 y Fp(0)1767 2425 y Fq(dt\016)1877 2440 y Fp(\010)1924 2424 y Fe(t)1952 2440 y Fp(\()p Fn(X)5 b Fp(\))2104 2425 y Fr(=)37 b Fq(\026)2256 2437 y Fn(S)s(RB)2407 2425 y Fr(\()p Fq(dX)7 b Fr(\))749 b(\(19\))498 2615 y(for)29 b(a)f(set)h(of)g Fq(X)36 b Fr(of)29 b(full)g(\(or)g(at)g(least)f(p)r(ositiv)n(e\))h(Leb)r(esgue)g (measure.)40 b(In)29 b(Eq.\(19\))f(the)h(limit)498 2715 y(is)f(to)f(b)r(e)h(understo)r(o)r(d)f(as)g(a)g(w)n(eak)g(limit)1779 2685 y Fn(d)1818 2715 y Fr(.)498 2928 y Fl(5.2)86 b(He)l(at)30 b(Flow)g(and)h(Entr)l(opy)f(Pr)l(o)l(duction)g(in)g(R)l(eservoirs.)498 3081 y Fr(Since)h(our)e(in)n(terest)h(here)g(is)g(sp)r(eci\014cally)g (in)g(F)-7 b(ourier's)29 b(la)n(w)h(our)f(next)i(question)e(ab)r(out)i (the)498 3181 y(stationary)25 b(state)h(of)h(the)f(system)h(coupled)f (to)g(t)n(w)n(o)g(reserv)n(oir)d(at)k(di\013eren)n(t)f(temp)r(eratures) g(is)498 3280 y(the)32 b(existence)g(and)g(nature)f(of)h(the)g(heat)g (\015ux)g(across)e(the)i(system.)50 b(It)32 b(is)g(clear)e(that)i(ev)n (en)498 3380 y(existence)23 b(is)g(not)g(automatic:)35 b(most)23 b(trivially)f(just)i(imagine)f(that)g(our)g(system)g(is)g (comp)r(osed)498 3480 y(of)28 b(t)n(w)n(o)f(nonin)n(teracting)f(parts)h (eac)n(h)g(coupled)g(to)g(a)h(single)f(reserv)n(oir.)623 3579 y(T)-7 b(o)23 b(study)i(the)f(heat)g(\015ux)g(through)g(the)g (system)g(w)n(e)g(\014rst)g(de\014ne)g(a)g(lo)r(cal)f(energy)g(densit)n (y)-7 b(.)498 3692 y(F)g(or)27 b(a)g(\015uid)h(let)g Fz(q)c Fm(2)f Fr(\003)g Fm(\032)g Fi(R)1417 3644 y Fn(d)1484 3692 y Fr(then)940 3931 y Fq(h)p Fr(\()p Fz(q)p Fr(;)14 b Fq(P)r(;)g(Q)p Fr(\))24 b(=)1439 3827 y Fn(N)1409 3852 y Fj(X)1415 4029 y Fn(i)p Fp(=1)1542 3931 y Fq(\016)s Fr(\()p Fz(q)c Fm(\000)e Fz(q)1817 3943 y Fn(i)1845 3931 y Fr(\))1891 3764 y Fj(2)1891 3913 y(4)1956 3874 y Fz(p)2009 3886 y Fn(i)2037 3844 y Fp(2)p 1956 3912 118 4 v 1958 3988 a Fr(2)p Fq(m)2102 3931 y Fr(+)2185 3852 y Fj(X)2193 4031 y Fn(j)s Fk(6)p Fp(=)p Fn(i)2319 3931 y Fq(\036)p Fr(\()p Fz(q)2450 3943 y Fn(j)2505 3931 y Fm(\000)g Fz(q)2638 3943 y Fn(i)2666 3931 y Fr(\))g(+)h Fq(u)p Fr(\()p Fz(q)2930 3943 y Fn(i)2957 3931 y Fr(\))2989 3764 y Fj(3)2989 3913 y(5)3339 3931 y Fr(\(20\))498 4174 y(where)29 b(the)h(square)e(brac)n (k)n(et)g(is)h(iden)n(tical)g(to)h(that)f(in)h(Eq.\(4\).)42 b(F)-7 b(or)29 b(a)g(crystal)f(with)i(nearest)498 4273 y(neigh)n(b)r(or)d(in)n(teractions)f(w)n(e)h(de\014ne)h(the)g(lo)r(cal) f(energy)g(densit)n(y)g(at)g(site)h Fz(i)f Fr(as)711 4465 y Fq(h)p Fr(\()p Fz(i)p Fr(;)14 b Fq(P)r(;)g(Q)p Fr(\))23 b(=)1177 4409 y Fz(p)1230 4379 y Fp(2)1230 4433 y Fh(i)p 1165 4446 115 4 v 1165 4522 a Fr(2)p Fq(m)1308 4465 y Fr(+)18 b Fq(U)9 b Fr(\()p Fz(q)1539 4477 y Fh(i)1565 4465 y Fr(\))18 b(+)1730 4409 y(1)p 1708 4446 85 4 v 1708 4522 a(2)p Fq(d)1817 4386 y Fj(X)1859 4565 y Fn(k)1951 4465 y Fr(\()p Fq(V)h Fr(\()p Fz(q)2132 4477 y Fh(i)2176 4465 y Fm(\000)f Fz(q)2309 4477 y Fh(i)p Fk(\000)p Fp(1)2415 4486 y Fe(k)2456 4465 y Fr(\))h(+)f Fq(V)h Fr(\()p Fz(q)2739 4477 y Fh(i)p Fp(+1)2844 4486 y Fe(k)2904 4465 y Fm(\000)f Fz(q)3037 4477 y Fh(i)3062 4465 y Fr(\)\))213 b(\(21\))498 4683 y(where)29 b(1)782 4695 y Fn(k)853 4683 y Fr(is)h(the)g Fq(d)p Fr(-dimensional)f(v)n(ector)g(with)h(all)g(comp)r(onen)n(ts)f(0) g(except)h(the)h Fq(k)s Fr(-th)f(equal)498 4783 y(to)e(1.)p 498 4846 1196 4 v 498 4903 a Fd(d)534 4926 y Fw(The)19 b(only)h(thermostated)f(\\ph)n(ysical")h(mo)r(del)e(for)g(whic)n(h)h(a) g(SNS)g(corresp)r(onding)g(to)h(an)f(SRB)g(measure)f(with)498 5005 y(the)31 b(desired)f(prop)r(erties)h(has)f(b)r(een)h(pro)n(v)n(en) g(is)f(the)h(Moran-Ho)r(o)n(v)n(er)f(mo)r(del)f(of)h(a)g(single)g (particle)g(mo)n(ving)498 5084 y(among)25 b(\014xed)h(p)r(erio)r(dic)f (scatterers)g(\(see)h(Section)h(1\))e(to)h(whic)n(h)f(is)f(added)i(an)g (external)f(electric)h(\014eld)f(and)h(a)498 5163 y(Gaussian)e (thermostat)1138 5139 y Fc(14)1204 5163 y Fw(.)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(12)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 13 13 13 12 bop 623 628 a Fr(Giv)n(en)33 b(the)i(lo)r(cal)e(energy)g(densit)n (y)h(w)n(e)g(can)f(de\014ne)i(a)e(lo)r(cal)h(microscopic)e(heat)i (\015o)n(w)f(\011)498 728 y(through)g(the)h(con)n(tin)n(uit)n(y)f (equation.)54 b(T)-7 b(o)33 b(a)n(v)n(oid)f(rep)r(etition)h(w)n(e)g (shall)h(do)f(so)g(only)g(for)g(the)498 827 y(crystal.)j(W)-7 b(riting)1636 955 y Fq(dh)p Fr(\()p Fz(i)p Fr(;)14 b Fq(P)r(;)g(Q)p Fr(\))p 1636 993 378 4 v 1788 1069 a Fq(dt)2046 1012 y Fr(=)2148 991 y(~)2134 1012 y Fm(r)p Fr(\011\()p Fz(i)p Fr(\))980 b(\(22\))498 1196 y(where)759 1175 y(~)745 1196 y Fm(r)p Fr(\011\()p Fz(i)p Fr(\))35 b(=)1104 1133 y Fj(P)1191 1221 y Fn(k)1256 1174 y Fr(~)1246 1196 y Fq(@)1290 1208 y Fn(i)1313 1217 y Fe(k)1354 1196 y Fr(\011)1419 1208 y Fn(k)1459 1196 y Fr(\()p Fz(i)p Fr(\))g(with)1792 1174 y(~)1781 1196 y Fq(@)1825 1208 y Fn(i)1848 1217 y Fe(k)1889 1196 y Fr(\011)1954 1208 y Fn(j)1989 1196 y Fr(\()p Fz(i)p Fr(\))f(=)g(\(\011)2310 1208 y Fn(j)2345 1196 y Fr(\()p Fz(i)23 b Fr(+)g(1)2557 1208 y Fn(k)2597 1196 y Fr(\))g Fm(\000)g Fr(\011)2805 1208 y Fn(j)2840 1196 y Fr(\()p Fz(i)p Fr(\)\))p Fq(=)p Fr(2.)57 b(It)35 b(easy)e(to)498 1295 y(v)n(erify)27 b(that)1402 1466 y(\011)1467 1478 y Fn(k)1507 1466 y Fr(\()p Fz(i)p Fr(\))c(=)g Fz(f)9 b Fr(\()p Fz(q)1829 1478 y Fh(i)1874 1466 y Fm(\000)18 b Fz(q)2007 1478 y Fh(i)p Fp(+1)2112 1487 y Fe(k)2153 1466 y Fr(\))2195 1410 y Fz(p)2248 1422 y Fh(i)2292 1410 y Fr(+)g Fz(p)2428 1422 y Fh(i)p Fp(+1)2533 1431 y Fe(k)p 2195 1447 379 4 v 2363 1523 a Fr(2)3339 1466 y(\(23\))498 1635 y(where)23 b Fz(f)9 b Fr(\()p Fz(q)p Fr(\))25 b(=)d Fm(\000r)p Fq(V)d Fr(\()p Fz(q)p Fr(\).)37 b(W)-7 b(e)24 b(will)g(usually)f(b)r(e)h(in)n(terested)f(in)h(the)h(heat)e(\015o)n(w) g(\010\()p Fq(j)5 b Fr(\))25 b(through)498 1734 y(the)j(plane)g Fm(f)p Fz(i)22 b Fm(2)h Fr(\003)14 b(;)27 b Fq(i)1183 1746 y Fp(1)1243 1734 y Fr(=)c Fq(j)5 b Fm(g)p Fr(.)36 b(It)28 b(is)g(clear)e(that)i(w)n(e)f(can)g(in)n(tegrate)g(eq.\(22\))g (and)g(obtain)1530 1892 y(\010\()p Fq(j)5 b Fr(\))37 b(=)1969 1813 y Fj(X)1841 1983 y Fg(i)p Ff(=\()p Fe(j;i)1989 1995 y Ff(2)2022 1983 y Fe(;:::;i)2139 1998 y(d)2174 1983 y Ff(\))p Fe(;)1912 2037 y Ff(1)p Fb(\024)p Fe(i)2007 2052 y(l)2031 2037 y Fb(\024)p Fe(N)2121 2052 y(l)2240 1892 y Fr(\011)2305 1904 y Fp(1)2341 1892 y Fr(\()p Fz(i)p Fr(\))p Fq(:)884 b Fr(\(24\))623 2167 y(Of)38 b(course)g(the)h(heat)f (curren)n(t)g(inside)h(the)g(system)f(in)h(the)g(steady)f(state)h(is)f (just)i(the)498 2267 y(energy)e(\015ux)h(from)f(one)g(reserv)n(oir)e (to)j(the)g(other,)i(presumably)d(from)g(the)h(one)f(with)i(the)498 2367 y(higher)25 b(temp)r(erature)f(to)i(the)f(one)g(with)h(the)f(lo)n (w)n(er)f(one.)36 b(Let)25 b Fq(S)2538 2337 y Fn(t)2592 2367 y Fr(b)r(e)h(the)f(time)h(ev)n(olution)f(for)498 2466 y(observ)-5 b(ables)24 b(\(a)n(v)n(eraged)e(o)n(v)n(er)h(the)j (realizations)d(if)j(the)f(ev)n(olution)f(is)h(sto)r(c)n(hastic\).)35 b(Since)26 b(the)498 2566 y(equations)34 b(of)g(motion)g(are)g (Hamiltonian)g(except)g(at)h(the)g(b)r(oundary)-7 b(,)35 b(one)f(\014nds)h(that)g(the)498 2666 y(time)28 b(deriv)-5 b(ativ)n(e)27 b(of)h(the)g(energy)e(is)h(giv)n(en)g(b)n(y)1433 2789 y Fq(d)p 1418 2826 74 4 v 1418 2902 a(dt)1501 2845 y(S)1557 2811 y Fn(t)1586 2845 y Fq(H)7 b Fr(\()p Fq(P)r(;)14 b(Q)p Fr(\))38 b(=)e Fm(\000)p Fq(S)2144 2811 y Fn(t)2187 2845 y Fr(\(\010)2279 2857 y Fn(L)2347 2845 y Fr(+)18 b(\010)2490 2857 y Fn(R)2544 2845 y Fr(\))763 b(\(25\))498 3014 y(where)32 b(\010)803 3026 y Fn(L)885 3014 y Fr(\(\010)977 3026 y Fn(R)1032 3014 y Fr(\))h(dep)r(ends)f(only)g(on)g(the)h(v)-5 b(ariables)31 b(of)i(the)f(left)h(\(righ)n(t\))g(b)r(oundary)e(of)i (the)498 3114 y(system.)41 b(It)29 b(is)g(natural)f(to)h(in)n(terpret)g (\010)1787 3126 y Fn(L)1865 3114 y Fr(as)g(the)g(\015o)n(w)f(of)h (energy)f(from)h(the)g(system)g(to)g(the)498 3214 y(left)22 b(heat)f(reserv)n(oir)d(and)j(similarly)f(for)h(\010)1815 3226 y Fn(R)1869 3214 y Fr(.)35 b(W)-7 b(e)22 b(supp)r(ose)f(that)g(a)g (stationary)e(state)i Fq(\026)g Fr(exists,)498 3313 y(and,)28 b(for)f(an)n(y)g(observ)-5 b(able)26 b Fq(f)9 b Fr(,)27 b(w)n(e)g(set)h Fq(\026)p Fr(\()p Fq(f)9 b Fr(\))23 b(=)1998 3246 y Fj(R)2067 3313 y Fq(f)9 b(d\026)p Fr(.)37 b(One)27 b(obtains)1244 3509 y Fm(\000)p Fq(\026)14 b Fr(\(\010)1465 3521 y Fn(L)1533 3509 y Fr(+)k(\010)1676 3521 y Fn(R)1730 3509 y Fr(\))38 b(=)e Fq(\026)1965 3392 y Fj(\022)2051 3453 y Fq(d)p 2036 3490 V 2036 3566 a(dt)2119 3509 y(S)2175 3474 y Fn(t)2204 3509 y Fq(H)7 b Fr(\()p Fq(p;)14 b(q)s Fr(\))2463 3392 y Fj(\023)2561 3509 y Fr(=)37 b(0)14 b Fq(;)597 b Fr(\(26\))498 3695 y(and)28 b(therefore)1646 3836 y Fq(\026)p Fr(\(\010)1788 3848 y Fn(L)1838 3836 y Fr(\))37 b(=)g