Content-Type: multipart/mixed; boundary="-------------0303071450522" This is a multi-part message in MIME format. ---------------0303071450522 Content-Type: text/plain; name="03-97.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-97.keywords" nonequilibrium statistical mechanics, anharmonic oscillators, heat conduction, fluctuation theorem, entropy production ---------------0303071450522 Content-Type: application/postscript; name="uab.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="uab.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Rey-BelletUAB2002.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Rey-BelletUAB2002 -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.10.29:1522 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind 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Fk(')p Fp(\()p Fk(x)p Fp(\))p Fk(\032)p Fp(\()p Fk(x)p Fp(\))-2532 b(\(3\))1351 1427 y(=)97 b Fk(H)1582 1439 y Fl(B)1639 1427 y Fp(\()p Fk(\036)p Fp(\))20 b(+)e Fk(H)1924 1439 y Fl(S)1972 1427 y Fp(\()p Fk(p;)c(q)s Fp(\))k(+)g Fk(q)k Fj(\001)c(h)p Fk(\036;)c(\013)p Fj(i)g Fk(;)456 1600 y Fp(where)k Fk(\032)p Fp(\()p Fk(x)p Fp(\))h(is)g(a)f(real)g(rotation)f(in)n(v)-5 b(arian)n(t)17 b(function)j(and)e Fk(\013)24 b Fp(=)e(\()p Fk(\013)2514 1570 y Fm(\(1\))2604 1600 y Fk(;)14 b Fj(\001)g(\001)g(\001)27 b Fk(;)14 b(\013)2855 1570 y Fm(\()p Fl(d)p Fm(\))2946 1600 y Fp(\))19 b(is,)i(in)d(F)-7 b(ourier)456 1703 y(space,)20 b(giv)n(en)f(b)n(y)28 b(^)-51 b Fk(\013)1063 1673 y Fm(\()p Fl(i)p Fm(\))1143 1703 y Fp(\()p Fk(k)s Fp(\))24 b(=)1364 1635 y Fi(\000)1402 1703 y Fp(0)p Fk(;)14 b Fj(\000)p Fk(ik)1621 1673 y Fm(\()p Fl(i)p Fm(\))1707 1703 y Fp(^)-50 b Fk(\032)p Fp(\()p Fk(k)s Fp(\))p Fk(=k)1940 1673 y Fm(2)1977 1635 y Fi(\001)2015 1703 y Fp(.)35 b(Let)20 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5216 y Fk(;)p eop %%Page: 4 4 4 3 bop 456 251 a Fm(4)1166 b(LUC)23 b(REY-BELLET)456 450 y Fp(where)29 b Fk(P)42 b Fp(is)30 b(a)g(p)r(olynomial.)44 b(As)30 b(a)g(consequence)f(there)h(is)g(a)g(p)r(olynomial)g Fk(p)p Fp(\()p Fk(u)p Fp(\))g(whic)n(h)g(is)g(a)456 550 y(real)c(function)i(of)g Fk(iu)f Fp(and)g(has)g(its)h(ro)r(ots)f(in)g (the)h(lo)n(w)n(er)e(half)i(plane)g(suc)n(h)f(that)456 744 y(\(10\))790 b Fk(C)1459 709 y Fm(\()p Fl(ii)p Fm(\))1562 744 y Fp(\()p Fk(t)p Fp(\))37 b(=)1795 631 y Fi(Z)1878 651 y Fh(1)1841 819 y(\0001)1977 744 y Fk(du)2189 687 y Fp(1)p 2092 725 238 4 v 2092 801 a Fj(j)p Fk(p)p Fp(\()p Fk(u)p Fp(\))p Fj(j)2292 777 y Fm(2)2339 744 y Fk(e)2378 709 y Fl(iut)2483 744 y Fk(:)456 949 y Fp(Note)20 b(that)i(this)f(is)f (a)h(Mark)n(o)n(vian)d(assumption)i([)p Fq(6)p Fp(]:)34 b(for)20 b(suc)n(h)g(couplings)g Fk(\030)t Fp(\()p Fk(t)p Fp(\))i(is)f(a)f(Mark)n(o)n(vian)456 1048 y(Gaussian)j(pro)r(cess:)34 b Fk(p)p Fp(\()p Fj(\000)p Fk(id=dt)p Fp(\))p Fk(\030)t Fp(\()p 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y(\(23\))1063 b Fk(W)1745 5228 y Fl(\022)1820 5216 y Fp(=)36 b(exp)14 b(\()p Fk(\022)r(G)p Fp(\))p eop %%Page: 6 6 6 5 bop 456 251 a Fm(6)1166 b(LUC)23 b(REY-BELLET)456 450 y Fp(will)j(b)r(e)g(used)g(rep)r(eatedly)-7 b(.)36 b(W)-7 b(e)26 b(denote)g(as)f Fj(j)15 b(\001)g(j)1947 462 y Fl(\022)2011 450 y Fp(the)26 b(w)n(eigh)n(ted)g(total)f(v)-5 b(ariation)25 b(norm)g(giv)n(en)456 550 y(b)n(y)456 731 y(\(24\))917 b Fj(j)p Fk(\031)s Fj(j)1617 743 y Fl(\022)1692 731 y Fp(=)87 b(sup)1793 805 y Fh(j)p Fl(f)7 b Fh(j\024)p Fl(W)1986 814 y Ff(\022)2033 611 y Fi(\014)2033 661 y(\014)2033 710 y(\014)2033 760 y(\014)2061 618 y(Z)2158 731 y Fk(f)i(d\031)2301 611 y Fi(\014)2301 661 y(\014)2301 710 y(\014)2301 760 y(\014)2356 731 y Fk(;)456 935 y Fp(for)23 b(an)n(y)h(\(signed\))g (measure)f Fk(\031)s Fp(.)37 b(W)-7 b(e)24 b(in)n(tro)r(duce)g(norms)f Fj(k)12 b(\001)g(k)2360 947 y Fl(\022)2421 935 y Fp(and)24 b(Banac)n(h)f(spaces)g Fj(H)3193 947 y Fl(\022)3255 935 y Fp(giv)n(en)456 1034 y(b)n(y)456 1214 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b(smo)r(oth\))h(con)n(trol.)46 b(The)31 b(Supp)r(ort)g(Theorem)456 550 y(asserts)23 b(in)j(particular)e(that)i(the)f(supp)r(ort)g(of)h(the)f(transition)g (probabilities)f Fk(P)2957 562 y Fl(t)2987 550 y Fp(\()p Fk(x;)14 b(dy)s Fp(\))26 b(is)f(the)456 649 y(same)32 b(as)h(the)h(closure)e(of)i(the)f(set)h(of)f(all)g(p)r(oin)n(ts)h(reac) n(hable)d(in)j(a)f(time)h Fk(t)f Fp(starting)g(from)g Fk(x)456 749 y Fp(with)26 b(a)f(smo)r(oth)h(con)n(trol.)35 b(One)25 b(can)h(sho)n(w)e([)p Fq(11)p Fp(])i(that)g(for)f(all)h Fk(t)f Fp(and)h(for)f(all)h Fk(x)g Fp(the)g(transition)456 849 y(probabilities)d(ha)n(v)n(e)h(full)h(supp)r(ort.)35 b(This)25 b(is)f(ac)n(hiev)n(ed)f(done)i(b)n(y)f(an)g(explicit)h (construction)f(of)456 948 y(the)h(p)r(ossible)f(con)n(trols)f(whic)n (h)h(driv)n(e)g(the)h(system)f(from)g Fk(x)i Fp(to)e Fk(y)j Fp(in)e(a)f(giv)n(en)g(time)h(span.)36 b(This)456 1048 y(con)n(trollabilit)n(y)26 b(prop)r(ert)n(y)g(together)h(with)h 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Fp(\))h(\(for)g(a)f(detailed)456 1892 y(exp)r(osition)28 b(of)h(the)h(sub)5 b(ject)29 b(see)f([)p Fq(28)p Fp(]\).)41 b(F)-7 b(or)29 b(this)g(problem)g(w)n(e)f(pro)n(v)n (ed)g(the)h(follo)n(wing.)40 b(Fix)456 1992 y Fk(t)23 b(>)f Fp(0)h(and)h Fk(\022)h(<)e Fp(\(max)p Fj(f)p Fk(T)1249 2004 y Fm(1)1285 1992 y Fk(;)14 b(T)1371 2004 y Fl(n)1416 1992 y Fj(g)p Fp(\))1490 1962 y Fh(\000)p Fm(1)1602 1992 y Fp(and)23 b(consider)g(the)h(function)g Fk(W)2617 2004 y Fl(\022)2655 1992 y Fp(\()p Fk(x)p Fp(\))h(giv)n(en)e(b)n(y)g(Eq.)35 b(\(23\).)456 2091 y(There)e(exist)g(a)g(constan)n(t)g Fk(E)1381 2103 y Fm(0)1452 2091 y Fp(and)g(functions)h Fk(\024)p Fp(\()p Fk(E)5 b Fp(\))34 b(and)g Fk(b)p Fp(\()p Fk(E)5 b Fp(\))33 b(de\014ned)h(on)g([)p Fk(E)3060 2103 y Fm(0)3097 2091 y Fk(;)14 b Fj(1)p Fp(\))34 b(with)456 2191 y(lim)571 2203 y Fl(E)s Fh(!1)773 2191 y Fk(\024)p Fp(\()p Fk(E)5 b Fp(\))24 b(=)e(0suc)n(h)27 b(that)h(for)f Fk(E)h(>)23 b(E)1836 2203 y Fm(0)456 2408 y Fp(\(28\))567 b Fk(T)1232 2374 y Fl(t)1260 2408 y Fk(W)1338 2420 y Fl(\022)1376 2408 y Fp(\()p Fk(x)p Fp(\))38 b Fj(\024)e Fk(\024)p Fp(\()p Fk(E)5 b Fp(\))p Fk(W)1882 2420 y Fl(\022)1921 2408 y Fp(\()p Fk(x)p Fp(\))19 b(+)f Fk(b)p Fp(\()p Fk(E)5 b Fp(\))p Fk(\037)2352 2423 y Fh(f)p Fl(G)p Fh(\024)p Fl(E)s Fh(g)2580 2408 y Fp(\()p Fk(x)p Fp(\))14 b Fk(:)456 2625 y Fp(This)22 b(means)g(that,)i(outside)e(the)h(compact)f(set)h Fj(f)p Fk(G)g Fj(\024)f Fk(E)5 b Fj(g)p Fp(,)24 b(the)e(dynamics)h(is)f (dissipativ)n(e)g(and)456 2725 y(since)c Fk(\024)p Fp(\()p Fk(E)5 b Fp(\))20 b(tends)f(to)g(zero)e(as)i Fk(E)24 b Fp(tends)19 b(to)g(in\014nit)n(y)-7 b(,)21 b(the)e(dissipation)f(at)h (high)g(energies)f(can)g(b)r(e)456 2825 y(made)j(arbitrarily)e(strong.) 