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Fn(Onderzo)r(eksleider) c(FW)n(O,)e(Flanders.)55 b(Email:)32 b(c)n(hristian.maes@fys.kuleuv)n (en.ac.b)r(e)1926 5255 y Fv(1)p eop %%Page: 2 2 2 1 bop 515 523 a Fv(is)33 b(a)g(di\013eomorphism)d(of)k(\012.)57 b(The)34 b(resulting)d(\(discrete\))i(time)g(ev)n(olution)e(is)i (obtained)515 623 y(b)n(y)c(iteration)e(and)j(the)g(rev)n(ersibilit)n (y)24 b(means)k(that)i(there)f(is)g(a)g(di\013eomorphism)d Fq(\022)32 b Fv(on)d(\012)515 722 y(with)35 b Fq(\022)753 692 y Fm(2)826 722 y Fv(=)g(1)g(and)g Fq(\022)26 b Fp(\016)d Fq(\036)h Fp(\016)f Fq(\022)38 b Fv(=)e Fq(\036)1667 692 y Fl(\000)p Fm(1)1756 722 y Fv(.)60 b(Consider)34 b(no)n(w)g(min)n(us)g(the)h(logarithm)c(of)36 b(the)515 822 y(Jacobian)31 b(determinan)n(t)g Fq(J)42 b Fv(whic)n(h)33 b(arises)e(from)h(the)i(c)n(hange)f(of)h(v)-5 b(ariables)30 b(implied)g(b)n(y)515 922 y(the)e(dynamics.)34 b(W)-7 b(e)28 b(write)1436 901 y(_)1413 922 y Fq(S)f Fp(\021)c(\000)14 b Fv(log)e Fq(J)c Fv(.)37 b(One)28 b(is)e(in)n(terested)g(in)h(the)h (\015uctuations)f(of)1377 1191 y Fq(w)1436 1203 y Fk(N)1500 1191 y Fv(\()p Fq(\030)t Fv(\))c Fp(\021)1824 1135 y Fv(1)p 1725 1172 239 4 v 1725 1258 a Fq(\032)p Fv(\()1823 1237 y(_)1800 1258 y Fq(S)5 b Fv(\))p Fq(N)2014 1080 y Fk(N)q(=)p Fm(2)2014 1112 y Fj(X)1988 1294 y Fl(\000)p Fk(N)q(=)p Fm(2)2197 1170 y Fv(_)2174 1191 y Fq(S)g Fv(\()p Fq(\036)2311 1157 y Fk(n)2357 1191 y Fv(\()p Fq(\030)t Fv(\)\))p Fq(;)692 b Fv(\(1.1\))515 1469 y(for)26 b(large)f(time)h Fq(N)9 b Fv(.)36 b(Here,)27 b Fq(\032)h Fv(is)e(the)h(stationary)e (probabilit)n(y)e(measure)i(\(SRB)j(measure\))515 1569 y(of)f(the)h(dynamics)d(with)j(exp)r(ectations)1502 1827 y Fq(\032)p Fv(\()p Fq(f)9 b Fv(\))23 b(=)g(lim)1798 1881 y Fk(N)1926 1771 y Fv(1)p 1909 1808 76 4 v 1909 1884 a Fq(N)2039 1724 y Fk(N)2009 1749 y Fj(X)2052 1924 y Fm(0)2142 1827 y Fq(f)9 b Fv(\()p Fq(\036)2273 1793 y Fk(n)2319 1827 y Fq(\030)t Fv(\))817 b(\(1.2\))515 2081 y(corresp)r(onding)28 b(to)i(time-a)n(v)n(erages)c(for)k(almost)f 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y(dition:)46 b(it)33 b(is)f(a)h(transitiv)n(e)e (Anoso)n(v)h(system.)53 b(This)32 b(ensures)h(that)h(the)f(system)f (allo)n(ws)515 3141 y(a)f(Mark)n(o)n(v)f(partition)g(\(and)i(the)h (represen)n(tation)c(via)i(some)g(sym)n(b)r(olic)d(dynamics\))i(and)515 3241 y(the)g(existence)f(of)h(the)g(SRB)g(measure)e Fq(\032)i Fv(in)f(\(1.2\).)43 b(This)29 b(tec)n(hnical)f(assumption)f(is)i(not) 515 3341 y(tak)n(en)37 b(ph)n(ysically)d(v)n(ery)i(serious)g(but)i (instead)f(it)g(is)f(supp)r(osed)i(to)g(guide)e(us)i(to)n(w)n(ards)515 3440 y(general)28 b(results)i(whic)n(h)f(are)h(true)h(in)f(a)h(broader) e(con)n(text)2382 3410 y Fm(1)2419 3440 y Fv(.)46 b(That)31 b(is)f(what)h(is)f(a\016rmed)515 3540 y(in)36 b(the)h(so)f(called)f(c)n (haotic)g(h)n(yp)r(othesis:)54 b(\\A)37 b(rev)n(ersible)c(man)n(y)i (particle)g(system)g(in)h(a)515 3640 y(stationary)24 b(state)j(can)g(b)r(e)h(regarded)d(as)i(a)f(transitiv)n(e)f(Anoso)n(v)h (system)f(for)i(the)h(purp)r(ose)515 3739 y(of)36 b(computing)f(the)h (macroscopic)d(prop)r(erties,")j(see)g(also)f(e.g.)63 b([10)o(,)37 b(11)o(,)f(7,)h(6)o(,)g(25)o(,)f(9].)515 3839 y(The)27 b(\015uctuation)g(theorem)f(then)i(states)f(that)h Fq(w)2092 3851 y Fk(N)2156 3839 y Fv(\()p Fq(\030)t Fv(\))g(has)f(a)g (distribution)e Fq(\032)3001 3851 y Fk(N)3064 3839 y Fv(\()p Fq(w)r Fv(\))k(with)515 3938 y(resp)r(ect)e(to)h(the)g (stationary)d(state)i Fq(\032)h Fv(suc)n(h)f(that)1439 4163 y(lim)1467 4217 y Fk(N)1708 4107 y Fv(1)p 1578 4144 301 4 v 1578 4231 a Fq(N)9 b(\032)p Fv(\()1752 4210 y(_)1729 4231 y Fq(S)c Fv(\))p Fq(w)1903 4163 y Fv(ln)2028 4107 y Fq(\032)2071 4119 y Fk(N)2134 4107 y Fv(\()p Fq(w)r Fv(\))p 1996 4144 297 4 v 1996 4220 a Fq(\032)2039 4232 y Fk(N)2102 4220 y Fv(\()p Fp(\000)p Fq(w)r Fv(\))2325 4163 y(=)23 b(1)753 b(\(1.4\))515 4399 y(alw)n(a)n(ys.)45 b(In)32 b(other)e(w)n(ords,)h(the)h(distribution)d(of)i(en)n(trop)n(y)f (pro)r(duction)g(o)n(v)n(er)g(long)g(time)515 4499 y(in)n(terv)-5 b(als)25 b(satis\014es)g(some)h(general)g(symmetry)f(prop)r(ert)n(y)-7 b(.)515 4599 y(This)24 b(theorem)h(originated)d(from)i(n)n(umerical)e (evidence,)j(e.g.)36 b(in)25 b([5)o(],)i(and)e(it)g(has)g(v)-5 b(arious)515 4698 y(in)n(teresting)17 b(consequences.)33 b(F)-7 b(or)19 b(example,)g(in)g([7)o(],)j(it)d(w)n(as)g(in)n (terpreted)f(as)h(extending)g(the)515 4798 y(Green-Kub)r(o)31 b(form)n(ulas)f(to)i(arbitrary)e(forcing)h(\014elds)h(for)g(a)g(class)f (of)i(non-equilibrium)515 4898 y(dynamics.)639 4997 y(In)k([16)o(],)i (Kurc)n(han)d(p)r(oin)n(ted)f(out)i(that)g(this)f(\015uctuation)g (theorem)f(also)f(holds)i(for)515 5097 y(certain)h(di\013usion)g(pro)r (cesses.)69 b(This)37 b(is)h(the)h(con)n(text)f(of)h(\014nite)f (systems)f(undergoing)p 515 5166 1146 4 v 607 5219 a Fi(1)642 5243 y Fn(The)26 b(situation)i(resem)n(bles)e(here)g(to)g (some)g(exten)n(t)i(that)e(for)f(the)i(ergo)r(dic)g(h)n(yp)r(othesis.) 38 b(Ergo)r(dicit)n(y)515 5322 y(is)26 b(lik)n(ely)i(to)e(b)r(e)g 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y(and)f(the)h(action)e(functional)g (satisfying)f(the)j(theorem.)639 1121 y(In)23 b(this)f(note)g(w)n(e)g (understand)h(the)g(\015uctuation)e(theorem)g(within)g(the)i(Gibbs)f (formal-)515 1220 y(ism.)50 b(Since)32 b(this)g(formalism)c(is)k(often) h(considered)e(as)h(giving)e(a)i(mathematical)d(struc-)515 1320 y(ture)36 b(to)g(the)h(theory)f(of)g(equilibrium)c(statistical)h (mec)n(hanics)g(and)k(in)e(order)g(to)i(a)n(v)n(oid)515 1420 y(misunderstanding,)25 b(w)n(e)j(insist)e(from)h(the)i(b)r (eginning)d(that)j(w)n(e)f(wish)f(to)h(see)g(this)g(Gibbs)515 1519 y(formalism)23 b(applied)j(here)i(to)f(nonequilibrium)c (conditions.)36 b(The)28 b(righ)n(t)e(w)n(a)n(y)h(of)h(lo)r(oking)515 1619 y(at)f(it,)h(is)e(to)i(consider)d(space-time)g(histories)g(dra)n (wn)i(from)f(a)i(Gibbs)f(measure.)35 b(In)28 b(other)515 1719 y(w)n(ords,)h(our)h(analysis)d(is)i(not)h(to)g(b)r(e)h(regarded)d (as)i(an)g(in)n(v)n(estigation)25 b(of)31 b(\015uctuations)e(in)515 1818 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(guides,)h(cannot)g(really)e(b)r(e)j(necessary)f(for)515 3412 y(a)24 b(\015uctuation)f(theorem)g(or)g(its)g(consequences)g(to)h (hold.)35 b(P)n(erhaps)23 b(it)g(is)g(more)g(natural)f(to)515 3512 y(assume)i(immediately)d(that)26 b(for)g(the)g(purp)r(ose)f(of)h (computing)e(macroscopic)e(prop)r(erties,)515 3611 y(a)k(man)n(y)f (particle)f(system)h(in)h(a)g(steady)g(state)g(should)g(b)r(e)h (regarded)d(as)i(a)g(Gibbs)g(system)515 3711 y(for)k(the)h(space-time)d (histories.)43 b(And)31 b(w)n(e)f(kno)n(w)g(that)g(the)h(reason)e(for)h (Gibbs)g(distribu-)515 3811 y(tions)c(has)h(little)f(to)h(do)h(with)f (the)h(detailed)e(prop)r(erties)f(of)j(the)g(system's)e(dynamics)f(but) 515 3910 y(instead)f(is)h(based)h(on)f(statistical)e(principles)1974 3880 y Fm(2)2012 3910 y Fv(.)36 b(There)26 b(is)e(\014nally)g(a)i (second,)f(practically)515 4010 y(sp)r(eaking,)32 b(more)g(imp)r(ortan) n(t)e(extension)i(of)h(the)g(earlier)d(results.)52 b(In)33 b(the)g(analysis)d(b)r(e-)515 4110 y(lo)n(w,)35 b(w)n(e)f(presen)n(t)h (a)f(lo)r(cal)e(v)n(ersion)h(of)h(the)i(\015uctuation)e(theorem.)57 b(A)35 b(mec)n(hanism)d(for)515 4209 y(the)h(v)-5 b(alidit)n(y)30 b(of)j(a)g(lo)r(cal)e(v)n(ersion)f(w)n(as)i(already)f(discussed)h(in)g (the)h(recen)n(t)g([8].)53 b(This)32 b(is)515 4309 y(crucial)e(b)r (ecause)j(it)f(is)g(only)g(a)h(lo)r(cal)d(\015uctuation)j(theorem)e (that)j(leads)d(to)i(observ)-5 b(able)515 4408 y(consequences)26 b(and)i(w)n(e)f(will)e(see)i(that)h(this)f(is)f(quite)h(natural)f(in)h (a)g(Gibbsian)e(setup.)515 4641 y Fy(1.2)112 b(Disclaimers.)515 4794 y Fv(Our)27 b(analysis)d(b)r(elo)n(w)i(is)h(limited)d(in)j(v)-5 b(arious)25 b(w)n(a)n(ys)h(including:)515 4894 y(1.)46 b(Time)30 b(\(and)h(space\))f(is)g(discrete:)42 b(a)31 b(regular)d(lattice)h(pla)n(ys)g(the)j(role)d(of)i(space-time.)p 515 4968 1146 4 v 607 5021 a Fi(2)642 5045 y Fn(W)-6 b(e)27 b(ha)n(v)n(e)g(in)g(mind)g(the)h(maxim)n(um)f(en)n(trop)n(y)g (principle)i(and)e(the)g(foundations)h(of)e(statistical)k(me-)515 5124 y(c)n(hanics)c(in)g(the)g(theory)g(of)e(large)i(deviations,)i(see) d(e.g.)35 b([17,)25 b(14].)34 b(This)26 b(m)n(ust)f(b)r(e)g(con)n (trasted)i(with)f(the)515 5203 y(approac)n(h)f(from)f(the)h(theory)g (of)f(dynamical)j(systems)d(\(as)h(summarized)g(for)f(example)i(in)f ([26]\).)33 b(Notice)515 5281 y(that)e(Mark)n(o)n(v)f(partitions)i(do)e (not)h(corresp)r(ond)g(to)f(a)g(statistical)k(pro)r(cedure;)f(they)e (fully)g(enco)r(de)h(the)515 5360 y(dynamics.)1926 5650 y Fv(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fv(W)-7 b(e)32 b(b)r(eliev)n(e)d(that)i(going)f(to)h (con)n(tin)n(uous)e(time)h(is)h(a)g(tec)n(hnical)e(step)i(\(whic)n(h)g (is)f(not)i(ex-)515 623 y(p)r(ected)k(to)g(b)r(e)h(v)n(ery)d (di\016cult\))h(and)h(that)h(this)e(is)g(irrelev)-5 b(an)n(t)33 b(for)i(the)h(purp)r(ose)g(of)g(the)515 722 y(pap)r(er.)515 822 y(2.)f(No)25 b(hard-core)d(conditions:)33 b(w)n(e)24 b(tak)n(e)g(a)g(smo)r(oth)g(p)r(oten)n(tial)e(and)j(all)d(transition)g (prob-)515 922 y(abilities)34 b(are)j(b)r(ounded)h(a)n(w)n(a)n(y)e (from)h(zero.)67 b(In)38 b(particular,)g(this)f(seems)g(problematic)515 1021 y(when)25 b(dealing)d(with)i(dynamics)e(sub)5 b(ject)25 b(to)f(certain)f(conserv)-5 b(ation)22 b(la)n(ws.)34 b(Again,)24 b(w)n(e)g(do)515 1121 y(not)f(think)g(that)h(this)f(is)g (essen)n(tial)d(b)r(ecause)j(the)h(Gibbs)f(prop)r(erties)f(w)n(e)h(use) h(also)d(hold)i(for)515 1220 y(hard-core)k(in)n(teractions.)41 b(Extra)28 b(care)h(and)g(conditions)f(w)n(ould)g(b)r(e)i(needed)g(for) f(writing)515 1320 y(do)n(wn)e(certain)f(form)n(ulae)e(but)29 b(w)n(e)e(b)r(eliev)n(e)e(they)j(do)f(not)h(mo)r(dify)e(the)i(main)d (result.)515 1420 y(3.)43 b(Discrete)28 b(spins:)40 b(w)n(e)29 b(deal)g(with)g(regular)e(lattice)h(spin)h(systems.)41 b(While)29 b(some)f(com-)515 1519 y(pactness)35 b(of)i(phase)f(space)f (is)g(nice)h(to)g(ha)n(v)n(e)f(around,)j(our)d(results)g(dep)r(end)i (solely)c(on)515 1619 y(ha)n(ving)25 b(a)h(large)f(deviation)f (principle)g(for)j(Gibbs)f(states.)36 b(The)27 b(exten)n(t)g(to)g(whic) n(h)f(suc)n(h)g(a)515 1719 y(principle)e(holds)j(decides)f(on)h(the)h (p)r(ossible)d(extensions)h(of)i(our)f(results.)515 1818 y(4.)56 b(No)34 b(phase)f(transitions:)47 b(while)33 b(the)h(\015uctuation)g(theorem)e(holds)h(quite)g(generally)-7 b(,)515 1918 y(its)32 b(con)n(ten)n(ts)f(can)i(b)r(e)g(empt)n(y)e(when) i(the)g(large)d(deviations)g(happ)r(en)j(on)f(another)g(scale)515 2017 y(than)k(linear)d(in)i(time)g(\(and)h(spatial)d(v)n(olume\).)60 b(In)36 b(other)f(w)n(ords,)i(the)f(corresp)r(onding)515 2117 y(rate)25 b(function)g(could)g(fail)f(to)h(b)r(e)h(strictly)e(con) n(v)n(ex)g(in)h(whic)n(h)g(case)g(instabilities)c(or)k(phase)515 2217 y(transitions)35 b(are)j(presen)n(t.)69 b(These)38 b(`violations')c(of)39 b(the)f(\015uctuation)g(theorem)f(can)h(of)515 2316 y(course)31 b(not)i(happ)r(en)g(when)f(the)h(spatial)d(v)n(olume)g (is)i(\014nite)g(\(for)g(a)g(su\016cien)n(tly)f(c)n(haotic)515 2416 y(dynamics)20 b(or)j(for)f(a)h(non-degenerated)e(sto)r(c)n(hastic) h(dynamics\))f(or,)i(for)g(in\014nite)f(systems,)515 2516 y(when)35 b(w)n(e)g(are)f(in)h(the)h(`high)e(temp)r(erature')g (regime.)57 b(Suc)n(h)35 b(scenario's)e(are)h(of)h(course)515 2615 y(w)n(ell-do)r(cumen)n(ted)24 b(for)j(Gibbs)g(states.)515 2715 y(5.)35 b(Steady)24 b(states)f(and)h(time-homogeneit)n(y:)30 b(w)n(e)23 b(do)h(not)f(consider)f(here)i(the)g(\(ph)n(ysically)515 2814 y(v)n(ery)j(relev)-5 b(an)n(t\))27 b(problem)g(of)h(forces)g(or)g (p)r(oten)n(tials)e(dep)r(ending)i(on)g(time)g(nor)f(do)i(w)n(e)f(in-) 515 2914 y(v)n(estigate)i(here)i(the)h(long)e(time)g(b)r(eha)n(vior)f (of)j(the)g(system)e(started)h(in)g(an)n(ything)f(other)515 3014 y(than)e(in)f(a)h(stationary)d(state.)41 b(In)30 b(these)f(cases,)f(w)n(e)h(m)n(ust)f(refer)g(to)h(the)h(study)f(of)g (Gibbs)515 3113 y(states)h(on)f(half-spaces)f(with)i(particular)d(b)r (oundary)j(conditions)d(but)k(the)g(main)d(p)r(oin)n(ts)515 3213 y(m)n(ust)e(remain)f(in)n(tact.)515 3313 y(W)-7 b(e)38 b(hop)r(e)f(to)h(include)e(in)g(a)i(future)g(publication)c(the)k (extensions)e(men)n(tioned)f(ab)r(o)n(v)n(e.)515 3412 y(In)d(particular,)f(all)f(examples)f(that)k(app)r(ear)e(in)h([19)o(])g (can)g(b)r(e)h(systematically)27 b(obtained)515 3512 y(using)e(the)i(one)f(and)g(same)f(algorithm)e(that)k(will)c(b)r(e)k (explained)e(b)r(elo)n(w.)35 b(W)-7 b(e)27 b(will)c(brie\015y)515 3611 y(illustrate)18 b(suc)n(h)k(a)g(result)f(\(for)h(a)f(con)n(tin)n (uous)f(time)h(dynamics)f(with)h(a)h(conserv)-5 b(ation)19 b(la)n(w\))515 3711 y(at)27 b(the)h(end)g(\(Section)f(3.3\).)515 3943 y Fy(1.3)112 b(Notation)37 b(and)i(de\014nitions.)515 4097 y Fv(W)-7 b(e)22 b(restrict)e(ourselv)n(es)f(here)j(to)f(lattice)f (spin)h(systems.)34 b(F)-7 b(or)21 b(lattice)f(w)n(e)h(tak)n(e)h(the)g (regular)515 4196 y Fq(d)15 b Fv(+)g(1-dimensional)21 b(set)p 1319 4198 45 4 v 1319 4142 43 4 v 1319 4196 3 3 v 1320 4194 V 1321 4192 V 1322 4190 V 1323 4189 V 1324 4187 V 1325 4185 V 1326 4183 V 1327 4181 V 1328 4180 V 1329 4178 V 1330 4176 V 1331 4174 V 1331 4172 V 1332 4170 V 1333 4168 V 1334 4166 V 1335 4165 V 1336 4163 V 1337 4161 V 1338 4159 V 1339 4157 V 1340 4155 V 1341 4153 V 1342 4152 V 1343 4150 V 1344 4148 V 1345 4146 V 1345 4144 V 1346 4142 V 1334 4196 V 1335 4194 V 1336 4192 V 1337 4190 V 1338 4189 V 1339 4187 V 1340 4185 V 1341 4183 V 1342 4181 V 1343 4180 V 1344 4178 V 1345 4176 V 1345 4174 V 1346 4172 V 1347 4170 V 1348 4168 V 1349 4166 V 1350 4165 V 1351 4163 V 1352 4161 V 1353 4159 V 1354 4157 V 1355 4155 V 1356 4153 V 1357 4152 V 1358 4150 V 1359 4148 V 1359 4146 V 1360 4144 V 1361 4142 V 1318 4152 4 12 v 1319 4148 4 8 v 1320 4146 4 6 v 1321 4144 4 4 v 1321 4142 4 2 v 1362 4196 4 12 v 1361 4196 4 8 v 1360 4196 4 6 v 1359 4196 4 4 v 1359 4196 4 2 v 1364 4165 a Fk(d)p Fm(+1)1487 4196 y Fq(;)14 b(d)23 b Fp(\025)g Fv(0.)36 b(The)26 b(reason)e(for)h(taking)g Fq(d)15 b Fv(+)g(1)26 b(is)e(that)j(the)f(extra)515 4296 y(dimension)36 b(refers)i(to)h(the)g(time-axis.)68 b(The)39 b(p)r(oin)n(ts)f(of)h(the)h(lattice)d(are)h(denoted)h(b)n(y)515 4396 y Fq(x;)14 b(y)s(;)g(:)g(:)g(:)31 b Fv(with)h Fq(x)e Fv(=)g(\()p Fq(i;)14 b(n)p Fv(\))p Fq(;)g(n)30 b Fp(2)p 1570 4398 45 4 v 1570 4341 43 4 v 1570 4396 3 3 v 1571 4394 V 1572 4392 V 1573 4390 V 1574 4389 V 1575 4387 V 1575 4385 V 1576 4383 V 1577 4381 V 1578 4379 V 1579 4377 V 1580 4375 V 1581 4373 V 1582 4371 V 1583 4369 V 1584 4368 V 1585 4366 V 1586 4364 V 1587 4362 V 1588 4360 V 1589 4358 V 1589 4356 V 1590 4355 V 1591 4353 V 1592 4351 V 1593 4349 V 1594 4347 V 1595 4345 V 1596 4343 V 1597 4341 V 1585 4396 V 1586 4394 V 1587 4392 V 1588 4390 V 1589 4389 V 1589 4387 V 1590 4385 V 1591 4383 V 1592 4381 V 1593 4379 V 1594 4377 V 1595 4375 V 1596 4373 V 1597 4371 V 1598 4369 V 1599 4368 V 1600 4366 V 1601 4364 V 1602 4362 V 1603 4360 V 1603 4358 V 1604 4356 V 1605 4355 V 1606 4353 V 1607 4351 V 1608 4349 V 1609 4347 V 1610 4345 V 1611 4343 V 1612 4341 V 1568 4351 4 12 v 1569 4347 4 8 v 1570 4345 4 6 v 1572 4343 4 4 v 1572 4341 4 2 v 1613 4396 4 12 v 1612 4396 4 8 v 1611 4396 4 6 v 1610 4396 4 4 v 1609 4396 4 2 v 89 w Fq(;)14 b(i)29 b Fp(2)p 1810 4398 45 4 v 1810 4341 43 4 v 1810 4396 3 3 v 1810 4394 V 1811 4392 V 1812 4390 V 1813 4389 V 1814 4387 V 1815 4385 V 1816 4383 V 1817 4381 V 1818 4379 V 1819 4377 V 1820 4375 V 1821 4373 V 1822 4371 V 1823 4369 V 1824 4368 V 1824 4366 V 1825 4364 V 1826 4362 V 1827 4360 V 1828 4358 V 1829 4356 V 1830 4355 V 1831 4353 V 1832 4351 V 1833 4349 V 1834 4347 V 1835 4345 V 1836 4343 V 1837 4341 V 1824 4396 V 1825 4394 V 1826 4392 V 1827 4390 V 1828 4389 V 1829 4387 V 1830 4385 V 1831 4383 V 1832 4381 V 1833 4379 V 1834 4377 V 1835 4375 V 1836 4373 V 1837 4371 V 1838 4369 V 1838 4368 V 1839 4366 V 1840 4364 V 1841 4362 V 1842 4360 V 1843 4358 V 1844 4356 V 1845 4355 V 1846 4353 V 1847 4351 V 1848 4349 V 1849 4347 V 1850 4345 V 1851 4343 V 1852 4341 V 1808 4351 4 12 v 1809 4347 4 8 v 1810 4345 4 6 v 1811 4343 4 4 v 1812 4341 4 2 v 1853 4396 4 12 v 1852 4396 4 8 v 1850 4396 4 6 v 1849 4396 4 4 v 1849 4396 4 2 v 1854 4364 a Fk(d)1893 4396 y Fv(.)50 b(W)-7 b(e)32 b(can)f(read)g(the)h(time)f(b)n(y)g(the)h (mapping)515 4495 y Fq(t)p Fv(\()p Fq(x)p Fv(\))25 b(=)f Fq(n)29 b Fv(if)f Fq(x)d Fv(=)f(\()p Fq(i;)14 b(n)p Fv(\).)40 b(The)28 b(distance)f(b)r(et)n(w)n(een)i(t)n(w)n(o)f(p)r(oin)n(ts)f Fq(x)e Fv(=)f(\()p Fq(i;)14 b(n)p Fv(\))p Fq(;)g(y)27 b Fv(=)d(\()p Fq(j;)14 b(m)p Fv(\))26 b Fp(2)p 529 4597 45 4 v 529 4540 43 4 v 529 4595 3 3 v 530 4593 V 530 4591 V 531 4589 V 532 4588 V 533 4586 V 534 4584 V 535 4582 V 536 4580 V 537 4578 V 538 4576 V 539 4574 V 540 4572 V 541 4571 V 542 4569 V 543 4567 V 544 4565 V 544 4563 V 545 4561 V 546 4559 V 547 4558 V 548 4556 V 549 4554 V 550 4552 V 551 4550 V 552 4548 V 553 4546 V 554 4544 V 555 4543 V 556 4541 V 544 4595 V 544 4593 V 545 4591 V 546 4589 V 547 4588 V 548 4586 V 549 4584 V 550 4582 V 551 4580 V 552 4578 V 553 4576 V 554 4574 V 555 4572 V 556 4571 V 557 4569 V 558 4567 V 558 4565 V 559 4563 V 560 4561 V 561 4559 V 562 4558 V 563 4556 V 564 4554 V 565 4552 V 566 4550 V 567 4548 V 568 4546 V 569 4544 V 570 4543 V 571 4541 V 527 4550 4 12 v 528 4546 4 8 v 529 4544 4 6 v 530 4543 4 4 v 531 4541 4 2 v 572 4595 4 12 v 571 4595 4 8 v 570 4595 4 6 v 568 4595 4 4 v 568 4595 4 2 v 573 4563 a Fk(d)p Fm(+1)721 4595 y Fv(is)e Fp(j)p Fq(x)13 b Fp(\000)g Fq(y)s Fp(j)23 b(\021)g Fv(max)n Fp(fj)p Fq(n)13 b Fp(\000)g Fq(m)p Fp(j)p Fq(;)h Fp(j)p Fq(i)f Fp(\000)g Fq(j)5 b Fp(jg)23 b Fv(with)i Fp(j)p Fq(i)13 b Fp(\000)g Fq(j)5 b Fp(j)22 b(\021)h Fv(max)n Fp(fj)p Fq(i)2654 4607 y Fm(1)2704 4595 y Fp(\000)13 b Fq(j)2816 4607 y Fm(1)2853 4595 y Fp(j)p Fq(;)h(:)g(:)g(:)f(;)h Fp(j)p Fq(i)3112 4607 y Fk(d)3164 4595 y Fp(\000)f Fq(j)3276 4607 y Fk(d)3314 4595 y Fp(jg)515 4694 y Fv(for)34 b(the)h(t)n(w)n(o)f (sites)g Fq(i)g Fv(=)h(\()p Fq(i)1380 4706 y Fm(1)1417 4694 y Fq(;)14 b(:)g(:)g(:)g(;)g(i)1631 4706 y Fk(d)1669 4694 y Fv(\))p Fq(;)g(j)40 b Fv(=)35 b(\()p Fq(j)1978 4706 y Fm(1)2015 4694 y Fq(;)14 b(:)g(:)g(:)g(;)g(j)2234 4706 y Fk(d)2273 4694 y Fv(\))35 b Fp(2)p 2444 4696 45 4 v 2444 4640 43 4 v 2444 4694 3 3 v 2445 4692 V 2446 4690 V 2447 4688 V 2448 4687 V 2449 4685 V 2450 4683 V 2451 4681 V 2452 4679 V 2453 4678 V 2454 4676 V 2455 4674 V 2456 4672 V 2457 4670 V 2457 4668 V 2458 4666 V 2459 4665 V 2460 4663 V 2461 4661 V 2462 4659 V 2463 4657 V 2464 4655 V 2465 4653 V 2466 4652 V 2467 4650 V 2468 4648 V 2469 4646 V 2470 4644 V 2471 4642 V 2472 4640 V 2459 4694 V 2460 4692 V 2461 4690 V 2462 4688 V 2463 4687 V 2464 4685 V 2465 4683 V 2466 4681 V 2467 4679 V 2468 4678 V 2469 4676 V 2470 4674 V 2471 4672 V 2472 4670 V 2472 4668 V 2473 4666 V 2474 4665 V 2475 4663 V 2476 4661 V 2477 4659 V 2478 4657 V 2479 4655 V 2480 4653 V 2481 4652 V 2482 4650 V 2483 4648 V 2484 4646 V 2485 4644 V 2486 4642 V 2486 4640 V 2443 4650 4 12 v 2444 4646 4 8 v 2445 4644 4 6 v 2446 4642 4 4 v 2447 4640 4 2 v 2488 4694 4 12 v 2486 4694 4 8 v 2485 4694 4 6 v 2484 4694 4 4 v 2484 4694 4 2 v 2489 4663 a Fk(d)2528 4694 y Fv(.)58 b(The)35 b(set)g(of)g(\014nite)f(and)515 4794 y(non-empt)n(y)24 b(subsets)h(of)p 1319 4796 45 4 v 1319 4740 43 4 v 1319 4794 3 3 v 1320 4792 V 1320 4790 V 1321 4788 V 1322 4787 V 1323 4785 V 1324 4783 V 1325 4781 V 1326 4779 V 1327 4777 V 1328 4775 V 1329 4774 V 1330 4772 V 1331 4770 V 1332 4768 V 1333 4766 V 1334 4764 V 1334 4762 V 1335 4761 V 1336 4759 V 1337 4757 V 1338 4755 V 1339 4753 V 1340 4751 V 1341 4749 V 1342 4747 V 1343 4746 V 1344 4744 V 1345 4742 V 1346 4740 V 1334 4794 V 1334 4792 V 1335 4790 V 1336 4788 V 1337 4787 V 1338 4785 V 1339 4783 V 1340 4781 V 1341 4779 V 1342 4777 V 1343 4775 V 1344 4774 V 1345 4772 V 1346 4770 V 1347 4768 V 1348 4766 V 1348 4764 V 1349 4762 V 1350 4761 V 1351 4759 V 1352 4757 V 1353 4755 V 1354 4753 V 1355 4751 V 1356 4749 V 1357 4747 V 1358 4746 V 1359 4744 V 1360 