Fm(\000)p Fq(\026)p Fr(\(\010)2216 3848 y Fn(R)2270 3836 y Fr(\))14 b Fq(:)1000 b Fr(\(27\))623 3977 y(T)-7 b(o)20 b(c)n(hec)n(k)g(that)h(the)g(heat)f(\015ux)h(is)g (indeed)g(as)f(exp)r(ected)h(it)g(is)g(useful)g(to)f(de\014ne)h(the)g (en)n(trop)n(y)498 4077 y(pro)r(duction)27 b Fq(\033)k Fr(of)d(the)g(reserv)n(oirs)d(as)1682 4047 y Fp(6)1697 4256 y Fq(\033)40 b Fr(=)1896 4200 y(\010)1956 4212 y Fn(L)p 1896 4237 110 4 v 1902 4313 a Fq(T)1951 4325 y Fn(L)2034 4256 y Fr(+)2127 4200 y(\010)2187 4212 y Fn(R)p 2127 4237 115 4 v 2133 4313 a Fq(T)2182 4325 y Fn(R)2265 4256 y Fq(;)1051 b Fr(\(28\))498 4433 y(i.e.,)24 b Fq(\033)j Fr(is)c(the)g(sum)g(of)g(the)g(energy)f(\015o)n(ws)g(in)n(to)h(the)g (reserv)n(oirs)d(divided)k(b)n(y)e(the)i(temp)r(eratures)498 4532 y(of)h(the)g(reserv)n(oirs.)33 b(This)24 b(\(microscopic\))g (de\014nition)h(of)f(the)h(reserv)n(oirs)d(en)n(trop)n(y)h(pro)r (duction)498 4632 y(is)j(in)g(accordance)e(with)j(our)e(notion)h(of)g (heat)g(reserv)n(oir)d(at)j(sp)r(eci\014ed)g(temp)r(erature.)36 b(It)26 b(do)r(es)498 4731 y(not)37 b(require)f(that)h(the)h(system)e (itself)i(b)r(e)f(close)f(to)h(equilibrium.)65 b(A)37 b(con)n(v)n(enien)n(t)f(w)n(a)n(y)g(of)498 4831 y(pro)n(ving)26 b(that)j(if)f Fq(T)1104 4843 y Fn(L)1177 4831 y Fq(>)23 b(T)1314 4843 y Fn(R)1395 4831 y Fr(then)29 b(heat)e(is)h(\015o)n(wing) f(through)g(the)h(system)g(from)g(left)g(to)g(righ)n(t)498 4931 y(is)g(to)f(sho)n(w)g(that)h(in)g(the)g(steady)f(state)1379 5117 y Fq(\026)p Fr(\()p Fq(\033)s Fr(\))38 b(=)1682 5000 y Fj(\022)1784 5061 y Fr(1)p 1753 5098 103 4 v 1753 5174 a Fq(T)1802 5186 y Fn(R)1884 5117 y Fm(\000)2006 5061 y Fr(1)p 1977 5098 99 4 v 1977 5174 a Fq(T)2026 5186 y Fn(L)2086 5000 y Fj(\023)2161 5117 y Fq(\026)p Fr(\(\010)2303 5129 y Fn(R)2357 5117 y Fr(\))g Fm(\025)e Fr(0)14 b Fq(;)732 b Fr(\(29\))p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(13)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 14 14 14 13 bop 498 628 a Fr(and)1308 778 y Fq(\026)p Fr(\()p Fq(\033)s Fr(\))38 b(=)e(0)110 b(if)35 b(and)27 b(only)i(if)90 b Fq(T)2378 790 y Fn(L)2450 778 y Fr(=)22 b Fq(T)2586 790 y Fn(R)2654 778 y Fq(:)662 b Fr(\(30\))498 929 y(\(One)30 b(ma)n(y)f(also)g(consider)g(the)h(heat)g(\015o)n(w)f(\010\()p Fq(i)p Fr(\))i(\(or)e(the)h(corresp)r(onding)e Fq(J)8 b Fr(\()p Fz(q)p Fr(\))31 b(for)f(a)f(\015uid\))498 1029 y(inside)38 b(the)g(system)f(and)g(formally)g(de\014ne)h(a)f(corresp)r (onding)e(en)n(trop)n(y)i(pro)r(duction)g Fq(\033)3355 1041 y Fn(i)3422 1029 y Fr(=)498 1128 y(\()p Fq(T)591 1093 y Fk(\000)p Fp(1)579 1153 y Fn(R)701 1128 y Fm(\000)21 b Fq(T)848 1093 y Fk(\000)p Fp(1)836 1153 y Fn(L)936 1128 y Fr(\)\010\()p Fq(i)p Fr(\).)50 b(One)31 b(ob)n(viously)g(has)g (that)h Fq(\026)p Fr(\(\010)2223 1140 y Fn(L)2273 1128 y Fr(\))e(=)g Fq(\026)p Fr(\(\010)2572 1140 y Fn(i)2600 1128 y Fr(\))i(in)g(the)g(stationary)e(state)498 1228 y(but)25 b Fq(\033)694 1240 y Fn(i)746 1228 y Fr(is)e(not)h(the)g (macroscopic)e(en)n(trop)n(y)h(pro)r(duction)g(densit)n(y)h(inside)g (the)g(system)3190 1198 y Fp(27)p Fn(;)p Fp(15)p Fn(;)p Fp(16)3432 1228 y Fr(.\))623 1327 y(If)29 b Fq(T)756 1339 y Fn(L)834 1327 y Fr(and)g Fq(T)1046 1339 y Fn(R)1129 1327 y Fr(are)f(close,)h(one)f(exp)r(ects)h(linear)f(resp)r(onse)g (theory)g(to)h(b)r(e)h(v)-5 b(alid.)41 b(Setting)498 1427 y Fq(T)34 b Fr(=)23 b(\()p Fq(T)750 1439 y Fn(L)818 1427 y Fr(+)18 b Fq(T)950 1439 y Fn(R)1004 1427 y Fr(\))p Fq(=)p Fr(2)27 b(and)g Fq(\016)s(T)34 b Fr(=)23 b(\()p Fq(T)1600 1439 y Fn(L)1668 1427 y Fm(\000)18 b Fq(T)1800 1439 y Fn(R)1854 1427 y Fr(\),)28 b(formal)f(p)r(erturbation)g(theory)g (giv)n(es)1128 1624 y Fq(\026)p Fr(\()p Fq(\033)s Fr(\))38 b(=)1432 1511 y Fj(Z)1515 1532 y Fk(1)1478 1700 y Fp(0)1599 1624 y Fq(dt)14 b(\026)1736 1636 y Fp(0)1773 1624 y Fr(\()p Fq(\033)s(S)1911 1590 y Fn(t)1906 1645 y Fp(0)1944 1624 y Fq(\033)s Fr(\))19 b(+)46 b(lo)n(w)n(er)26 b(orders)g(in)i Fq(\016)s(T)d(;)482 b Fr(\(31\))498 1825 y(where)21 b Fq(\026)782 1837 y Fp(0)841 1825 y Fr(is)h(the)g(equilibrium)g(Gibbs)g (state)g(at)f(temp)r(erature)h Fq(T)33 b Fr(and)21 b Fq(S)2780 1795 y Fn(t)2775 1846 y Fp(0)2834 1825 y Fr(the)i(time)f(ev)n (olution)498 1925 y(for)27 b(observ)-5 b(ables)26 b(with)i Fq(T)1300 1937 y Fn(L)1373 1925 y Fr(=)22 b Fq(T)1509 1937 y Fn(R)1586 1925 y Fr(=)h Fq(T)12 b Fr(.)36 b(F)-7 b(or)27 b(the)h(heat)f(\015ux)h(\010)23 b(=)g(\010\()p Fq(j)5 b Fr(\))28 b(one)f(obtains)946 2122 y Fq(\026)p Fr(\(\010\))37 b(=)1269 2066 y Fq(T)1318 2078 y Fn(L)1385 2066 y Fm(\000)18 b Fq(T)1517 2078 y Fn(R)p 1269 2103 303 4 v 1371 2179 a Fq(T)1432 2155 y Fp(2)1595 2009 y Fj(Z)1678 2030 y Fk(1)1641 2198 y Fp(0)1762 2122 y Fq(dt)c(\026)1899 2134 y Fp(0)1937 2122 y Fr(\(\010)p Fq(S)2085 2088 y Fn(t)2080 2143 y Fp(0)2117 2122 y Fr(\010\))19 b(+)45 b(lo)n(w)n(er)26 b(orders)g(in)i Fq(\016)s(T)d(:)300 b Fr(\(32\))498 2319 y(It)25 b(is)g(imp)r(ortan)n(t)f(to)h(note)g(that) g(in)g(Eqs.\(31\))e(and)i(\(32\),)g(the)g(reserv)n(oirs)d(are)h(still)i (presen)n(t)g(via)498 2419 y(the)31 b(time)g(ev)n(olution)f Fq(S)1255 2389 y Fn(t)1250 2439 y Fp(0)1287 2419 y Fr(.)45 b(Although)31 b(v)n(ery)e(similar)h(this)h(is)f(not)h(the)f(Green-Kub)r (o)g(form)n(ula)498 2518 y(whic)n(h)e(will)f(b)r(e)h(discussed)g(b)r (elo)n(w.)498 2735 y Fl(5.3)86 b Fr(F)-7 b(ourier's)27 b(La)n(w.)498 2889 y(Assuming)20 b(that)g(1.)34 b(and)19 b(2.)34 b(ha)n(v)n(e)19 b(b)r(een)h(pro)n(v)n(ed,)g(w)n(e)f(can)g(then) h(de\014ne)g(the)h(heat)e(conductivit)n(y)498 2988 y Fq(\024)546 3000 y Fk(L)618 2988 y Fr(as)i(in)g(Section)h(1,)g(where)f Fm(L)h Fr(the)g(length)f(of)h(the)g(system)f(\(\015uid)h(or)f (crystal\))g(in)h(microscopic)498 3088 y(units)37 b(and)f Fq(A)h Fr(the)g(area)e(of)h(its)h(crossection.)61 b(Since)37 b Fq(\016)s(T)9 b(=)p Fm(L)36 b Fr(is)g(the)h(a)n(v)n(erage)c(temp)r (erature)498 3187 y(gradien)n(t)27 b(the)h(heat)f(conductivit)n(y)g(at) h(temp)r(erature)f Fq(T)39 b Fr(should)27 b(then)h(b)r(e)g(giv)n(en)f (b)n(y)1410 3372 y Fq(\024)36 b Fr(=)68 b(lim)1596 3426 y Fk(L!1)1788 3372 y Fm(L)46 b Fr(lim)1859 3427 y Fn(\016)r(T)9 b Fk(!)p Fp(0)2092 3316 y Fr(1)p 2063 3353 101 4 v 2063 3429 a Fq(\016)s(T)2186 3372 y Fr(\()q Fq(\026)p Fr(\(\010\))p Fq(=)-5 b(A)p Fr(\))28 b Fq(;)764 b Fr(\(33\))498 3556 y(i.e.,)32 b Fq(\024)c Fr(=)g Fq(\024)p Fr(\()p Fq(T)12 b Fr(\))30 b(is)h(the)g(heat)g(\015ux)g(p)r(er)f(unit)i(area)d(divided) i(b)n(y)g(the)g(temp)r(erature)f(gradien)n(t.)498 3656 y(One)g(migh)n(t)g(also)f(ha)n(v)n(e)f(tak)n(en)i(the)g(limit)h Fq(A)c Fm(!)g(1)j Fr(in)h(Eq.\(33\).)43 b(As)30 b(migh)n(t)g(b)r(e)g (exp)r(ected)h(it)498 3756 y(is)d(the)g(limit)g Fm(L)23 b(!)g(1)28 b Fr(whic)n(h)g(is)f(the)h(crux)f(of)h(the)g(matter.)498 3981 y Fz(6)95 b(Summary)30 b(of)i(Exact)h(Results)498 4163 y Fr(W)-7 b(e)32 b(no)n(w)f(summarize)g(brie\015y)h(the)g(limited) h(n)n(um)n(b)r(er)e(of)h(results)f(relating)g(to)h(p)r(oin)n(ts)f(1-3)g (of)498 4263 y(the)d(last)f(section.)498 4480 y Fl(6.1)86 b(Fluid)31 b(systems.)498 4647 y Fr(Consider)18 b(a)h(system)g(of)g Fq(N)28 b Fr(particle)19 b(in)g(\003)k Fm(\032)g Fi(R)1966 4599 y Fn(d)2004 4647 y Fr(,)e Fq(d)j Fm(\025)e Fr(2,)f(with)f (Hamiltonian)f(giv)n(en)f(in)h(Eq.\(3\),)498 4746 y(suc)n(h)29 b(that)h Fq(u)p Fr(\()p Fz(q)999 4758 y Fn(i)1027 4746 y Fr(\))d(=)f(0)j(and)g(the)h(pair)f(p)r(oten)n(tial)g Fq(\036)p Fr(\()p Fm(j)p Fz(q)p Fm(j)p Fr(\))i(is)f(p)r(ositiv)n(e,)g (with)g Fq(\036)p Fr(\(0\))d(=)e Fq(C)3262 4758 y Fp(1)3300 4746 y Fr(,)30 b(and)498 4846 y Fm(\000)p Fq(C)622 4858 y Fp(2)687 4846 y Fq(<)d(\036)828 4816 y Fk(0)852 4846 y Fr(\()p Fm(j)p Fz(q)p Fm(j)p Fr(\))i Fq(<)e Fr(0,)k(0)c Fq(<)g(C)1449 4858 y Fp(1)1487 4846 y Fq(;)14 b(C)1583 4858 y Fp(2)1648 4846 y Fq(<)27 b Fm(1)p Fr(,)32 b(for)e Fm(j)p Fq(q)s Fm(j)d Fq(>)h Fr(0.)45 b(Then)30 b(it)h(w)n(as)e(sho)n (wn)h(in)h([36,37,35])498 4946 y(that,)22 b(using)e(Maxw)n(ell)f(b)r (oundary)g(conditions)g(Eq.