33 b(This)22 b(particular)e(prop)r(ert)n(y)g(of)h(the)g(Liapuno)n(v)g (function)g(together)456 2924 y(with)31 b(the)f(smo)r(othness)g(of)h (the)f(transition)g(probabilities)g(implies)g(that)h(the)g(semigroup)e Fk(T)3416 2894 y Fl(t)456 3024 y Fp(is)e(compact)g(on)g(the)h(Banac)n (h)f(spaces)f Fj(H)1750 3036 y Fl(\022)1788 3024 y Fp(.)605 3123 y(One)i(should)g(think)h(of)f(the)g(dynamics)g(as)g(follo)n(ws:)37 b(to)28 b(the)g(conserv)-5 b(ativ)n(e)27 b(Hamiltonian)456 3223 y(dynamics)34 b(t)n(w)n(o)g(forces)g(are)h(added,)h(a)f (dissipativ)n(e)f(force)h(\(the)g(terms)g Fj(\000)p Fk(\015)5 b(r)2929 3235 y Fl(i)2991 3223 y Fp(in)36 b(\(19\)\))f(and)456 3323 y(a)30 b(random)g(force)g(\(the)h(white)g(noises\))f(whic)n(h)h (are)f(prop)r(ortional)f(to)h(the)h(temp)r(erature.)46 b(A)n(t)456 3422 y(high)25 b(energies,)g(i.e.,)h(at)g(energies)e(m)n (uc)n(h)i(bigger)e(than)i(the)g(temp)r(eratures)f(of)g(the)h(reserv)n (oirs,)456 3522 y(dissipation)e(dominates)g(and)g(there)g(is)h(a)f (strong)f(drift)i(whic)n(h)f(let)h(the)g(energy)e(of)i(the)g(system)456 3622 y(decrease)j(and)i(the)h(\015uctuating)f(forces)f(are)g (negligible)g(compared)g(to)h(the)g(dissipation.)44 b(A)n(t)456 3721 y(lo)n(w)24 b(energies,)h(on)g(the)g(con)n(trary)e(the)j (\015uctuations)f(are)f(not)h(dominated)h(b)n(y)e(the)i(dissipation)456 3821 y(an)n(ymore.)605 3920 y(Since)k(b)r(oth)g(dissipation)f(and)h (noise)f(act)h(only)f(at)h(the)g(b)r(oundary)f(of)h(the)g(c)n(hain,)g (a)f(k)n(ey)456 4020 y(ingredien)n(t)g(in)h(the)g(analysis)e(is)i(a)f (b)r(ound)h(on)g(propagation)d(of)j(energy)f(in)g(anharmonic)g(lat-)456 4120 y(tices.)35 b(Think)25 b(for)f(example)h(of)f(an)h(initial)g (condition)f(in)h(whic)n(h)g(the)g(energy)e(is)i(concen)n(trated)456 4219 y(in)k(one)h(oscillator)e(far)h(a)n(w)n(a)n(y)e(from)i(the)h(b)r (oundary)-7 b(.)43 b(T)-7 b(o)29 b(pro)n(v)n(e)f(Eq.)42 b(\(28\))30 b(one)f(m)n(ust)g(ha)n(v)n(e)g(a)456 4319 y(lo)n(w)n(er)19 b(b)r(ound)i(on)g(ho)n(w)f(m)n(uc)n(h)h(of)g(the)g (energy)f(propagates)e(through)j(the)g(c)n(hain)f(to)h(the)g(b)r(ound-) 456 4419 y(ary)26 b(to)h(get)g(dissipated.)37 b(F)-7 b(or)27 b(this)h(b)r(ound)f(the)h(condition)f Fq(H2)g Fp(is)h(crucial.)36 b(It)27 b(is)h(w)n(ell-kno)n(wn)456 4518 y([)p Fq(40)o(,)e(25)o(,)g(3)p Fp(])c(that)h(in)g(net)n(w)n(orks)e (of)i(anharmonic)e(oscillators)g(there)h(are)g(breathers)f(whic)n(h)h (are)456 4618 y(\(Nekhoroshev-stable\))k(time)i(p)r(erio)r(dic)g(exp)r (onen)n(tially)f(lo)r(calized)g(solutions)h(of)f(the)i(Hamil-)456 4717 y(tonian)22 b(equations)f(of)i(motions.)34 b(A)23 b(simple)g(scaling)e(argumen)n(t)g(sho)n(ws)h(that)g(the)h(high)g (energy)456 4817 y(b)r(eha)n(vior)30 b(of)i(breathers)e(is)i(v)n(ery)e (di\013eren)n(t)i(dep)r(ending)g(on)g(whether)g(the)g(condition)f Fq(H2)h Fp(is)456 4917 y(satis\014ed)c(or)g(not.)41 b(If)29 b(it)g(is)g(not)g(satis\014ed,)g(the)g(higher)f(the)h(energy)-7 b(,)29 b(the)g(more)f(lo)r(calized)g(the)456 5016 y(breather)d(tends)j (to)e(b)r(e:)37 b(at)27 b(high)g(energy)f(w)n(e)g(can)h(ha)n(v)n(e)f (states)g(with)i(oscillators)d(oscillating)456 5116 y(v)n(ery)j(fast)h (and)g(barely)g(in)n(teracting)f(with)i(their)f(neigh)n(b)r(ors)g(whic) n(h)g(are)f(essen)n(tially)h(at)g(rest.)456 5216 y(On)23 b(the)h(con)n(trary)-7 b(,)23 b(if)h(Condition)g Fq(H2)f Fp(is)h(satis\014ed,)g(then)g(one)g(can)f(sho)n(w)g(that)h(for)f(an)n (y)g(initial)p eop %%Page: 8 8 8 7 bop 456 251 a Fm(8)1166 b(LUC)23 b(REY-BELLET)456 450 y Fp(condition)29 b(of)h(su\016cien)n(tly)f(large)f(energy)h Fk(E)5 b Fp(,)30 b(the)g(kinetic)g(energy)e(of)i(the)g(oscillator)e(on) h(the)456 550 y(b)r(oundary)d(will)i(b)r(e)g(at)g(least)f(of)g(order)g Fk(E)1773 520 y Fm(2)p Fl(=k)1875 528 y Fg(2)1939 550 y Fp(on)h(a)f(time)h(in)n(terv)-5 b(al)27 b(of)h(order)e(1.)605 649 y(T)-7 b(o)31 b(pro)n(v)n(e)e(\(28\),)j(in)g(a)e(\014rst)h(step)h (one)f(sets)f Fk(T)2059 661 y Fm(1)2125 649 y Fp(=)f Fk(T)2268 661 y Fl(n)2342 649 y Fp(=)f(0)j(in)h(\(19\))e(and)h(one)g (obtains)g(a)456 749 y(set)g(of)h(deterministic)g(ODE's)f(equations.)48 b(Using)31 b(the)h(b)r(ound)g(on)g(propagation)d(of)j(energy)456 849 y(one)27 b(sho)n(ws)f(that)i(for)f(large)f(enough)h Fk(G)p Fp(\(0\))d(=)e Fk(E)33 b Fp(w)n(e)27 b(ha)n(v)n(e)456 1000 y(\(29\))897 b Fk(G)p Fp(\(1\))19 b Fj(\000)f Fk(G)p Fp(\(0\))24 b Fj(\024)e(\000)p Fk(cE)2223 966 y Fm(2)p Fl(=k)2325 974 y Fg(2)2376 1000 y Fk(:)456 1144 y Fp(Note)31 b(that)g(this)g(corresp)r(onds)e(to)i(the)g(ph)n(ysical)g(situation)f (where)h(the)g(reserv)n(oirs,)e(at)i(time)456 1244 y Fk(t)45 b Fp(=)h(0)40 b(are)h(at)g(energy)f(0.)77 b(In)42 b(this)f(case)f(the)i(system)f(simply)g(radiates)f(all)h(its)h(energy) 456 1344 y(in)n(to)25 b(the)h(reserv)n(oirs)d(and)i(relax)g(in)n(to)g (a)g(state)h(corresp)r(onding)e(to)h(a)g(stationary)g(p)r(oin)n(t)g(of) h(the)456 1443 y(Hamiltonian)20 b Fk(H)7 b Fp(.)35 b(If)21 b(the)g(temp)r(eratures)g(are)f(non-zero,)g(one)h(sho)n(ws)f(that)h(on) g(a)f(suitable)h(time)456 1543 y(in)n(terv)-5 b(al)25 b(the)i(random)e(solution)h(of)g(\(19\))g(do)g(follo)n(w)g(closely)f (the)i(deterministic)f(tra)5 b(jectories)456 1643 y(with)19 b(v)n(ery)g(high)g(probabilit)n(y)-7 b(.)33 b(So)19 b(for)g(most)g(tra) 5 b(jectories,)19 b(an)g(estimate)g(of)g(the)h(t)n(yp)r(e)g(\(29\))f (also)456 1742 y(holds.)38 b(T)-7 b(o)28 b(conclude)g(of)h(the)f(pro)r (of)g(of)h(\(28\),)f(one)g(considers)f(the)i(function)f Fk(W)2981 1754 y Fl(\022)3044 1742 y Fp(=)c(exp)13 b(\()p Fk(\022)r(G)p Fp(\))456 1842 y(and)27 b(uses)g(some)g(sto)r(c)n(hastic) g(analysis)f(\(see)i([)p Fq(31)o Fp(])g(for)f(details\).)1145 2039 y Fq(4.)47 b(Heat)32 b(Flo)m(w)f(and)h(En)m(trop)m(y)h(Pro)s (duction)605 2189 y Fp(T)-7 b(o)27 b(de\014ne)g(the)h(heat)f(\015o)n(w) g(and)g(the)g(en)n(trop)n(y)f(pro)r(duction)h(w)n(e)g(write)g(the)h (energy)e(of)h(the)456 2288 y(c)n(hain)g Fk(H)34 b Fp(as)27 b(a)g(sum)h(of)g(lo)r(cal)f(energies)f Fk(H)k Fp(=)1916 2226 y Fi(P)2004 2246 y Fl(n)2004 2313 y(i)p Fm(=1)2130 2288 y Fk(H)2199 2300 y Fl(i)2254 2288 y Fp(where)774 2490 y Fk(H)843 2502 y Fm(1)977 2490 y Fp(=)1148 2434 y Fk(p)1190 2403 y Fm(2)1190 2454 y(1)p 1148 2471 80 4 v 1167 2547 a Fp(2)1256 2490 y(+)18 b Fk(U)1405 2455 y Fm(\(1\))1493 2490 y Fp(\()p Fk(q)1562 2502 y Fm(1)1600 2490 y Fp(\))h(+)1744 2434 y(1)p 1744 2471 42 4 v 1744 2547 a(2)1795 2490 y Fk(U)1861 2455 y Fm(\(2\))1950 2490 y Fp(\()p Fk(q)2019 2502 y Fm(1)2075 2490 y Fj(\000)f Fk(q)2195 2502 y Fm(1)2232 2490 y Fp(\))c Fk(;)783 2704 y(H)852 2716 y Fl(i)977 2704 y Fp(=)1148 2647 y Fk(p)1190 2617 y Fm(2)1190 2669 y Fl(i)p 1148 2685 80 4 v 1167 2761 a Fp(2)1256 2704 y(+)k Fk(U)1405 2669 y Fm(\(1\))1493 2704 y Fp(\()p Fk(q)1562 2716 y Fl(i)1590 2704 y Fp(\))h(+)1734 2647 y(1)p 1734 2685 42 4 v 1734 2761 a(2)1799 2611 y Fi(\020)1849 2704 y Fk(U)1915 2669 y Fm(\(2\))2004 