4742 V 1361 4740 V 1317 4749 4 12 v 1318 4746 4 8 v 1319 4744 4 6 v 1320 4742 4 4 v 1321 4740 4 2 v 1362 4794 4 12 v 1361 4794 4 8 v 1360 4794 4 6 v 1358 4794 4 4 v 1358 4794 4 2 v 1363 4762 a Fk(d)p Fm(+1)1512 4794 y Fv(is)f(denoted)h(b)n(y)g Fp(S)6 b Fv(.)37 b(F)-7 b(or)24 b(general)f(elemen)n(ts)g(of)j Fp(S)31 b Fv(w)n(e)25 b(write)515 4894 y(\003)p Fq(;)14 b(A;)g(:)g(:)g(:)o Fv(;)34 b(they)e(corresp)r(ond)e(to)h(\(\014nite\))h (space-time)d(regions.)46 b(\003)2665 4864 y Fk(c)2728 4894 y Fv(=)p 2836 4896 45 4 v 2836 4839 43 4 v 2836 4894 3 3 v 2837 4892 V 2838 4890 V 2839 4888 V 2839 4887 V 2840 4885 V 2841 4883 V 2842 4881 V 2843 4879 V 2844 4877 V 2845 4875 V 2846 4873 V 2847 4871 V 2848 4869 V 2849 4868 V 2850 4866 V 2851 4864 V 2852 4862 V 2853 4860 V 2853 4858 V 2854 4856 V 2855 4855 V 2856 4853 V 2857 4851 V 2858 4849 V 2859 4847 V 2860 4845 V 2861 4843 V 2862 4841 V 2863 4840 V 2851 4894 V 2852 4892 V 2853 4890 V 2854 4888 V 2854 4887 V 2855 4885 V 2856 4883 V 2857 4881 V 2858 4879 V 2859 4877 V 2860 4875 V 2861 4873 V 2862 4871 V 2863 4869 V 2864 4868 V 2865 4866 V 2866 4864 V 2867 4862 V 2868 4860 V 2868 4858 V 2869 4856 V 2870 4855 V 2871 4853 V 2872 4851 V 2873 4849 V 2874 4847 V 2875 4845 V 2876 4843 V 2877 4841 V 2878 4840 V 2834 4849 4 12 v 2835 4845 4 8 v 2836 4843 4 6 v 2837 4841 4 4 v 2838 4840 4 2 v 2879 4894 4 12 v 2878 4894 4 8 v 2877 4894 4 6 v 2876 4894 4 4 v 2875 4894 4 2 v 2881 4862 a Fk(d)p Fm(+1)3024 4894 y Fp(n)21 b Fv(\003)31 b(is)g(the)515 4993 y(complemen)n(t)24 b(of)k(\003;)f Fp(j)p Fv(\003)p Fp(j)h Fv(is)e(the)i(cardinalit)n(y)23 b(of)28 b(\003.)639 5093 y(A)21 b(space-time)c(con\014guration)g(of)j (our)f(lattice)f(spin)h(system)g(is)g(denoted)h(b)n(y)f Fq(\033)n(;)14 b(\021)s(;)g(\030)t(;)g(:)g(:)g(:)q Fv(.)515 5193 y(This)30 b(is)g(a)h(mapping)f Fq(\033)i Fv(:)p 1360 5195 45 4 v 1360 5138 43 4 v 1360 5193 3 3 v 1361 5191 V 1362 5189 V 1363 5187 V 1364 5186 V 1364 5184 V 1365 5182 V 1366 5180 V 1367 5178 V 1368 5176 V 1369 5174 V 1370 5172 V 1371 5170 V 1372 5168 V 1373 5166 V 1374 5165 V 1375 5163 V 1376 5161 V 1377 5159 V 1378 5157 V 1378 5155 V 1379 5153 V 1380 5152 V 1381 5150 V 1382 5148 V 1383 5146 V 1384 5144 V 1385 5142 V 1386 5140 V 1387 5138 V 1375 5193 V 1376 5191 V 1377 5189 V 1378 5187 V 1378 5186 V 1379 5184 V 1380 5182 V 1381 5180 V 1382 5178 V 1383 5176 V 1384 5174 V 1385 5172 V 1386 5170 V 1387 5168 V 1388 5166 V 1389 5165 V 1390 5163 V 1391 5161 V 1392 5159 V 1392 5157 V 1393 5155 V 1394 5153 V 1395 5152 V 1396 5150 V 1397 5148 V 1398 5146 V 1399 5144 V 1400 5142 V 1401 5140 V 1402 5138 V 1358 5148 4 12 v 1359 5144 4 8 v 1360 5142 4 6 v 1362 5140 4 4 v 1362 5138 4 2 v 1403 5193 4 12 v 1402 5193 4 8 v 1401 5193 4 6 v 1400 5193 4 4 v 1399 5193 4 2 v 1405 5161 a Fk(d)p Fm(+1)1557 5193 y Fp(!)d Fq(S)37 b Fv(with)30 b(v)-5 b(alues)30 b Fq(\033)s Fv(\()p Fq(x)p Fv(\))h Fp(2)f Fq(S)36 b Fv(in)31 b(the)h(single)d(site)h(state)515 5292 y(space)j Fq(S)38 b Fv(whic)n(h)33 b(is)g(tak)n(en)g(\014nite.)55 b(Ising)32 b(spins)h(ha)n(v)n(e)f Fq(S)38 b Fv(=)33 b Fp(f)p Fv(+1)p Fq(;)14 b Fp(\000)p Fv(1)p Fp(g)p Fv(.)53 b(The)34 b(set)g(of)f(all)1926 5650 y(4)p eop %%Page: 5 5 5 4 bop 515 531 a Fv(con\014gurations)25 b(is)i(\012)1195 543 y Fk(d)p Fm(+1)1342 531 y Fv(=)d Fq(S)p 1503 502 45 4 v 1503 446 43 4 v 1503 501 3 3 v 1504 499 V 1505 497 V 1506 495 V 1507 493 V 1508 491 V 1509 489 V 1510 488 V 1510 486 V 1511 484 V 1512 482 V 1513 480 V 1514 478 V 1515 476 V 1516 474 V 1517 473 V 1518 471 V 1519 469 V 1520 467 V 1521 465 V 1522 463 V 1523 461 V 1524 460 V 1524 458 V 1525 456 V 1526 454 V 1527 452 V 1528 450 V 1529 448 V 1530 446 V 1518 501 V 1519 499 V 1520 497 V 1521 495 V 1522 493 V 1523 491 V 1524 489 V 1524 488 V 1525 486 V 1526 484 V 1527 482 V 1528 480 V 1529 478 V 1530 476 V 1531 474 V 1532 473 V 1533 471 V 1534 469 V 1535 467 V 1536 465 V 1537 463 V 1538 461 V 1538 460 V 1539 458 V 1540 456 V 1541 454 V 1542 452 V 1543 450 V 1544 448 V 1545 446 V 1501 456 4 12 v 1502 452 4 8 v 1504 450 4 6 v 1505 448 4 4 v 1505 446 4 2 v 1546 501 4 12 v 1545 501 4 8 v 1544 501 4 6 v 1543 501 4 4 v 1542 501 4 2 v 1548 469 a Fh(d)p Fg(+1)1658 531 y 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(y)i(the)g(\()p Fq(\033)s Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)25 b Fp(2)e Fv(\003\).)37 b(W)-7 b(e)27 b(write)e Fp(F)31 b(\021)23 b(F)p 2287 1162 45 4 v 2287 1106 43 4 v 2287 1161 3 3 v 2288 1159 V 2289 1157 V 2290 1155 V 2291 1153 V 2292 1151 V 2292 1149 V 2293 1147 V 2294 1146 V 2295 1144 V 2296 1142 V 2297 1140 V 2298 1139 V 2299 1137 V 2300 1135 V 2301 1133 V 2302 1131 V 2303 1129 V 2304 1127 V 2305 1126 V 2306 1124 V 2306 1122 V 2307 1120 V 2308 1118 V 2309 1116 V 2310 1114 V 2311 1112 V 2312 1110 V 2313 1108 V 2314 1106 V 2302 1161 V 2303 1159 V 2304 1157 V 2305 1155 V 2306 1153 V 2307 1151 V 2307 1149 V 2308 1147 V 2309 1146 V 2310 1144 V 2311 1142 V 2312 1140 V 2313 1139 V 2314 1137 V 2315 1135 V 2316 1133 V 2317 1131 V 2318 1129 V 2319 1127 V 2320 1126 V 2321 1124 V 2321 1122 V 2322 1120 V 2323 1118 V 2324 1116 V 2325 1114 V 2326 1112 V 2327 1110 V 2328 1108 V 2329 1106 V 2285 1116 4 12 v 2286 1112 4 8 v 2287 1110 4 6 v 2289 1108 4 4 v 2289 1106 4 2 v 2330 1161 4 12 v 2329 1161 4 8 v 2328 1161 4 6 v 2327 1161 4 4 v 2326 1161 4 2 v 61 w Fh(d)p Fg(+1)8 b Fv(;)27 b(\(\012)p Fq(;)14 b Fp(F)8 b Fv(\))27 b(is)e(the)i(measurable)515 1228 y(space)g(of)g(space-time)e(con\014gurations.)515 1328 y(Lo)r(cal)30 b(functions)h(on)g(\012)1285 1340 y Fk(d)p Fm(+1)1439 1328 y Fv(are)g(real-v)-5 b(alued)28 b(functions)j Fq(f)40 b Fv(whic)n(h)31 b(are)f Fp(F)2895 1340 y Fm(\003)2945 1328 y Fv(-measurable)515 1427 y(for)24 b(some)f(\003)g Fp(2)g(S)6 b Fv(.)37 b(The)25 b(\(\014nite\))g(dep)r (endence)g(set)g(of)g(suc)n(h)f(a)g(lo)r(cal)f Fq(f)33 b Fv(is)24 b(denoted)h(b)n(y)f Fq(D)3313 1439 y Fk(f)3356 1427 y Fv(.)515 1527 y(A)31 b(con)n(tin)n(uous)d(function)i(is)g(ev)n (ery)f(function)h(on)h(\012)2170 1539 y Fk(d)p Fm(+1)2323 1527 y Fv(whic)n(h)f(is)f(the)i(uniform)e(limit)e(of)515 1627 y(lo)r(cal)e(functions.)36 b(The)28 b(uniform)d(norm)h(is)h (denoted)g(b)n(y)h Fp(k)p Fq(f)9 b Fp(k)21 b(\021)i Fv(sup)2681 1647 y Fk(\033)2739 1627 y Fp(j)p Fq(f)9 b Fv(\()p Fq(\033)s Fv(\))p Fp(j)p Fv(.)515 1726 y(Finally)-7 b(,)36 b(con\014gurations)e (and)i(functions)g(on)g(\012)2091 1738 y Fk(d)p Fm(+1)2251 1726 y Fv(can)g(b)r(e)h(translated)e(o)n(v)n(er)g Fq(\034)3159 1738 y Fk(x)3201 1726 y Fq(;)14 b(x)39 b Fp(2)p 529 1828 45 4 v 529 1772 43 4 v 529 1826 3 3 v 530 1824 V 530 1822 V 531 1820 V 532 1819 V 533 1817 V 534 1815 V 535 1813 V 536 1811 V 537 1809 V 538 1807 V 539 1805 V 540 1803 V 541 1802 V 542 1800 V 543 1798 V 544 1796 V 544 1794 V 545 1792 V 546 1790 V 547 1789 V 548 1787 V 549 1785 V 550 1783 V 551 1781 V 552 1779 V 553 1777 V 554 1775 V 555 1774 V 556 1772 V 544 1826 V 544 1824 V 545 1822 V 546 1820 V 547 1819 V 548 1817 V 549 1815 V 550 1813 V 551 1811 V 552 1809 V 553 1807 V 554 1805 V 555 1803 V 556 1802 V 557 1800 V 558 1798 V 558 1796 V 559 1794 V 560 1792 V 561 1790 V 562 1789 V 563 1787 V 564 1785 V 565 1783 V 566 1781 V 567 1779 V 568 1777 V 569 1775 V 570 1774 V 571 1772 V 527 1781 4 12 v 528 1777 4 8 v 529 1775 4 6 v 530 1774 4 4 v 531 1772 4 2 v 572 1826 4 12 v 571 1826 4 8 v 570 1826 4 6 v 568 1826 4 4 v 568 1826 4 2 v 573 1794 a Fk(d)p Fm(+1)725 1826 y Fv(:)28 b Fq(\034)812 1838 y Fk(x)854 1826 y Fq(\033)s Fv(\()p Fq(y)s Fv(\))h(=)f Fq(\033)s Fv(\()p Fq(y)c Fv(+)c Fq(x)p Fv(\))p Fq(;)14 b(\034)1518 1838 y Fk(x)1561 1826 y Fq(f)9 b Fv(\()p Fq(\033)s Fv(\))29 b(=)f Fq(f)9 b Fv(\()p Fq(\034)1965 1838 y Fk(x)2007 1826 y Fq(\033)s Fv(\).)47 b(If)31 b(clear)e(from)g(the)j(con)n(text,)f(w)n(e)f(also)515 1926 y(write)c Fq(f)768 1938 y Fk(x)837 1926 y Fv(for)h Fq(\034)1000 1938 y Fk(x)1042 1926 y Fq(f)9 b Fv(.)639 2125 y(W)-7 b(e)32 b(consider)d(families)e(of)k(lo)r(cal)e(one-to-one)g (\(in)n(v)n(ertible\))f(transformations)f Fq(\031)3210 2137 y Fm(\003)3291 2125 y Fv(on)515 2224 y Fq(S)571 2194 y Fm(\003)647 2224 y Fv(where)f(\003)h(will)d(v)-5 b(ary)26 b(in)h(some)e(large)f(enough)j(subset)g(of)g Fp(S)33 b Fv(\(whic)n(h)26 b(will)e(b)r(e)k(sp)r(eci\014ed)515 2324 y(later)e(on\).)37 b(As)27 b(maps)f(on)i(\012)1404 2336 y Fk(d)p 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Fe(and)k Fq(\031)2521 4710 y Fm(\003)2571 4698 y Fv(\()p Fq(\033)s Fv(\)\()p Fq(x)p Fv(\))d(=)e Fp(\000)p Fq(\033)s Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)24 b Fp(2)f Fv(\003)515 4798 y Fe(c)l(orr)l(esp)l (onding)31 b(to)f(a)g(spin-\015ip)g(in)g Fv(\003)23 b Fp(2)g(S)6 b Fe(.)639 4981 y Fv(Probabilit)n(y)20 b(measures)h(on)j (\(\012)1627 4993 y Fk(d)p Fm(+1)1750 4981 y Fq(;)14 b Fp(F)8 b Fv(\))24 b(are)f(denoted)h(b)n(y)g Fq(\026;)14 b(\027)q(;)g(:)g(:)g(:)o Fv(.)36 b(The)24 b(corresp)r(ond-)515 5080 y(ing)34 b(random)f(\014eld)i(is)f(written)h(as)g Fq(X)42 b Fv(=)36 b(\()p Fq(X)7 b Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)37 b Fp(2)p 2318 5082 45 4 v 2318 5026 43 4 v 2318 5080 3 3 v 2319 5078 V 2320 5076 V 2321 5074 V 2322 5073 V 2323 5071 V 2324 5069 V 2325 5067 V 2326 5065 V 2327 5064 V 2327 5062 V 2328 5060 V 2329 5058 V 2330 5056 V 2331 5054 V 2332 5052 V 2333 5050 V 2334 5049 V 2335 5047 V 2336 5045 V 2337 5043 V 2338 5041 V 2339 5039 V 2340 5037 V 2341 5036 V 2341 5034 V 2342 5032 V 2343 5030 V 2344 5028 V 2345 5026 V 2333 5080 V 2334 5078 V 2335 5076 V 2336 5074 V 2337 5073 V 2338 5071 V 2339 5069 V 2340 5067 V 2341 5065 V 2341 5064 V 2342 5062 V 2343 5060 V 2344 5058 V 2345 5056 V 2346 5054 V 2347 5052 V 2348 5050 V 2349 5049 V 2350 5047 V 2351 5045 V 2352 5043 V 2353 5041 V 2354 5039 V 2355 5037 V 2355 5036 V 2356 5034 V 2357 5032 V 2358 5030 V 2359 5028 V 2360 5026 V 2316 5036 4 12 v 2318 5032 4 8 v 2319 5030 4 6 v 2320 5028 4 4 v 2320 5026 4 2 v 2361 5080 4 12 v 2360 5080 4 8 v 2359 5080 4 6 v 2358 5080 4 4 v 2358 5080 4 2 v 2363 5049 a Fk(d)p Fm(+1)2486 5080 y Fv(\).)60 b(The)36 