\(9\))h(with)g(temp)r(erature)g Fq(T)12 b Fr(\()p Fz(r)p Fr(\))23 b Fq(>)g Fr(0,)e Fz(r)i Fm(2)h Fq(@)5 b Fr(\003)498 5045 y(\(\003)36 b(a)g(regular)f(domain\))h (there)g(exists)g(a)g(unique)g(stationary)f Fq(\026)p Fr(.)63 b(F)-7 b(urthermore)35 b(this)i Fq(\026)f Fr(is)498 5163 y(absolutely)27 b(con)n(tin)n(uous)f(with)i(resp)r(ect)g(to)f(Leb) r(esgue)g(measure)f(on)h(\012)c(=)g(\003)2909 5133 y Fn(N)2990 5163 y Fm(\002)18 b Fi(R)3145 5115 y Fn(dN)3270 5163 y Fr(and)27 b(is)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(14)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 15 15 15 14 bop 498 632 a Fr(approac)n(hed,)26 b(as)h Fq(t)c Fm(!)g(1)p Fr(,)k(from)h(almost)e(an)n(y)h(initial)h(\()p Fq(P)r(;)14 b(Q)p Fr(\),)28 b Fo(i.e.)g Fr(the)g(set)f(\012)2956 602 y Fk(0)3007 632 y Fr(for)g(whic)n(h)h(the)498 732 y(approac)n(h)e(ma)n(y)h(fail)g(has)g(Leb)r(esgue)h(measure)e(zero.)623 832 y(The)f(argumen)n(t)g(mak)n(es)f(use)h(of)h(the)f(b)r(oundedness)h (of)f(the)h(force)f(acting)g(on)g(an)n(y)g(particle)498 931 y(in)33 b(the)f(in)n(terior)f(of)i(\003.)50 b(This)32 b(assures)f(that)i(an)n(y)e(particle)h(with)h(a)e(su\016cien)n(tly)i (high)f(sp)r(eed)498 1031 y(will)27 b(hit)g(the)h(b)r(oundary)e(with)h (only)f(little)i(deviation)e(from)h(a)f(straigh)n(t)g(path.)36 b(This)27 b(and)g(the)498 1130 y(fact)e(that)g(the)g(force)f Fm(\000)p Fq(\036)1288 1100 y Fk(0)1312 1130 y Fr(\()p Fm(j)p Fz(q)p Fm(j)p Fr(\))h(is)g(ev)n(erywhere)e(p)r(ositiv)n(e)h (insures)g(e\013ectiv)n(e)h(con)n(tact)f(b)r(et)n(w)n(een)498 1230 y(the)i(system)g(and)g(the)g(sto)r(c)n(hastic)f(b)r(oundaries)g (whic)n(h,)h(according)e(to)i(Eq.\(9\),)g(\\spread")e(the)498 1330 y(v)n(elo)r(cit)n(y)31 b(of)h(particles)f(whic)n(h)h(hit)g(it.)50 b(This)32 b(yields)f(something)h(lik)n(e)f(a)h(\\Harris)e(condition") 498 1429 y(whic)n(h)e(guaran)n(tees)d(existence,)j(uniqueness)f(and)g (approac)n(h)f(to)h(the)h(stationary)e(state.)623 1529 y(Using)35 b(general)f(tec)n(hnique)i(dev)n(elop)r(ed)f(in)h([6])g (Eq.\(29\))e(for)h(systems)h(in)g(con)n(tact)e(with)498 1629 y(sto)r(c)n(hastic)f(reserv)n(oirs)d(satisfying)j(detailed)g (balance)f(is)i(immediate.)54 b(It)34 b(is)f(probably)f(also)498 1728 y(p)r(ossible)26 b(to)h(pro)n(v)n(e)d(inequalit)n(y)j(\(30\))f (for)g(suc)n(h)g(systems)g(but)h(the)g(latter)f(has)g(not)g(b)r(een)h (done)498 1828 y(generally)f(as)h(far)g(as)g(for)g(w)n(e)g(kno)n(w;)g (see)g(b)r(elo)n(w.)498 2040 y Fl(6.2)86 b(Harmonic)31 b(Crystal.)498 2194 y Fr(A)c(system)f(with)h(Hamiltonian)f(giv)n(en)f (b)n(y)h(Eq.\(8\))g(in)g(whic)n(h)h(b)r(oth)f Fq(V)46 b Fr(and)26 b Fq(U)2947 2206 y Fh(i)2998 2194 y Fr(\(when)h(it)f(do)r (es)498 2293 y(not)e(v)-5 b(anish\))25 b(are)e(quadratic)g(functions)i (of)f(their)g(argumen)n(ts)f(is)h(an)g(ideal)g(harmonic)f(crystal.)498 2393 y(When)j(suc)n(h)f(a)g(system)g(is)h(placed)f(in)g(con)n(tact)g (with)h(sto)r(c)n(hastic)e(reserv)n(oirs)f(of)i(the)h(Langevin)498 2492 y(t)n(yp)r(e)35 b(the)g(resulting)e(pro)r(cess)h(and)g(th)n(us)g (also)f(the)i(stationary)e(measure)g(is)i(Gaussian)e(and)498 2592 y(one)25 b(only)g(needs)h(to)f(compute)h(the)f(co)n(v)-5 b(ariances.)34 b(This)26 b(w)n(as)e(done)h(essen)n(tially)g(explicitly) h(for)498 2692 y(a)32 b(c)n(hain)f(in)h([75].)49 b(The)32 b(most)g(imp)r(ortan)n(t)g(di\013erence)f(with)i(the)f(equilibrium)g (state)g(is)g(that)498 2791 y(there)22 b(are)f(no)n(w)h(non-v)-5 b(anishing)21 b(co)n(v)-5 b(ariances)20 b(b)r(et)n(w)n(een)i(p)r (osition)g(and)g(momen)n(tum)g(v)-5 b(ariables)498 2891 y(prop)r(ortional)26 b(to)i Fq(\016)s(T)12 b Fr(.)36 b(One)28 b(\014nds)g(uniqueness)f(and)h(approac)n(h)e(to)i(the)g (stationary)e(measure)498 2991 y Fq(\026)i Fr(that)g(satis\014es)f (Eqs.\(29\))f(and)h(\(30\))623 3090 y(As)i(already)e(men)n(tioned)i(ho) n(w)n(ev)n(er)e(the)i(heat)g(\015ux)g Fq(\026)p Fr(\(\010\))h(is)e (essen)n(tially)g(indep)r(enden)n(t)i(of)498 3190 y Fm(L)36 b Fr(and)f Fq(\024)808 3202 y Fk(L)893 3190 y Fr(de\014ned)g(in)h (Eq.\(33\))e(gro)n(ws)f(as)i Fm(L)p Fr(.)60 b(\(F)-7 b(or)35 b(the)g(case)f(of)i(\\random)d(masses")h Fq(\024)3437 3202 y Fk(L)498 3289 y Fr(gro)n(ws)21 b(as)825 3219 y Fm(p)p 894 3219 58 4 v 70 x(L)951 3259 y Fp(13)p Fn(;)p Fp(69)p Fn(;)p Fp(76)1193 3289 y Fr(\).)36 b(The)23 b(solution)g(of)g ([75])g(w)n(as)f(extended)h(to)g Fq(d)g(>)g Fr(1)g(in)g([67])g(where)f (there)498 3389 y(w)n(ere)j(also)g(considered)f(v)-5 b(arious)25 b(p)r(ossibilities)h(for)f Fq(U)2167 3401 y Fh(i)2192 3389 y Fr(,)h Fo(e.g.)f Fr(pinned)i(do)n(wn)e(ev)n (erywhere,)f(only)498 3489 y(at)h(the)g(b)r(oundary)g(or)f(no)n(where.) 34 b(Lo)r(oking)24 b(at)h(the)g(in)n(v)-5 b(arian)n(t)24 b(measure)g(in)h(the)h(limit)f Fm(L)f(!)f(1)498 3588 y Fr(one)f(\014nds)h(that)g(the)h(deca)n(y)d(of)i(the)g(p)r (osition-momen)n(tum)f(co)n(v)-5 b(ariance)21 b(is)i(rapid)f(when)h Fq(U)3310 3600 y Fh(i)3358 3588 y Fm(6)p Fr(=)f(0)498 3688 y(\(at)28 b(least)f(on)g(the)h(b)r(oundary\))f(but)i(do)r(es)e (not)h(deca)n(y)e(at)i(all)f(when)h Fq(U)2699 3700 y Fh(i)2747 3688 y Fr(=)22 b(0.)623 3788 y(The)35 b(case)f(of)h(an)g (in\014nite)h(harmonic)e(c)n(hain)g(with)i(left)g(and)e(righ)n(t)h(p)r (ortions)f(acting)h(as)498 3887 y(Hamiltonian)e(reserv)n(oirs)d(w)n(as) j(in)n(v)n(estigated)f(in)h([80].)53 b(The)33 b(results)g(are)f (qualitativ)n(ely)h(the)498 3987 y(same)d(as)f(for)h(the)h(sto)r(c)n (hastic)e(reserv)n(oirs:)39 b(the)31 b(heat)f(curren)n(t)f(remain)h (prop)r(ortional)f(to)h(the)498 4086 y(initial)h(temp)r(erature)g (di\013erence)f(as)g Fq(t)f Fm(!)f(1)j Fr(and)f(the)i(system)e(approac) n(hes)f(its)i(stationary)498 4186 y(state)d(whic)n(h)f(is)h(again)e(a)h (Gaussian)g(measure)1967 4156 y Fn(e)2001 4186 y Fr(.)498 4399 y Fl(6.3)86 b(A)n(nharmonic)31 b(Crystals.)498 4552 y Fr(The)25 b(anharmonic)e(crystal)g(coupled)i(to)f(Hamiltonian/Sto)r (c)n(hastic)f(reserv)n(oirs)f(describ)r(ed)i(b)n(y)498 4651 y(Eq.\(11\))k(has)g(b)r(een)g(in)n(v)n(estigated)g(in)g ([21,22,20].)37 b(T)-7 b(ec)n(hnical)28 b(conditions)g(on)g(the)h(gro)n (wth)e(at)498 4751 y(in\014nit)n(y)k(of)g(the)h(p)r(oten)n(tial)f(are)f (needed:)43 b(either)2082 4721 y Fp(21)2184 4751 y Fr(that)31 b Fm(V)38 b Fr(is)30 b(quadratic)g(at)h(in\014nit)n(y)h(or)3417 4721 y Fp(20)498 4851 y Fr(more)37 b(general)f(p)r(olynomial)g(gro)n (wth.)65 b(\(In)38 b(the)f(latter)g(case,)i(the)f(one-b)r(o)r(dy)f(p)r (oten)n(tial)g Fq(U)p 498 4934 1196 4 v 498 4981 a Fd(e)532 5005 y Fw(F)-6 b(or)29 b(an)i(in\014nite)f(quan)n(tum)h(harmonic)e(c)n (hain)h(with)g(a)g(sp)r(ecial)h(particle)f(sub)t(ject)g(to)h(a)f (su\016cien)n(tly)h(small)498 5084 y(non)g(harmonic)f(p)r(oten)n(tial,) j(the)e(existence)h(of)e(stationary)h(states)h(and)f(their)f(mixing)f (prop)r(ert)n(y)i(has)g(b)r(een)498 5163 y(established,)24 b(b)r(oth)h(for)e(KMS)g(states)1537 5139 y Fc(29)p Fd(;)p Fc(65)1705 5163 y Fw(and)h(for)f(SNS)2105 5139 y Fc(29)2170 5163 y Fw(.)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(15)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 16 16 16 15 bop 498 632 a Fr(gro)n(ws)31 b(more)g(slo)n(wly)h(at)g(in\014nit) n(y)h(then)g(the)g(t)n(w)n(o-b)r(o)r(dy)e(p)r(oten)n(tial)i Fq(V)19 b Fr(\).)51 b(One)33 b(assumes)e(also)498 732 y(that)f(the)h(t)n(w)n(o-b)r(o)r(dy)e(p)r(oten)n(tial)h(is)g(strictly)f (con)n(v)n(ex,)g(and)h(this)h(condition)e(alone)g(implies)3417 702 y Fp(22)498 831 y Fr(that)35 b(the)g(stationary)e(state)h(is)h (unique.)58 b(Under)34 b(these)h(conditions)f(the)h(follo)n(wing)f (results)498 931 y(hold:)558 1100 y(1.)41 b(Existence)35 b(and)g(uniqueness)g(of)g(the)h(stationary)d(state)i Fq(\026)p Fr(.)60 b(The)35 b(stationary)f(state)h(is)664 1199 y(mixing,)j Fo(i.e.)f Fr(an)n(y)e(initial)h(distribution)h(will)f (con)n(v)n(erges)e(to)i(the)g(stationary)f(state)h(as)664 1299 y Fq(t)h Fm(!)f(1)p Fr(.)61 b(The)36 b(stationary)e(state)h(has)g (a)h Fm(C)2093 1269 y Fk(1)2198 1299 y Fr(densit)n(y)g(whic)n(h)f(deca) n(ys)g(at)g(in\014nit)n(y)h(at)664 1399 y(least)29 b(as)g(fast)h(as)f (a)g(Gibbs)h(state)g(with)g(temp)r(erature)f(equal)g(to)h(the)g(maxim)n (um)f(of)h(the)664 1498 y(temp)r(erature)d(of)h(the)g(reserv)n(oirs.) 558 1669 y(2.)41 b(The)26 b(stationary)d(state)i(is)h(conducting:)35 b(One)25 b(has)g Fq(\026)p Fr(\(\010)2424 1681 y Fn(R)2479 1669 y Fr(\))e(=)g(0)h(if)i(and)f(only)g(if)h Fq(T)3224 1681 y Fn(L)3296 1669 y Fr(=)d Fq(T)3433 1681 y Fn(R)664 1769 y Fr(and)30 b Fq(\026)p Fr(\(\010)970 1781 y Fn(R)1024 1769 y Fr(\))d Fq(>)f Fr(0)j(if)h Fq(T)1372 1781 y Fn(L)1448 1769 y Fq(>)c(T)1588 1781 y Fn(R)1642 1769 y Fr(.)43 b(Linear)29 b(resp)r(onse)f(theory)h(is)h(v)-5 b(alid:)41 b(F)-7 b(or)29 b(a)g(large)f(class)664 1869 y(of)i(observ)-5 b(ables)28 b Fq(f)9 b Fr(,)31 b(the)f(exp)r(ectation)g(v)-5 b(alue)30 b Fq(\026)p Fr(\()p Fq(f)9 b Fr(\))30 b(is)g(a)f (real-analytic)f(function)j(of)f(the)664 1968 y(temp)r(erature)g (di\013erence)g Fq(\016)s(T)12 b Fr(.)44 b(In)31 b(particular,)f(near)f (equilibrium,)i(one)f(obtains,)g(with)664 2068 y Fq(T)k Fr(=)23 b(\()p Fq(T)916 2080 y Fn(L)984 2068 y Fr(+)18 b Fq(T)1116 2080 y Fn(R)1170 2068 y Fr(\))p Fq(=)p Fr(2)1593 2266 y Fq(\026)p Fr(\(\010)1735 2278 y Fn(R)1790 2266 y Fr(\))37 b(=)1971 2210 y Fq(\016)s(T)p 1971 2247 101 4 v 1972 2323 a(T)2033 2299 y Fp(2)2081 2266 y Fq(D)20 b Fr(+)e Fq(O)r Fr(\()p Fq(\016)s(T)2451 2231 y Fp(2)2489 2266 y Fr(\))c Fq(;)781 b Fr(\(34\))664 2444 y(Eq.32)24 b(for)g(the)h(co)r(e\016cien)n(t)g Fq(D)i Fr(has)d(not)h(b)r(een)g(pro) n(v)n(ed,)f(but)i(rather)d(the)j(sligh)n(tly)e(w)n(eak)n(er)664 2543 y(form)1671 2698 y Fq(D)h Fr(=)e Fq(\026)1903 2710 y Fp(0)1940 2698 y Fr(\(\010)2032 2710 y Fn(R)2087 2698 y Fr(\()p Fq(L)2176 2662 y Fk(\000)p Fp(1)2176 2720 y(0)2264 2698 y Fr(\010)2324 2710 y Fn(R)2379 2698 y Fr(\)\))14 b Fq(;)859 b Fr(\(35\))664 2852 y(where)26 b Fq(\026)953 2864 y Fp(0)1016 2852 y Fr(is)f(the)i(Gibbs)f(state)g(Eq.