2704 y Fp(\()p Fk(q)2073 2716 y Fl(i)p Fh(\000)p Fm(1)2204 2704 y Fj(\000)f Fk(q)2324 2716 y Fl(i)2352 2704 y Fp(\))h(+)f Fk(U)2552 2669 y Fm(\(2\))2641 2704 y Fp(\()p Fk(q)2710 2716 y Fl(i)2756 2704 y Fj(\000)g Fk(q)2876 2716 y Fl(i)p Fm(+1)2988 2704 y Fp(\))3020 2611 y Fi(\021)3097 2704 y Fk(;)-2664 b Fp(\(30\))766 2918 y Fk(H)835 2930 y Fl(n)977 2918 y Fp(=)1148 2861 y Fk(p)1190 2831 y Fm(2)1190 2882 y Fl(n)p 1148 2898 87 4 v 1171 2974 a Fp(2)1263 2918 y(+)19 b Fk(U)1413 2883 y Fm(\(1\))1501 2918 y Fp(\()p Fk(q)1570 2930 y Fl(n)1616 2918 y Fp(\))g(+)1760 2861 y(1)p 1760 2898 42 4 v 1760 2974 a(2)1811 2918 y Fk(U)1877 2883 y Fm(\(2\))1966 2918 y Fp(\()p Fk(q)2035 2930 y Fl(n)2099 2918 y Fj(\000)f Fk(q)2219 2930 y Fl(n)p Fh(\000)p Fm(1)2349 2918 y Fp(\))c Fk(:)456 3085 y Fp(Di\013eren)n(tiating)27 b(with)h(resp)r(ect)g(to)f(time)h(one)f(\014nds)456 3268 y(\(31\))1490 3212 y Fk(d)p 1475 3249 74 4 v 1475 3325 a(dt)1558 3268 y(T)1619 3234 y Fl(t)1647 3268 y Fk(H)1716 3280 y Fl(i)1781 3268 y Fp(=)37 b Fk(T)1944 3234 y Fl(t)1972 3268 y Fp(\(\010)2064 3280 y Fl(i)p Fh(\000)p Fm(1)2195 3268 y Fj(\000)18 b Fp(\010)2338 3280 y Fl(i)2366 3268 y Fp(\))c Fk(;)456 3436 y Fp(where)1245 3580 y(\010)1305 3592 y Fm(0)1439 3580 y Fp(=)96 b Fj(\000)p Fk(\025r)1750 3592 y Fm(1)1788 3580 y Fk(p)1830 3592 y Fm(1)1881 3580 y Fk(;)1254 3748 y Fp(\010)1314 3760 y Fl(i)1439 3748 y Fp(=)1610 3691 y(\()p Fk(p)1684 3703 y Fl(i)1730 3691 y Fp(+)18 b Fk(p)1855 3703 y Fl(i)p Fm(+1)1967 3691 y Fp(\))p 1610 3728 389 4 v 1784 3805 a(2)2009 3748 y Fj(r)p Fk(U)2144 3713 y Fm(\(2\))2233 3748 y Fp(\()p Fk(q)2302 3760 y Fl(i)2348 3748 y Fj(\000)h Fk(q)2469 3760 y Fl(i)p Fm(+1)2580 3748 y Fp(\))14 b Fk(;)-2193 b Fp(\(32\))1237 3895 y(\010)1297 3907 y Fl(n)1439 3895 y Fp(=)96 b Fk(\025r)1685 3907 y Fl(n)1731 3895 y Fk(p)1773 3907 y Fl(n)1832 3895 y Fk(:)456 4040 y Fp(It)36 b(is)g(natural)f(in)n(terpret)h(\010)1358 4052 y Fl(i)1385 4040 y Fp(,)j Fk(i)d Fp(=)h(1)p Fk(;)14 b Fj(\001)g(\001)g(\001)27 b Fk(;)14 b(n)24 b Fj(\000)f Fp(1)36 b(as)f(the)i(heat)f(\015o)n(w)f(from)h(the)g Fk(i)3115 4010 y Fl(th)3219 4040 y Fp(to)g(the)456 4139 y(\()p Fk(i)20 b Fp(+)g(1\))696 4109 y Fl(th)794 4139 y Fp(particle,)31 b(\010)1181 4151 y Fm(0)1248 4139 y Fp(as)f(the)h(\015o)n(w)e(from)h(the)h(left)g(reserv)n(oir)d(in)n(to)i (the)h(c)n(hain,)f(and)h(\010)3295 4151 y Fl(n)3370 4139 y Fp(as)456 4239 y(the)g(\015o)n(w)e(from)i(the)g(c)n(hain)f(in)n(to)g (the)h(righ)n(t)f(reserv)n(oir.)42 b(W)-7 b(e)31 b(de\014ne)g(corresp)r (onding)e(en)n(trop)n(y)456 4339 y(pro)r(ductions)e(b)n(y)456 4528 y(\(33\))949 b Fk(\033)1600 4540 y Fl(i)1665 4528 y Fp(=)1766 4411 y Fi(\022)1859 4472 y Fp(1)p 1837 4509 86 4 v 1837 4585 a Fk(T)1886 4597 y Fm(1)1951 4528 y Fj(\000)2070 4472 y Fp(1)p 2044 4509 94 4 v 2044 4585 a Fk(T)2093 4597 y Fl(n)2148 4411 y Fi(\023)2223 4528 y Fp(\010)2283 4540 y Fl(i)2324 4528 y Fk(:)456 4717 y Fp(There)31 b(are)g(other)h(p)r(ossible)f(de\014nitions)i(of)f(heat)g (\015o)n(ws)f(and)h(corresp)r(onding)e(en)n(trop)n(y)h(pro-)456 4817 y(duction)k(that)g(one)g(migh)n(t)g(w)n(an)n(t)g(to)g(consider.)58 b(One)35 b(migh)n(t,)i(for)d(example,)j(consider)d(the)456 4917 y(\015o)n(ws)f(at)i(the)g(b)r(oundary)f(of)g(the)h(c)n(hains,)h (and)e(de\014ne)h Fk(\033)2305 4929 y Fl(b)2374 4917 y Fp(=)f(\010)2533 4929 y Fm(1)2570 4917 y Fk(=T)2661 4929 y Fm(1)2721 4917 y Fj(\000)23 b Fp(\010)2869 4929 y Fl(n)2914 4917 y Fk(=T)3005 4929 y Fl(n)3049 4917 y Fp(.)58 b(Also)34 b(our)456 5016 y(c)n(hoice)h(of)i(lo)r(cal)f(energy)g (is)g(somewhat)g(arbitrary)-7 b(,)37 b(other)f(c)n(hoices)g(are)g(p)r (ossible)g(but)h(this)456 5116 y(do)r(es)27 b(not)g(c)n(hange)g(the)h (subsequen)n(t)f(analysis.)605 5216 y(Our)g(results)g(on)g(the)h(heat)g (\015o)n(w)f(are)f(summarized)h(in)p eop %%Page: 9 9 9 8 bop 1018 251 a Fm(ST)-5 b(A)g(TISTICAL)29 b(MECHANICS)g(OF)g (ANHARMONIC)h(LA)-5 b(TTICES)529 b(9)605 450 y Fe(Theorem)32 b Fp(4.1)p Fe(.)40 b Fq(:)i(En)m(trop)m(y)33 b(pro)s(duction)456 597 y Fo(\(a\))27 b(The)i(exp)l(e)l(ctation)e(of)h(the)g(entr)l(opy)f (pr)l(o)l(duction)h Fk(\033)2166 609 y Fl(j)2229 597 y Fo(in)g(the)f(stationary)h(state)f(is)h(indep)l(en-)456 697 y(dent)h(of)i Fk(j)j Fo(and)d(nonne)l(gative)456 902 y Fp(\(34\))1705 789 y Fi(Z)1802 902 y Fk(\033)1849 914 y Fl(j)1885 902 y Fk(d\026)37 b Fj(\025)f Fp(0)14 b Fk(;)456 1108 y Fo(and)30 b(it)g(is)g(p)l(ositive)h(away)g(fr)l(om)f (e)l(quilibrium)456 1313 y Fp(\(35\))1117 1200 y Fi(Z)1214 1313 y Fk(\033)1261 1325 y Fl(j)1296 1313 y Fk(d\026)37 b Fp(=)g(0)199 b(if)36 b(and)30 b(only)g(if)206 b Fk(T)2505 1325 y Fm(1)2565 1313 y Fp(=)23 b Fk(T)2702 1325 y Fl(n)2760 1313 y Fk(:)456 1571 y Fo(\(b\))29 b(The)i(er)l(go)l(dic)g(aver)l(ages) 456 1796 y Fp(\(36\))1710 1740 y(1)p 1710 1777 42 4 v 1716 1853 a Fk(t)1776 1683 y Fi(Z)1859 1704 y Fl(t)1822 1872 y Fm(0)1902 1796 y Fk(\033)1949 1808 y Fl(j)1984 1796 y Fp(\()p Fk(x)p Fp(\()p Fk(s)p Fp(\)\))456 2006 y Fo(satisfy)j(the)g(lar)l(ge)h(deviation)g(principle:)49 b(Ther)l(e)34 b(exist)g(a)g(neighb)l(orho)l(o)l(d)i Fk(O)g Fo(of)f(the)f(interval)456 2105 y Fp([)p Fj(\000)558 2039 y Fi(R)626 2105 y Fk(\033)673 2117 y Fl(j)709 2105 y Fk(d\026;)839 2039 y Fi(R)908 2105 y Fk(\033)955 2117 y Fl(j)990 2105 y Fk(d\026)p Fp(])i Fo(and)g(a)f(r)l(ate)g(function)h Fk(e)p Fp(\()p Fk(w)r Fp(\))g Fo(\(b)l(oth)f(indep)l(endent)h(of)g(j\)) 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3147 y Fl(j)1810 3135 y Fp(giv)n(en)f(b)n(y)456 3405 y(\(39\))220 b Fk(R)887 3417 y Fl(j)959 3405 y Fp(=)1092 3349 y(1)p 1070 3386 86 4 v 1070 3462 a Fk(T)1119 3474 y Fm(1)1180 3263 y Fi( )1256 3349 y Fk(r)1295 3319 y Fm(2)1293 3370 y(1)p 1256 3386 78 4 v 1273 3462 a Fp(2)1361 3405 y(+)1489 3298 y Fl(j)1444 3326 y Fi(X)1444 3505 y Fl(k)q Fm(=1)1579 3405 y Fk(H)1648 3417 y Fl(k)1688 3405 y Fp(\()p Fk(p;)14 b(q)s Fp(\))1871 3263 y Fi(!)1956 3405 y Fp(+)2075 3349 y(1)p 2049 3386 94 4 v 2049 3462 a Fk(T)2098 3474 y Fl(n)2166 3238 y Fi(0)2166 3388 y(@)2320 3301 y Fl(n)2280 3326 y Fi(X)2239 3505 y Fl(k)q Fm(=)p Fl(j)s Fm(+1)2455 3405 y Fk(H)2524 3417 y Fl(k)2565 3405 y Fp(\()p Fk(p;)g(q)s Fp(\))19 b(+)2860 3349 y Fk(r)2899 3319 y Fm(2)2897 3370 y Fl(n)p 2860 3386 83 4 v 2881 3462 a Fp(2)2953 3238 y Fi(1)2953 3388 y(A)3053 3405 y Fk(;)456 3673 y Fp(so)28 b(that)i(exp)13 b(\()p Fj(\000)p Fk(R)1041 3685 y Fl(j)1076 3673 y Fp(\))30 b(is)f(a)g(\\t)n(w)n(o-temp) r(eratures")d(Gibbs)k(state.)41 b(W)-7 b(e)30 b(denote)f(b)n(y)g Fk(J)37 b Fp(the)30 b(time)456 3772 y(rev)n(ersal)16 b(op)r(erator)i(whic)n(h)g(c)n(hanges)g(the)h(sign)g(of)g(the)g(momen)n (ta)f(of)h(all)g(particles)f Fk(J)8 b(f)h Fp(\()p Fk(p;)14 b(q)s(;)g(r)r Fp(\))24 b(=)456 3872 y Fk(f)9 b Fp(\()p Fj(\000)p Fk(p;)14 b(q)s(;)g(r)r Fp(\))26 