b(exp)r(ectation)e(of)h(a)515 5180 y(function)29 b Fq(f)39 b Fv(is)29 b(written)g(as)g Fq(\026)p Fv(\()p Fq(f)9 b Fv(\))27 b Fp(\021)1687 5113 y Fj(R)1757 5180 y Fq(f)9 b Fv(\()p Fq(\033)s Fv(\))p Fq(\026)p Fv(\()p Fq(d\033)s Fv(\).)45 b(As)30 b Fe(a)i(priori)g Fv(measure)c(w)n(e)i(tak)n(e)f(the)515 5280 y(uniform)h(pro)r(duct)j (measure)d Fq(d\033)36 b Fv(with)c(normalized)d(coun)n(ting)h(measure)h (as)h(marginals,)515 5379 y(for)27 b(whic)n(h)879 5312 y Fj(R)949 5379 y Fq(f)9 b Fv(\()p Fq(\033)s Fv(\))p Fq(d\033)27 b Fp(\021)22 b Fv(1)p Fq(=)p Fp(j)p Fq(S)5 b Fp(j)1503 5349 y Fl(j)p Fm(\003)p Fl(j)1605 5317 y Fj(P)1693 5404 y Fk(\033)1731 5412 y Fg(\003)1791 5379 y Fq(f)k Fv(\()p Fq(\033)1920 5391 y Fm(\003)1970 5379 y Fv(\))28 b(when)g Fq(f)36 b Fv(is)27 b Fp(F)2468 5391 y Fm(\003)2517 5379 y Fv(-measurable.)1926 5650 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fv(W)-7 b(e)31 b(will)e(b)r(e)i(dealing)e(with)h (Gibbs)h(states)f Fq(\026)h Fv(in)g(what)g(follo)n(ws;)f Fq(\026)h Fv(is)f(a)h(Gibbs)f(measure)515 623 y(with)e(resp)r(ect)h(to) g(the)g(Hamiltonian)c Fq(H)35 b Fv(at)29 b(in)n(v)n(erse)e(temp)r (erature)g Fq(\014)34 b Fv(\(and)29 b(alw)n(a)n(ys)d(with)515 722 y(resp)r(ect)f(to)g(the)g(coun)n(ting)f(measure)f(as)i Fe(a)j(priori)f Fv(measure\))c(when)j(for)e(ev)n(ery)g(\003)f Fp(2)g(S)32 b Fv(and)515 822 y(for)27 b(eac)n(h)g(pair)e(of)j (con\014gurations)d Fq(\033)1677 834 y Fm(\003)1727 822 y Fq(;)14 b(\021)1805 834 y Fm(\003)1878 822 y Fp(2)23 b Fq(S)2012 792 y Fm(\003)537 992 y Fq(\026)p Fv([)p Fq(X)7 b Fv(\()p Fq(x)p Fv(\))24 b(=)f Fq(\033)s Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)24 b Fp(2)g Fv(\003)p Fp(j)p Fq(X)7 b Fv(\()p Fq(x)p Fv(\))23 b(=)g Fq(\030)t Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)24 b Fp(2)g Fv(\003)2032 962 y Fk(c)2065 992 y Fv(])p 537 1029 1551 4 v 540 1105 a Fq(\026)p Fv([)p Fq(X)7 b Fv(\()p Fq(x)p Fv(\))24 b(=)f Fq(\021)s Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)24 b Fp(2)g Fv(\003)p Fp(j)p Fq(X)7 b Fv(\()p Fq(x)p Fv(\))23 b(=)f Fq(\030)t Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)25 b Fp(2)e Fv(\003)2028 1081 y Fk(c)2062 1105 y Fv(])2121 1048 y(=)g(exp[)p Fp(\000)p Fq(\014)t Fv(\()p Fq(H)7 b Fv(\()p Fq(\033)2662 1060 y Fm(\003)2712 1048 y Fq(\030)2748 1060 y Fm(\003)2793 1044 y Fh(c)2829 1048 y Fv(\))19 b Fp(\000)f Fq(H)7 b Fv(\()p Fq(\021)3112 1060 y Fm(\003)3162 1048 y Fq(\030)3198 1060 y Fm(\003)3243 1044 y Fh(c)3279 1048 y Fv(\)\)])3208 1196 y(\(1.9\))515 1296 y(for)39 b Fq(\026)p Fp(\000)g Fv(almost)e(ev)n(ery)h Fq(\030)47 b Fp(2)d Fv(\012)1561 1308 y Fk(d)p Fm(+1)1683 1296 y Fv(.)73 b(The)40 b(Hamiltonian)35 b Fq(H)50 b Fv(=)2678 1233 y Fj(P)2765 1321 y Fk(A)2833 1296 y Fq(U)2890 1308 y Fk(A)2984 1296 y Fv(is)38 b(formally)515 1395 y(written)27 b(as)h(a)g(sum)g(of)g(\(in)n(teraction\))e(p)r(oten)n (tials)g Fq(U)2175 1407 y Fk(A)2229 1395 y Fv(\()p Fq(\033)s Fv(\))f(=)f Fq(U)2514 1407 y Fk(A)2568 1395 y Fv(\()p Fq(\033)2647 1407 y Fk(A)2702 1395 y Fv(\))k(with)g(w)n(ell-de\014ned) 515 1495 y(relativ)n(e)17 b(energies)h Fq(H)7 b Fv(\()p Fq(\033)s Fv(\))t Fp(\000)t Fq(H)g Fv(\()p Fq(\021)s Fv(\))23 b(for)d Fp(f)p Fq(x)j Fp(2)p 1904 1497 45 4 v 1904 1441 43 4 v 1904 1495 3 3 v 1905 1493 V 1906 1491 V 1907 1489 V 1908 1488 V 1909 1486 V 1910 1484 V 1911 1482 V 1912 1480 V 1913 1478 V 1914 1476 V 1915 1474 V 1915 1473 V 1916 1471 V 1917 1469 V 1918 1467 V 1919 1465 V 1920 1463 V 1921 1461 V 1922 1459 V 1923 1458 V 1924 1456 V 1925 1454 V 1926 1452 V 1927 1450 V 1928 1448 V 1929 1446 V 1929 1445 V 1930 1443 V 1931 1441 V 1919 1495 V 1920 1493 V 1921 1491 V 1922 1489 V 1923 1488 V 1924 1486 V 1925 1484 V 1926 1482 V 1927 1480 V 1928 1478 V 1929 1476 V 1929 1474 V 1930 1473 V 1931 1471 V 1932 1469 V 1933 1467 V 1934 1465 V 1935 1463 V 1936 1461 V 1937 1459 V 1938 1458 V 1939 1456 V 1940 1454 V 1941 1452 V 1942 1450 V 1943 1448 V 1943 1446 V 1944 1445 V 1945 1443 V 1946 1441 V 1903 1450 4 12 v 1904 1446 4 8 v 1905 1445 4 6 v 1906 1443 4 4 v 1906 1441 4 2 v 1947 1495 4 12 v 1946 1495 4 8 v 1945 1495 4 6 v 1944 1495 4 4 v 1944 1495 4 2 v 1949 1463 a Fk(d)p Fm(+1)2095 1495 y Fv(:)g Fq(\033)s Fv(\()p Fq(x)p Fv(\))i Fp(6)p Fv(=)d Fq(\021)s Fv(\()p Fq(x)p Fv(\))p Fp(g)i(2)f(S)28 b Fv(if)2860 1433 y Fj(P)2947 1520 y Fk(A)p Fl(3)p Fk(x)3098 1495 y Fp(k)p Fq(U)3197 1507 y Fk(A)3250 1495 y Fp(k)22 b Fq(<)515 1603 y Fp(1)p Fq(;)14 b(x)31 b Fp(2)g Fq(Z)862 1573 y Fk(d)p Fm(+1)985 1603 y Fv(.)51 b(Other)31 b(w)n(eak)n(er)g (conditions)f(than)i(uniform)e(absolute)h(summabilit)n(y)c(of)515 1703 y(the)d(p)r(oten)n(tial)f(are)g(p)r(ossible.)33 b(The)25 b(essen)n(tial)c(Gibbs)j(prop)r(ert)n(y)f(is)g(\(1.9\))h(whic) n(h)f(iden)n(ti\014es)515 1802 y(the)38 b(existence)f(of)i(a)e(w)n (ell-de\014ned)f(relativ)n(e)f(energy)j(go)n(v)n(erning)c(the)39 b(relativ)n(e)c(w)n(eigh)n(ts)515 1902 y(of)d(con\014gurations)d(that)j (lo)r(cally)c(di\013er.)49 b(\(1.9\))32 b(is)e(the)j(in\014nite)d(v)n (olume)f(v)n(ersion)h(of)i(the)515 2002 y(equiv)-5 b(alen)n(t)25 b(statemen)n(t)i(for)g(\014nite)g(v)n(olume)e(Gibbs)i(states)1084 2219 y Fq(\026)1134 2231 y Fm(\003)1184 2219 y Fv(\()p Fq(\033)1263 2231 y Fm(\003)1313 2219 y Fp(j)p Fq(\021)1377 2231 y Fm(\003)1422 2215 y Fh(c)1458 2219 y Fv(\))c(=)1698 2163 y(1)p 1611 2200 217 4 v 1611 2290 a Fq(Z)1674 2250 y Fk(\014)1668 2314 y Fm(\003)1718 2290 y Fv(\()p Fq(\021)s Fv(\))1851 2219 y(exp[)p Fp(\000)p Fq(\014)2183 2140 y Fj(X)2130 2322 y Fk(A)p Fl(\\)p Fm(\003)p Fl(6)p Fm(=)p Fl(;)2369 2219 y Fq(U)2426 2231 y Fk(A)2480 2219 y Fv(\()p Fq(\033)2559 2231 y Fm(\003)2609 2219 y Fq(\021)2650 2231 y Fm(\003)2695 2215 y Fh(c)2731 2219 y Fv(\)])p Fq(;)358 b Fv(\(1.10\))515 2513 y(with)31 b Fq(Z)771 2473 y Fk(\014)765 2537 y Fm(\003)815 2513 y Fv(\()p Fq(\021)s Fv(\))h(the)g(normalizing)26 b(factor)31 b(\(partition)e (function)i(with)g Fq(\021)k Fv(b)r(oundary)c(condi-)515 2613 y(tions\).)515 2712 y(T)-7 b(raditionally)g(,)27 b(Gibbs)k(measures)e(giv)n(e)h(the)i(distribution)d(of)i(the)h (microscopic)27 b(degrees)515 2812 y(of)c(freedom)f(for)h(a)g (macroscopic)c(system)j(in)h(thermo)r(dynamic)d(equilibrium.)30 b(The)24 b(c)n(hoice)515 2911 y(of)i(the)g(ensem)n(ble)e(is)h (determined)f(b)n(y)i(the)g(exp)r(erimen)n(tal)c(situation)i(and)i(is)f (\014xed)h(b)n(y)g(the)515 3011 y(c)n(hoice)j(of)h(the)i(relev)-5 b(an)n(t)29 b(macro-v)-5 b(ariables.)40 b(There)30 b(is)g(ho)n(w)n(ev)n (er)f(no)h Fe(a)j(priori)g Fv(reason)c(to)515 3111 y(exclude)f (nonequilibrium)d(situations)i(from)i(the)h(Gibbs)f(formalism)c(if)k (considered)e(as)i(a)515 3210 y(pro)r(cedure)20 b(of)h(statistical)c (inference.)34 b(Then)21 b(the)g(information)d(concerning)g(the)k (nonequi-)515 3310 y(librium)f(state)k(\(lik)n(e)e(obtained)g(from)h (measuring)e(the)j(curren)n(ts\))f(is)g(incorp)r(orated)e(in)j(the)515 3410 y(ensem)n(ble.)60 b(Moreo)n(v)n(er,)36 b(as)g(w)n(e)f(will)f(use)i (in)f(Section)h(3,)i(one)e(can)f(in)h(man)n(y)e(cases)i(ex-)515 3509 y(plicitly)24 b(construct)j(the)h(Gibbs)f(states)g(go)n(v)n (erning)d(the)k(space-time)d(distribution)g(as)i(the)515 3609 y(path-space)34 b(measure)g(for)g(the)i(dynamics.)58 b(The)36 b(fact)f(that)h(these)f(examples)e(concern)515 3708 y(sto)r(c)n(hastic)26 b(dynamics)f(should)h(not)i(b)r(e)g (regarded)e(as)h(a)g(return)g(to)h(the)g(strongly)d(c)n(haotic)515 3808 y(regime)h(but)k(rather)e(as)h(the)g(prop)r(er)f(w)n(a)n(y)g(to)h (deal)f(with)h(incomplete)d(kno)n(wledge)h(ab)r(out)515 3908 y(the)d(microscopic)c(con\014guration)h(of)j(a)g(system)f(comp)r (osed)g(of)h(a)f(h)n(uge)h(n)n(um)n(b)r(er)f(of)h(lo)r(cally)515 4007 y(in)n(teracting)g(comp)r(onen)n(ts.)515 4240 y Fy(1.4)112 b(Main)39 b(observ)-6 b(ation.)515 4393 y Fv(W)f(e)28 b(start)f(with)h(the)g(simplest)d(observ)-5 b(ation.)35 b(The)28 b(rest)f(will)e(follo)n(w)g(as)j(immediate)c(gen-) 515 4493 y(eralizations)f(\(with)k(p)r(erhaps)g(a)g(sligh)n(tly)d(more) i(complicated)e(notation\).)515 4592 y(Lo)r(ok)j(at)h(\(1.9\).)37 b(This)27 b(Gibbs)h(prop)r(ert)n(y)e(implies)f(that)j(the)g(image)d (measure)h(of)i Fq(\026)g Fv(under)515 4692 y(a)h(transformation)c (that)30 b(a\013ects)f(only)f(the)i(spins)e(in)h(\003)g(is)g (absolutely)d(con)n(tin)n(uous)i(with)515 4792 y(resp)r(ect)35 b(to)g Fq(\026)g Fv(with)f(the)i(Boltzmann-Gibbs)31 b(factor)j(as)h (Radon-Nik)n(o)r(dym)d(deriv)-5 b(ativ)n(e.)515 4891 y(Putting)29 b(it)h(simpler,)e(it)i(is)f(an)h(immediate)d(consequence)i (of)i(the)g(Gibbs)e(prop)r(ert)n(y)h(that)515 4991 y(for)d(all)e(con)n (tin)n(uous)h(functions)h Fq(f)1615 5173 y(\026)p Fv(\()p Fq(\031)1744 5185 y Fm(\003)1794 5173 y Fq(f)9 b Fv(\))23 b(=)g Fq(\026)p Fv(\()p Fq(f)9 b(W)2197 5185 y Fm(\003)2247 5173 y Fv(\))888 b(\(1.11\))1926 5650 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fv(with)29 b Fq(W)784 535 y Fm(\003)859 523 y Fp(\021)d Fv(exp[)p Fp(\000)p Fq(\014)1230 461 y Fj(P)1317 548 y Fk(A)p Fl(\\)p Fm(\003)p Fl(6)p Fm(=)p Fl(;)1546 523 y Fv(\()p Fq(\031)1625 535 y Fm(\003)1676 523 y Fq(U)1733 535 y Fk(A)1806 523 y Fp(\000)19 b Fq(U)1947 535 y Fk(A)2001 523 y Fv(\)].)42 b(But)30 b(no)n(w)f(the)g(road)f(is)h (straigh)n(t:)38 b(tak)n(e)515 643 y Fq(f)31 b Fv(=)23 b Fq(W)765 608 y Fk(\025)p Fl(\000)p Fm(1)753 668 y(\003)921 643 y Fv(in)k(\(1.11\))g(and)g(compute)1525 832 y Fq(\026)p Fv(\()p Fq(W)1697 798 y Fk(\025)1685 853 y Fm(\003)1741 832 y Fv(\))d(=)e Fq(\026)p Fv(\()p Fq(W)2056 797 y Fk(\025)p Fl(\000)p Fm(1)2044 857 y(\003)2185 832 y Fq(W)2263 844 y Fm(\003)2313 832 y Fv(\))p Fq(:)799 b Fv(\(1.12\))515 1012 y(F)-7 b(rom)26 b(\(1.11\))h(this)g(is)f(equal)g(to)1510 1192 y Fq(\026)p Fv(\()p Fq(\031)1639 1204 y Fm(\003)1689 1192 y Fq(W)1779 1156 y Fk(\025)p Fl(\000)p Fm(1)1767 1216 y(\003)1908 1192 y Fv(\))d(=)g Fq(\026)p Fv(\()p Fq(W)2223 1156 y Fm(1)p Fl(\000)p Fk(\025)2211 1216 y Fm(\003)2351 1192 y Fv(\))784 b(\(1.13\))515 1381 y(where)26 b(the)h(last)e(equalit)n(y)g(follo)n(ws)e(from)j Fq(\031)1882 1393 y Fm(\003)1932 1381 y Fv(\()p Fq(W)2054 1345 y