\(12\))f(with) h(temp)r(erature)f Fq(T)37 b Fr(and)26 b Fq(L)2938 2864 y Fp(0)3001 2852 y Fr(is)g(the)g(genera-)664 2952 y(tor)h(of)g(the)h (Mark)n(o)n(vian)d(semi-group)h Fq(S)1929 2922 y Fn(t)1924 2973 y Fp(0)1989 2952 y Fr(asso)r(ciated)g(with)i(the)g(sto)r(c)n (hastic)e(di\013eren)n(tial)664 3052 y(equations)h(\(11\))g(with)h Fq(T)1449 3064 y Fn(L)1522 3052 y Fr(=)22 b Fq(T)1658 3064 y Fn(R)1735 3052 y Fr(=)h Fq(T)12 b Fr(.)36 b(Notice)27 b(that,)h(formally)-7 b(,)27 b(one)g(has)1783 3253 y Fq(L)1840 3218 y Fk(\000)p Fp(1)1840 3275 y(0)1952 3253 y Fr(=)2039 3140 y Fj(Z)2122 3161 y Fk(1)2086 3329 y Fp(0)2207 3253 y Fq(dtS)2336 3219 y Fn(t)2331 3274 y Fp(0)3339 3253 y Fr(\(36\))664 3454 y(and,)g(inserting)f(Eq.\(36\))g (in)n(to)h(Eq.\(35\))f(yields)h(Eqs.\(32\).)35 b(In)27 b(order)f(to)h(pro)n(v)n(e)e(Eq.\(36\),)664 3554 y(one)f(needs)g (presumably)g(some)g(information)f(on)h(the)h(deca)n(y)e(of)i (correlations.)33 b(This)25 b(has)664 3654 y(not)k(b)r(een)g(obtained)g (so)f(far.)39 b(Nothing)29 b(is)g(kno)n(wn)f(ab)r(out)g(the)h(dep)r (endence)h(of)e Fq(D)j Fr(on)e Fm(L)664 3753 y Fr(and)f(th)n(us)f(ipso) g(facto)h(ab)r(out)f(the)h(v)-5 b(alidit)n(y)28 b(of)f(F)-7 b(ourier's)27 b(la)n(w.)498 3983 y Fz(7)95 b(The)32 b(Green-Kub)s(o)f (F)-8 b(orm)m(ula)498 4166 y Fr(It)27 b(is)g(clear)f(that,)i(aside)e (from)h(the)g(case)f(of)h(the)h(harmonic)e(crystal,)g(whic)n(h)h(do)r (es)f(not)h(satisfy)498 4266 y(F)-7 b(ourier's)35 b(la)n(w,)i(none)f (of)g(the)g(exact)f(results)h(quoted)g(in)g(the)g(last)g(section)f(sa)n (ys)g(an)n(ything)498 4366 y(ab)r(out)c(the)h(lo)r(cal)f(structure,)g Fo(e.g.)g Fr(ab)r(out)h(lo)r(cal)e(equilibrium)i(in)f(the)h(SNS.)g (This)f(means)g(in)498 4465 y(particular)38 b(that)h(at)g(this)h(time)g (w)n(e)e(ha)n(v)n(e)g(no)h(rigorous)e(w)n(a)n(y)h(of)h(relating)f(the)i (lo)r(cal)e(heat)498 4565 y(\015ux)c Fq(\026)p Fr(\(\010\))g(to)f(the)h (gradien)n(t)e(of)i(the)g(lo)r(cal)f(temp)r(erature)g(as)f(de\014ned)i (in)g(Eq.\(4\).)54 b(Ev)n(en)33 b(in)498 4665 y(the)k(absence)f(of)g(L) -7 b(TE)36 b(one)h(can)f(de\014ne)h(a)f(lo)r(cal)g(kinetic)h(temp)r (erature)f(b)n(y)g(means)g(of)h(the)498 4764 y(a)n(v)n(erage)e(lo)r (cal)i(kinetic)g(energy)-7 b(.)66 b(Th)n(us)37 b(for)g(the)g(crystal)g (at)g(site)h Fz(i)p Fr(,)h Fq(T)12 b Fr(\()p Fz(i)p Fr(\))39 b(=)g Fq(\026)p Fr(\()p Fz(p)3195 4734 y Fp(2)3195 4788 y Fh(i)3232 4764 y Fq(=m)p Fr(\))p Fq(=d)p Fr(.)498 4864 y(F)-7 b(or)38 b(the)i(harmonic)e(crystal)g Fq(T)12 b Fr(\()p Fz(i)p Fr(\))38 b(in)h(found)g(to)g(b)r(e)g(uniform)g(a)n(w)n (a)n(y)e(from)i(the)g(ends,)j Fo(i.e.)498 4963 y Fr(there)28 b(is)g(no)f(temp)r(erature)h(gradien)n(t.)36 b(Ev)n(en)27 b(accepting)g(suc)n(h)h(a)f(de\014nition)i(of)e(temp)r(erature)498 5063 y(\(in)38 b(n)n(umerical)e(sim)n(ulation,)j(to)e(b)r(e)h (discussed)e(later,)j Fq(T)12 b Fr(\()p Fz(i)p Fr(\))37 b(is)g(one)g(of)g(the)g(most)g(directly)498 5163 y(measured)29 b(quan)n(tities\))g(w)n(e)h(are)e(completely)i(lac)n(king)e(at)i(this)g (p)r(oin)n(t)g(an)n(y)f(rigorous)e(or)h(ev)n(en)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(16)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 17 17 17 16 bop 498 632 a Fr(formal)38 b(connection)h(b)r(et)n(w)n(een)g(the) g Fq(\024)g Fr(de\014ned)g(in)g(Eq.\(33\))f(and)h(the)g(usual)g (Green-Kub)r(o)498 732 y(form)n(ula)29 b(for)h(the)g(conductivit)n(y)g (whic)n(h)g(is)f(de\014ned)i(in)f(terms)g(of)f(the)i(time)f(ev)n (olution)f(of)h(an)498 831 y(isolated)i(system)f(in)i(equilibrium.)50 b(Denoting)33 b(b)n(y)2172 810 y(^)2158 831 y Fq(S)2214 801 y Fn(t)2209 852 y Fp(0)2279 831 y Fr(the)f(Hamiltonian)g(ev)n (olution)g(of)g(the)498 931 y(isolated)27 b(system)g(the)h(thermal)g (conductivit)n(y)f Fq(\024)2056 943 y Fn(GK)2199 931 y Fr(is)h(giv)n(en)e(b)n(y)2615 901 y Fp(53)p Fn(;)p Fp(66)1376 1120 y Fq(\024)1424 1132 y Fn(GK)1576 1120 y Fr(=)68 b(lim)1678 1174 y Fn(L)p Fk(!1)1882 1064 y Fq(L)p 1879 1101 63 4 v 1879 1177 a(A)1989 1064 y Fr(1)p 1961 1101 98 4 v 1961 1177 a Fq(T)2022 1153 y Fp(2)2083 1007 y Fj(Z)2166 1028 y Fk(1)2129 1196 y Fp(0)2250 1120 y Fq(dt)14 b Fm(h)p Fr(\010)2443 1099 y(^)2429 1120 y Fq(S)2485 1086 y Fn(t)2480 1141 y Fp(0)2517 1120 y Fr(\010)p Fm(i)730 b Fr(\(37\))498 1314 y(where)24 b Fm(h\001i)g Fr(denotes)g(the)h(micro)r(canonical)d(a)n(v)n(erage)f(and)j(the)g (energy)f(densit)n(y)h(is)g(c)n(hosen)f(suc)n(h)498 1414 y(that)31 b(it)g(corresp)r(onds)d(to)i(the)h(thermo)r(dynamic)f(energy) g(at)g(temp)r(erature)g Fq(T)41 b Fr(\(since)31 b(w)n(e)f(are)498 1513 y(in)c(equilibrium)f(this)g(is)g(the)h(same)e(as)h(the)g(kinetic)h (temp)r(erature\).)36 b(If)25 b(the)h(total)f(momen)n(tum)498 1613 y(\005)i(is)g(conserv)n(ed)e(it)i(has)f(to)h(b)r(e)g(set)g(equal)f (to)g(zero.)36 b(Alternativ)n(ely)26 b(one)g(ma)n(y)g(use)h(in)g (Eq.\(37\))498 1713 y(truncated)21 b(correlation)e(functions)i Fm(h)p Fr(\010)1736 1692 y(^)1722 1713 y Fq(S)1778 1682 y Fn(t)1773 1733 y Fp(0)1810 1713 y Fr(\010)p Fm(i)1902 1682 y Fn(T)1902 1735 y(E)s(;)p Fp(\005)2050 1713 y Fr(=)i Fm(h)p Fr(\010)2244 1692 y(^)2230 1713 y Fq(S)2286 1682 y Fn(t)2281 1733 y Fp(0)2318 1713 y Fr(\010)p Fm(i)2410 1725 y Fn(E)s(;)p Fp(\005)2540 1713 y Fm(\000)5 b(h)p Fr(\010)p Fm(i)2734 1682 y Fp(2)2734 1735 y Fn(E)s(;)p Fp(\005)2880 1713 y Fr(and)21 b(then)g(a)n(v)n(erage)498 1825 y(o)n(v)n(er)27 b Fq(E)33 b Fr(and)28 b(\005)h(using)e(the)i (canonical)e(distribution)2175 1795 y Fn(f)2218 1825 y Fr(.)39 b(One)28 b(exp)r(ects)g(that)h(the)f(equiv)-5 b(alence)498 1925 y(of)28 b(equilibrium)f(ensem)n(bles)g(will)h(extend) g(also)f(to)g(this)h(case.)623 2025 y(The)34 b(Green-Kub)r(o)g(form)n (ula,)i(Eq.\(37\),)f(also)f(mak)n(es)g(sense)g(for)g(a)h(system)f(with) h(a)g(few)498 2124 y(degree)28 b(of)g(freedom)g(where)g(it)h(can)g(b)r (e)g(related)f(to)g(the)h(v)-5 b(ariance)28 b(in)g(the)h(\015uctuation) g(of)g(the)498 2224 y(curren)n(t;)i(see)g(the)g(article)f(of)g(Bunimo)n (vic)n(h)g(and)g(Sp)r(ohn)2336 2194 y Fp(11)2437 2224 y Fr(for)g(a)h(discussion.)45 b(There)30 b(is)g(no)498 2324 y(clear)j(connection)g(ho)n(w)n(ev)n(er)e(b)r(et)n(w)n(een)j(the)g (in)n(tegral)e(in)i(Eq.\(37\))f(for)g(a)g(small)h(system)f(and)498 2423 y(the)27 b Fq(\024)g Fr(in)g(F)-7 b(ourier's)26 b(la)n(w.)36 b(F)-7 b(rom)27 b(a)f(mathematical)h(p)r(oin)n(t)g(of)g (view)f(it)i(is)e(not)h(ev)n(en)g(clear)f(ho)n(w)498 2523 y(to)g(pro)n(v)n(e)e(equiv)-5 b(alence)26 b(for)f(macroscopic)f (systems,)i Fo(i.e.)g Fr(sho)n(w)f(that)h Fq(\024)d Fr(=)g Fq(\024)2898 2535 y Fn(GK)3013 2523 y Fr(.)37 b(It)26 b(w)n(ould)g(b)r(e)498 2622 y(nice)i(to)f(\014nd)h(ev)n(en)f(a)h (formal)e(argumen)n(t)h(establishing)g(the)h(equiv)-5 b(alence.)498 2846 y Fz(8)95 b(En)m(trop)m(y)33 b(Pro)s(duction)f(and)g (Large)h(Deviations)498 3028 y Fr(The)j(prop)r(er)g(de\014nition)g(of)g (microscopic)f(en)n(trop)n(y)g(pro)r(duction)g(has)h(attracted)g(m)n (uc)n(h)f(at-)498 3128 y(ten)n(tion)30 b(in)g(recen)n(t)f(y)n(ears.)42 b(The)30 b(in)n(terest)f(comes)g(from)h(the)g(observ)-5 b(ation)28 b(of)i(an)g(in)n(teresting)498 3227 y(symmetry)d(prop)r(ert) n(y)g(in)h(the)g(large)e(deviation)i(functional)f(asso)r(ciated)g(to)g (the)i(phase)e(space)498 3327 y(v)n(olume)e(con)n(traction)e(rate)h(in) i(thermostated)e(systems.)36 b(This)25 b(prop)r(ert)n(y)f(w)n(as)g (\014rst)h(observ)n(ed)498 3427 y(n)n(umerically)32 b(in)i([26])e(and)h (then)h(pro)n(v)n(ed)d(under)i(strong)f(h)n(yp)r(erb)r(olicit)n(y)g (condition)h(in)h([31].)498 3526 y(F)-7 b(or)26 b(suc)n(h)h(systems)g (the)g(phase)f(space)g(v)n(olume)h(con)n(traction)e(has)i(strong)e (connection)i(to)g(the)498 3626 y(en)n(trop)n(y)22 b(pro)r(duction)i (in)g(Section)f(5.)35 b(Using)24 b(this)g(connection)f(the)h (\015uctuation)f(theorem)h(has)498 3725 y(b)r(een)32 b(extended)g(to)f(large)f(deviations)h(of)h(the)g(en)n(trop)n(y)e(pro)r (duction)h(of)h(v)-5 b(arious)30 b(sto)r(c)n(hastic)498 3825 y(systems)779 3795 y Fp(52)p Fn(;)p Fp(58)935 3825 y Fr(.)62 b(F)-7 b(or)35 b(crystals)f(with)j(sto)r(c)n(hastic)e(reserv) n(oirs)d(or)j(for)h(the)g(mo)r(del)g(considered)498 3925 y(in)29 b(Section)g(4.3,)f(only)g(formal)g(pro)r(ofs)g(of)g(the)h (Galla)n(v)n(otti-Cohen)e(\015uctuation)i(theorem)f(are)498 4024 y(a)n(v)-5 b(ailable)26 b(so)h(far.)623 4124 y(The)41 b(Galla)n(v)n(otti-Cohen)e(\015uctuation)i(theorem)f(can)h(b)r(e)g (form)n(ulated)g(as)f(follo)n(ws:)63 b(In)498 4224 y(b)r(oth)27 b(deterministic)f(and)g(sto)r(c)n(hastic)g(systems)f(one)h(iden)n (ti\014es)h(an)f(observ)-5 b(able)24 b Fq(\033)s Fr(\()p Fq(X)7 b Fr(\))27 b(as)f(the)498 4323 y(\\en)n(trop)n(y)i(pro)r (duction")h(although)g(its)h(iden)n(ti\014cation)g(with)g(the)g(thermo) r(dynamic)f(en)n(trop)n(y)498 4423 y(pro)r(duction)e(is)h(not)f(alw)n (a)n(ys)f(clear.)36 b(If)28 b(one)f(no)n(w)g(considers)f(the)i(ergo)r (dic)f(a)n(v)n(erage,)1520 4623 y Fq(\033)1567 4635 y Fn(t)1610 4623 y Fr(\()p Fq(X)7 b Fr(\))23 b(=)1871 4567 y(1)p 1871 4604 42 4 v 1877 4680 a Fq(t)1936 4510 y Fj(Z)2019 4530 y Fn(t)1982 4699 y Fp(0)2062 4623 y Fq(ds\033)17 b Fr(\()q Fq(X)7 b Fr(\()p Fq(s)p Fr(\)\))887 b(\(38\))498 4816 y(where)22 b(for)g(deterministic)h(dynamics)f Fq(X)7 b Fr(\()p Fq(t)p Fr(\))22 b(is)h(tra)5 b(jectory)20 b(of)j(the)g (system)f(while)g(for)g(sto)r(c)n(has-)498 4916 y(tic)31 b(dynamics)g Fq(X)7 b Fr(\()p Fq(t)p Fr(\))31 b(is)g(a)f(particular)g (realization)g(of)g(the)i(random)e(pro)r(cess.)46 b(Then)31 b(b)n(y)g(the)p 498 5004 1196 4 v 498 5060 a Fd(f)538 5084 y Fw(If)e(one)h(do)r(es)g(not)h(\014x)f(\005)f(in)g(an)h (equilibrium)d(ensem)n(ble)i(for)g(whic)n(h)h(the)g(total)g(momen)n (tum)e(is)h(conserv)n(ed)498 5163 y(then)c(the)f(in)n(tegral)g(in)g (Eq.