b(and)f(w)n(e)h(denote)f(as)g Fk(L)1558 3842 y Fl(T)1636 3872 y Fp(the)h(formal)f(adjoin)n(t)h(of)g (the)g(generator)d Fk(L)j Fp(\(the)g(F)-7 b(okk)n(er-)456 3971 y(Planc)n(k)26 b(op)r(erator\).)35 b(The)28 b(follo)n(wing)f (\(formal\))g(op)r(erator)f(iden)n(tities)i(are)e(easily)h(v)n (eri\014ed)456 4143 y(\(40\))880 b Fk(e)1523 4108 y Fl(R)1573 4116 y Ff(j)1608 4143 y Fk(J)8 b(L)1719 4108 y Fl(T)1771 4143 y Fk(J)g(e)1864 4108 y Fh(\000)p Fl(R)1966 4116 y Ff(j)2037 4143 y Fp(=)37 b Fk(L)18 b Fj(\000)g Fk(\033)2344 4155 y Fl(j)2393 4143 y Fk(;)456 4310 y Fp(and)27 b(also)g(for)g(an)n (y)f(constan)n(t)h Fk(\013)456 4478 y Fp(\(41\))599 b Fk(e)1242 4444 y Fh(\000)p Fl(R)1344 4452 y Ff(j)1378 4478 y Fk(J)8 b Fp(\()p Fk(L)1521 4444 y Fl(T)1592 4478 y Fj(\000)18 b Fk(\013\033)1775 4490 y Fl(j)1811 4478 y Fp(\))p Fk(J)8 b(e)1936 4444 y Fl(R)1986 4452 y Ff(j)2058 4478 y Fp(=)36 b Fk(L)18 b Fj(\000)g Fp(\(1)g Fj(\000)g Fk(\013)p Fp(\))p Fk(\033)2624 4490 y Fl(j)2674 4478 y Fk(:)456 4646 y Fp(These)32 b(iden)n(tities)g(are)g(the)h(k)n(ey)e (elemen)n(t)i(to)f(pro)n(v)n(e)f(b)r(oth)i(Eqs.)50 b(\(34\))33 b(and)f(\(38\).)51 b(The)32 b(fact)456 4746 y(that)g(the)h(en)n(trop)n (y)f(pro)r(duction)g(is)g(strictly)h(p)r(ositiv)n(e)f(a)n(w)n(a)n(y)e (from)j(equilibrium)f(do)h(require)456 4845 y(more)26 b(w)n(ork)h([)p Fq(11)o Fp(].)605 4945 y(Let)35 b(us)g(sk)n(etc)n(h)g (the)g(pro)r(of)g(of)g(Eqs.)59 b(\(34\).)g(W)-7 b(e)35 b(write)g(the)h(p)r(ositiv)n(e)e(densit)n(y)h Fk(\032)p Fp(\()p Fk(x)p Fp(\))i(of)456 5044 y Fk(\026)p Fp(\()p Fk(dx)p Fp(\))24 b(=)f Fk(\032)p Fp(\()p Fk(x)p Fp(\))p Fk(dx)29 b Fp(as)456 5216 y(\(42\))1052 b Fk(\032)23 b Fp(=)g Fk(J)8 b(e)1903 5181 y Fh(\000)p Fl(R)2005 5189 y Ff(j)2039 5216 y Fk(e)2078 5181 y Fh(\000)p Fl(F)2172 5189 y Ff(j)2221 5216 y Fk(:)p eop %%Page: 10 10 10 9 bop 456 251 a Fm(10)1133 b(LUC)23 b(REY-BELLET)456 450 y Fp(Let)28 b Fk(L)662 420 y Fh(\003)727 450 y Fp(denote)g(the)h (adjoin)n(t)e(of)h Fk(L)g Fp(on)g Fk(L)1773 420 y Fm(2)1809 450 y Fp(\()p Fk(\026)p Fp(\),)h(it)g(is)f(giv)n(en)f(b)n(y)g Fk(L)2532 420 y Fh(\003)2594 450 y Fp(=)c Fk(\032)2725 420 y Fh(\000)p Fm(1)2814 450 y Fk(L\032)28 b Fp(and)f(using)h(Eq.)456 550 y(\(40\))f(a)g(simple)h(computation)f(sho)n(ws)g(that)998 689 y Fk(J)8 b(L)1109 655 y Fh(\003)1147 689 y Fk(J)105 b Fp(=)96 b Fk(e)1498 655 y Fl(F)1540 663 y Ff(j)1575 689 y Fp(\()p Fk(L)19 b Fj(\000)f Fk(\033)1813 701 y Fl(j)1848 689 y Fp(\))p Fk(e)1919 655 y Fh(\000)p Fl(F)2013 663 y Ff(j)1298 833 y Fp(=)96 b Fk(L)18 b Fj(\000)g Fk(\033)1664 845 y Fl(j)1718 833 y Fj(\000)g Fp(\()p Fk(LF)1943 845 y Fl(j)1978 833 y Fp(\))h Fj(\000)f Fp(2)2223 754 y Fi(X)2168 936 y Fl(i)p Fh(2f)p Fm(1)p Fl(;n)p Fh(g)2412 833 y Fk(T)2461 845 y Fl(i)2488 833 y Fp(\()p Fj(r)2589 845 y Fl(r)2620 853 y Ff(i)2650 833 y Fk(F)2703 845 y Fl(j)2739 833 y Fp(\))p Fj(r)2840 845 y Fl(r)2871 853 y Ff(i)456 833 y Fp(\(43\))1459 1079 y(+)1593 1000 y Fi(X)1538 1182 y Fl(i)p Fh(2f)p Fm(1)p Fl(;n)p Fh(g)1768 1079 y Fp(\()p Fk(T)1849 1091 y Fl(i)1876 1079 y Fj(jr)1968 1091 y Fl(r)1999 1099 y Ff(i)2030 1079 y Fk(F)2083 1091 y Fl(j)2118 1079 y Fj(j)2141 1045 y Fm(2)2179 1079 y Fp(\))c Fk(:)456 1307 y Fp(The)34 b(op)r(erator)f Fk(J)8 b(L)1086 1277 y Fh(\003)1124 1307 y Fk(J)43 b Fp(is)35 b(the)g(generator)e(of)h(the)h (time-rev)n(ersed)f(pro)r(cess)f(and)i(using)f(that)456 1407 y Fk(L)513 1377 y Fh(\003)550 1407 y Fp(1)23 b(=)f(0)28 b(w)n(e)f(\014nd)h(the)g(iden)n(tit)n(y)456 1555 y(\(44\))790 b Fk(\033)1441 1567 y Fl(i)1506 1555 y Fp(=)1663 1476 y Fi(X)1608 1658 y Fl(i)p Fh(2f)p Fm(1)p Fl(;n)p Fh(g)1852 1555 y Fk(T)1901 1567 y Fl(i)1928 1555 y Fj(jr)2020 1567 y Fl(r)2051 1575 y Ff(i)2081 1555 y Fk(F)2134 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2398 y Fl(i)1396 2452 y(x)1438 2432 y Fp(\()p Fk(t;)14 b(\013)p Fp(\))37 b(=)g Fq(E)1824 2444 y Fl(x)1879 2340 y Fi(h)1918 2432 y Fk(e)1957 2398 y Fh(\000)p Fl(\013)2064 2351 y Fd(R)2108 2371 y Ff(t)2096 2420 y Fg(0)2147 2398 y Fl(\033)2185 2406 y Ff(i)2212 2398 y Fm(\()p Fl(x)p Fm(\()p Fl(s)p Fm(\)\))11 b Fl(ds)2466 2340 y Fi(i)2533 2432 y Fk(:)456 2604 y Fp(F)-7 b(ormally)34 b(the)h(F)-7 b(eynman-Kac)34 b(form)n(ula)g(giv)n(es)f(\000)2074 2574 y Fl(i)2074 2625 y(x)2116 2604 y Fp(\()p Fk(t;)14 b(\013)p Fp(\))36 b(=)f Fk(e)2475 2574 y Fl(t)p Fm(\()p Fl(L)p Fh(\000)p Fl(\013\033)2705 2582 y Ff(i)2731 2574 y Fm(\))2761 2604 y Fp(1\()p Fk(x)p Fp(\))h(and)f(the)g(large)456 2704 y(deviation)27 b(functional)g Fk(e)p Fp(\()p Fk(w)r Fp(\))i(is)e(giv)n(en)g(b)n(y)h(the)g(Legendre)e(transform)h(of)g(the)h (function)456 2878 y(\(46\))838 b Fk(e)p Fp(\()p Fk(\013)p Fp(\))24 b Fj(\021)44 b Fp(lim)1710 2928 y Fl(t)p Fh(!1)1881 2878 y Fj(\000)1956 2822 y Fp(1)p 1956 2859 42 4 v 1962 2935 a Fk(t)2021 2878 y Fp(log)14 b(\000)2194 2844 y Fl(i)2194 2899 y(x)2236 2878 y Fp(\()p Fk(t;)g(\013)p Fp(\))g Fk(:)456 3043 y Fp(By)28 b(a)h(P)n(erron-F)-7 b(ro)r(eb)r(enius)26 b(argumen)n(t)i Fk(e)p Fp(\()p Fk(\013)p Fp(\))i(is)e(the)i(largest)d(eigen)n(v)-5 b(alue)28 b(of)h Fk(L)19 b Fj(\000)g Fk(\013\033)3163 3055 y Fl(i)3191 3043 y Fp(.)41 b(Since)456 3143 y Fk(L)10 b Fj(\000)g Fk(\013\033)698 3155 y Fl(i)748 3143 y Fp(is)23 b(conjugated)g(to)g Fk(L)1399 3113 y Fl(T)1460 3143 y Fj(\000)10 b Fp(\(1)g Fj(\000)g Fk(\013)p Fp(\))p Fk(\033)1826 3155 y Fl(i)1877 3143 y Fp(b)n(y)23 b(Eq.)35 b(\(41\))22 b(and)i(since)f Fk(L)10 b Fj(\000)g Fk(\013\033)2916 3155 y Fl(i)2966 3143 y Fp(and)23 b Fk(L)3180 3113 y Fl(T)3242 3143 y Fj(\000)10 b Fk(\013\033)3417 3155 y Fl(i)456 3242 y Fp(should)23 b(ha)n(v)n(e)f(the)i(same)f(sp)r(ectrum)g(w)n(e)g (conclude)g(that)h Fk(e)p Fp(\()p Fk(\013)p Fp(\))g(=)e Fk(e)p Fp(\(1)10 b Fj(\000)g Fk(\013)p Fp(\).)36 b(T)-7 b(aking)23 b(Legendre)456 3342 y(transform)j(w)n(e)h(obtain)h(Eq.)36 b(\(38\).)605 3442 y(Ho)n(w)30 b(do)h(w)n(e)g(mak)n(e)f(this)h(argumen) n(t)f(rigorous?)44 b(F)-7 b(rom)31 b(the)g(form)g(of)g(the)g(en)n(trop) n(y)f(pro-)456 3541 y(duction)g Fk(\033)k Fp(\(it)c(is)g(an)g(un)n(b)r (ounded)h(function\),)g(one)f(sees)g(that)g Fk(L)20 b Fj(\000)f Fk(\013\033)2743 3553 y Fl(i)2802 3541 y Fp(is)30 b(not)g(a)g(relativ)n(ely)456 3641 y(b)r(ounded)25 b(p)r(erturbation)f (of)g Fk(L)p Fp(.)36 b(A)25 b(priori)f(it)h(is)f(not)h(ev)n(en)f(ob)n (vious)f Fk(L)13 b Fj(\000)g Fk(\013\033)2830 3653 y Fl(i)2882 3641 y Fp(is)25 b(the)g(generator)456 3741 y(of)k(a)h(semigroup,)f(i.e.,)i(that)f(the)g(function)g(\000)1913 3710 y Fl(i)1913 3761 y(x)1955 3741 y Fp(\()p Fk(t;)14 b(\013)p Fp(\))31 b(is)f(\014nite)g(for)f Fk(\013)f Fj(6)p Fp(=)e(0.)43 b(T)-7 b(o)30 b(mak)n(e)f(things)456 3840 y(w)n(ork)d(w)n(e)h(will)h(use)f(the)h(follo)n(wing)f(iden)n(tit)n(y)h (whic)n(h)f(can)g(b)r(e)h(c)n(hec)n(k)n(ed)f(easily)456 3976 y(\(47\))875 b Fk(L)18 