Fk(\025)p Fl(\000)p Fm(1)2042 1405 y(\003)2182 1381 y Fv(\))e(=)e Fq(W)2415 1345 y Fm(1)p Fl(\000)p Fk(\025)2403 1405 y Fm(\003)2544 1381 y Fv(.)36 b(Th)n(us,)27 b(it)f(is)g(immediate)515 1480 y(that)i(Gibbs)e(measures)g(satisfy)1310 1660 y Fq(\026)p Fv(\()p Fq(e)1431 1626 y Fl(\000)p Fk(\025\014)s(R)1613 1634 y Fg(\003)1659 1660 y Fv(\))d(=)g Fq(\026)p Fv(\()p Fq(e)1923 1626 y Fl(\000)p Fm(\(1)p Fl(\000)p Fk(\025)p Fm(\))p Fk(\014)s(R)2242 1634 y Fg(\003)2288 1660 y Fv(\))p Fq(;)14 b(\025)23 b Fp(2)h Fv(I)-14 b(R)583 b(\(1.14\))515 1840 y(with)27 b(relativ)n(e)d(energy)j Fq(R)1328 1852 y Fm(\003)1400 1840 y Fp(\021)c Fq(\031)1535 1852 y Fm(\003)1585 1840 y Fq(H)i Fp(\000)18 b Fq(H)35 b Fv(corresp)r(onding)24 b(to)k(the)g(transformation)c Fq(\031)3257 1852 y Fm(\003)3307 1840 y Fv(:)1478 2032 y Fq(R)1541 2044 y Fm(\003)1614 2032 y Fv(=)1754 1953 y Fj(X)1701 2135 y Fk(A)p Fl(\\)p Fm(\003)p Fl(6)p Fm(=)p Fl(;)1927 2032 y Fv([)p Fq(\031)1997 2044 y Fm(\003)2046 2032 y Fq(U)2103 2044 y Fk(A)2176 2032 y Fp(\000)18 b Fq(U)2316 2044 y Fk(A)2369 2032 y Fv(])p Fq(:)752 b Fv(\(1.15\))639 2315 y(W)-7 b(e)29 b(no)n(w)e(imagine)e(the)j(ab)r(o)n(v)n(e)f(for)g(a)h(sequence)f(of)h (v)n(olumes)e(\003)h(gro)n(wing)e(to)p 3114 2317 45 4 v 3114 2261 43 4 v 3114 2315 3 3 v 3115 2313 V 3116 2311 V 3117 2309 V 3118 2308 V 3119 2306 V 3120 2304 V 3121 2302 V 3122 2300 V 3122 2298 V 3123 2297 V 3124 2295 V 3125 2293 V 3126 2291 V 3127 2289 V 3128 2287 V 3129 2285 V 3130 2283 V 3131 2282 V 3132 2280 V 3133 2278 V 3134 2276 V 3135 2274 V 3136 2272 V 3136 2270 V 3137 2269 V 3138 2267 V 3139 2265 V 3140 2263 V 3141 2261 V 3129 2315 V 3130 2313 V 3131 2311 V 3132 2309 V 3133 2308 V 3134 2306 V 3135 2304 V 3136 2302 V 3136 2300 V 3137 2298 V 3138 2297 V 3139 2295 V 3140 2293 V 3141 2291 V 3142 2289 V 3143 2287 V 3144 2285 V 3145 2283 V 3146 2282 V 3147 2280 V 3148 2278 V 3149 2276 V 3150 2274 V 3151 2272 V 3151 2270 V 3152 2269 V 3153 2267 V 3154 2265 V 3155 2263 V 3156 2261 V 3112 2270 4 12 v 3114 2267 4 8 v 3115 2265 4 6 v 3116 2263 4 4 v 3116 2261 4 2 v 3157 2315 4 12 v 3156 2315 4 8 v 3155 2315 4 6 v 3154 2315 4 4 v 3154 2315 4 2 v 3159 2283 a Fk(d)p Fm(+1)3310 2315 y Fv(in)515 2415 y(a)h(su\016cien)n(tly)e(regular)g (manner)h(\(e.g.)36 b(increasing)24 b(cub)r(es\).)37 b(Supp)r(ose)26 b(no)n(w)g(furthermore)515 2514 y(that)k Fq(\026)f Fv(is)g(a)g(Gibbs)g(measure)f(for)h(a)g(translation-in)n(v)-5 b(aria)o(n)n(t)24 b(in)n(teraction)j(p)r(oten)n(tial)g(and)515 2614 y(that)1426 2714 y Fq(R)1489 2726 y Fm(\003)1539 2714 y Fv(\()p Fq(\033)s Fv(\))d(=)1769 2635 y Fj(X)1765 2813 y Fk(x)p Fl(2)p Fm(\003)1906 2714 y Fq(\034)1942 2726 y Fk(x)1984 2714 y Fq(J)8 b Fv(\()p Fq(\033)s Fv(\))20 b(+)e Fq(h)2303 2726 y Fm(\003)2352 2714 y Fv(\()p Fq(\033)s Fv(\))701 b(\(1.16\))515 2944 y(with)23 b Fq(J)31 b Fv(a)24 b(b)r(ounded)g(con)n(tin)n(uous)d(function)i(and)h Fp(jj)p Fq(h)2162 2956 y Fm(\003)2211 2944 y Fp(jj)p Fq(=)p Fp(j)p Fv(\003)p Fp(j)f(!)g Fv(0)g(as)g(\003)g(b)r(ecomes)f(in\014nite.)515 3044 y(This)k(will)f(b)r(e)j(made)f(explicit)d(later)i(on.)37 b(Then,)28 b(the)g(follo)n(wing)23 b(limit)i(exists:)1189 3258 y Fq(p)p Fv(\()p Fq(\025J)8 b Fp(j)p Fq(\026)p Fv(\))24 b Fp(\021)e(\000)14 b Fv(lim)1695 3312 y Fm(\003)1830 3202 y Fv(1)p 1799 3239 104 4 v 1799 3315 a Fp(j)p Fv(\003)p Fp(j)1927 3258 y Fv(ln)f Fq(\026)p Fv([exp\()p Fp(\000)p Fq(\014)t(\025)2424 3179 y Fj(X)2420 3357 y Fk(x)p Fl(2)p Fm(\003)2562 3258 y Fq(J)2608 3270 y Fk(x)2649 3258 y Fv(\)])463 b(\(1.17\))515 3521 y(with)27 b Fq(J)750 3533 y Fk(x)815 3521 y Fp(\021)22 b Fq(\034)938 3533 y Fk(x)980 3521 y Fq(J)8 b Fv(,)28 b(and,)g(from)e(\(1.14\),)h(it)g(satis\014es) 1494 3701 y Fq(p)p Fv(\()p Fq(\025J)8 b Fp(j)p Fq(\026)p Fv(\))24 b(=)f Fq(p)p Fv(\(\(1)18 b Fp(\000)g Fq(\025)p Fv(\))p Fq(J)8 b Fp(j)p Fq(\026)p Fv(\))p Fq(:)769 b Fv(\(1.18\))515 3881 y(As)27 b(a)h(consequence,)e(its)h(Legendre)g (transform)1422 4061 y Fq(i)1451 4073 y Fk(J)1497 4061 y Fv(\()p Fq(w)r Fp(j)p Fq(\026)p Fv(\))d Fp(\021)f Fv(sup)1850 4131 y Fk(\025)1932 4061 y Fv([)p Fq(p)p Fv(\()p Fq(\025J)8 b Fp(j)p Fq(\026)p Fv(\))19 b Fp(\000)f Fq(\025w)r Fv(])697 b(\(1.19\))515 4285 y(satis\014es)1460 4385 y Fq(i)1489 4397 y Fk(J)1535 4385 y Fv(\()p Fq(w)r Fp(j)p Fq(\026)p Fv(\))19 b Fp(\000)f Fq(i)1864 4397 y Fk(J)1910 4385 y Fv(\()p Fp(\000)p Fq(w)r Fp(j)p Fq(\026)p Fv(\))24 b(=)f Fp(\000)p Fq(w)r(:)733 b Fv(\(1.20\))515 4533 y(It)35 b(is)f(not)h(necessary)e(\(but)j(it)e(is)g(p)r(ossible\))f(to)i(emplo)n (y)d(the)j(whole)f(mac)n(hinery)e(of)j(the)515 4632 y(theory)27 b(of)g(large)e(deviations)g(for)i(Gibbs)g(states)g(to)h(understand)f (what)g(this)g(means:)36 b(the)515 4732 y(probabilit)n(y)17 b(la)n(w)i Fq(P)1129 4744 y Fm(\003)1179 4732 y Fv(\()p Fq(w)r Fv(\))j(for)e(the)i(random)d(v)-5 b(ariable)2185 4670 y Fj(P)2273 4757 y Fk(x)p Fl(2)p Fm(\003)2419 4732 y Fq(J)2465 4744 y Fk(x)2507 4732 y Fv(\()p Fq(X)7 b Fv(\))p Fq(=)p Fp(j)p Fv(\003)p Fp(j)20 b Fv(as)g(induced)g(from)515 4842 y(the)26 b(random)e(\014eld)i(\()p Fq(X)7 b Fv(\()p Fq(x)p Fv(\))p Fq(;)14 b(x)24 b Fp(2)p 1557 4844 45 4 v 1557 4787 43 4 v 1557 4842 3 3 v 1558 4840 V 1559 4838 V 1560 4836 V 1561 4835 V 1562 4833 V 1563 4831 V 1563 4829 V 1564 4827 V 1565 4825 V 1566 4823 V 1567 4821 V 1568 4819 V 1569 4817 V 1570 4816 V 1571 4814 V 1572 4812 V 1573 4810 V 1574 4808 V 1575 4806 V 1576 4804 V 1577 4802 V 1577 4801 V 1578 4799 V 1579 4797 V 1580 4795 V 1581 4793 V 1582 4791 V 1583 4789 V 1584 4788 V 1572 4842 V 1573 4840 V 1574 4838 V 1575 4836 V 1576 4835 V 1577 4833 V 1577 4831 V 1578 4829 V 1579 4827 V 1580 4825 V 1581 4823 V 1582 4821 V 1583 4819 V 1584 4817 V 1585 4816 V 1586 4814 V 1587 4812 V 1588 4810 V 1589 4808 V 1590 4806 V 1591 4804 V 1591 4802 V 1592 4801 V 1593 4799 V 1594 4797 V 1595 4795 V 1596 4793 V 1597 4791 V 1598 4789 V 1599 4788 V 1555 4797 4 12 v 1556 4793 4 8 v 1557 4791 4 6 v 1559 4789 4 4 v 1559 4788 4 2 v 1600 4842 4 12 v 1599 4842 4 8 v 1598 4842 4 6 v 1597 4842 4 4 v 1596 4842 4 2 v 1602 4810 a Fk(d)p Fm(+1)1725 4842 y Fv(\))i(with)g(distribution)d Fq(\026)p Fv(,)j(b)r(eha)n(v)n(es)f(\(for)h(large)e(\003\))i(as)1563 5022 y Fq(P)1616 5034 y Fm(\003)1666 5022 y Fv(\()p Fq(w)r Fv(\))e Fp(\030)f Fq(e)1942 4987 y Fl(\000)p Fk(i)2017 4995 y Fh(J)2057 4987 y Fm(\()p Fk(w)r Fl(j)p Fk(\026)p Fm(\))p Fl(j)p Fm(\003)p Fl(j)2308 5022 y Fq(:)836 b Fv(\(1.21\))515 5201 y(and)34 b(the)g(rate)g(function)g Fq(i)1371 5213 y Fk(J)1417 5201 y Fv(\()p Fq(w)r Fp(j)p Fq(\026)p Fv(\))h(satis\014es)e(\(1.20\).)56 b(Comparing)31 b(this)i(with)h(\(1.4\),)h(w)n(e)515 5301 y(see)j(w)n(e)g(ha)n(v)n(e)f (obtained)g(exactly)g(the)i(same)e(structure)h(as)g(in)f(the)i(Galla)n (v)n(otti-Cohen)515 5401 y(\015uctuation)26 b(theorem)h(with)g (practically)c(no)k(e\013ort.)1926 5650 y(7)p eop %%Page: 8 8 8 7 bop 515 523 a Fy(1.5)112 b(Plan.)515 676 y Fv(W)-7 b(e)40 b(\014rst)g(presen)n(t)f(the)h(\015uctuation)f(theorem)g(in)g(a) h(Gibbsian)d(con)n(text)j(without)f(to)r(o)515 776 y(m)n(uc)n(h)29 b(reference)h(to)g(an)h(underlying)d(dynamics)g(through)h(whic)n(h,)i (p)r(ossibly)-7 b(,)29 b(the)i(Gibbs)515 876 y(states)j(are)h(obtained) f(as)g(space-time)f(measures.)57 b(Y)-7 b(et,)38 b(to)d(a)n(v)n(oid)e (misunderstanding,)515 975 y(w)n(e)28 b(rep)r(eat)g(that)g(w)n(e)g (think)g(of)g(these)g(Gibbs)g(measures)e(here)i(as)f(describing)f(the)j (steady)515 1075 y(states)36 b(or)g(sym)n(b)r(olic)d(dynamics)h(for)i (some)f(spatially-extended)e(non-equilibrium)e(dy-)515 1174 y(namics.)j(They)26 b(are)g(to)h(b)r(e)g(though)n(t)f(of)g(as)g (distributions)e(for)i(the)h(space-time)d(histories.)515 1274 y(Via)g(standard)h(thermo)r(dynamic)e(relations,)g(w)n(e)i(giv)n (e)f(the)i(relation)d(b)r(et)n(w)n(een)i(the)i(action)515 1374 y(functional)22 b(satisfying)g(the)j(large)d(deviation)f (principle)h(\(\015uctuation)h(theorem\))h(and)g(the)515 1473 y(relativ)n(e)k(en)n(trop)n(y)h(b)r(et)n(w)n(een)i(the)g(forw)n (ard)f(and)g(the)i(bac)n(kw)n(ard)d(ev)n(olution.)44 b(In)31 b(particu-)515 1573 y(lar,)23 b(in)i(quadratic)d(appro)n (ximation,)f(the)k(Green-Kub)r(o)f(form)n(ula)e(app)r(ears.)35 b(Time)23 b(en)n(ters)515 1673 y(explicitly)g(in)k(Section)f(3)h(where) g(via)e(the)j(example)d(of)i(probabilistic)c(cellular)g(automata)515 1772 y(the)28 b(general)d(philosoph)n(y)f(is)j(illustrated.)515 2047 y Fr(2)134 b(Fluctuation)44 b(theorem)h(for)g(Gibbs)e(states.)515 2229 y Fv(In)29 b(the)h(presen)n(t)f(setup,)h(w)n(e)f(ha)n(v)n(e)g(no)g Fe(a)j(priori)f Fv(reason)d(to)h(prefer)g(one)g(lattice)f(direction)515 2328 y(o)n(v)n(er)23 b(another)h(and)g(w)n(e)g(\014x)h(the)g(family)d (of)j(increasing)c(cub)r(es)k(\003)2536 2340 y Fk(n)2606 2328 y Fv(of)g(side)e(length)h Fq(n)f Fp(2)g Fv(I)-14 b(N)3342 2340 y Fm(0)515 2428 y Fv(cen)n(tered)28 b(around)g(the)h (origin)c(in)j(whic)n(h)g(w)n(e)g(are)g(going)f(to)h(apply)g(the)h (transformations)515 2527 y Fq(\031)562 2539 y Fm(\003)607 2547 y Fh(n)681 2527 y Fp(\021)f Fq(\031)821 2539 y Fk(n)897 2527 y Fv(ha)n(ving)h(the)i(prop)r(erties)e(describ)r(ed)g(in)i (Section)e(1.3.)46 b(F)-7 b(or)30 b(ev)n(ery)g Fq(A)e Fp(2)h(S)37 b Fv(w)n(e)515 2627 y(write)26 b Fq(A)789 2639 y Fk(n)862 2627 y Fv(for)h(the)h(smallest)c(cub)r(e)29 b(\003)1706 2639 y Fk(n)1778 2627 y Fv(\(with)e Fq(n)c Fv(=)g Fq(n)p Fv(\()p Fq(A)p Fv(\)\))29 b(for)e(whic)n(h)f Fq(A)e Fp(\032)e Fv(\003)2992 2639 y Fk(n)3037 2627 y Fv(.)515 2859 y Fy(2.1)112 b(Symmetry)38 b(breaking)h(p)s(oten)m(tial.) 515 3013 y Fv(In)23 b(what)g(follo)n(ws)d(w)n(e)i(simply)f(set)i(\012)g (=)f(\012)1819 3025 y Fk(d)p Fm(+1)1942 3013 y Fv(.)35 b(A)24 b(p)r(oten)n(tial)d Fq(U)31 b Fv(is)22 b(a)h(real-v)-5 b(alued)19 b(function)515 3112 y(on)j Fp(S)15 b(\002)9 b Fv(\012)22 b(suc)n(h)h(that)g Fq(U)1261 3124 y Fk(A)1337 3112 y Fp(2)h(F)1476 3124 y Fk(A)1552 3112 y Fv(\(i.e.)35 b(only)21 b(dep)r(ending)h(on)h(the)g(spins)f(inside)e Fq(A)p Fv(\))k(for)e(eac)n(h)515 3212 y Fq(A)h Fp(2)g(S)31 b Fv(\(put)24 b Fq(U)996 3227 y Fl(;)1057 3212 y