\(37\))g(is)f(div)n(ergen)n(t)1727 5139 y Fc(74)1817 5163 y Fw(but)i(this)e(do)r(es)h(not)h(sa)n(y)f(an)n(ything)h(ab)r(out) g Ft(\024)3028 5174 y Fd(GK)3134 5163 y Fw(.)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(17)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 18 18 18 17 bop 498 632 a Fr(large)30 b(deviation)g(principle)h(\(assumed)g (to)g(hold\))g(there)g(exists)g(an)g Fq(e)p Fr(\()p Fq(p)p Fr(\))g(suc)n(h)g(that)g(for)g(an)n(y)498 732 y(in)n(terv)-5 b(al)27 b Fq(I)1371 981 y Fr(lim)1349 1031 y Fn(t)p Fk(!1)1521 981 y Fr(log)14 b(Prob)n(\()p Fq(\033)1897 993 y Fn(t)1927 981 y Fr(\()p Fq(x)p Fr(\))25 b Fm(2)e Fq(I)7 b Fr(\))23 b(=)29 b(inf)2327 1034 y Fn(p)p Fk(2)p Fn(I)2454 981 y Fq(e)p Fr(\()p Fq(p)p Fr(\))14 b Fq(:)703 b Fr(\(39\))498 1274 y(The)30 b(\015uctuation)g(theorem)f(asserts)f(that)i(the)g(o)r (dd)g(part)f(of)h Fq(e)p Fr(\()p Fq(p)p Fr(\))g(is)f(linear)g(with)h (slop)r(e)g Fm(\000)p Fr(1,)498 1374 y Fo(i.e.)e Fq(e)p Fr(\()p Fq(p)p Fr(\))18 b Fm(\000)g Fq(e)p Fr(\()p Fm(\000)p Fq(p)p Fr(\))23 b(=)g Fm(\000)p Fq(p)p Fr(.)623 1473 y(In)e(the)h(deterministic)g(case)e(the)i(en)n(trop)n(y)e(pro)r (duction)h(v)-5 b(ariable)21 b(that)h(one)f(consider)f(is,)j(as)498 1573 y(already)g(men)n(tioned,)j(the)f(phase)f(space)g(con)n(traction)f (rate)h(whic)n(h,)h(in)g(the)g(case)f(of)h(Gaussian)498 1673 y(thermostats,)j(coincides)g(with)h(en)n(trop)n(y)e(pro)r(duction) h(as)f(w)n(e)h(de\014ned)h(it)g(in)f(Subsection)h(5.2.)498 1772 y(This)35 b(can)f(b)r(e)h(easily)f(seen)h(b)n(y)f(computing)h(the) g(div)n(ergence)e(of)i(the)g(equations)f(of)g(motion)498 1872 y(quoted)26 b(at)g(the)g(end)h(of)f(Subsection)g(4.4.)35 b(Already)25 b(for)h(the)g(case)g(of)g(the)g(Nos)n(\023)-39 b(e-Ho)r(o)n(v)n(er)22 b(ther-)498 1971 y(mostat)31 b(the)g(phase)g (space)f(con)n(traction)f(and)i(the)h(en)n(trop)n(y)d(pro)r(duction)i (are)f(not)h(the)h(same)498 2071 y(quan)n(tities)e(although)f(their)h (a)n(v)n(erage)d(v)-5 b(alue)29 b(in)h(the)h(SNS)f(are)f(the)h(same.)43 b(F)-7 b(or)29 b(anharmonic)498 2171 y(c)n(hains)24 b(with)h(a)f(t)n(w) n(o)g(b)r(o)r(dy)g(p)r(oten)n(tial)h(giv)n(en)f Fq(V)18 b Fr(\()p Fq(x)p Fr(\))25 b(=)d Fq(x)2257 2141 y Fp(2)2307 2171 y Fr(+)12 b Fq(\014)t(x)2482 2141 y Fp(4)2545 2171 y Fr(the)24 b(v)-5 b(alidit)n(y)25 b(of)f(the)h(\015uctua-)498 2270 y(tion)i(theorem)g(has)f(b)r(een)i(c)n(hec)n(k)n(ed)e(n)n (umerically)g(in)h([59])g(for)f(the)i(en)n(trop)n(y)d(pro)r(duction)i (\(but)498 2370 y(not)h(for)f(the)h(phase)f(space)g(con)n(traction\).) 623 2470 y(The)h(connection)g(b)r(et)n(w)n(een)g(the)g(Galla)n(v)n (otti-Cohen)f(\015uctuation)h(theorem)g(and)g(its)g(v)-5 b(ar-)498 2569 y(ious)33 b(generalizations)e(can)i(b)r(e)h(related)f (to)g(the)h(follo)n(wing)e(observ)-5 b(ation:)47 b(F)-7 b(or)33 b(a)g(sto)r(c)n(hastic)498 2669 y(mo)r(del)27 b(describ)r(ed)f(b)n(y)h(a)f(Mark)n(o)n(v)e(pro)r(cess)i(one)g(can)h (consider)e(the)i(measure)f Fq(P)38 b Fr(on)27 b(the)g(path)498 2768 y(space)21 b(induced)h(b)n(y)f(the)h(ev)n(olution.)35 b(Let)21 b Fq(\036)i Fr(b)r(e)f(a)f(path)h(leading)f(from)g Fq(X)28 b Fr(to)21 b Fq(Y)41 b Fr(in)22 b(phase)f(space)498 2868 y(in)j(the)g(time)g(in)n(terv)-5 b(al)23 b([0)p Fq(;)14 b(t)p Fr(])23 b(and)g(consider)g(no)n(w)g(the)h(transformation) d(whic)n(h)j(maps)f(the)h(path)498 2968 y Fq(\036)k Fr(in)n(to)e(the)i (time)f(rev)n(ersed)e(path)i(leading)f(from)h Fq(I)7 b(Y)46 b Fr(to)26 b Fq(I)7 b(X)g Fr(,)27 b(where)f Fq(I)34 b Fr(is)27 b(the)g(in)n(v)n(olution)f(on)498 3067 y(phase)e(space)g (whic)n(h)g(rev)n(erses)e(the)j(v)n(elo)r(cities)f(of)g(all)g (particles.)35 b(This)25 b(transformation)d(maps)498 3167 y(the)30 b(measure)f Fq(P)42 b Fr(in)n(to)29 b(a)h(new)f(measure)p 1804 3100 65 4 v 29 w Fq(P)42 b Fr(whic)n(h)30 b(is)f(absolutely)g(con) n(tin)n(uous)g(with)h(resp)r(ect)498 3267 y(to)e Fq(P)39 b Fr(with)28 b(Radon-Nyk)n(o)r(dym)e(deriv)-5 b(ativ)n(e)27 b(giv)n(en)g(b)n(y)1045 3512 y Fq(dP)p 1045 3550 109 4 v 1045 3633 a(d)p 1088 3566 65 4 v(P)1163 3569 y Fr(\()p Fq(\036)p Fr(\))38 b(=)e(exp)1556 3452 y Fj(\022)1617 3569 y Fq(R)q Fr(\()p Fq(\036)p Fr(\()p Fq(t)p Fr(\)\))20 b Fm(\000)e Fq(R)q Fr(\()p Fq(\036)p Fr(\(0\)\))h(+)2376 3456 y Fj(Z)2459 3476 y Fn(t)2422 3644 y Fp(0)2502 3569 y Fq(\033)s Fr(\()p Fq(\036)p Fr(\()p Fq(s)p Fr(\)\))p Fq(ds)2850 3452 y Fj(\023)2927 3569 y Fq(;)389 b Fr(\(40\))498 3868 y(where)36 b Fq(\033)j Fr(is)d(the)g(en)n(trop)n(y)f(pro)r (duction)h(de\014ned)g(in)g(Eq.\(28\).)61 b(Eq.)h(\(40\))36 b(states,)h(roughly)498 3967 y(sp)r(eaking,)30 b(that)g(the)h (probabilit)n(y)e(of)h(a)f(time)i(rev)n(ersed)d(path)i(is)g(equal)g (\(up)g(to)g(a)g(b)r(oundary)498 4067 y(term\))21 b(to)f(the)h (probabilit)n(y)e(of)i(tah)n(t)f(path)g(times)h(the)g(exp)r(onen)n (tial)f(of)g(the)h(in)n(tegrated)e(en)n(trop)n(y)498 4166 y(pro)r(duction)30 b(along)e(this)i(path.)44 b(This)30 b(prop)r(ert)n(y)f(can)g(b)r(e)h(assumed)g(as)f(a)g(general)g (de\014nition)498 4266 y(of)d(en)n(trop)n(y)e(pro)r(duction)1286 4236 y Fp(64)1381 4266 y Fr(and)i(can)f(b)r(e)h(seen)f(as)g(a)g (generalization)e(of)j(detailed)f(balance.)36 b(In)498 4366 y(fact)25 b(in)g(the)g(equilibrium)g(case)f(\()p Fq(T)1585 4378 y Fn(L)1657 4366 y Fr(=)f Fq(T)1794 4378 y Fn(R)1871 4366 y Fr(=)g Fq(T)12 b Fr(\),)25 b(the)g(righ)n(t)f(hand)g (side)h(of)g(Eq.\(40\))f(dep)r(ends)498 4465 y(on)g Fq(\036)i Fr(only)e(through)g(its)g(endp)r(oin)n(ts)h(and)f(is)h(equal)f(to)g (exp)2357 4398 y Fj(\000)2395 4465 y Fq(T)2456 4435 y Fk(\000)p Fp(1)2544 4465 y Fr(\()p Fq(H)7 b Fr(\()p Fq(\036)p Fr(\()p Fq(t)p Fr(\)\))20 b Fm(\000)e Fq(H)7 b Fr(\()p Fq(\036)p Fr(\(0\)\)\))3289 4398 y Fj(\001)3353 4465 y Fr(and)498 4565 y(this)28 b(is)g(precisely)e(detailed)i(balance.)623 4665 y(F)-7 b(or)35 b(thermostated)f(systems)h(the)h(ab)r(o)n(v)n(e)e (calculation)h(is)g(quite)h(delicate)g(b)r(ecause,)h(al-)498 4764 y(though)j(one)f(can)h(still)g(consider)f(the)i(measure)e Fq(P)51 b Fr(on)40 b(the)h(path)f(space,)i(it)e(will)h(not)f(b)r(e)498 4864 y(absolutely)31 b(con)n(tin)n(uous)g(with)i(resp)r(ect)e(to)h(the) g(time)h(rev)n(ersed)d(measure)p 2902 4797 V 31 w Fq(P)12 b Fr(.)50 b(The)32 b(original)498 4963 y(pro)r(of)23 b(of)h(the)g(\015uctuation)g(theorem)f(in)h([31])f(can)h(b)r(e)g (though)n(t)f(of)h(as)f(based)h(on)f(the)h(construc-)498 5063 y(tion)g(of)f(a)g(series)f(of)h(appro)n(ximan)n(ts)e(to)j(the)f (measure)g Fq(P)35 b Fr(whic)n(h)23 b(satis\014es)g(a)g(relation)f (formally)498 5163 y(similar)27 b(to)g(Eq.\(40\).)