b Fj(\000)g Fk(\013\033)1737 3988 y Fl(i)1802 3976 y Fp(=)37 b Fk(e)1943 3942 y Fl(\013R)2036 3950 y Ff(i)p 2066 3909 57 4 v 2066 3976 a Fk(L)2123 3988 y Fl(\013)2170 3976 y Fk(e)2209 3942 y Fh(\000)p Fl(\013R)2354 3950 y Ff(i)2398 3976 y Fk(;)456 4112 y Fp(where)456 4247 y(\(48\))p 933 4181 V 329 w Fk(L)989 4259 y Fl(\013)1074 4247 y Fp(=)f Fk(L)18 b Fp(+)g Fk(\015)1450 4168 y Fi(X)1395 4350 y Fl(i)p Fh(2f)p Fm(1)p Fl(;n)p Fh(g)1639 4180 y Fi(\000)1677 4247 y Fp(2)p Fk(\013r)1809 4259 y Fl(i)1837 4247 y Fj(r)1906 4259 y Fl(r)1937 4267 y Ff(i)1986 4247 y Fj(\000)g Fp(\()p Fk(\013)h Fj(\000)f Fk(\013)2309 4213 y Fm(2)2346 4247 y Fp(\))p Fk(T)2439 4212 y Fh(\000)p Fm(1)2427 4270 y Fl(i)2528 4247 y Fk(r)2567 4213 y Fm(2)2565 4268 y Fl(i)2605 4180 y Fi(\001)2661 4247 y Fp(+)g(2)p Fk(d\015)5 b(\013)14 b(:)456 4491 y Fp(This)25 b(sho)n(ws)g(that)h(all)f(the)h(op)r(erators)d Fk(L)14 b Fj(\000)g Fk(\013\033)1927 4503 y Fl(i)1982 4491 y Fp(are)24 b(conjugated)h(to)h(the)g(same)f(op)r(erator)p 3318 4424 V 24 w Fk(L)3374 4503 y Fl(\013)3421 4491 y Fp(.)456 4591 y(It)20 b(is)g(not)h(hard)e(to)h(see)g(that)p 1335 4524 V 21 w Fk(L)1391 4603 y Fl(\013)1459 4591 y Fp(is)g(relativ)n(ely)f(b)r(ounded)h(p)r(erturbation)g(of)g Fk(L)p Fp(.)34 b(Using)20 b(the)h(same)456 4690 y(tec)n(hniques)h(as)h (the)g(one)g(used)g(in)g(the)g(pro)r(of)g(of)g(Theorem)f(3.1)g(one)g (can)h(sho)n(w)f(that)h(exp)14 b(\()p Fk(t)p 3308 4623 V(L)3365 4702 y Fl(\013)3412 4690 y Fp(\))456 4790 y(de\014nes)25 b(a)f(quasib)r(ounded)h(compact)f(semigroup)g(on)h Fj(H)2223 4802 y Fl(\022)2285 4790 y Fp(pro)n(vided)f(that)i Fj(\000)p Fk(\013)d(<)f(\022)r(T)3120 4802 y Fl(i)3171 4790 y Fk(<)g Fp(1)13 b Fj(\000)g Fk(\013)456 4889 y Fp(and)34 b(using)h(Eq.)58 b(\(48\))35 b(this)g(sho)n(ws)f(that)h(the)h(function)f Fk(e)p Fp(\()p Fk(\013)p Fp(\))h(exists)e(and)h(is)g(real)f(analytic) 456 4989 y(pro)n(vided)456 5170 y(\(49\))587 b Fk(\013)23 b Fj(2)1346 5053 y Fi(\022)1407 5170 y Fj(\000)1620 5114 y Fk(T)1669 5126 y Fm(min)p 1481 5151 440 4 v 1481 5227 a Fk(T)1530 5239 y Fm(max)1675 5227 y Fj(\000)18 b Fk(T)1807 5239 y Fm(min)1944 5170 y Fk(;)28 b Fp(1)18 b(+)2286 5114 y Fk(T)2335 5126 y Fm(min)p 2148 5151 V 2148 5227 a Fk(T)2197 5239 y Fm(max)2342 5227 y Fj(\000)g Fk(T)2474 5239 y Fm(min)2597 5053 y Fi(\023)2686 5170 y Fk(:)p eop %%Page: 11 11 11 10 bop 1018 251 a Fm(ST)-5 b(A)g(TISTICAL)29 b(MECHANICS)g(OF)g (ANHARMONIC)h(LA)-5 b(TTICES)496 b(11)456 450 y Fp(Using)27 b(Gaertner-Ellis)f(Theorem)g(concludes)i(the)f(pro)r(of)g(of)h(Theorem) f(4.1.)605 609 y Fe(Remark)35 b Fp(4.2)p Fe(.)42 b Fq(Time)33 b(rev)m(ersal)i(and)h(en)m(trop)m(y)g(pro)s(duction)f(I.)d Fp(It)f(is)f(instructiv)n(e)456 709 y(to)37 b(giv)n(e)g(a)h(in)n (terpretation)e(of)i(Eq.)67 b(\(43\))38 b(in)g(terms)f(of)h(the)g(path) g(space)f(measure)g(of)h(the)456 808 y(pro)r(cess)30 b(\(see)h([)p Fq(26)o Fp(]\).)48 b(A)n(t)31 b(equilibrium)h(\()p Fk(T)1812 820 y Fm(1)1877 808 y Fp(=)d Fk(T)2020 820 y Fl(n)2094 808 y Fp(=)f Fk(T)12 b Fp(\))31 b(this)g(is)g(simply)g (detailed)g(balance,)456 908 y Fk(R)519 920 y Fl(i)583 908 y Fp(=)36 b Fk(G)p Fp(,)i Fk(J)8 b(L)921 878 y Fh(\003)959 908 y Fk(J)44 b Fp(=)37 b Fk(L)e Fp(or)g Fk(p)1395 920 y Fl(t)1424 908 y Fp(\()p Fk(J)8 b(y)s(;)14 b(J)8 b(x)p Fp(\))37 b(=)f Fk(p)1904 920 y Fl(t)1933 908 y Fp(\()p Fk(x;)14 b(y)s Fp(\))g(exp)g(\()p Fj(\000)p Fp(\()p Fk(G)p Fp(\()p Fk(y)s Fp(\))19 b Fj(\000)f Fk(G)p Fp(\()p Fk(x)p Fp(\)\))p Fk(=T)12 b Fp(\))q(.)61 b(If)36 b(w)n(e)f(are)456 1008 y(a)n(w)n(a)n(y)f(from)j(equilibrium)f(w)n(e)h(can)f(in)n(terpret) g(it)h(as)f(follo)n(ws.)63 b(Let)37 b Fk(P)2719 1020 y Fl(st)2817 1008 y Fp(denotes)f(the)h(path)456 1107 y(space)d(measure)f(of)i(the)g(stationary)f(pro)r(cess)f(starting)h(in) h(the)g(state)g Fk(\026)g Fp(and)f(let)h Fk(\022)j Fp(denote)456 1207 y(the)25 b(op)r(erator)e(de\014ned)i(b)n(y)f Fk(\022)r(x)p Fp(\()p Fk(t)p Fp(\))g(=)f Fk(x)p Fp(\()p Fj(\000)p Fk(t)p Fp(\).)37 b(The)24 b(path)h(space)f(measure)g(of)g(the)h (\(stationary\))456 1307 y(time)36 b(rev)n(ersed)e(pro)r(cess)h(is)h (giv)n(en)f(b)n(y)h Fk(P)1789 1276 y Fl(r)r(ev)1777 1327 y(st)1929 1307 y Fp(=)h Fk(J)8 b(P)2138 1319 y Fl(st)2223 1307 y Fj(\016)23 b Fk(\022)2329 1276 y Fh(\000)p Fm(1)2455 1307 y Fp(and)36 b(its)g(generator)e(is)i Fk(J)8 b(L)3330 1276 y Fh(\003)3367 1307 y Fk(J)g Fp(.)456 1406 y(Using)27 b(the)h(relation)f(\(43\))g(and)g(F)-7 b(eynman-Kac)27 b(form)n(ula)f(w)n(e)h(\014nd)456 1601 y(\(50\))1005 1545 y Fk(dP)1113 1514 y Fl(r)r(ev)1101 1565 y(st)p 1005 1582 212 4 v 1032 1658 a Fk(dP)1128 1670 y Fl(st)1227 1601 y Fp(\()p Fk(x)p Fp(\()p Fk(t)p Fp(\)\))38 b(=)f Fk(e)1611 1566 y Fh(\000)1673 1520 y Fd(R)1718 1540 y Ff(t)1706 1588 y Fg(0)1768 1566 y Fl(ds\033)1872 1574 y Ff(j)1904 1566 y Fm(\()p Fl(x)p Fm(\()p Fl(s)p Fm(\)\))2099 1601 y Fj(\002)18 b Fk(e)2221 1566 y Fh(\000)p Fm(\()p Fl(F)2341 1574 y Ff(j)2372 1566 y Fm(\()p Fl(x)p Fm(\()p Fl(t)p Fm(\)\))p Fh(\000)p Fl(F)2633 1574 y Ff(j)2663 1566 y Fm(\()p Fl(x)p Fm(\(0\)\)\))2882 1601 y Fk(:)456 1792 y Fp(As)23 b(p)r(oin)n(ted)g(out)g(in)h([)p Fq(26)o Fp(],)h(this)e(giv)n(es)f(a)h(microscopic)f(de\014nition)h(of)g(the)h (en)n(trop)n(y)e(pro)r(duction)456 1892 y(and)27 b(relates)g(it)h (directly)f(to)g(the)h(action)g(of)f(time-rev)n(ersal.)605 2051 y Fe(Remark)i Fp(4.3)p Fe(.)37 b Fq(Time)27 b(rev)m(ersal)i(and)g (en)m(trop)m(y)h(pro)s(duction)e(I)s(I.)e Fp(If)g(w)n(e)e(go)g(bac)n(k) g(to)456 2151 y(the)i(original)e(Hamiltonian)h(description)g(of)h(the)g (system,)g(w)n(e)f(note)g(that)h(the)g(dynamics)g(and)456 2250 y(the)k(in)f(ital)h(conditions)f(of)h(the)g(reserv)n(oirs)c(are)j (in)n(v)-5 b(arian)n(t)29 b(under)g(time)h(rev)n(ersal:)39 b(c)n(hanging)456 2350 y Fk(t)27 b Fp(in)n(to)h Fj(\000)p Fk(t)p Fp(,)f(and)g(c)n(hanging)g(the)h(signs)e(of)i(the)g(momen)n(ta)f Fk(p)g Fp(of)h(the)g(crystal)e(and)i(of)f(the)h(\014elds)456 2450 y(\005)40 b(lea)n(v)n(es)f(the)i(equations)f(of)g(motion)h(unc)n (hanged.)75 b(Also)40 b(the)h(initial)f(condition)h(of)f(the)456 2549 y(reserv)n(oirs)26 b(are)i(distributed)h(according)e(to)i(the)h (Gibbs)f(measure)f(and)h(are)f(in)n(v)-5 b(arian)n(t)28 b(under)456 2649 y(rev)n(ersal)23 b(of)i(the)g(v)n(elo)r(cities)g(in)g (the)h(reserv)n(oirs.)33 b(Consequen)n(tly)24 b(w)n(e)h(can)g(study)g (the)h(b)r(eha)n(vior)456 2748 y(of)e(the)i(system)e(as)g Fk(t)f Fj(!)g(\0001)i Fp(simply)g(b)n(y)f(c)n(hanging)g Fk(p)g Fp(in)n(to)h Fj(\000)p Fk(p)p Fp(,)g Fk(\031)j Fp(in)n(to)c Fj(\000)p Fk(\031)k Fp(and)d(considering)456 2848 y Fk(t)35 b Fj(!)h(1)p Fp(.)59 b(If)36 b(w)n(e)e(do)h(this)g(and)g (reduce)g(the)g(dynamics)g(as)f(in)i(the)f(section)g(2)f(w)n(e)h (obtain)g(a)456 2948 y(Mark)n(o)n(v)22 b(pro)r(cess)h(with)h(a)g (generator)e(giv)n(en)i(b)n(y)g