Fp(\021)f Fv(0\).)35 b(It)24 b(describ)r(es)e(the)i(in)n(teraction)d(b)r(et)n(w)n (een)i(the)h(spins)f(in)f(the)i(region)515 3312 y Fq(A)p Fv(.)48 b(W)-7 b(e)32 b(consider)d(a)i(family)e(of)i Fq(m)21 b Fv(+)f(1)31 b(in)n(teraction)e(p)r(oten)n(tials)g(\()p Fq(U)2710 3281 y Fk(\013)2701 3334 y(A)2757 3312 y Fv(\))2789 3324 y Fk(A)2843 3312 y Fq(;)14 b(\013)30 b Fv(=)f(0)p Fq(;)14 b(:)g(:)g(:)f(;)h(m)p Fv(.)515 3411 y(W)-7 b(e)28 b(assume)e(translation-in)n(v)-5 b(ari)o(ance,)21 b(meaning)k(that)1576 3594 y Fq(U)1642 3560 y Fk(\013)1633 3614 y(A)1689 3594 y Fv(\()p Fq(\021)s Fv(\))f(=)e Fq(U)1974 3560 y Fk(\013)1965 3614 y(A)p Fm(+)p Fk(x)2108 3594 y Fv(\()p Fq(\034)2176 3606 y Fk(x)2218 3594 y Fq(\021)s Fv(\))p Fq(;)850 b Fv(\(2.22\))515 3777 y(for)21 b(all)e Fq(A)k Fp(2)g(S)6 b Fq(;)14 b(x)24 b Fp(2)p 1165 3779 45 4 v 1165 3722 43 4 v 1165 3777 3 3 v 1165 3775 V 1166 3773 V 1167 3771 V 1168 3770 V 1169 3768 V 1170 3766 V 1171 3764 V 1172 3762 V 1173 3760 V 1174 3758 V 1175 3756 V 1176 3754 V 1177 3752 V 1178 3750 V 1179 3749 V 1179 3747 V 1180 3745 V 1181 3743 V 1182 3741 V 1183 3739 V 1184 3737 V 1185 3735 V 1186 3734 V 1187 3732 V 1188 3730 V 1189 3728 V 1190 3726 V 1191 3724 V 1192 3722 V 1179 3777 V 1180 3775 V 1181 3773 V 1182 3771 V 1183 3770 V 1184 3768 V 1185 3766 V 1186 3764 V 1187 3762 V 1188 3760 V 1189 3758 V 1190 3756 V 1191 3754 V 1192 3752 V 1193 3750 V 1193 3749 V 1194 3747 V 1195 3745 V 1196 3743 V 1197 3741 V 1198 3739 V 1199 3737 V 1200 3735 V 1201 3734 V 1202 3732 V 1203 3730 V 1204 3728 V 1205 3726 V 1206 3724 V 1207 3722 V 1163 3732 4 12 v 1164 3728 4 8 v 1165 3726 4 6 v 1166 3724 4 4 v 1167 3722 4 2 v 1208 3777 4 12 v 1207 3777 4 8 v 1205 3777 4 6 v 1204 3777 4 4 v 1204 3777 4 2 v 1209 3745 a Fk(d)p Fm(+1)1332 3777 y Fq(;)14 b(\021)26 b Fp(2)e Fv(\012.)35 b(As)21 b(usual)f(w)n(e)h(also)e(tak)n(e)i(it)g(that)g(the)h(total)e(in)n (teraction)515 3876 y(of)32 b(a)g(\014nite)g(region)e(with)i(the)h (rest)f(of)g(the)h(lattice)d(is)i(\014nite,)h(i.e.)50 b(w)n(e)32 b(assume)f(that)i(the)515 3976 y(p)r(oten)n(tial)25 b(is)i(uniformly)d(absolutely)h(summable:)1674 4092 y Fj(X)1670 4270 y Fk(A)p Fl(3)p Fm(0)1811 4170 y Fp(k)p Fq(U)1919 4136 y Fk(\013)1910 4191 y(A)1966 4170 y Fp(k)d Fq(<)h Fp(1)p Fq(:)943 b Fv(\(2.23\))515 4436 y(\(This)34 b(assumption)e(of)j(uniformit)n(y)d(is)h(not)i(strictly)e(needed)i(but) g(it)f(a)n(v)n(oids)e(irrelev)-5 b(an)n(t)515 4536 y(tec)n (hnicalities.)35 b(Similarly)-7 b(,)23 b(hard)28 b(core)g(in)n (teractions)d(are)i(also)g(not)h(excluded)g(but)h(extra)515 4635 y(care)i(and)h(assumptions)e(w)n(ould)g(b)r(e)j(needed.\))52 b(Giv)n(en)31 b(the)h(family)e(of)i(transformations)515 4735 y Fq(\031)562 4747 y Fk(n)607 4735 y Fv(,)c(w)n(e)f(de\014ne)h (the)g(relativ)n(e)c(energies)1199 4929 y Fq(R)1263 4895 y Fk(\013)1262 4950 y(n)1333 4929 y Fp(\021)1494 4851 y Fj(X)1421 5033 y Fk(A)p Fl(\\)p Fm(\003)1561 5041 y Fh(n)1602 5033 y Fl(6)p Fm(=)p Fl(;)1687 4929 y Fv(\()p Fq(\031)1766 4941 y Fk(n)1812 4929 y Fq(U)1878 4895 y Fk(\013)1869 4950 y(A)1943 4929 y Fp(\000)18 b Fq(U)2092 4895 y Fk(\013)2083 4950 y(A)2139 4929 y Fv(\))p Fq(;)c(\013)24 b Fv(=)e(0)p Fq(;)14 b(:)g(:)g(:)g(;)g(m:)472 b Fv(\(2.24\))515 5211 y(W)-7 b(e)28 b(mak)n(e)e(a)i(di\013erence)f(b)r(et)n(w)n(een)h (the)g(p)r(oten)n(tial)e Fq(U)2198 5181 y Fm(0)2263 5211 y Fv(and)i(the)g Fq(U)2634 5181 y Fk(\013)2681 5211 y Fq(;)14 b(\013)24 b Fv(=)f(1)p Fq(;)14 b(:)g(:)g(:)g(;)g(m)27 b Fv(from)515 5311 y(their)j(b)r(eha)n(vior)e(under)j(the)h Fq(\031)1492 5323 y Fk(n)1537 5311 y Fv(.)48 b(W)-7 b(e)31 b(assume)e(that)j Fq(U)2294 5281 y Fm(0)2362 5311 y Fv(is)e(in)n(v)-5 b(arian)n(t)28 b(under)j(the)g Fq(\031)3233 5323 y Fk(n)3310 5311 y Fv(in)1926 5650 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fv(the)28 b(sense)f(that)h Fq(\031)1098 535 y Fk(n)1143 523 y Fq(U)1209 493 y Fm(0)1200 546 y Fk(A)1277 523 y Fv(=)23 b Fq(U)1431 493 y Fm(0)1422 546 y Fk(A)1503 523 y Fv(whenev)n(er)k Fq(n)c Fp(\025)f Fq(n)p Fv(\()p Fq(A)p Fv(\))29 b(implying)23 b(that)1689 761 y(lim)1726 811 y Fk(n)1828 705 y Fp(k)p Fq(R)1934 675 y Fm(0)1933 725 y Fk(n)1978 705 y Fp(k)p 1828 742 192 4 v 1849 818 a(j)p Fv(\003)1930 830 y Fk(n)1975 818 y Fp(j)2052 761 y Fv(=)g(0)p Fq(:)962 b Fv(\(2.25\))515 988 y(The)26 b(reason)e(for)h(taking)f Fq(m)f(>)g Fv(1)i(is)g(to)h (allo)n(w)c(for)k(and)f(to)h(distinguish)d(b)r(et)n(w)n(een)i(p)r (ossibly)515 1087 y(di\013eren)n(t)c(mec)n(hanisms)e(for)j(breaking)e (the)j(symmetry)d(of)j(the)g(reference)e(in)n(teraction)f Fq(U)3319 1057 y Fm(0)3356 1087 y Fv(.)639 1187 y(W)-7 b(e)19 b(de\014ne)g(the)f(curren)n(t)g(asso)r(ciated)e(to)i(the)h (symmetry)d(breaking)g(in)n(teraction)f Fq(U)3212 1157 y Fk(\013)3259 1187 y Fq(;)f(\013)24 b Fv(=)515 1287 y(1)p Fq(;)14 b(:)g(:)g(:)f(;)h(m)27 b Fv(to)h(b)r(e)1245 1421 y Fq(J)1299 1386 y Fk(\013)1291 1441 y(x)1369 1421 y Fp(\021)23 b Fv(lim)1494 1470 y Fk(n)1661 1342 y Fj(X)1586 1520 y Fk(x)p Fl(2)p Fk(A)p Fl(\032)p Fm(\003)1816 1528 y Fh(n)1934 1364 y Fv(1)p 1880 1402 150 4 v 1880 1478 a(2)p Fp(j)p Fq(A)p Fp(j)2040 1421 y Fv(\()p Fq(\031)2119 1436 y Fk(n)p Fm(\()p Fk(A)p Fm(\))2266 1421 y Fq(U)2332 1386 y Fk(\013)2323 1441 y(A)2398 1421 y Fp(\000)18 b Fq(U)2547 1386 y Fk(\013)2538 1441 y(A)2594 1421 y Fv(\))p Fq(:)518 b Fv(\(2.26\))515 1657 y Fq(J)569 1627 y Fk(\013)561 1677 y(x)642 1657 y Fv(is)25 b(a)h(con)n(tin)n(uous)e(function)i(on)g (\012)g(and,)h(from)e(\(2.22\),)g Fq(J)2424 1627 y Fk(\013)2416 1677 y(x)2472 1657 y Fv(\()p Fq(\021)s Fv(\))f(=)e Fq(J)2745 1627 y Fk(\013)2737 1677 y Fm(0)2792 1657 y Fv(\()p Fq(\034)2860 1669 y Fk(x)2903 1657 y Fq(\021)s Fv(\).)37 b(The)26 b(term)515 1756 y(`curren)n(t')19 b(is)g(suggestiv)n(e)e(for)j(in)n (terpreting)d(\(2.26\))i(as)g(the)h(real)e(curren)n(t)i(at)f(the)i (space-time)515 1856 y(p)r(oin)n(t)f Fq(x)j Fv(asso)r(ciated)c(to)i (some)f(driving)f(of)i(a)h(reference)e(steady)h(state)h Fq(\027)k Fv(thereb)n(y)21 b(breaking)515 1956 y(the)33 b(time-rev)n(ersal)c(symmetry)h(in)j(the)h(case)e(of)h(Example)e(1.1,)j (see)f(next)g(section.)53 b(W)-7 b(e)515 2055 y(tak)n(e)28 b Fq(\027)35 b Fv(to)29 b(b)r(e)h(a)e(Gibbs)h(state)g(with)g(resp)r (ect)g(to)g(the)g(in)n(teraction)e Fq(U)2718 2025 y Fm(0)2755 2055 y Fv(,)i(i.e.)41 b(with)29 b(formal)515 2155 y(Hamiltonian)1701 2255 y Fq(H)1777 2220 y Fm(0)1837 2255 y Fp(\021)1925 2176 y Fj(X)1960 2354 y Fk(A)2059 2255 y Fq(U)2125 2220 y Fm(0)2116 2275 y Fk(A)2169 2255 y Fq(;)975 b Fv(\(2.27\))515 2483 y(\(see)27 b(\(1.9\)\))h(for)f(whic)n(h)f(the)i Fq(\031)s Fp(\000)p Fv(symmetry)d(is)i(un)n(brok)n(en:)1751 2665 y Fq(\027)c Fp(\016)18 b Fq(\031)1922 2677 y Fk(n)1991 2665 y Fv(=)k Fq(\027)q(:)1024 b Fv(\(2.28\))515 2848 y(As)27 b(a)h(consequence,)e(the)i(curren)n(ts)f(\(2.26\))g(v)-5 b(anish)26 b(iden)n(tically)d(in)k(that)h(state:)1503 3030 y Fq(\027)5 b Fv(\()p Fq(J)1635 2996 y Fk(\013)1627 3051 y(x)1683 3030 y Fv(\))23 b(=)g(0)p Fq(;)14 b(\013)23 b Fv(=)g(1)p Fq(;)14 b(:)g(:)g(:)f(;)h(m:)776 b Fv(\(2.29\))515 3213 y(The)37 b(p)r(erturb)r(ed)h(or)f(driv)n(en)f(state)h(is)g (denoted)h(b)n(y)f Fq(\026)p Fv(.)67 b(It)38 b(is)f(a)g(translation-in) n(v)-5 b(ari)o(an)n(t)515 3313 y(Gibbs)27 b(state)g(at)g(in)n(v)n(erse) f(temp)r(erature)g Fq(\014)32 b Fv(with)27 b(resp)r(ect)g(to)h(the)g (formal)d(Hamiltonian)1546 3557 y Fq(H)30 b Fp(\021)23 b Fq(H)1809 3523 y Fm(0)1864 3557 y Fv(+)1981 3454 y Fk(m)1951 3478 y Fj(X)1947 3654 y Fk(\013)p Fm(=1)2088 3557 y Fq(E)2154 3523 y Fk(\013)2201 3557 y Fq(H)2277 3523 y Fk(\013)2325 3557 y Fq(;)819 b Fv(\(2.30\))515 3816 y(where)33 b(the)g Fq(H)985 3786 y Fk(\013)1066 3816 y Fv(are)f(built)h(\(as)g(in)f(\(2.27\)\))h(from)f(the)i(in)n (teraction)d(p)r(oten)n(tials)g Fq(U)3165 3786 y Fk(\013)3245 3816 y Fv(and)515 3916 y(where)40 b(the)h Fq(E)990 3886 y Fk(\013)1077 3916 y Fv(are)f(real)f(n)n(um)n(b)r(ers)f (parameterizing)e(the)41 b(strength)f(of)g(a)h(symmetry)515 4015 y(breaking)21 b(or)h(driving)e(force.)35 b(As)23 b(b)r(efore,)h(in)f(the)g(de\014nition)f(of)h(Gibbs)f(states,)i(w)n(e)f (alw)n(a)n(ys)515 4115 y(tak)n(e)k(the)h(normalized)23 b(coun)n(ting)j(measure)g(as)h Fe(a)j(priori)f Fv(measure,)d(see)h (\(1.9\).)515 4347 y Fy(2.2)112 b(Fluctuation)38 b(theorem.)515 4501 y Fs(Theorem)30 b(2.1)41 b Fe(Supp)l(ose)i(that)h Fq(\026)f Fe(is)h(a)f(tr)l(anslation-invariant)h(Gibbs)h(state)e(for)h (the)515 4600 y(tr)l(anslation-invariant)33 b(p)l(otential)h Fv(\()p Fq(U)1726 4612 y Fk(A)1808 4600 y Fv(=)28 b Fq(U)1967 4570 y Fm(0)1958 4623 y Fk(A)2033 4600 y Fv(+)2118 4538 y Fj(P)2206 4559 y Fk(m)2206 4625 y(\013)p Fm(=1)2351 4600 y Fq(E)2417 4570 y Fk(\013)2464 4600 y Fq(U)2530 4570 y Fk(\013)2521 4623 y(A)2577 4600 y Fv(\))2609 4612 y Fk(A)2696 4600 y Fe(as)33 b(in)g(the)g(pr)l(e)l(c)l(e)l(ding)515 4700 y(subse)l(ction.)38 b(The)31 b(limit)1089 4901 y Fq(p)p Fv(\()p Fq(\025;)14 b(E)5 b Fv(\))24 b Fp(\021)e(\000)14 b Fv(lim)1573 4951 y Fk(n)1729 4845 y Fv(1)p 1675 4882 149 4 v 1675 4958 a Fp(j)p Fv(\003)1756 4970 y Fk(n)1801 4958 y Fp(j)1848 4901 y Fv(ln)f Fq(\026)p Fv([)p Fq(e)2043 4853 y Fl(\000)p Fk(\014)2146 4796 y Fj(P)2234 4884 y Fh(x)p Ff(2)p Fg(\003)2344 4892 y Fh(n)2400 4796 y Fj(P)2488 4817 y Fh(m)2488 4884 y(\013)p Fg(=1)2612 4853 y Fk(\025)2651 4861 y Fh(\013)2693 4853 y Fk(J)2735 4827 y Fh(\013)2730 4869 y(x)2782 4901 y Fv(])362 b(\(2.31\))515 5128 y Fe(exists)29 b(and)h(satis\014es)1529 5227 y Fq(p)p Fv(\()p Fq(\025;)14 b(E)5 b Fv(\))24 b(=)f Fq(p)p Fv(\(2)p Fq(E)g Fp(\000)18 b Fq(\025;)c(E)5 b Fv(\))803 b(\(2.32\))515 5377 y Fe(for)30 b(every)h Fq(\025)23 b Fp(\021)g Fv(\()p Fq(\025)1106 5389 y Fm(1)1144 5377 y Fq(;)14 b(:)g(:)g(:)g(;)g(\025)1377 5389 y Fk(m)1440 5377 y Fv(\))30 b Fe(and)g Fq(E)e Fp(\021)23 b Fv(\()p Fq(E)1938 5347 y Fm(1)1976 5377 y Fq(;)14 b(:)g(:)g(:)f(;)h (E)2226 5347 y Fk(m)2289 5377 y Fv(\))24 b Fp(2)f Fv(I)-14 b(R)2500 5340 y Fk(m)2563 5377 y Fe(.)1926 5650 y Fv(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fs(Pro)s(of:)39 b Fv(The)29 