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(18)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 19 19 19 18 bop 498 628 a Fz(9)95 b(Lo)s(cal)32 b(equilibrium)498 810 y Fr(Since)d(w)n(e)g(ha)n(v)n(e)f(nothing)h(to)g(sa)n(y)f(ab)r(out) h(L)-7 b(TE)29 b(in)h(Hamiltonian)f(heat)g(conducting)f(systems)498 909 y(w)n(e)39 b(discuss)f(brie\015y)g(some)h(mo)r(dels)f(where)h(the)g (Hamiltonian)f(dynamics)h(in)g(the)g(bulk)g(is)498 1009 y(mo)r(di\014ed)23 b(b)n(y)g(the)g(addition)g(of)g(sto)r(c)n(hastic)f (forces.)34 b(These)22 b(mo)r(dels)h(sho)n(w)f(that)h(a)f(mec)n(hanism) 498 1109 y(su\016cien)n(tly)e(strong)e(to)i(destro)n(y)e(the)i (coherence)e(of)i(the)g(phonons)f(do)r(es)g(indeed)h(pro)r(duce)g(b)r (oth)498 1208 y(a)i(temp)r(erature)f(gradien)n(t)g(and)h(a)f(normal)g (thermal)g(conductivit)n(y)-7 b(,)23 b(ev)n(en)f(in)g(one)g (dimensional)498 1308 y(systems.)623 1408 y(The)d(\014rst)g(suc)n(h)f (mo)r(del)1364 1377 y Fp(7)p Fn(;)p Fp(76)1507 1408 y Fr(is)h(a)f(harmonic)g(c)n(hain)h(coupled)g(to)g(self-consisten)n(t)f (reserv)n(oirs.)498 1507 y(Eac)n(h)25 b(particle)g(of)h(the)h(c)n(hain) e(is)h(coupled)g(to)g(its)g(o)n(wn)f(sto)r(c)n(hastic)g(reserv)n(oir)f (at)h(temp)r(erature)498 1607 y Fq(T)547 1619 y Fn(i)574 1607 y Fr(,)d Fq(i)h Fr(=)g(1)p Fq(;)14 b Fm(\001)g(\001)g(\001)f Fq(;)h(N)9 b Fr(,)21 b(mo)r(deled)g(b)n(y)f(Langevin)g(dynamics.)34 b(The)20 b(reserv)n(oirs)e(coupled)i(to)g(the)h(par-)498 1706 y(ticles)26 b(in)h(the)g(bulk)f(supp)r(osedly)g(sim)n(ulate)g(the) g(e\013ect)h(of)f(strong)f(anharmonic)g(in)n(teractions.)498 1806 y(The)f(temp)r(erature)g(of)g(the)h(reserv)n(oirs)c(coupled)j(to)g (the)h(\014rst)f(and)g(last)f(particles)h(are)f(at)h(\014xed)498 1906 y(temp)r(erature)e Fq(T)1015 1918 y Fp(1)1075 1906 y Fr(=)h Fq(T)1212 1918 y Fn(L)1283 1906 y Fr(and)f Fq(T)1488 1918 y Fn(N)1574 1906 y Fr(=)h Fq(T)1711 1918 y Fn(R)1765 1906 y Fr(.)35 b(The)22 b(temp)r(erature)g(of)h(the)f(remaining)g (reserv)n(oirs)d(is)498 2005 y(\014xed)j(b)n(y)f(a)h(self-consistency)e (condition:)34 b(one)22 b(requires,)f(that)h(in)g(the)g(stationary)f (state)g(there)498 2105 y(is)30 b(no)f(net)h(energy)f(exc)n(hange)f(b)r (et)n(w)n(een)i(the)g(reserv)n(oirs)d(and)i(the)h(particles)f(in)h(the) g(bulk.)44 b(It)498 2205 y(is)25 b(argued)848 2174 y Fp(7)910 2205 y Fr(that)g(in)g(the)g(limit)g(of)f(large)g Fq(N)9 b Fr(,)25 b(the)g(system)f(exhibits)h(a)f(temp)r(erature)g (gradien)n(t)498 2304 y(and)k(F)-7 b(ourier's)26 b(la)n(w)h(is)g (indeed)h(satis\014ed)1791 2274 y Fp(76)1861 2304 y Fr(.)623 2404 y(Another)18 b(mo)r(del)1178 2374 y Fp(17)1267 2404 y Fr(is)h(a)f(one-dimensional)g(c)n(hain)g(of)h(quan)n(tum)g(mec)n (hanical)f(atoms)h(\(eac)n(h)498 2504 y(of)29 b(whic)n(h)f(has)h(a)f (\014nite)h(n)n(um)n(b)r(er)g(of)f(energy)g(lev)n(els\).)40 b(The)28 b(\014rst)h(and)f(the)i(last)e(atoms)g(of)h(the)498 2603 y(c)n(hain)23 b(are)f(eac)n(h)g(coupled)h(to)g(heat)g(reserv)n (oirs)e(at)i(temp)r(eratures)f Fq(T)2620 2615 y Fn(L)2693 2603 y Fr(and)h Fq(T)2899 2615 y Fn(R)2953 2603 y Fr(.)35 b(The)23 b(atoms)g(in)498 2703 y(the)g(c)n(hains)g(are)e(not)i (directly)g(coupled)g(to)f(eac)n(h)g(other,)i(but)f(eac)n(h)f(pair)h (of)f(nearest)g(neigh)n(b)r(ors)498 2802 y(in)g(the)h(c)n(hain)e(is)h (coupled)g(to)g(a)g(heat)g(reserv)n(oir.)32 b(Since)22 b(eac)n(h)f(atom)h(has)f(only)h(a)g(\014nite)g(n)n(um)n(b)r(er)498 2902 y(of)27 b(energy)f(lev)n(els,)h(it)h(is)f(p)r(ossible)g(to)g(c)n (ho)r(ose)f(the)i(coupling)e(to)h(the)h(in)n(termediate)f(reserv)n (oirs)498 3002 y(in)32 b(suc)n(h)g(a)f(w)n(a)n(y)g(that)h(no)g(energy)f (is)h(exc)n(hanged)e(b)r(et)n(w)n(een)i(the)g(particles)g(in)g(the)g (bulk)g(and)498 3101 y(the)j(reserv)n(oirs.)53 b(Compared)33 b(to)g(the)i(previous)e(mo)r(del,)j(no)d(self-consistency)g(condition)h (is)498 3201 y(needed)41 b(and)g(the)g(transfer)f(of)h(energy)f(b)r(et) n(w)n(een)h(the)g(in)n(termediate)g(reserv)n(oirs)d(and)j(the)498 3301 y(system)c(alw)n(a)n(ys)e(v)-5 b(anishes,)39 b(not)e(only)f(in)i (the)f(stationary)f(state.)64 b(The)38 b(mo)r(del)f(is)g(studied)498 3400 y(in)31 b(the)g(w)n(eak)f(coupling)g(\(or)g(V)-7 b(an)31 b(Ho)n(v)n(e\))f(limit,)i(where)e(the)h(coupling)f(b)r(et)n(w)n (een)h(the)g(c)n(hains)498 3500 y(and)g(all)g(reserv)n(oirs)e(go)r(es)h (to)h(zero)f(and)h(the)h(time)g(is)f(suitably)g(rescaled.)47 b(In)31 b(this)h(limit)g(the)498 3599 y(ev)n(olution)27 b(is)h(describ)r(ed)g(b)n(y)g(a)g(quan)n(tum)g(semigroup.)37 b(F)-7 b(or)28 b(a)f(suitable)h(c)n(hoice)g(of)g(couplings)498 3699 y(the)20 b(mo)r(del)g(can)f(b)r(e)h(solv)n(ed)f(exactly)g(and)h (it)g(exhibits)g(a)f(temp)r(erature)g(gradien)n(t)g(and)g(F)-7 b(ourier's)498 3799 y(la)n(w.)623 3898 y(Another)26 b(mo)r(del)i (considered)1600 3868 y Fp(48)1697 3898 y Fr(consists)e(of)h(a)g(c)n (hain)f(of)h(uncoupled)g(harmonic)f(oscilla-)498 3998 y(tors)j(\()p Fq(U)756 4010 y Fn(i)784 3998 y Fr(\()p Fq(q)853 4010 y Fn(i)881 3998 y Fr(\))e(=)g Fq(q)1072 3968 y Fp(2)1069 4020 y Fn(i)1109 3998 y Fq(=)p Fr(2,)j Fq(V)19 b Fr(\()p Fq(q)1382 4010 y Fn(i)1430 3998 y Fm(\000)h Fq(q)1552 4010 y Fn(j)1587 3998 y Fr(\))27 b(=)g(0\).)44 b(The)30 b(oscillators)e(at)i(the)h(b)r(oundary)e(are)g(coupled)498 4098 y(to)34 b(heat)f(reserv)n(oirs)d(mo)r(deled)k(b)n(y)f(Glaub)r(er)h (pro)r(cesses)e(whic)n(h)h(thermalize)g(the)h(oscillators)498 4197 y(according)h(to)i(the)g(Gibbs)g(distribution)f(at)h(temp)r (eratures)f Fq(T)2530 4209 y Fn(L)2616 4197 y Fr(and)h Fq(T)2836 4209 y Fn(R)2890 4197 y Fr(.)64 b(The)37 b(energy)e(is)498 4297 y(exc)n(hanged)30 b(b)r(et)n(w)n(een)i(the)g(oscillators)e(in)i (the)g(c)n(hain)f(according)f(to)h(the)h(follo)n(wing)f(\(micro-)498 4396 y(canonical)f(pro)r(cedure\):)43 b(at)30 b(eac)n(h)h(pair)f(of)h (nearest)f(neigh)n(b)r(or)g(sites,)h(there)g(is)g(a)f(clo)r(c)n(k)g (with)498 4496 y(exp)r(onen)n(tial)35 b(la)n(w,)i(when)f(it)g(rings)e (the)i(energy)f(of)g(the)h(pair)f(of)h(particle)f(is)g(redistributed) 498 4596 y(in)c(a)f(uniform)h(w)n(a)n(y)-7 b(,)30 b(k)n(eeping)g(the)g (total)h(energy)e(of)i(the)g(system)f(constan)n(t.)45 b(This)30 b(mo)r(del)h(is)498 4695 y(exactly)c(solv)-5 b(able)27 b(and)g(satis\014es)g(F)-7 b(ourier's)27 b(la)n(w.)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(19)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 20 20 20 19 bop 498 628 a Fz(10)95 b(Numerical)30 b(results)498 810 y Fr(The)h(a)n(v)-5 b(ailabilit)n(y)29 b(of)i(fast)g(computers)f (has)g(p)r(ermitted)h(the)g(in)n(v)n(estigation)f(of)g(heat)h(curren)n (t)498 909 y(carrying)k(SNS)h(via)g(n)n(umerical)g(sim)n(ulations.)62 b(Man)n(y)36 b(suc)n(h)g(sim)n(ulations)f(ha)n(v)n(e)h(b)r(een)g(p)r (er-)498 1009 y(formed,)24 b(see)f(e.g.)35 b([12,73,59,60,42,40,1,34]) 17 b(for)23 b(recen)n(t)g(w)n(orks)f(on)h(v)-5 b(arious)22 b(mo)r(dels.)36 b(Earlier)498 1109 y(w)n(orks)27 b(w)n(ere)h(not)g(alw) n(a)n(ys)f(consisten)n(t)h(but)h(recen)n(tly)f(careful)g(sim)n (ulations)g(of)h(b)r(oth)g(one)f(and)498 1208 y(t)n(w)n(o)22 b(dimensional)g(crystals)g(ha)n(v)n(e)f(b)r(ecome)i(a)n(v)-5 b(ailable)2195 1178 y Fp(62)2264 1208 y Fr(.)35 b(A)23 b(coheren)n(t)f(picture)h(no)n(w)f(seems)g(to)498 1308 y(emerge)28 b(from)g(the)h(n)n(umerical)f(results)1741 1278 y Fp(62)1811 1308 y Fr(.)40 b(W)-7 b(e)29 b(describ)r(e)g(this)g (brie\015y)f(and)g(refer)g(the)h(reader)498 1408 y(to)f([61])e(for)h(a) h(more)e(complete)i(o)n(v)n(erview.)623 1507 y(F)-7 b(or)27 b(ob)n(vious)g(reasons,)f(it)j(is)f(con)n(v)n(enien)n(t)f(in)h(n)n (umerical)g(w)n(orks)e(to)i(k)n(eep)g(the)g(n)n(um)n(b)r(er)g(of)498 1607 y(degrees)34 b(of)i(freedom)f(as)g(lo)n(w)g(as)g(p)r(ossible.)60 b(On)35 b(the)h(other)f(hand)h(sto)r(c)n(hastic)e(di\013eren)n(tial)498 1706 y(equations,)42 b(lik)n(e)e(the)g(ones)f(used)h(in)g(the)g (Langevin)f(sto)r(c)n(hastic)g(reserv)n(oirs)d(require)j(v)n(ery)498 1806 y(go)r(o)r(d)32 b(random)f(n)n(um)n(b)r(er)h(generators)d(and)j (presen)n(t)g(n)n(umerical)f(problems)h(connected)f(with)498 1906 y(the)26 b(singularit)n(y)d(of)j(the)f(co)n(v)-5 b(ariance)23 b(of)i(the)h(white)f(noise.)36 b(F)-7 b(or)24 b(this)i(reason)d(the)j(sim)n(ulations)498 2005 y(done)39 b(using)g(the)g(deterministic)g(thermostat)g(are)f(easier)g(and)h(lik)n (ely)f(more)g(reliable.)71 b(In)498 2105 y(an)n(y)29 b(case)h(there)f(is)h(go)r(o)r(d)g(agreemen)n(t)f(\(in)h(the)h(data)e (if)i(not)f(alw)n(a)n(ys)e(in)i(the)h(in)n(terpretation\))498 2205 y(b)r(et)n(w)n(een)d(sim)n(ulations)e(with)j(di\013eren)n(t)e (thermostats.)623 2304 y(Since)32 b(it)h(is)g(not)f(easy)g(to)g(lo)r (ok)g(for)g(in)n(v)-5 b(arian)n(t)31 b(measures)g(with)i(n)n(umerical)f (sim)n(ulations)498 2404 y(one)i(t)n(ypically)f(computes)h(the)g(time)g (a)n(v)n(erage)e(of)h(a)h(few)g(in)n(teresting)f(observ)-5 b(ables)33 b(along)f(a)498 2504 y(giv)n(en)j(tra)5 b(jectory)-7 b(.)60 b(Assuming)36 b(the)g(v)-5 b(alidit)n(y)36 b(of)f(Eq.