Fk(J)8 b(LJ)32 b Fp(\(the)25 b(v)-5 b(ariables)23 b Fk(r)k Fp(are)c(left)i(unc)n(hanged\))456 3047 y(and)i(the)h(system)f(relaxes)g(in)n(to)g(the)h(stationary)e (state)h Fk(J)8 b(\026)p Fp(.)605 3147 y(Since)30 b Fk(\033)871 3159 y Fl(j)936 3147 y Fp(is)g(an)f(o)r(dd)h(function)g(of)g Fk(p)p Fp(,)g(w)n(e)f(ha)n(v)n(e)2142 3080 y Fi(R)2211 3147 y Fk(\033)2258 3159 y Fl(j)2294 3147 y Fk(d\026)d Fj(\024)h Fp(0.)42 b(One)30 b(migh)n(t)f(b)r(e)i(tempted)456 3247 y(to)37 b(dra)n(w)g(the)h(conclusion)e(that,)41 b(in)d(the)g(far)f(distan)n(t)g(past,)j(heat)e(w)n(as)e(\015o)n(wing)h (from)g(the)456 3346 y(cold)c(reserv)n(oir)f(in)n(to)i(the)g(hot)h (one.)56 b(This)34 b(is)g(incorrect)f(since)h(the)g(v)n(ery)f (de\014nition)i(of)f(the)456 3446 y(heat)24 b(\015o)n(ws)g(in)n(v)n (olv)n(es)e(a)i(time-deriv)-5 b(ativ)n(e,)25 b(if)g(w)n(e)f(consider)f Fk(t)g Fj(!)g(\0001)h Fp(one)h(should)f(c)n(hange)f(the)456 3546 y(de\014nition)28 b(of)f(the)h(\015o)n(ws)f(accordingly)-7 b(.)605 3705 y Fe(Remark)26 b Fp(4.4)p Fe(.)35 b Fq(Green-Kub)s(o)24 b(form)m(ula.)35 b Fp(As)22 b(noted)g(in)g([)p Fq(15,)j(22)o Fp(])d(one)g(can)f(deriv)n(e)g(the)456 3804 y(Green-Kub)r(o)26 b(from)i(the)g(\015uctuation)g(theorem.)38 b(Here)27 b(the)i(external)e(\\\014eld")g(is)h(the)g(in)n(v)n(erse)456 3904 y(temp)r(erature)g(di\013erence)h(\001)p Fk(\014)h Fp(=)25 b(\()p Fk(\014)1618 3916 y Fl(n)1683 3904 y Fj(\000)19 b Fk(\014)1814 3916 y Fm(1)1851 3904 y Fp(\))30 b(and)f(w)n(e)f(ha)n(v) n(e)g Fk(\033)2439 3916 y Fl(j)2500 3904 y Fp(=)d(\001)p Fk(\014)t(\036)2759 3916 y Fl(j)2795 3904 y Fp(.)42 b(W)-7 b(e)29 b(consider)f(the)456 4003 y(function)g Fk(f)9 b Fp(\()p Fk(a;)14 b Fp(\001)p Fk(\014)t Fp(\))28 b(giv)n(en)f(b)n(y) 456 4194 y(\(51\))521 b Fk(f)9 b Fp(\()p Fk(a;)14 b Fp(\001)p Fk(\014)t Fp(\))23 b Fj(\021)44 b Fp(lim)1551 4244 y Fl(t)p Fh(!1)1722 4194 y Fj(\000)1797 4138 y Fp(1)p 1797 4175 42 4 v 1803 4251 a Fk(t)1862 4194 y Fp(log)14 b Fq(E)2046 4206 y Fl(\026)2104 4102 y Fi(h)2143 4194 y Fk(e)2182 4160 y Fh(\000)p Fl(a)2281 4113 y Fd(R)2326 4133 y Ff(t)2314 4181 y Fg(0)2365 4160 y Fl(\036)2405 4168 y Ff(i)2431 4160 y Fm(\()p Fl(x)p Fm(\()p Fl(s)p Fm(\)\))d Fl(ds)2685 4102 y Fi(i)2752 4194 y Fk(;)456 4370 y Fp(where)39 b Fk(a)44 b Fp(=)f Fk(\013)p Fp(\001)p Fk(\014)i Fp(and)40 b(the)g(second)g(v)-5 b(ariable)39 b(in)h Fk(f)48 b Fp(indicates)40 b(the)h(dep)r(endence)f(of)g(the)456 4469 y(dynamics)24 b(of)h(the)h(stationary)e(state)h Fk(\026)g Fp(on)g(\001)p Fk(\014)t Fp(.)36 b(F)-7 b(rom)25 b(the)h(compactness)e(prop)r(erties)g(of)h(the)456 4569 y(semigroups)34 b(in)n(v)n(olv)n(ed)g(it)i(is)f(easy)g(to)g(see)g(that) h Fk(f)9 b Fp(\()p Fk(a;)14 b Fp(\001)p Fk(\014)t Fp(\))36 b(is)g(a)f(real-analytic)e(function)k(of)456 4669 y(b)r(oth)28 b(v)-5 b(ariables)26 b Fk(a)h Fp(and)h(\001)p Fk(\014)t Fp(.)37 b(The)28 b(relation)f Fk(e)p Fp(\()p Fk(\013)p Fp(\))c(=)g Fk(e)p Fp(\(1)18 b Fj(\000)g Fk(\013)p Fp(\))28 b(no)n(w)f(reads)456 4820 y(\(52\))832 b Fk(f)9 b Fp(\()p Fk(a;)14 b Fp(\001)p Fk(\014)t Fp(\))37 b(=)g Fk(f)9 b Fp(\(\001)p Fk(\014)23 b Fj(\000)18 b Fk(a;)c Fp(\001)p Fk(\014)t Fp(\))g Fk(:)456 4971 y Fp(Di\013eren)n(tiating)27 b(this)h(relation)f(w)n(e)g(\014nd)1101 5115 y Fk(@)1150 5085 y Fm(2)1187 5115 y Fk(f)p 1006 5152 327 4 v 1006 5228 a(@)5 b(a@)g Fp(\(\001)p Fk(\014)t Fp(\))1342 5171 y(\(0)p Fk(;)14 b Fp(0\))36 b(=)h Fj(\000)1835 5115 y Fk(@)1884 5085 y Fm(2)1921 5115 y Fk(f)p 1740 5152 V 1740 5228 a(@)5 b(a@)g Fp(\(\001)p Fk(\014)t Fp(\))2076 5171 y(\(0)p Fk(;)14 b Fp(0\))k Fj(\000)2372 5115 y Fk(@)2421 5085 y Fm(2)2457 5115 y Fk(f)p 2372 5152 136 4 v 2375 5228 a(@)5 b(a)2468 5204 y Fm(2)2517 5171 y Fp(\(0)p Fk(;)14 b Fp(0\))g Fk(:)-2283 b Fp(\(53\))p eop %%Page: 12 12 12 11 bop 456 251 a Fm(12)1133 b(LUC)23 b(REY-BELLET)456 450 y Fp(W)-7 b(e)28 b(ha)n(v)n(e)1067 572 y Fk(@)5 b(f)p 1067 609 99 4 v 1070 685 a(@)g(a)1176 628 y Fp(\(0)p Fk(;)14 b Fp(\001)p Fk(\014)t Fp(\))97 b(=)1697 515 y Fi(Z)1794 628 y Fk(\036)1843 640 y Fl(j)1879 628 y Fk(d\026)14 b(;)-1553 b Fp(\(54\))1109 800 y Fk(@)1158 770 y Fm(2)1195 800 y Fk(f)p 1109 837 136 4 v 1112 914 a(@)5 b(a)1205 890 y Fm(2)1255 857 y Fp(\(0)p Fk(;)14 b Fp(0\))96 b(=)110 b Fj(\000)1786 800 y Fp(1)p 1786 837 42 4 v 1786 914 a(2)1851 744 y Fi(Z)1934 764 y Fh(1)1897 932 y Fm(0)2018 740 y Fi(\022)2079 744 y(Z)2162 857 y Fp(\()p Fk(T)2255 822 y Fl(t)2243 877 y Fm(0)2284 857 y Fk(\036)2333 869 y Fl(j)2368 857 y Fp(\))p Fk(\036)2449 869 y Fl(j)2485 857 y Fk(d\026)2578 869 y Fl(O)2635 740 y Fi(\023)2724 857 y Fk(ds)14 b(;)-2387 b Fp(\(55\))456 1066 y(where)39 b Fk(T)769 1035 y Fl(t)757 1086 y Fm(0)838 1066 y Fp(is)h(the)g (semigroup)f(at)h(equilibrium)g(\(\001)2167 1078 y Fl(\014)2256 1066 y Fp(=)k(0\))c(and)f Fk(\026)2702 1078 y Fm(0)2780 1066 y Fp(is)h(the)g(equilibrium)456 1165 y(measure)26 b Fk(Z)842 1135 y Fh(\000)p Fm(1)945 1165 y Fp(exp)13 b(\()p Fj(\000)p Fk(\014)t(G)p Fp(\))q(.)37 b(W)-7 b(e)28 b(obtain)456 1371 y(\(56\))1035 1314 y Fk(@)p 943 1351 234 4 v 943 1427 a(@)5 b Fp(\(\001)p Fk(\014)t Fp(\))1210 1253 y Fi(\022)1272 1258 y(Z)1368 1371 y Fk(\036)1417 1383 y Fl(j)1453 1371 y Fk(d\026)1546 1253 y Fi(\023)1607 1250 y(\014)1607 1300 y(\014)1607 1350 y(\014)1607 1400 y(\014)1635 1454 y Fm(\001)p Fl(\014)s Fm(=0)1855 1371 y Fp(=)1967 1314 y(1)p 1967 1351 42 4 v 1967 1427 a(2)2032 1258 y Fi(Z)2115 1278 y Fh(1)2078 1446 y Fm(0)2199 1253 y Fi(\022)2260 1258 y(Z)2343 1371 y Fp(\()p Fk(T)2436 1336 y Fl(t)2465 1371 y Fk(\036)2514 1383 y Fl(j)2549 1371 y Fp(\))p Fk(\036)2630 1383 y Fl(j)2666 1371 y Fk(d\026)2759 1253 y Fi(\023)2848 1371 y Fk(ds)14 b(;)456 1587 y Fp(and)27 b(this)h(is)f(the)h(familiar)f(Green-Kub)r(o)g(form)n(ula.)844 1816 y Fq(5.)47 b(Langevin)32 b(equations)f(and)i(other)e(lattice)h(of) g(oscillators)605 1966 y(5.1.)46 b(Hyp)s(ercub)s(es.)41 b Fp(The)25 b(reader)f(ma)n(y)h(w)n(onder)f(wh)n(y)i(w)n(e)f(are)f (only)h(considering)f(one-)456 2065 y(dimensional)d(lattice)i(of)f (oscillators.)33 b(Although)22 b(it)h(is)f(not)g(di\016cult)h(to)f (construct)g(Hamilton-)456 2165 y(ian)33 b(mo)r(dels)g(of)g(higher)f (dimensional)h(lattice)g(of)g(oscillators)e(in)n(teracting)i(with)g (reserv)n(oirs,)456 2265 y(the)28 b(reduction)g(to)h(a)f(tractable)f 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b(ha)n(v)n(e,)e(similarly)h(to) g(Theorem)g(3.1)456 4770 y Fq(Ergo)s(dic)i(prop)s(erties:)34 b Fp(The)26 b(Mark)n(o)n(v)e(pro)r(cess)h Fk(x)p Fp(\()p Fk(t)p Fp(\))i(has)f(a)f(unique)h(in)n(v)-5 b(arian)n(t)25 b(measure)g Fk(\026)p Fp(.)456 4869 y(The)33 b(measure)f Fk(\026)h Fp(is)g(ergo)r(dic)e(and)i(mixing)g(and)g(has)f(a)h Fj(C)2317 4839 y Fh(1)2420 4869 y Fp(ev)n(erywhere)e(p)r(ositiv)n(e)i (densit)n(y)-7 b(.)456 4969 y(The)27 b(con)n(v)n(ergence)e(to)j(the)g (stationary)e(state)h(o)r(ccurs)g(exp)r(onen)n(tially)g(fast.)605 5116 y(W)-7 b(e)31 b(can)f(also)f(pro)n(v)n(e)f(a)i(result)g(analogous) e(to)i(Theorem)g(4.1)f(b)n(y)h(considering)f(the)i(heat)456 5216 y(\015o)n(w)e(through)h(an)g(h)n(yp)r(ersurface)f Fj(f)p Fk(i)1625 5228 y Fm(1)1690 5216 y Fp(=)e Fk(k)s Fj(g)p Fp(.)45 b(W)-7 b(e)31 b(de\014ne)f(the)h(energy)f(of)g(the)h (oscillators)d(in)p eop %%Page: 13 13 13 12 bop 1018 251 a Fm(ST)-5 b(A)g(TISTICAL)29 b(MECHANICS)g(OF)g (ANHARMONIC)h(LA)-5 b(TTICES)496 b(13)456 450 y 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Fp(1.)73 b(One)40 b(can)f(sho)n(w)g(\([)p Fq(27)p Fp(])h(for)f (details\))h(that)h(if)f Fk(d)2862 4227 y Fm(2)2899 4257 y Fk(U)2965 4227 y Fm(\()2995 4257 y Fp(2\))p Fk(=dq)3194 4227 y Fm(2)3271 4257 y Fp(is)g Fk(n)p Fp(-)456 4357 y(nondegenerate)25 b(for)h Fk(n)h Fp(su\016cien)n(tly)g(large)e(then)i (the)g(en)n(trop)n(y)f(pro)r(duction)g(is)h(non-negativ)n(e.)456 4456 y(Ho)n(w)j(large)g Fk(n)h Fp(should)g(b)r(e)h(dep)r(ends)f(on)g (the)h(graph,)f(in)g(particular)f(on)h(ho)n(w)g(man)n(y)f(nearest)456 4556 y(neigh)n(b)r(ors)22 b(an)h(oscillator)e(has)i(and)g(if)h(there)f (are)g(man)n(y)g(lo)r(ops)f(in)i(the)g(graph.)34 b(W)-7 b(e)24 b(conjecture)456 4656 y(that)i(suc)n(h)h(a)f(condition)g(should) g(also)g(imply)h(existence)f(and)g(uniqueness)h(of)f(the)h(stationary) 456 4755 y(state.)1726 5000 y Fq(References)491 5133 y Fs([1])35 b(Bergmann,)21 b(P)-6 b(.G.)21 b(and)g(Leb)r(o)n(witz,)i (J.)e(L.:)30 b(New)21 b(approac)n(h)i(to)f(nonequilibrium)e(pro)r (cesses.)h(Ph)n(ys.)g(Rev.)601 5216 y(\(2\))j Fr(99)p Fs(,)f(578{587)i(\(1955\))p eop %%Page: 15 15 15 14 bop 1018 251 a Fm(ST)-5 b(A)g(TISTICAL)29 b(MECHANICS)g(OF)g (ANHARMONIC)h(LA)-5 b(TTICES)496 b(15)491 450 y Fs([2])35 b(Bonetto,)29 b(F.,)f(Leb)r(o)n(witz)g(J.L.,)f(and)h(Rey-Bellet,)g(L.:) 38 b(F)-6 b(ourier)27 b(La)n(w:)38 b(A)27 b(c)n(hallenge)i(to)e (Theorists.)g(In:)601 533 y Fu(Mathematic)l(al)f(Physics)g(2000)p Fs(,)f(Imp.)e(Coll.)f(Press,)h(London)i(2000,)f(pp.)g(128{150)491 616 y([3])35 b(Bam)n(busi,)23 b(D.:)33 b(Exp)r(onen)n(tial)26 b(stabilit)n(y)f(of)g(breathers)g(in)g(Hamiltonian)f(net)n(w)n(orks)h (of)g(w)n(eakly)g(coupled)601 699 y(oscillators.)d(Nonlinearit)n(y)i Fr(9)p Fs(,)f(433-457)i(\(1996\))491 782 y([4])35 b(Cherno)n(v,)21 b(N.I.,)f(Eyink,)h(G.L.,)f(Leb)r(o)n(witz,)i(J.L.,)f(and)g(Sinai,)f (Y.G.:)29 b(Steady-state)23 b(electric)e(conduction)601 865 y(in)i(the)i(p)r(erio)r(dic)e(Loren)n(tz)i(gas.)e(Comm)n(un.)f (Math.)i(Ph)n(ys.)f Fr(154)p Fs(,)f(569{601)k(\(1993\))491 948 y([5])35 b(Da)n(vies,)23 b(E.)g(B.:)31 b(Mark)n(o)n(vian)24 b(master)e(equations.)j(Comm.)c(Math.)j(Ph)n(ys.)f Fr(39)p Fs(,)g(91{110)i(\(1974\).)491 1031 y([6])35 b(Dym)27 b(H.)h(and)h(McKean,)h(H.P)-6 b(.:)40 b Fu(Gaussian)32 b(pr)l(o)l(c)l(esses,)h(function)d(the)l(ory,)i(and)f(the)f(inverse)g (sp)l(e)l(ctr)l(al)601 1114 y(pr)l(oblem)p Fs(.)i(Probabilit)n(y)g(and) g(Mathematical)g(Statistics,)i(V)-6 b(ol.)31 b Fr(31)p Fs(,)h(New)g(Y)-6 b(ork{London:)49 b(Academic)601 1197 y(Press,)22 b(1976)491 1280 y([7])35 b(Ec)n(kmann,)18 b(J.-P)-6 b(.:)27 b(Non-equilibrium)14 b(steady)19 b(states.)f(In:)28 b Fu(Pr)l(o)l(c)l(e)l(e)l(dings)22 b(of)d(the)h(International)h(Congr)l (ess)601 1363 y(of)k(Mathematicians,)i(Beijing)p Fs(,)22 b(V)-6 b(ol.)23 b Fr(I)r(I)r(I)p Fs(,)i(Higher)f(Education)h(Press,)d (2002,)j(pp.)e(409{418)491 1446 y([8])35 b(Ec)n(kmann,)18 b(J.-P)-6 b(.)16 b(and)i(Hairer,)f(M.:)27 b(Non-equilibrium)15 b(statistical)j(mec)n(hanics)f(of)g(strongly)h(anharmonic)601 1529 y(c)n(hains)24 b(of)f(oscillators.)g(Comm)n(un.)e(Math.)j(Ph)n (ys.)f Fr(212)p Fs(,)g(105{164)i(\(2000\))491 1612 y([9])35 b(Ec)n(kmann,)20 b(J.-P)-6 b(.)18 b(and)i(Hairer,)f(M.:)29 b(Sp)r(ectral)20 b(prop)r(erties)g(of)f(h)n(yp)r(oelliptic)h(op)r (erators.)g(Preprin)n(t)f(\(2002\))601 1695 y Fc(h)n(ttp://mp)r (ej.unige.c)n(h/)p Fb(\030)p Fc(ec)n(kmann/ps)p 1815 1695 24 4 v 26 w(\014les/hairer5.ps)456 1778 y Fs([10])35 b(Ec)n(kmann,)30 b(J.-P)-6 b(.,)30 b(Pillet)f(C.-A.,)g(and)i (Rey-Bellet,)g(L.:)42 b(Non-equilibrium)28 b(statistical)i(mec)n (hanics)g(of)601 1861 y(anharmonic)h(c)n(hains)i(coupled)g(to)g(t)n(w)n (o)g(heat)g(baths)h(at)e(di\013eren)n(t)h(temp)r(eratures.)f(Comm)n (un.)e(Math.)601 1944 y(Ph)n(ys.)23 b Fr(201)p Fs(,)f(657{697)k (\(1999\))456 2028 y([11])35 b(Ec)n(kmann,)17 b(J.-P)-6 b(.,)16 b(Pillet,)h(C.-A.,)e(and)i(Rey-Bellet,)g(L.:)27 b(En)n(trop)n(y)17 b(pro)r(duction)g(in)f(non-linear,)g(thermally)601 2111 y(driv)n(en)23 b(Hamiltonian)g(systems.)g(J.)g(Stat.)h(Ph)n(ys.)g Fr(95)p Fs(,)e(305{331)k(\(1999\))456 2194 y([12])35 b(Ev)l(ans,)17 b(D.J.,)g(Cohen,)h(E.G.D.,)e(and)h(Morriss,)e(G.P)-6 b(.:)27 b(Probabilit)n(y)16 b(of)g(second)h(la)n(w)f(violation)g(in)g (shearing)601 2277 y(steady)25 b(\015o)n(ws.)e(Ph)n(ys.)g(Rev.)h(Lett.) h Fr(71)p Fs(,)d(2401{2404)k(\(1993\))456 2360 y([13])35 b(F)-6 b(armer,)18 b(J.,)h(Goldstein,)i(S.,)f(and)f(Sp)r(eer,)i(E.R.:) 28 b(In)n(v)l(arian)n(t)20 b(states)h(of)e(a)g(thermally)f(conducting)k (barrier.)601 2443 y(J.)h(Stat.)i(Ph)n(ys.)e Fr(34)p Fs(,)f(263{277)k(\(1984\))456 2526 y([14])35 b(F)-6 b(ord,)18 b(G.W.,)g(Kac,)h(M.)e(and)h(Mazur,)g(P)-6 b(.:)28 b(Statistical)19 b(mec)n(hanics)e(of)h(assem)n(blies)e(of)h(coupled)i(oscillators.)601 2609 y(J.)k(Math.)h(Ph)n(ys.)f Fr(6)p Fs(,)g(504{515)i(\(1965\))456 2692 y([15])35 b(Galla)n(v)n(otti,)22 b(G.:)30 b(Chaotic)22 b(h)n(yp)r(othesis:)31 b(Onsager)22 b(recipro)r(cit)n(y)g(and)g (\015uctuation-dissipation)h(theorem.)601 2775 y(J.)g(Stat.)i(Ph)n(ys.) e Fr(84)p Fs(,)f(899{925)k(\(1996\))456 2858 y([16])35 b(Galla)n(v)n(otti,)27 b(G.)e(and)i(Cohen)f(E.G.D.:)35 b(Dynamical)24 b(ensem)n(bles)i(in)f(stationary)i(states.)f(J.)g(Stat.) g(Ph)n(ys.)601 2941 y Fr(80)p Fs(,)c(931{970)k(\(1995\))456 3024 y([17])35 b(Goldstein,)e(S.,)f(Kipnis,)g(C.,)g(and)f(Ianiro,)i (N.:)45 b(Stationary)32 b(states)g(for)f(a)g(mec)n(hanical)g(system)f (with)601 3107 y(sto)r(c)n(hastic)25 b(b)r(oundary)f(conditions.)h(J.)e (Stat.)h(Ph)n(ys.)g Fr(41)p Fs(,)e(915{939)k(\(1985\))456 3190 y([18])35 b(H)n(\023)-33 b(erau,)23 b(F.)h(and)h(Nier,)d(F.:)32 b(Isotropic)25 b(h)n(yp)r(o)r(ellipticit)n(y)f(and)h(trend)g(to)f (equilibrium)e(for)h(F)-6 b(okk)n(er-Planc)n(k)601 3273 y(equation)25 b(with)f(high)f(degree)i(p)r(oten)n(tial.)g(Preprin)n(t)e (\(2002\))601 3356 y Fc(h)n(ttp://www.maths.univ-rennes1.fr/)p Fb(\030)k Fc(nier/rec)n(herc)n(he/F)-6 b(okk)n(erPlanc)n(k.ps)456 3439 y Fs([19])35 b(H\177)-35 b(ormander,)16 b(L.:)28 b Fu(The)19 b(A)n(nalysis)h(of)f(line)l(ar)h(p)l(artial)h(di\013er)l (ential)e(op)l(er)l(ators)p Fs(.)f(V)-6 b(ol)17 b Fr(I)r(I)r(I)p Fs(,)h(Berlin:)27 b(Springer,)601 3522 y(1985)456 3605 y([20])35 