b(existence)e(of)i(the)h (limit)25 b(is)j(a)h(standard)f(result)f(of)i(the)h(Gibbs)e(formalism,) 515 623 y(see)k(e.g.)52 b([12)o(,)33 b(4,)g(27)o(].)52 b(As)33 b(announced)f(via)g(\(1.11\))g(the)h(main)e(observ)-5 b(ation)30 b(leading)g(to)515 722 y(\(2.32\))d(is)f(that)1145 916 y Fq(\026)p Fv(\()p Fq(\031)1274 928 y Fk(n)1320 916 y Fq(f)9 b Fv(\))23 b(=)g Fq(\026)p Fv(\(exp[)p Fp(\000)p Fq(\014)1948 838 y Fj(X)1875 1020 y Fk(A)p Fl(\\)p Fm(\003)2015 1028 y Fh(n)2055 1020 y Fl(6)p Fm(=)p Fl(;)2140 916 y Fv(\()p Fq(\031)2219 928 y Fk(n)2265 916 y Fq(U)2322 928 y Fk(A)2394 916 y Fp(\000)18 b Fq(U)2534 928 y Fk(A)2588 916 y Fv(\)])p Fq(f)9 b Fv(\))p Fq(;)419 b Fv(\(2.33\))515 1188 y(simply)31 b(b)r(ecause)k Fq(\026)f Fv(is)g(a)g(Gibbs)g(state)g (for)g(the)h(p)r(oten)n(tial)e(\()p Fq(U)2542 1200 y Fk(A)2596 1188 y Fv(\))i(at)f(in)n(v)n(erse)f(temp)r(era-)515 1287 y(ture)d Fq(\014)t Fv(.)47 b(Therefore,)30 b(taking)f(n)n(um)n(b)r (ers)h Fq(h)1863 1299 y Fk(\013)1910 1287 y Fq(;)14 b(\013)28 b Fv(=)g(1)p Fq(;)14 b(:)g(:)g(:)f(;)h(m)31 b Fv(and)f Fq(f)37 b Fv(=)28 b(exp[)p Fq(\014)3001 1225 y Fj(P)3089 1245 y Fk(m)3089 1312 y(\013)p Fm(=1)3220 1287 y Fv(\(1)20 b Fp(\000)515 1387 y Fq(h)563 1399 y Fk(\013)610 1387 y Fv(\))p Fq(E)708 1357 y Fk(\013)756 1387 y Fq(R)820 1357 y Fk(\013)819 1407 y(n)867 1387 y Fv(])27 b(in)g(that)h(form)n (ula,)1000 1631 y Fq(\026)p Fv(\(exp[)p Fp(\000)p Fq(\014)t(R)1412 1597 y Fm(0)1411 1651 y Fk(n)1474 1631 y Fp(\000)18 b Fq(\014)1657 1527 y Fk(m)1626 1552 y Fj(X)1623 1728 y Fk(\013)p Fm(=1)1764 1631 y Fq(h)1812 1643 y Fk(\013)1859 1631 y Fq(E)1925 1597 y Fk(\013)1972 1631 y Fq(R)2036 1597 y Fk(\013)2035 1651 y(n)2083 1631 y Fv(]\))24 b(=)1000 1906 y Fq(\026)p Fv(\(exp[)p Fp(\000)p Fq(\014)t(R)1412 1872 y Fm(0)1411 1927 y Fk(n)1474 1906 y Fp(\000)18 b Fq(\014)1657 1803 y Fk(m)1626 1828 y Fj(X)1623 2003 y Fk(\013)p Fm(=1)1764 1906 y Fq(E)1830 1872 y Fk(\013)1877 1906 y Fq(R)1941 1872 y Fk(\013)1940 1927 y(n)1988 1906 y Fv(])c(exp[)p Fq(\014)2274 1803 y Fk(m)2244 1828 y Fj(X)2240 2003 y Fk(\013)p Fm(=1)2367 1906 y Fv(\(1)k Fp(\000)g Fq(h)2590 1918 y Fk(\013)2638 1906 y Fv(\))p Fq(E)2736 1872 y Fk(\013)2783 1906 y Fq(R)2847 1872 y Fk(\013)2846 1927 y(n)2894 1906 y Fv(]\))24 b(=)1000 2182 y Fq(\026)p Fv(\(exp[)p Fp(\000)p Fq(\014)1396 2078 y Fk(m)1365 2103 y Fj(X)1361 2279 y Fk(\013)p Fm(=1)1489 2182 y Fv(\(1)18 b Fp(\000)g Fq(h)1712 2194 y Fk(\013)1759 2182 y Fv(\))p Fq(E)1857 2148 y Fk(\013)1905 2182 y Fq(R)1969 2148 y Fk(\013)1968 2203 y(n)2016 2182 y Fv(]\))p Fq(:)1073 b Fv(\(2.34\))515 2430 y(No)n(w,)1491 2542 y Fq(R)1555 2508 y Fk(\013)1554 2562 y(n)1625 2542 y Fv(=)23 b(2)1793 2463 y Fj(X)1769 2641 y Fk(x)p Fl(2)p Fm(\003)1897 2649 y Fh(n)1951 2542 y Fq(J)2005 2508 y Fk(\013)1997 2562 y(x)2070 2542 y Fp(\000)18 b Fq(I)2189 2554 y Fm(1)2246 2542 y Fv(+)g Fq(I)2365 2554 y Fm(2)3167 2542 y Fv(\(2.35\))515 2770 y(where)27 b(b)r(oth)1190 2888 y Fq(I)1226 2900 y Fm(1)1287 2888 y Fp(\021)1399 2809 y Fj(X)1375 2988 y Fk(x)p Fl(2)p Fm(\003)1503 2996 y Fh(n)1706 2809 y Fj(X)1557 2991 y Fk(A)p Fl(3)p Fk(x;A)p Fl(\\)p Fm(\003)1850 2975 y Fh(c)1850 3008 y(n)1890 2991 y Fl(6)p Fm(=)p Fl(;)2032 2832 y Fv(1)p 1999 2869 109 4 v 1999 2945 a Fp(j)p Fq(A)p Fp(j)2117 2888 y Fv(\()p Fq(\031)2196 2903 y Fk(n)p Fm(\()p Fk(A)p Fm(\))2344 2888 y Fq(U)2410 2854 y Fk(\013)2401 2909 y(A)2475 2888 y Fp(\000)18 b Fq(U)2624 2854 y Fk(\013)2615 2909 y(A)2671 2888 y Fv(\))464 b(\(2.36\))515 3129 y(and)1379 3229 y Fq(I)1415 3241 y Fm(2)1476 3229 y Fp(\021)1737 3150 y Fj(X)1564 3332 y Fk(A)p Fl(\\)p Fm(\003)1704 3340 y Fh(n)1744 3332 y Fk(;A)p Fl(\\)p Fm(\003)1904 3315 y Fh(c)1904 3348 y(n)1945 3332 y Fl(6)p Fm(=)p Fl(;)2030 3229 y Fv(\()p Fq(\031)2109 3241 y Fk(n)2155 3229 y Fq(U)2221 3194 y Fk(\013)2212 3249 y(A)2286 3229 y Fp(\000)18 b Fq(U)2435 3194 y Fk(\013)2426 3249 y(A)2482 3229 y Fv(\))653 b(\(2.37\))515 3474 y(are)23 b(small)e(of)j(order)f Fq(o)p Fv(\()p Fp(j)p Fv(\003)1322 3486 y Fk(n)1367 3474 y Fp(j)p Fv(\))i(b)r(ecause)f(of)g(\(2.23\):)35 b Fp(jj)p Fq(I)2195 3486 y Fk(i)2223 3474 y Fp(jj)p Fq(=)p Fp(j)p Fv(\003)2392 3486 y Fk(n)2436 3474 y Fp(j)23 b(!)h Fv(0)f(as)h Fq(n)g Fv(go)r(es)f(to)i(in\014nit)n(y)-7 b(,)515 3574 y Fq(i)24 b Fv(=)h(1)p Fq(;)14 b Fv(2.)40 b(Up)r(on)29 b(inserting)d(\(2.35\))j (in)n(to)e(\(2.34\))h(and)h(taking)e Fq(h)2532 3586 y Fk(\013)2604 3574 y Fv(=)e Fq(\025)2742 3586 y Fk(\013)2790 3574 y Fq(=)p Fv(2)p Fq(E)2940 3543 y Fk(\013)3015 3574 y Fv(\(for)k Fq(E)3242 3543 y Fk(\013)3314 3574 y Fp(6)p Fv(=)515 3673 y(0\),)e(w)n(e)g(get)1130 3778 y(1)p 1077 3816 149 4 v 1077 3892 a Fp(j)p Fv(\003)1158 3904 y Fk(n)1203 3892 y Fp(j)1236 3835 y(j)14 b Fv(ln)1519 3769 y Fq(\026)p Fv(\(exp[)p Fp(\000)p Fq(\014)1881 3707 y Fj(P)1969 3794 y Fk(x)p Fl(2)p Fm(\003)2097 3802 y Fh(n)2155 3707 y Fj(P)2243 3727 y Fk(m)2243 3794 y(\013)p Fm(=1)2388 3769 y Fq(\025)2436 3781 y Fk(\013)2484 3769 y Fq(J)2538 3739 y Fk(\013)2530 3790 y(x)2585 3769 y Fv(]\))p 1365 3816 1429 4 v 1365 3892 a Fq(\026)p Fv(\(exp)q([)p Fp(\000)p Fq(\014)1727 3830 y Fj(P)1815 3917 y Fk(x)p Fl(2)p Fm(\003)1943 3925 y Fh(n)2002 3830 y Fj(P)2089 3851 y Fk(m)2089 3917 y(\013)p Fm(=1)2221 3892 y Fv(\(2)p Fq(E)2361 3868 y Fk(\013)2426 3892 y Fp(\000)k Fq(\025)2557 3904 y Fk(\013)2605 3892 y Fv(\))p Fq(J)2691 3868 y Fk(\013)2683 3913 y(x)2739 3892 y Fv(]\))2804 3835 y Fp(j)340 b Fv(\(2.38\))515 4038 y(going)25 b(to)j(zero)e(as)h Fq(n)c Fp(")f(1)p Fv(.)37 b(This)27 b(is)g(exactly)e(what)j(w)n(as)f(needed.)p 2631 4038 48 48 v 515 4186 a Fs(Remark)36 b(1:)45 b Fv(Gibbs)31 b(states)h(satisfy)e(a)i(large)e(deviation)f(principle,)h(see)i(e.g.)50 b([17)o(])32 b(and)515 4285 y([4)o(])24 b(for)f(additional)d (references.)35 b(As)23 b(a)g(result,)h(\(2.32\))e(implies)e (\(1.21\)-\(1.20\).)34 b(W)-7 b(e)24 b(do)f(not)515 4385 y(add)k(a)g(more)f(precise)g(form)n(ulation)e(here.)515 4484 y Fs(Remark)34 b(2:)43 b Fv(Related)30 b(to)h(this,)g(as)g(is)f (clear)e(from)i(the)h(pro)r(of,)h(the)f(essen)n(tial)d(prop)r(ert)n(y) 515 4593 y(is)h(that)i(the)h(functionals)c Fp(f)p Fv(log)1529 4552 y Fk(d)p Fm(\()p Fk(\026)p Fl(\016)p Fk(\031)1703 4560 y Fg(\003)1745 4552 y Fm(\))p 1529 4574 242 4 v 1612 4621 a Fk(d\026)1809 4593 y Fv(:)g(\003)g Fp(2)h(S)6 b(g)31 b Fv(satisfy)e(a)i(large)d(deviation)g(principle)515 4703 y(under)i Fq(\026)p Fv(.)47 b(W)-7 b(e)31 b(sp)r(eak)f(ab)r(out)h (the)g(\(somewhat)e(more)g(restricted\))g(Gibbs)h(prop)r(ert)n(y)g(b)r (e-)515 4803 y(cause,)23 b(in)g(all)e(cases)i(w)n(e)g(ha)n(v)n(e)f(in)h (mind,)g(the)h(large)d(deviations)g(arise)g(from)i(Gibbsianness)515 4903 y(of)k(the)h(random)e(\014eld.)515 5002 y Fs(Remark)43 b(3:)58 b Fv(The)39 b(theorem)e(ab)r(o)n(v)n(e)g(pro)n(vides)f(a)i Fs(lo)s(cal)g Fv(v)n(ersion)e(of)i(the)h(\015uctuation)515 5102 y(theorem)32 b(since)g(the)i(measure)e Fq(\026)h Fv(liv)n(es)e(on)j(a)f(m)n(uc)n(h)f(larger)f(\(in)i(fact,)i (in\014nite\))e(v)n(olume)515 5201 y(than)25 b(the)g(size)f(of)g(the)i (observ)-5 b(ation)22 b(windo)n(w)i(\003)2029 5213 y Fk(n)2073 5201 y Fv(.)36 b(The)25 b(relations)d(\(1.14\))i(and)h (\(2.34\))f(are)515 5301 y(iden)n(tities)h(exactly)i(v)n(eri\014ed)f (for)h(the)i(\014nite)f(v)n(olumes)d(\003)2312 5313 y Fk(n)2357 5301 y Fv(.)38 b(This)27 b(is)g(similar)c(to)28 b(the)h(lo)r(cal)515 5401 y(\015uctuation)24 b(theorem)g(of)h([8)o(].) 37 b(Notice)24 b(also)f(that)j(the)f(limit)d Fq(p)p Fv(\()p Fq(\025;)14 b(E)5 b Fv(\))26 b(exists)e(and)h(remains)1905 5650 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fv(unc)n(hanged)36 b(if)g(instead)g(of)g(taking)f (the)j(sequence)e(of)g(cub)r(es)h(\003)2591 535 y Fk(n)2673 523 y Fv(w)n(e)f(tak)n(e)g(v)n(olumes)e(\003)515 623 y(gro)n(wing)k(to)p 969 625 45 4 v 969 568 43 4 v 969 623 3 3 v 970 621 V 971 619 V 972 617 V 973 616 V 973 614 V 974 612 V 975 610 V 976 608 V 977 606 V 978 604 V 979 602 V 980 600 V 981 598 V 982 597 V 983 595 V 984 593 V 985 591 V 986 589 V 987 587 V 987 585 V 988 583 V 989 582 V 990 580 V 991 578 V 992 576 V 993 574 V 994 572 V 995 570 V 996 569 V 984 623 V 985 621 V 986 619 V 987 617 V 987 616 V 988 614 V 989 612 V 990 610 V 991 608 V 992 606 V 993 604 V 994 602 V 995 600 V 996 598 V 997 597 V 998 595 V 999 593 V 1000 591 V 1001 589 V 1001 587 V 1002 585 V 1003 583 V 1004 582 V 1005 580 V 1006 578 V 1007 576 V 1008 574 V 1009 572 V 1010 570 V 1011 569 V 967 578 4 12 v 968 574 4 8 v 969 572 4 6 v 970 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Fq(:)790 b Fv(\(2.40\))515 2205 y(Suc)n(h)36 b(a)h(scaling)c(is)j(applied)e(in)i(the)h(study)g(of)g(large)d (deviations)g(in)i(the)h(phase)f(co)r(ex-)515 2304 y(istence)f(regime)e (where)j(the)g(probabilit)n(y)c(of)k(a)g(droplet)f(of)h(the)g(wrong)f (phase)h(is)e(only)515 2404 y(exp)r(onen)n(tially)24 b(small)g(in)j(the)h(surface)f(of)g(that)h(droplet.)515 2635 y Fy(2.3)112 b(Thermo)s(dynamic)40 b(relation.)515 2789 y Fv(As)19 b(men)n(tioned)e(in)i(the)g(in)n(tro)r(duction,)g(the)g (original)c(con)n(text)k(of)g(the)g(\015uctuation)g(theorem)515 2888 y(concerned)32 b(the)i(large)d(deviations)g(in)i(the)h(en)n(trop)n (y)e(pro)r(duction)g(rate)h(of)h(a)f(dynamical)515 2988 y(system.)55 b(Since)33 b(w)n(e)h(ha)n(v)n(e)e(not)i(sp)r(eci\014ed)g (an)n(y)f(dynamics)e(here,)36 b(w)n(e)d(m)n(ust)g(p)r(ostp)r(one)h(a) 515 3088 y(related)c(discussion)f(to)i(the)h(next)g(section.)48 b(Y)-7 b(et,)33 b(w)n(e)e(can)g(compare)f(with)h(the)h(thermo-)515 3187 y(dynamic)25 b(p)r(oten)n(tials.)639 3287 y(T)-7 b(o)27 b(start)h(de\014ne)f(the)h(energy)f(function)1642 3503 y(\010)1702 3515 y Fm(0)1739 3503 y Fv(\()p Fq(U)9 b Fv(\))23 b Fp(\021)1984 3424 y Fj(X)1980 3602 y Fk(A)p Fl(3)p Fm(0)2131 3447 y Fq(U)2188 3459 y Fk(A)p 2131 3484 111 4 v 2132 3560 a Fp(j)p Fq(A)p Fp(j)3167 3503 y Fv(\(2.41\))515 3764 y(and)k(its)g(translations)d(\010)1305 3776 y Fk(x)1346 3764 y Fv(\()p Fq(U)9 b Fv(\)\()p Fq(\021)s Fv(\))25 b(=)d(\010)1756 3776 y Fm(0)1793 3764 y Fv(\()p Fq(U)9 b Fv(\)\()p Fq(\034)1991 3776 y Fk(x)2034 