\(19\))g (this)h(represen)n(ts)f(a)n(v)n(erages)d(with)498 2603 y(resp)r(ect)f(to)h(the)g(in)n(v)-5 b(arian)n(t)30 b(measure.)48 b(This)31 b(p)r(ermits)h(in)n(v)n(estigation)e(of)i(questions)e(lik)n (e)i(the)498 2703 y(temp)r(erature)d(pro\014le)g(in)h(a)f(c)n(hain)g (in)g(the)h(steady)f(state)g(or)g(the)h(v)-5 b(alue)29 b(of)g(the)h(conductivit)n(y)498 2802 y(for)d(whic)n(h,)g(as)f(w)n(e)g (ha)n(v)n(e)g(seen,)h(there)g(is)f(no)h(analytical)f(result)g(at)h (this)g(time.)38 b(W)-7 b(e)27 b(will)g(fo)r(cus)498 2902 y(mainly)h(on)f(this)h(last)f(question.)558 3055 y(1.)41 b(In)d(one)e(dimension,)k(where)d(most)g(of)g(the)h(sim)n (ulations)e(ha)n(v)n(e)g(b)r(een)i(conducted,)i(the)664 3154 y(conductivit)n(y)-7 b(,)31 b(when)g Fq(U)1438 3166 y Fh(i)1463 3154 y Fr(\()p Fz(q)p Fr(\))e(=)e(0,)k(app)r(ears)f(to)g(b) r(eha)n(v)n(e,)g(as)g(a)g(function)i(of)e(the)h(length)664 3254 y Fm(L)d Fr(of)g(the)g(c)n(hain,)g(as)f Fm(L)1389 3224 y Fn(\013)1465 3254 y Fr(with)h Fq(\013)g Fr(a)g(p)r(ositiv)n(e)f (exp)r(onen)n(t,)h Fq(\013)c Fr(=)f(0)p Fq(:)p Fr(4)k(for)g(the)h (anharmonic)664 3354 y(c)n(hain.)36 b(On)27 b(the)g(other)f(hand)h(if)h Fq(U)1751 3366 y Fh(i)1776 3354 y Fr(\()p Fz(q)p Fr(\))c Fm(6)p Fr(=)e(0)27 b(\(t)n(ypically)f(one)h(consider)e Fq(U)2971 3366 y Fh(i)2996 3354 y Fr(\()p Fz(q)p Fr(\))f(=)3232 3321 y Fp(1)p 3232 3335 34 4 v 3232 3382 a(2)3275 3354 y Fq(!)3330 3324 y Fp(2)3367 3354 y Fz(q)3417 3324 y Fp(2)3455 3354 y Fr(\))664 3453 y(one)36 b(\014nds)h(a)f(\014nite)h (conductivit)n(y)g(if)g(some)f(nonlinearit)n(y)f(is)i(presen)n(t)e(in)i (the)g(system)664 3553 y(\(for)i(the)g(linear)f(case)g Fq(\013)k Fr(=)f(1)1661 3523 y Fp(75)1731 3553 y Fr(\).)70 b(In)39 b(this)g(situation)g(the)g(exact)f(form)g(of)h(the)g(non)664 3653 y(linearit)n(y)25 b(seems)g(irrelev)-5 b(an)n(t,)24 b(analogous)f(result)i(are)g(found)g(adding)g(a)g(4-th)g(order)f(term) 664 3752 y(to)34 b Fq(U)9 b Fr(\()p Fz(q)p Fr(\))35 b(as)e(w)n(ell)h (as)f(to)h Fq(V)19 b Fr(\()p Fz(q)p Fr(\).)57 b(Similar)33 b(results)h(are)f(obtained)h(if)g(one)g(compacti\014es)664 3852 y(the)28 b(con\014guration)e(space)g(of)i Fq(Q)p Fr(,)f Fo(e.g.)g Fr(b)n(y)g(considering)f(eac)n(h)h Fz(q)g Fr(as)g(a)g(p)r(oin)n(t)h(on)f(a)g(torus,)664 3951 y(obtaining)j(what)h (can)f(b)r(e)h(called)g(a)f(c)n(hain)g(of)h(rotators.)44 b(In)31 b(all)f(cases)g(the)h(system)g(has)664 4051 y(a)h(w)n(ell)f (de\014ned,)j(appro)n(ximately)c(linear,)i(temp)r(erature)g(pro\014le,) g(although)g(there)f(can)664 4151 y(b)r(e)i(a)f(\014nite)g(jump)h(b)r (et)n(w)n(een)g(the)f(temp)r(erature)g(of)g(the)h(\014rst)f(and)g(last) g(oscillator)e(and)664 4250 y(the)g(temp)r(erature)f(of)g(the)h(resp)r (ectiv)n(e)f(thermostats.)41 b(Moreo)n(v)n(er)27 b(for)i(one)g (dimensional)664 4350 y(systems)h(sim)n(ulations)f(using)h(di\013eren)n (t)h(kinds)f(of)g(reserv)n(oirs)d(app)r(ear)i(to)i(yield)f(similar)664 4450 y(v)-5 b(alues)27 b(of)h(the)g(exp)r(onen)n(t)f Fq(\013)p Fr(.)558 4609 y(2.)41 b(Recen)n(t)34 b(sim)n(ulations)e(seem) i(to)f(sho)n(w)g(that)g(the)h(conductivit)n(y)g(in)f(t)n(w)n(o)g (dimensions)g(is)664 4709 y(logarithmically)d(div)n(ergen)n(t)f(if)j Fq(U)1740 4721 y Fh(i)1765 4709 y Fr(\()p Fz(q)p Fr(\))e(=)f(0.)47 b(Although)31 b(it)h(is)f(not)g(easy)f(to)h(see)g(a)g(loga-)664 4808 y(rithm)20 b(is)f(suc)n(h)g(a)g(situation)g(the)h(sim)n(ulation)e (in)i([62])e(strongly)g(suggests)g(this)i(conclusion.)664 4908 y(Moreo)n(v)n(er)h(in)j(this)g(case)f(one)g(can)h(try)f(to)h (compute)f(the)h(thermal)g(conductivit)n(y)f Fq(\024)3273 4920 y Fn(GK)3413 4908 y Fr(as)664 5007 y(giv)n(en)30 b(in)h(Eq.\(37\))1256 4977 y Fn(g)1324 5007 y Fr(and)g(compare)f(it)h (with)g(the)g Fq(\024)g Fr(obtained)f(from)h(Eq.\(33\))e(\(without)p 498 5091 1196 4 v 498 5139 a Fd(g)534 5163 y Fw(In)24 b(this)g(case)g(the)h(equilibrium)c(ensem)n(ble)j(used)g(is)f Fu(h\001i)2012 5174 y Fd(E)s(;\031)2144 5163 y Fw(describ)r(ed)i(in)e (the)h(commen)n(ts)f(after)h(Eq.\(37\))p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(20)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 21 21 21 20 bop 664 632 a Fr(the)29 b(limit)h Fm(L)c(!)f(1)p Fr(\).)41 b(The)29 b(agreemen)n(t)e(obtained)i(from)f(the)i(n)n (umerical)e(data)g(supp)r(ort)664 732 y(the)33 b(v)-5 b(alidit)n(y)32 b(of)h(the)g(equalit)n(y)f Fq(\024)f Fr(=)g Fq(\024)1906 744 y Fn(GK)2021 732 y Fr(.)52 b(Although)32 b(w)n(e)g(do)h(not)f(kno)n(w)g(an)n(y)g(results)664 831 y(on)f(this)h(direction)f(w)n(e)g(b)r(eliev)n(e)g(that)h(adding)f(a)g (con\014ning)g Fq(U)9 b Fr(\()p Fz(q)p Fr(\))32 b(to)g(this)g(system)f (will)664 931 y(mak)n(e)c(the)h(conductivit)n(y)f(\014nite.)558 1097 y(3.)41 b(W)-7 b(e)23 b(further)f(exp)r(ect)g(that)h(the)f (conductivit)n(y)g(in)h(three)f(dimensions)g(will)g(b)r(e)h(\014nite,)g (with)664 1196 y(or)k(without)h(an)n(y)f(on)g(site)h(p)r(oten)n(tial.) 623 1361 y(The)34 b(ab)r(o)n(v)n(e)f(picture)h(can)g(b)r(e)h(in)n (terpreted)f(in)g(term)g(of)h(the)f(P)n(eierls)f(theory)h(whic)n(h,)i (as)498 1461 y(men)n(tioned)26 b(in)g(Section)g(4)f(relies)g(on)h (umklap)g(pro)r(cesses)e(to)i(pro)r(duce)f(a)h(\014nite)g(conductivit)n (y)-7 b(.)498 1560 y(W)g(e)29 b(also)e(note)i(that)g(in)f(one)g(and)h (t)n(w)n(o)e(dimension,)i(there)f(is)h(no)f(stable)g(crystal)f(without) i(an)498 1660 y(on)g(site)h(p)r(oten)n(tial,)g(but)g(there)g(is)f(lo)r (calization)f(in)i Fq(d)d Fm(\025)f Fr(3.)42 b(The)29 b(v)-5 b(ariance)29 b(in)h(the)g(deviation)498 1760 y(of)24 b(an)f(atom)g(near)g(the)h(cen)n(ter)f(from)g(its)h(equilibrium)g(p)r (osition,)g(when)g(the)g(b)r(oundary)f(atoms)498 1859 y(are)k(tied)h(do)n(wn,)f(gro)n(ws)f(lik)n(e)h Fm(L)h Fr(in)g Fq(d)23 b Fr(=)f(1,)28 b(lik)n(e)f(log)14 b Fm(L)28 b Fr(in)g Fq(d)23 b Fr(=)g(2)k(and)g(is)h(\014nite)g(in)g Fq(d)23 b Fr(=)g(3)3287 1829 y Fp(8)3323 1859 y Fr(.)498 2084 y Fz(Ac)m(kno)m(wledgmen)m(ts)498 2266 y Fr(L.)39 b(R.-B.)f(is)h(supp)r(orted)g(in)g(part)f(b)n(y)g(the)h(Swiss)g (National)f(Science)h(foundation.)70 b(F.)39 b(B.)498 2365 y(and)32 b(J.L)f(w)n(ere)g(supp)r(orted)g(b)n(y)h(NSF)g(gran)n(t)e (DMR-9813268)f(and)i(AF)n(OSR)h(gran)n(t)e(F49620-)498 2465 y(98-1-0207.)65 b(They)39 b(ac)n(kno)n(wledge)d(the)i(hospitalit)n (y)g(of)h(the)f(IHES)h(where)e(this)i(w)n(ork)e(w)n(as)498 2565 y(completed.)498 2789 y Fz(References)558 2971 y Fr(1.)k(K.)27 b(Aoki)h(and)f(D.)h(Kusnezo)n(v,)f(Preprin)n(t)f(1999)664 3071 y Fa(http://xxx.lanl.gov/ps/chao-dyn/99100)o(15)p Fr(.)558 3170 y(2.)41 b(N.W.)28 b(Ashcroft)e(and)h(N.D.)g(Mermin,)g Fl(Solid)j(State)f(Physics)p Fr(,)f(Saunders)e(College)g(1988.)558 3270 y(3.)41 b(V.)28 b(Bac)n(h,)f(J.)g(F)-7 b(ro)r(ehlic)n(h,)28 b(and)f(I.M.)h(Sigal,)f Fl(A)l(dv.)39 b(Math.)p Fr(,)29 b(137)d(\(205\),)h(1998.)558 3369 y(4.)41 b(V.)28 b(Bac)n(h,)f(J.)g(F) -7 b(ro)r(ehlic)n(h,)28 b(and)f(I.M.)h(Sigal,)f Fl(A)l(dv.)39 b(Math.)p Fr(,)29 b(137)d(\(299\),)h(1998.)558 3469 y(5.)41 b(V.)c(Bac)n(h,)h(J.)e(F)-7 b(ro)r(ehlic)n(h,)38 b(and)f(I.M.)g(Sigal,) 76 b Fl(Commun.)63 b(Math.)h(Phys.)p Fr(,)41 b(207)35 b(\(249\),)664 3569 y(1999.)558 3668 y(6.)41 b(P)-7 b(.G.)28 b(Bergman)e(and)h(J.L.)h(Leb)r(o)n(witz,)f Fl(Phys.)40 b(R)l(ev.)p Fr(,)28 b(99)e(\(2\),)i(1955.)558 3768 y(7.)41 b(M.)28 b(Bolsterli,)f(M.)h(Ric)n(h,)f(and)h(W.M.)g(Vissc)n(her,)f Fl(Phys.)39 b(R)l(ev.)g(A)p Fr(,)27 b(1)g(\(1086\),)g(1970.)558 3868 y(8.)41 b(H.J.)f(Brascamp,)g(E.H.)f(Lieb)h(and)f(J.L.)g(Leb)r(o)n (witz,)j Fl(Bul)t(letin)f(of)g(the)g(International)664 3967 y(Statistic)30 b(Institute)p Fr(,)c(4)h(\(393\),)g(1975.)558 4067 y(9.)41 b(S.)28 b(Brusc)n(h,)f Fl(The)j(kind)h(of)f(motion)h(we)f (c)l(al)t(l)g(he)l(at)p Fr(,)f(North)e(Holland,)g(1976.)517 4166 y(10.)40 b(L.)28 b(Bunimo)n(vic)n(h)f(and)g(Y)-7 b(a.)37 b(G.)28 b(Sina)-9 b(\025)-32 b(\020,)26 b Fl(Commun.)39 b(Math.)g(Phys.)p Fr(,)30 b(78)c(\(479\),)h(1981.)517 4266 y(11.)40 b(L.)28 b(Bunimo)n(vic)n(h)f(and)g(H.)h(Sp)r(ohn,)g Fl(Commun.)39 b(Math.)g(Phys.)p Fr(,)29 b(176)e(\(661\),)f(1996.)517 4366 y(12.)40 b(G.)23 b(Casati,)f(J.)g(F)-7 b(ord,)23 b(F.Viv)-5 b(aldi,)24 b(and)e(W.M.)h(Vissc)n(her,)g(Ph)n(ys.)34 b(Rev.)h(Lett.,)24 b(52)d(\(1861\),)664 4465 y(1984.)517 4565 y(13.)40 b(A.)28 b(Casher)f(and)g(J.L.)g(Leb)r(o)n(witz,)h Fl(J.)i(Math.)39 b(Phys.)p Fr(,)29 b(12)e(\(1701\),)f(1971.)517 4665 y(14.)40 b(N.I.)30 b(Cherno)n(v,)f(G.L.)g(Eyink,)h(J.L.)f(Leb)r(o) n(witz,)h(and)f(Y)-7 b(a.)42 b(G.)30 b(Sina)-9 b(\025)-32 b(\020,)29 b Fl(Commun.)44 b(Math.)664 4764 y(Phys.)p Fr(,)29 b(154)e(\(569\),)f(1993.)517 4864 y(15.)40 b(N.I.)28 b(Cherno)n(v,)f(and)g(J.L.)g(Leb)r(o)n(witz,)h Fl(Phys.)39 b(R)l(ev.)g(L)l(ett.)p Fr(,)27 b(71)g(\(2831\),)f(1995)517 4963 y(16.)40 b(N.I.)28 b(Cherno)n(v,)f(and)g(J.L.)g(Leb)r(o)n(witz,)h Fl(J.)h(Stat.)38 b(Phys.)p Fr(,)30 b(86)c(\(953\),)h(1997.)517 5063 y(17.)40 b(E.B.)27 b(Da)n(vies,)g(J.)h(Stat.)37 b(Ph)n(ys.,)26 b(18)h(\(161\),)g(1978.)517 5163 y(18.)40 b(P)-7 b(.)27 b(Deb)n(y)n(e,)h Fl(V)-6 b(ortr\177)-42 b(age)31 b(\177)-43 b(ub)l(er)29 b(die)i(Kinetische)g(The)l(orie)g(der) f(W\177)-42 b(arme)p Fr(,)29 b(T)-7 b(eubner,)27 b(1914.)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(21)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 22 22 22 21 bop 517 628 a Fr(19.)40 b(J.)28 b(Derezinski)f(and)g(V.)h (Jaksic,)f(Preprin)n(t)f(1998.)517 728 y(20.)40 b(J.-P)-7 b(.)27 b(Ec)n(kmann)f(and)i(M.)g(Hairer,)e(Preprin)n(t)h(1999)664 827 y Fa(http://xxx.lanl.gov/ps/chao-dyn/99090)o(35)p Fr(.)517 927 y(21.)40 b(J.-P)-7 b(.)31 b(Ec)n(kmann,)h(C.-A.)f(Pillet,) i(and)f(L.)f(Rey-Bellet,)i Fl(Commun.)49 b(Math.)i(Phys.)p Fr(,)34 b(201)664 1027 y(\(657\),)27 b(1999.)517 1126 y(22.)40 b(J.