b(Ishihara,)20 b(K.)g(and)h(Kunita,)g(H.:)28 b(A)20 b(classi\014cation)h(of)f(the)h(second)g(order)f(degenerate)j (elliptic)d(op)r(erators)601 3688 y(and)k(its)f(probabilistic)h(c)n (haracterization.)h(Z.)e(W)-6 b(ahrsc)n(h.)24 b(und)g(V)-6 b(erw.)23 b(Geb.)h Fr(39)p Fs(,)f(235{254)i(\(1974\))456 3771 y([21])35 b(Jaksic)26 b(V.and)h(Pillet)f(C-A.:)35 b(Non-equilibrium)24 b(steady)j(states)h(of)e(\014nite)h(quan)n(tum)g (systems)e(coupled)601 3854 y(to)f(thermal)f(reserv)n(oirs.)f(Comm)n (un.)f(Math.)j(Ph)n(ys.)f Fr(226)p Fs(,)g(131{162)i(\(2002\))456 3937 y([22])35 b(Kurc)n(han,)23 b(J:)h(Fluctuation)h(theorem)e(for)g (sto)r(c)n(hastic)j(dynamics.)c(J.)i(Ph)n(ys.)p Fr(A)j(31)p Fs(,)22 b(3719{3729)k(\(1998\))456 4020 y([23])35 b(Leb)r(o)n(witz,)f (J.L.)e(and)g(Sp)r(ohn,)j(H.:)47 b(A)31 b(Galla)n(v)n(otti-Cohen-t)n (yp)r(e)k(symmetry)30 b(in)h(the)i(large)f(deviation)601 4103 y(functional)24 b(for)f(sto)r(c)n(hastic)i(dynamics.)e(J.)g(Stat.) i(Ph)n(ys.)e Fr(95)p Fs(,)f(333-365)j(\(1999\))456 4186 y([24])35 b(Lepri,)25 b(S.,)h(Livi,)f(R.,)h(and)g(P)n(oliti,)g(A.:)34 b(Thermal)25 b(conduction)j(in)d(classical)h(lo)n(w-dimensional)e (lattices.)601 4269 y(Submitted)g(to)g(Ph)n(ysics)g(Rep)r(orts)601 4352 y Fc(h)n(ttp://xxx.lanl.go)n(v/abs/cond-mat/0112193)456 4435 y Fs([25])35 b(MacKa)n(y)-6 b(,)22 b(R.S.)e(and)i(Aubry)-6 b(,)21 b(S.:)30 b(Pro)r(of)20 b(of)h(existence)i(of)e(breathers)h(for)e (time-rev)n(ersible)g(or)h(Hamilton-)601 4518 y(ian)i(net)n(w)n(orks)i (of)e(w)n(eakly)h(coupled)h(oscillators.)e(Nonlinearit)n(y)h Fr(7)p Fs(,)e(1623{1643)27 b(\(1994\))456 4601 y([26])35 b(Maes,)23 b(C.:)30 b(The)24 b(\015uctuation)i(theorem)e(as)f(a)h (Gibbs)g(prop)r(ert)n(y)-6 b(.)24 b(J.)f(Stat.)i(Ph)n(ys.)e Fr(95)p Fs(,)g(367{392)i(\(1999\))456 4684 y([27])35 b(Maes,)23 b(C.,)g(Neto)r(cn)n(y)-6 b(,)25 b(K.,)d(and)i(V)-6 b(ersc)n(h)n(uere,)24 b(M.:)30 b(Heat)25 b(conduction)h(net)n(w)n (orks.)e(Preprin)n(t)f(\(2002\))601 4767 y Fc(h)n (ttp://tfdec1.fys.kuleuv)n(en.ac.b)r(e/)p Fb(\030)p Fc(c)n(hrist/)456 4850 y Fs([28])35 b(Meyn,)21 b(S.P)-6 b(.)20 b(and)h(Tw)n(eedie,)g (R.L.:)29 b Fu(Markov)23 b(Chains)g(and)h(Sto)l(chastic)f(Stability.)c Fs(Comm)n(unication)h(and)601 4933 y(Con)n(trol)k(Engineering)g (Series,)f(London:)32 b(Springer-V)-6 b(erlag)23 b(London,)h(1993)456 5016 y([29])35 b(Norriss,)20 b(J.:)29 b(Simpli\014ed)20 b(Mallia)n(vin)h(Calculus.)g(In)h Fu(S)n(\023)-35 b(eminair)l(e)25 b(de)f(pr)l(ob)l(abilit)n(\023)-35 b(es)26 b(XX)p Fs(,)21 b(Lectures)h(Note)h(in)601 5099 y(Math.)g Fr(1204)p Fs(,)f(Berlin:)30 b(Springer,)23 b(1986,)i(pp.)e(101{130)p eop %%Page: 16 16 16 15 bop 456 251 a Fm(16)1133 b(LUC)23 b(REY-BELLET)456 450 y Fs([30])35 b(Rey-Bellet,)29 b(L.)g(and)g(Thomas,)g(L.E.:)40 b(Asymptotic)29 b(b)r(eha)n(vior)g(of)f(thermal)g(non-equilibrium)f (steady)601 533 y(states)d(for)f(a)h(driv)n(en)g(c)n(hain)g(of)f (anharmonic)h(oscillators.)e(Comm)n(un.)g(Math.)i(Ph)n(ys.)f Fr(215)p Fs(,)f(1{24)j(\(2000\))456 616 y([31])35 b(Rey-Bellet,)e(L.)f (and)g(Thomas,)g(L.E.:)47 b(Exp)r(onen)n(tial)33 b(con)n(v)n(ergence)h (to)e(non-equilibrium)e(stationary)601 699 y(states)24 b(in)g(classical)f(statistical)h(mec)n(hanics.)f(Comm)n(un.)f(Math.)i (Ph)n(ys.)f Fr(225)p Fs(,)f(305{329)k(\(2002\))456 782 y([32])35 b(Rey-Bellet,)f(L.)e(and)h(Thomas,)g(L.E.:)48 b(Fluctuations)33 b(of)f(the)i(en)n(trop)n(y)f(pro)r(duction)h(in)e (anharmonic)601 865 y(c)n(hains.)23 b(Ann.)h(H.)f(P)n(oinc.)g Fr(3)p Fs(,)g(483{502)j(\(2002\))456 948 y([33])35 b(T)-6 b(ropp)r(er,)28 b(M.)f(M.:)39 b(Ergo)r(dic)29 b(and)f (quasideterministic)g(prop)r(erties)g(of)f(\014nite-dimensional)h(sto)r (c)n(hastic)601 1031 y(systems.)22 b(J.)i(Stat.)g(Ph)n(ys)g Fr(17)p Fs(,)e(491{509)k(\(1977\))456 1114 y([34])35 b(Rieder,)c(Z.,)f(Leb)r(o)n(witz,)j(J.L.,)d(and)h(Lieb,)g(E.:)43 b(Prop)r(erties)30 b(of)g(a)g(harmonic)f(crystal)h(in)g(a)g(stationary) 601 1197 y(non-equilibrium)21 b(state.)k(J.)e(Math.)h(Ph)n(ys.)f Fr(8)p Fs(,)g(1073{1085)j(\(1967\))456 1280 y([35])35 b(Ruelle,)30 b(D.:)42 b(Smo)r(oth)30 b(dynamics)f(and)h(new)g (theoretical)h(ideas)f(in)f(non-equilibrium)f(statistical)i(me-)601 1363 y(c)n(hanics.)24 b(J.)f(Stat.)i(Ph)n(ys.)e Fr(95)p Fs(,)f(393{468)k(\(1999\))456 1446 y([36])35 b(Ruelle,)25 b(D.:)35 b(En)n(trop)n(y)26 b(pro)r(duction)h(in)e(quan)n(tum)h(spin)f (systems.)g(Comm)n(un.)e(Math.)j(Ph)n(ys)g Fr(224)p Fs(,)e(3{16)601 1529 y(\(2001\))456 1612 y([37])35 b(Ruelle,)23 b(D.:)30 b(Statistical)24 b(mec)n(hanics:)31 b(a)23 b(departure)h(from)e (equilibrium)f(Nature)i Fr(414)p Fs(,)f(263{265)j(\(2001\))456 1695 y([38])35 b(Sp)r(ohn,)25 b(H.:)32 b Fu(L)l(ar)l(ge)27 b(sc)l(ale)g(dynamics)g(of)f(inter)l(acting)g(p)l(articles)p Fs(.)f(T)-6 b(exts)25 b(and)g(monographs)f(in)g(ph)n(ysics.)601 1778 y(Berlin,)e(Springer-V)-6 b(erlag,)23 b(1991)456 1861 y([39])35 b(Sp)r(ohn,)19 b(H.)d(and)i(Leb)r(o)n(witz,)h(J.L.:)27 b(Stationary)19 b(non-equilibrium)c(states)j(of)f(in\014nite)g (harmonic)f(systems.)601 1944 y(Comm)n(un.)21 b(Math.)j(Ph)n(ys.)f Fr(54)p Fs(,)g(97{120)i(\(1977\))456 2028 y([40])35 b(Siev)n(ers,)29 b(A.J.)e(and)i(T)-6 b(ak)n(eno,)31 b(S.:)40 b(In)n(trinsic)28 b(lo)r(calized)h(mo)r(des)e(in)h(anharmonic)g(crystals.)g(Ph)n(ys.)g (Rev.)601 2111 y(Lett.)c Fr(61)f Fs(970{973)j(\(1988\))456 2194 y([41])35 b(Stro)r(o)r(c)n(k,)21 b(D.W.)e(and)i(V)-6 b(aradhan,)21 b(S.R.S.:)28 b(On)20 b(the)g(supp)r(ort)h(of)e (di\013usion)h(pro)r(cesses)h(with)e(applications)601 2277 y(to)25 b(the)i(strong)e(maxim)n(um)d(principle.)j(In)g Fu(Pr)l(o)l(c.)j(6-th)f(Berkeley)g(Symp.)g(Math.)g(Stat.)g(Pr)l(ob.)p Fs(,)f(V)-6 b(ol)25 b Fr(I)r(I)r(I)p Fs(,)601 2360 y(Berk)n(eley:)31 b(Univ.)23 b(California)g(Press,)f(1972,)j(pp.)e(361{368)456 2443 y([42])35 b(Zab)r(ey)-6 b(,)33 b(E.:)44 b(Etats)32 b(stationnaires)f(et)g(pro)r(duction)h(d'en)n(tropie)f(d'un)g(syst)n (\022)-33 b(eme)30 b(harmonique)h(hors)f(de)601 2526 y(l')n(\023)-33 b(equilibre.)22 b(T)-6 b(ra)n(v)l(ail)23 b(de)h(Dipl^)-35 b(ome,)22 b(Univ)n(ersit)n(\023)-33 b(e)23 b(de)i(Gen)n(\022)-33 b(ev)n(e)25 b(\(2001\),)g(unpublished)605 2681 y Ft(Dep)l(ar)l(tment)32 b(of)g(Ma)l(thema)l(tics,)h(University)g (of)f(Vir)o(ginia,)i(Ker)o(chof)e(Hall,)i(Charlottes-)456 2764 y(ville,)25 b(V)-8 b(A)25 b(22903)605 2847 y Fu(Curr)l(ent)31 b(addr)l(ess)5 b Fs(:)43 b(Departmen)n(t)29 b(of)f(Mathematics)h(and)h (Statistics,)g(Univ)n(ersit)n(y)f(of)f(Massac)n(h)n(usetts,)456 2930 y(Lederle)c(Graduate)h(Researc)n(h)f(T)-6 b(o)n(w)n(er,)24 b(Amherst,)e(MA)h(01003)605 3013 y Fu(E-mail)j(addr)l(ess)5 b Fs(:)33 b Fa(lr7q@math.umass.edu)605 3096 y Fu(URL)p Fs(:)23 b Fa(http://www.math.umass.ed)q(u/~l)q(r7q)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF 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