-P)-7 b(.)25 b(Ec)n(kmann,)g(C.-A.)h(Pillet,)g(and)f(L.)h (Rey-Bellet,)g Fl(J.)i(Stat.)37 b(Phys.)p Fr(,)28 b(95)d(\(305\),)g (1999.)517 1226 y(23.)40 b(R.)32 b(Esp)r(osito,)g(J.L.)f(Leb)r(o)n (witz)h(and)f(R.)h(Marra,)g Fl(Commun.)50 b(Math.)h(Phys.)p Fr(,)34 b(160)c(\(49\),)664 1325 y(1994.)517 1425 y(24.)40 b(R.)28 b(Esp)r(osito,)f(J.L.)g(Leb)r(o)n(witz)g(and)h(R.)f(Marra,)f Fl(J.)k(Stat.)38 b(Phys.)p Fr(,)29 b(78)e(\(389\),)g(1995.)517 1525 y(25.)40 b(R.)28 b(Esp)r(osito,)f(J.L.)g(Leb)r(o)n(witz)g(and)h (R.)f(Marra,)f Fl(J.)k(Stat.)38 b(Phys.)p Fr(,)29 b(90)e(\(1129\),)f (1998.)517 1624 y(26.)40 b(D.)31 b(J.)f(Ev)-5 b(ans,)30 b(E.)f(G.)i(D.)f(Cohen,)h(and)f(G.)g(P)-7 b(.)30 b(Morris,)f Fl(Phys.)46 b(R)l(ev.)g(L)l(ett.)p Fr(,)30 b(71)f(\(2401\),)664 1724 y(1993.)517 1824 y(27.)40 b(D.J.)29 b(Ev)-5 b(ans)27 b(and)i(G.P)-7 b(.)28 b(Morris,)f Fl(Statistic)l(al)k(Me)l(chanics)h (of)g(None)l(quilibrium)f(Liquids)p Fr(,)664 1923 y(Academic)d(Press,)e (San)h(Diego)g(1990.)517 2023 y(28.)40 b(J.)28 b(F)-7 b(armer,)26 b(S.)i(Goldstein,)g(and)f(E.R.)h(Sp)r(eer,)f Fl(J.)j(Stat.)38 b(Phys.)p Fr(,)29 b(34)e(\(263\),)g(1984.)517 2122 y(29.)40 b(F.)28 b(Fidaleo)f(and)h(C.)f(Liv)n(erani)g Fl(J.)i(Stat.)38 b(Phys.)p Fr(,)57 b(\(98\),)27 b(2000.)517 2222 y(30.)40 b(J.)34 b(F)-7 b(ritz,)37 b(T.)e(F)-7 b(umaki,)36 b(and)e(J.L.)h(Leb)r(o)n(witz,)h(Probab.)56 b(Theory)33 b(Related)h(Fields,)j(99)664 2322 y(\(211\),)27 b(1994.)517 2421 y(31.)40 b(G.)28 b(Galla)n(v)n(otti)e(and)i(E.G.D.)g(Cohen,)f Fl(J.)j(Stat.)38 b(Phys.)p Fr(,)29 b(80)e(\(931\),)g(1995.)517 2521 y(32.)40 b(G.)28 b(Galla)n(v)n(otti,)e Fl(Physic)l(a)32 b(D)p Fr(,)27 b(105)g(\(163\),)f(1997)517 2621 y(33.)40 b(G.)28 b(Galla)n(v)n(otti,)e Fl(Chaos)p Fr(,)k(8)d(\(384\),)g(1998)517 2720 y(34.)40 b(C.)28 b(Giardina,)f(R.)h(Livi,)f(A.)h(P)n(oliti,)f(and) g(M.)h(V)-7 b(assalli,)27 b(Preprin)n(t)f(1999.)517 2820 y(35.)40 b(S.)28 b(Goldstein,)g(C.)f(Kipnis)h(and)f(N.)h(Ianiro,)f Fl(J.)i(Stat.)38 b(Phys.)p Fr(,)30 b(41)c(\(915\),)h(1985.)517 2919 y(36.)40 b(S.)20 b(Goldstein,)h(J.L.)e(Leb)r(o)n(witz)h(and)f(E.)g (Presutti,)i(in)e Fl(Col)t(lo)l(quia)25 b(Mathematic)l(a)f(So)l (cietati)664 3019 y(J\023)-42 b(anos)30 b(Bolay)f Fr(27)e(\(403\),)g (1979.)517 3119 y(37.)40 b(S.)20 b(Goldstein,)h(J.L.)e(Leb)r(o)n(witz)h (and)f(E.)g(Presutti,)i(in)e Fl(Col)t(lo)l(quia)25 b(Mathematic)l(a)f (So)l(cietati)664 3218 y(J\023)-42 b(anos)30 b(Bolay)f Fr(27)e(\(421\),)g(1979.)517 3318 y(38.)40 b(S.)32 b(Goldstein,)g(J.L.) f(Leb)r(o)n(witz)g(and)h(K.)f(Ra)n(vishank)-5 b(ar,)30 b Fl(Commun.)49 b(Math.)h(Phys.)p Fr(,)34 b(85)664 3418 y(\(419\),)27 b(1982.)517 3517 y(39.)40 b(S.)22 b(Goldstein,)g(J.L.)f (Leb)r(o)n(witz)g(and)g(K.)g(Ra)n(vishank)-5 b(ar,)21 b Fl(J.)j(Stat.)36 b(Phys.)p Fr(,)24 b(43)c(\(303\),)i(1986.)517 3617 y(40.)40 b(T.)28 b(Hatano.)36 b Fl(Phys.)k(R)l(ev.)e(E)p Fr(,)28 b(59)f(\(R1\),)h(1999.)517 3716 y(41.)40 b(W.G.)28 b(Ho)r(o)n(v)n(er,)e Fl(Computational)32 b(Statistic)l(al)e(Me)l (chanics)p Fr(,)f(Elsevier)d(1991)517 3816 y(42.)40 b(B.)28 b(Hu,)g(B.)f(Li)h(and)f(H.)h(Zhao,)f(Ph)n(ys.)36 b(Rev.)h(E,)27 b(57)g(\(2992\),)f(1998.)517 3916 y(43.)40 b(V.)28 b(Jaksic)f(and)g (C.-A.)h(Pillet,)f Fl(Commun.)39 b(Math.)g(Phys.)p Fr(,)30 b(176)c(\(619\),)h(1996.)517 4015 y(44.)40 b(V.)28 b(Jaksic)f(and)g (C.-A.)h(Pillet,)f Fl(Commun.)39 b(Math.)g(Phys.)p Fr(,)30 b(178)c(\(627\),)h(1996.)517 4115 y(45.)40 b(V.)28 b(Jaksic)f(and)g (C.-A.)h(Pillet,)f Fl(A)l(cta)j(Math.)p Fr(,)f(181)d(\(245\),)h(1998.) 517 4215 y(46.)40 b(E.A.)28 b(Jac)n(kson,)d Fl(R)l(o)l(cky)30 b(Mount.)39 b(J.)29 b(Math.)p Fr(,)h(8)d(\(127\),)g(1978.)517 4314 y(47.)40 b(C.)51 b(Kipnis)f(and)g(C.)g(Landim,)56 b Fl(Sc)l(aling)51 b(limits)g(of)h(inter)l(acting)e(p)l(article)i (systems)p Fr(,)664 4414 y(Springer,)27 b(Berlin)g(1999.)517 4513 y(48.)40 b(C.)28 b(Kipnis,)f(C.)h(Marc)n(hioro)d(and)j(E.)f (Presutti,)g(J.)g(Stat.)38 b(Ph)n(ys.,)26 b(27)h(\(65\),)g(1982.)517 4613 y(49.)40 b(A.)28 b(Komec)n(h)f(and)g(H.)h(Sp)r(ohn,)g Fl(Nonline)l(ar)i(A)n(nal.)p Fr(,)f(33)d(\(13\),)i(1998.)517 4713 y(50.)40 b(A.)23 b(Komec)n(h,)f(H.)g(Sp)r(ohn)h(and)f(M.)g(Kunze)g (M)g Fl(Comm.)38 b(Partial)26 b(Di\013er)l(ential)f(Equations)p Fr(,)664 4812 y(22)i(\(307\),)g(1997.)517 4912 y(51.)40 b(R.)28 b(Kub)r(o,)f Fl(A)l(cta)j(Phys.)39 b(A)n(ustr.,)30 b(Suppl.)p Fr(,)e(10)f(\(301\),)g(1973.)517 5012 y(52.)40 b(J.)28 b(Kurc)n(han,)e Fl(J.)k(Phys.)39 b(A)p Fr(,)28 b(31)e(\(3719\),)h(1998.)517 5111 y(53.)40 b(R.)31 b(Kub)r(o,)g(M.)g(T) -7 b(o)r(da)30 b(and)g(N.)h(Hashitsume,)g Fl(Statistic)l(al)i(Physics)h (II)p Fr(,)c(Springer)g(Series)p 498 5429 3046 4 v 498 5552 4 124 v 526 5511 a Fl(luc10:)40 b(submitte)l(d)29 b(to)h Fz(W)-8 b(orld)31 b(Scien)m(ti\014c)f Fl(on)g(F)-6 b(ebruary)30 b(15,)h(2000)698 b Fz(22)p 3540 5552 V 498 5555 3046 4 v eop %%Page: 23 23 23 22 bop 664 628 a Fr(in)28 b(Solid)g(State)f(Sciences,)h(V)-7 b(ol)27 b(31,)g(Springer,)g(Berlin)g(1991.)517 728 y(54.)40 b(J.L.)28 b(Leb)r(o)n(witz,)f Fl(Pr)l(o)l(g.)39 b(The)l(or.)h(Phys.)g (Suppl.)p Fr(,)28 b(64)f(\(35\),)g(1978.)517 827 y(55.)40 b(J.L.)28 b(Leb)r(o)n(witz)f(and)g(P)-7 b(.G.)28 b(Bergman,)e Fl(A)n(nn.)38 b(Physics)p Fr(,)29 b(1)e(\(1\),)h(1957.)517 927 y(56.)40 b(J.L.)28 b(Leb)r(o)n(witz)f(and)g(H.)h(Sp)r(ohn,)g Fl(J.)i(Stat.)38 b(Phys.)p Fr(,)29 b(19)e(\(633\),)g(1978.)517 1027 y(57.)40 b(J.L.)28 b(Leb)r(o)n(witz,)f(E.)g(Presutti)g(and)h(H.)g (Sp)r(ohn,)g Fl(J.)h(Stat.)38 b(Phys.)p Fr(,)30 b(51)c(\(841\),)h (1988.)517 1126 y(58.)40 b(J.L.)28 b(Leb)r(o)n(witz)f(and)g(H.)h(Sp)r (ohn,)g Fl(J.)i(Stat.)38 b(Phys.)p Fr(,)29 b(95)e(\(333\),)g(1999.)517 1226 y(59.)40 b(S.)28 b(Lepri,)f(R.)h(Livi,)g(and)f(A.)h(P)n(oliti,)f Fl(Phys.)40 b(R)l(ev.)e(L)l(ett.)p Fr(,)27 b(78)g(\(1896\),)f(1997.)517 1325 y(60.)40 b(S.)28 b(Lepri,)f(R.)h(Livi,)g(and)f(A.)h(P)n(oliti,)f Fl(Eur)l(ophys.)40 b(L)l(ett.)p Fr(,)27 b(43)g(\(271\),)g(1998.)517 1425 y(61.)40 b(S.)28 b(Lepri,)f(R.)h(Livi,)g(and)f(A.)h(P)n(oliti,)f (In)h(preparation.)517 1525 y(62.)40 b(S.)28 b(Lippi)g(and)f(R.)h (Livi,)g(Preprin)n(t)e(1999)664 1624 y Fa (http://xxx.lanl.gov/ps/chao-dyn/99100)o(34)p Fr(.)517 1724 y(63.)40 b(C.)28 b(Liv)n(erani)e(and)i(S.)f(Olla,)g Fl(Commun.)39 b(Math.)g(Phys.)p Fr(,)30 b(189)c(\(481\),)h(1997)517 1824 y(64.)40 b(C.)28 b(Maes,)f Fl(J.)j(Stat.)38 b(Phys.)p Fr(,)29 b(95)d(\(367\),)h(1999.)517 1923 y(65.)40 b(H.)28 b(Maassen,)f(M.)g(Guta)h(and)g(D.)g(Botvic)n(h,)f(Preprin)n(t)f(1999) 664 2023 y Fa(http://www.ma.utexas.edu/mp)p 1757 2036 28 4 v 25 w(a)n(rch/99/99-220.ps.gz)p Fr(.)517 2122 y(66.)40 b(J.)22 b(A.)g(McLennan,)g Fl(Intr)l(o)l(duction)i(to)g(none)l (quilibrium)h(statistic)l(al)g(me)l(chanics)p Fr(,)f(Pren)n(tice)664 2222 y(Hall,)k(1989.)517 2322 y(67.)40 b(H.)g(Nak)-5 b(aza)n(w)n(a,)40 b Fl(Supplement)h(of)g(the)g(Pr)l(o)l(gr)l(ess)g(of)h (The)l(or)l(etic)l(al)g(Physics)p Fr(,)i(45)39 b(\(231\),)664 2421 y(1970.)517 2521 y(68.)h(S.)28 b(Nos)n(\023)-39 b(e,)26 b Fl(J.)k(Chem.)40 b(Phys.)p Fr(,)29 b(81)d(\(511\),)h(1984.) 517 2621 y(69.)40 b(A.J.)28 b(O'Connor)e(and)h(J.L.)h(Leb)r(o)n(witz,)f Fl(J.)j(Math.)39 b(Phys.)p Fr(,)30 b(15)c(\(692\),)h(1974.)517 2720 y(70.)40 b(S.)21 b(Olla,)h(S.R.S.)g(V)-7 b(arahadan,)21 b(H.-T.)g(Y)-7 b(ao,)22 b Fl(Commun.)36 b(Math.)i(Phys.)p Fr(,)24 b(155)19 b(\(523\),)j(1993.)517 2820 y(71.)40 b(R.E.)28 b(P)n(eierls,)e Fl(Quantum)i(The)l(ory)j(of)g(Solids)p Fr(,)d(Univ)n(ersit)n(y)f(press)g(1956.)517 2919 y(72.)40 b(R.E.)20 b(P)n(eierls,)h(In)f Fl(The)l(or)l(etic)l(al)25 b(Physics)g(in)e(the)g(Twentieth)h(The)l(ory)p Fr(,)f(In)n(terscience)d (1960.)517 3019 y(73.)40 b(T.)28 b(Prosen)e(and)h(M.)h(Robnik,)g Fl(J.)h(Phys.)40 b(A:)30 b(Math.)39 b(Gen.)g Fr(,)28 b(25)f(\(3449\),)f(1992.)517 3119 y(74.)40 b(T.)28 b(Prosen)e(and)h (D.K.)h(Campb)r(ell,)g(Preprin)n(t)e(1999)664 3218 y Fa(http://xxx.lanl.gov/ps/chao-dyn/99100)o(24)p Fr(.)517 3318 y(75.)40 b(Z.)28 b(Rieder,)f(J.L.)g(Leb)r(o)n(witz,)h(and)f(E.H.)h (Lieb,)f Fl(J.)j(Math.)40 b(Phys.)p Fr(,)29 b(8)e(\(1073\),)f(1967.)517 3418 y(76.)40 b(M.)28 b(Ric)n(h)g(and)f(W.M.)h(Vissc)n(her,)f Fl(Phys.)40 b(R)l(ev.)e(B)p Fr(,)28 b(11)f(\(2164\),)f(1975.)517 3517 y(77.)40 b(R.J.)28 b(Rubin)g(and)f(W.L.)h(Greer,)f Fl(J.)j(Math.)39 b(Phys.)p Fr(,)30 b(12)c(\(1686\),)h(1971.)517 3617 y(78.)40 b(D.)28 b(Ruelle,)g(Preprin)n(t)f(1999)664 3716 y Fa(http://www.ma.utexas.edu/mp)p 1757 3729 V 25 w(a)n(rch/99/99-220.ps.gz)p Fr(.)517 3816 y(79.)40 b(H.)22 b(Sp)r(ohn,)h Fl(L)l(ar)l(ge)i(Sc)l(ale)f(Dynamics)h(of)g(Inter)l (acting)f(Particle)p Fr(,)g(Springer)c(V)-7 b(erlag,)22 b(1991.)517 3916 y(80.)40 b(H.)28 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