3764 y Fq(\021)s Fv(\).)37 b(W)-7 b(e)28 b(de\014ne)g(the)g(free)f(energy)f (densit)n(y)515 3864 y(for)h(the)h(in)n(teraction)c Fq(U)37 b Fv(as)1089 4060 y Fq(P)12 b Fv(\()p Fq(U)d Fv(\))23 b Fp(\021)g Fv(lim)1432 4110 y Fk(n)1588 4004 y Fv(1)p 1534 4041 149 4 v 1534 4117 a Fp(j)p Fv(\003)1615 4129 y Fk(n)1660 4117 y Fp(j)1707 4060 y Fv(ln)1838 3981 y Fj(X)1790 4159 y Fk(\033)r Fl(2)p Fm(\012)1922 4167 y Fg(\003)1960 4175 y Fh(n)2019 4060 y Fv(exp[)p Fp(\000)p Fq(\014)2333 3981 y Fj(X)2299 4159 y Fk(A)p Fl(\032)p Fm(\003)2446 4167 y Fh(n)2500 4060 y Fq(U)2557 4072 y Fk(A)2611 4060 y Fv(\()p Fq(\033)s Fv(\)]\))p Fq(:)364 b Fv(\(2.42\))515 4329 y(This)26 b(coincides)f(with)1469 4447 y Fq(P)12 b Fv(\()p Fq(U)d Fv(\))23 b(=)f(lim)1812 4497 y Fk(n)1967 4391 y Fv(1)p 1914 4428 V 1914 4504 a Fp(j)p Fv(\003)1995 4516 y Fk(n)2039 4504 y Fp(j)2086 4447 y Fv(ln)13 b Fq(Z)2232 4407 y Fk(\014)2226 4471 y Fm(\003)2271 4479 y Fh(n)2316 4447 y Fv(\()p Fq(\021)s Fv(\))743 b(\(2.43\))515 4642 y(of)27 b(\(1.10\))g(for)g(all)f(b)r (oundary)g(conditions)f Fq(\021)s Fv(.)639 4742 y(Finally)-7 b(,)23 b(the)j(en)n(trop)n(y)d(densit)n(y)h(of)i(a)e(translation-in)n (v)-5 b(aria)o(n)n(t)20 b(probabilit)n(y)h(measure)i Fq(\026)515 4842 y Fv(is)1189 4960 y Fq(s)p Fv(\()p Fq(\026)p Fv(\))g Fp(\021)g(\000)14 b Fv(lim)1568 5009 y Fk(n)1724 4903 y Fv(1)p 1670 4940 V 1670 5016 a Fp(j)p Fv(\003)1751 5028 y Fk(n)1796 5016 y Fp(j)1891 4881 y Fj(X)1843 5059 y Fk(\033)r Fl(2)p Fm(\012)1975 5067 y 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Fv(lim)1541 700 y Fk(n)1696 594 y Fv(1)p 1643 631 149 4 v 1643 707 a Fp(j)p Fv(\003)1724 719 y Fk(n)1769 707 y Fp(j)1863 571 y Fj(X)1816 749 y Fk(\033)r Fl(2)p Fm(\012)1948 757 y Fg(\003)1986 765 y Fh(n)2045 650 y Fq(\026)2095 662 y Fk(n)2140 650 y Fv([)p Fq(\033)s Fv(])14 b(ln)2343 594 y Fq(\026)2393 606 y Fk(n)2439 594 y Fv([)p Fq(\033)s Fv(])p 2343 631 192 4 v 2347 707 a Fq(\032)2390 719 y Fk(n)2435 707 y Fv([)p Fq(\033)s Fv(])2568 650 y Fp(\025)23 b Fv(0)p Fq(:)446 b Fv(\(2.45\))515 932 y(If)26 b Fq(\026)g Fv(is)f(a)h(translation-in)n(v)-5 b(aria)o(n)n(t)20 b(Gibbs)26 b(measure)e(\(at)i(in)n(v)n(erse)e(temp)r(erature)g Fq(\014)t Fv(\))j(for)f(the)515 1031 y(in)n(teraction)e Fq(U)9 b Fv(,)28 b(then)1458 1131 y Fq(P)12 b Fv(\()p Fq(U)d Fv(\))23 b(=)g Fq(s)p Fv(\()p Fq(\026)p Fv(\))c Fp(\000)f Fq(\014)t(\026)p Fv(\(\010)2212 1143 y Fm(0)2250 1131 y Fv(\()p Fq(U)9 b Fv(\)\))p Fq(:)732 b Fv(\(2.46\))515 1280 y(F)-7 b(or)30 b(a)g(giv)n(en)f(in)n(teraction)f(\()p Fq(U)1469 1292 y 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3007 y Fv(.)54 b(Clearly)-7 b(,)32 b Fq(P)12 b Fv(\()p Fq(\031)s(U)d Fv(\))33 b(=)f Fq(P)12 b Fv(\()p Fq(U)d Fv(\))34 b(b)n(y)f(the)g(assumed)f Fq(\031)s Fp(\000)515 3107 y Fv(in)n(v)-5 b(ariance)22 b(of)k(the)g(coun)n(ting)f(measure,)f(see)h(\(1.3\).)36 b(\(This)26 b(also)d(follo)n(ws)g(from)i(observing)515 3206 y(that)k Fp(\000)p Fq(P)12 b Fv(\()p Fq(\031)s(U)d Fv(\))19 b(+)g Fq(P)12 b Fv(\()p Fq(U)d Fv(\))25 b(=)f Fq(p)p Fv(\(2)p Fq(E)5 b(;)14 b(E)5 b Fv(\))25 b(=)g Fq(p)p Fv(\(0)p Fq(;)14 b(E)5 b Fv(\))25 b(=)f(0)29 b(b)n(y)f(\(2.32\)) g(and)h(\(2.31\).\))40 b(F)-7 b(or)28 b(the)515 3306 y(same)21 b(reason,)h Fq(s)p Fv(\()p Fq(\031)s(\026)p Fv(\))h(=)g Fq(s)p Fv(\()p Fq(\026)p Fv(\))g(and)f(if)g Fq(\026)h Fv(is)e(a)h(Gibbs)g(measure)e(for)i Fq(U)9 b Fv(,)24 b(then)f Fq(\031)s(\026)f Fv(is)g(a)g(Gibbs)515 3406 y(measure)31 b(for)i Fq(\031)s(U)43 b Fv(\(and)34 b(vice)e(v)n(ersa\).)54 b(\(T)-7 b(o)33 b(a)n(v)n(oid)e(trivialities,)f (it)j(is)g(understo)r(o)r(d)g(that)515 3505 y(the)24 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Fm(0)1394 4253 y Fp(j)p Fq(A)p Fp(jjj)p Fq(U)1614 4218 y Fm(0)1605 4273 y Fk(A)1659 4253 y Fp(jj)p Fq(;)1777 4149 y Fk(m)1746 4174 y Fj(X)1742 4350 y Fk(\013)p Fm(=1)1883 4253 y Fq(E)1949 4218 y Fk(\013)2015 4174 y Fj(X)2011 4352 y Fk(A)p Fl(3)p Fm(0)2152 4253 y Fp(j)p Fq(A)p Fp(jjj)p Fq(U)2372 4218 y Fk(\013)2363 4273 y(A)2420 4253 y Fp(jj)23 b Fq(<)g Fv(1)p Fq(:)525 b Fv(\(2.51\))515 4514 y Fs(Prop)s(osition)30 b(2.1)40 b Fe(F)-6 b(or)30 b(the)g(Gibbs)h(me)l(asur)l(e)e Fq(\026)24 b Fp(6)p Fv(=)e Fq(\031)s(\026)30 b Fe(of)h(The)l(or)l(em)g (2.1,)1195 4758 y Fq(s)p Fv(\()p Fq(\031)s(\026)p Fp(j)p Fq(\026)p Fv(\))24 b(=)e Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))i(=)f(2)p Fq(\014)2111 4654 y Fk(m)2080 4679 y Fj(X)2077 4855 y Fk(\013)p Fm(=1)2218 4758 y Fq(E)2284 4724 y Fk(\013)2331 4758 y Fq(\026)p Fv(\()p Fq(J)2467 4724 y Fk(\013)2459 4779 y Fm(0)2515 4758 y Fv(\))g Fq(>)g Fv(0)467 b(\(2.52\))515 5017 y Fe(and,)30 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Fq(\014)t(\031)s(\026) p Fv(\(\010)2727 706 y Fm(0)2765 694 y Fv(\()p Fq(U)9 b Fv(\)\))p Fq(:)217 b Fv(\(2.54\))515 865 y(As)28 b(is)g(w)n(ell)e (kno)n(wn)h(the)i(relativ)n(e)c(en)n(trop)n(y)j(\(2.45\))f(can)h(b)r(e) h(rewritten)e(as)h(a)g(di\013erence)f(of)515 965 y(free)g(energies:)35 b Fq(s)p Fv(\()p Fq(\031)s(\026)p Fp(j)p Fq(\026)p Fv(\))24 b(=)e Fq(P)12 b Fv(\()p Fq(U)d Fv(\))19 b Fp(\000)f Fq(F)12 b Fv(\()p Fq(U;)i(\031)s(\026)p Fv(\).)37 b(W)-7 b(e)28 b(can)f(no)n(w)g(use)h(that)963 1179 y Fq(\014)t(\026)p Fv(\(\010)1156 1191 y Fm(0)1194 1179 y Fv(\()p Fq(\031)s(U)9 b Fv(\))19 b Fp(\000)f Fv(\010)1536 1191 y Fm(0)1573 1179 y Fv(\()p Fq(U)9 b Fv(\)\))24 b(=)e Fp(\000)1936 1123 y Fq(@)p 1921 1160 79 4 v 1921 1236 a(@)5 b(t)2010 1179 y(P)12 b Fv(\()p Fq(U)27 b Fv(+)18 b Fq(t)p Fv(\()p Fq(\031)s(U)28 b Fp(\000)18 b Fq(U)9 b Fv(\)\)\()p Fq(t)23 b Fv(=)g(0\))236 b(\(2.55\))515 1383 y(is)26 b(exactly)g(equal)g(to)i (2)p Fq(\014)1311 1321 y Fj(P)1399 1341 y Fk(m)1399 1408 y(\013)p Fm(=1)1544 1383 y Fq(E)1610 1353 y Fk(\013)1658 1383 y Fq(\026)p Fv(\()p Fq(J)1794 1353 y Fk(\013)1786 1403 y Fm(0)1841 1383 y Fv(\),)g(as)f(required.)p 2349 1383 48 48 v 515 1528 a Fs(Remark)47 b(1:)64 b Fv(The)42 b(p)r(ositivit)n(y)c(of)k(\(2.52\))e(should)h(b)r(e)h(compared)d(with)i (\(1.3\).)79 b(The)515 1627 y(p)r(ositivit)n(y)36 b(of)k(the)g(en)n (trop)n(y)e(pro)r(duction)g(is)h(discussed)f(in)h([23)o(,)h(24)o(].)73 b(The)39 b(p)r(ositivit)n(y)515 1727 y(of)g(\(2.52\))g(just)h(follo)n (ws)c(here)j(from)f(the)i(Gibbs')f(v)-5 b(ariational)35 b(principle:)58 b(with)51 b(_)-36 b Fq(s)3226 1739 y Fk(n)3314 1727 y Fp(\021)515 1764 y Fj(P)616 1827 y Fq(E)682 1796 y Fk(\013)743 1764 y Fj(P)831 1851 y Fk(x)p Fl(2)p Fm(\003)959 1859 y Fh(n)1018 1827 y Fq(J)1072 1796 y Fk(\013)1064 1847 y(x)1119 1827 y Fv(,)32 b(for)f Fq(\026)p Fp(\000)g Fv(almost)e(ev)n(ery)h Fq(\033)n(;)d Fv(_)-36 b Fq(s)2067 1839 y Fk(n)2113 1827 y Fv(\()p Fq(\033)s Fv(\))p Fq(=)p Fp(j)p Fv(\003)2350 1839 y Fk(n)2395 1827 y Fp(j)29 b(!)g Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))p Fq(=)p Fv(2)p Fq(\014)34 b(>)29 b Fv(0)i(where)515 1926 y(the)24 b(almost)e(sure)h(con)n(v)n(ergence)f(assumes)g(that)i Fq(\026)g Fv(is)f(a)h(phase.)35 b(That)24 b Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))h(has)f(some-)515 2026 y(thing)g(to)i(do)f(with)g(en)n(trop)n(y)f(pro)r(duction)g(will)e (b)r(ecome)j(clear)e(in)i(the)g(next)h(section)e(when)515 2125 y(a)j(dynamics)e(and)i(the)h(time-rev)n(ersal)23 b(op)r(eration)i(is)i(considered.)515 2225 y Fs(Remark)39 b(2:)51 b Fv(Thinking)32 b(ab)r(out)j Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))h(as)e(en)n(trop)n(y)g(pro)r (duction,)h(\(2.52\))f(giv)n(es)f(the)515 2325 y(usual)38 b(bilinear)e(expression)h(in)h(terms)h(of)g(thermo)r(dynamic)d (\015uxes)j(and)h(forces.)71 b(Re-)515 2424 y(mem)n(b)r(er)32 b(that)j(the)g(dep)r(endence)h(of)e Fq(p)p Fv(\()p Fq(\025;)14 b(E)5 b Fv(\))36 b(on)e Fq(E)40 b Fv(in)34 b(\(2.31\))g(comes)f(from)g (the)i(state)515 2524 y Fq(\026)p Fv(.)58 b(The)35 b Fq(E)890 2494 y Fk(\013)972 2524 y Fv(corresp)r(ond)f(to)g(\014eld)g (strengths)h(or)e(amplitudes)f(pro)r(ducing)i(energy-)f(or)515 2624 y(particle)24 b(\015o)n(w.)36 b(Of)26 b(course,)g(on)g(the)h (formal)d(lev)n(el)g(ab)r(o)n(v)n(e,)h(the)i(distinction)d(m)n(ust)i (remain)515 2723 y(arbitrary)e(and)k(one)f(can)g(of)h(course)e(include) g(the)i Fq(E)2178 2693 y Fk(\013)2253 2723 y Fv(in)f(the)h(p)r(oten)n (tials)d Fq(U)2942 2693 y Fk(\013)2933 2746 y(A)2989 2723 y Fv(.)515 2954 y Fy(2.4)112 b(Green-Kub)s(o)39 b(form)m(ula.)515 3107 y Fv(It)32 b(has)g(b)r(een)h(observ)n(ed)d(in)i (other)f(places,)h([16)o(,)h(19)o(,)f(7],)h(that)g(the)f(\015uctuation) g(theorem)515 3206 y(quite)26 b(directly)f(giv)n(es)g(rise)h(to)h(v)-5 b(arious)25 b(familiar)e(form)n(ulae)h(of)j(linear)d(resp)r(onse.)36 b(W)-7 b(e)28 b(will)515 3306 y(not)k(pursue)g(this)f(matter)h(here)f (v)n(ery)g(far)h(except)g(for)g(rep)r(eating)f(the)i(simplest)c(deriv) -5 b(a-)515 3406 y(tions.)639 3505 y(Assuming)17 b(smo)r(othness)f(of)j (the)g(free)f(energy)f(in)h(the)h(external)e(\014elds,)i(w)n(e)f (di\013eren)n(tiate)515 3605 y(\(2.32\))27 b(with)g(resp)r(ect)g(to)h Fq(E)1396 3575 y Fk(\015)1466 3605 y Fv(and)f Fq(\025)1675 3617 y Fk(\013)1723 3605 y Fq(;)14 b(\013;)g(\015)28 b Fv(=)23 b(1)p Fq(;)14 b(:)g(:)g(:)f(;)h(m)27 b Fv(at)h Fq(E)g Fv(=)22 b Fq(\025)i Fv(=)e(0:)891 3763 y Fq(@)p 837 3800 158 4 v 837 3876 a(@)5 b(E)952 3852 y Fk(\015)1062 3763 y Fq(@)p 1014 3800 145 4 v 1014 3876 a(@)g(\025)1111 3888 y Fk(\013)1169 3819 y Fq(p)p Fv(\(0)p Fq(;)14 b Fv(0\))22 b(=)h Fp(\000)1635 3763 y Fq(@)p 1581 3800 158 4 v 1581 3876 a(@)5 b(E)1696 3852 y Fk(\015)1805 3763 y Fq(@)p 1758 3800 145 4 v 1758 3876 a(@)g(\025)1855 3888 y Fk(\013)1912 3819 y Fq(p)p Fv(\(0)p Fq(;)14 b Fv(0\))k Fp(\000)g Fv(2)2337 3763 y Fq(@)p 2292 3800 140 4 v 2292 3876 a(@)5 b(\025)2389 3888 y Fk(\015)2499 3763 y Fq(@)p 2451 3800 145 4 v 2451 3876 a(@)g(\025)2548 3888 y Fk(\013)2605 3819 y Fq(p)p Fv(\(0)p Fq(;)14 b Fv(0\))p Fq(:)312 b Fv(\(2.56\))515 4038 y(On)27 b(the)h(other)f(hand,) 1608 4120 y Fq(@)p 1560 4157 V 1560 4233 a(@)5 b(\025)1657 4245 y Fk(\013)1715 4176 y Fq(p)p Fv(\(0)p Fq(;)14 b(E)5 b Fv(\))23 b(=)f Fq(\014)t(\026)p Fv(\()p Fq(J)2263 4142 y Fk(\013)2255 4196 y Fm(0)2311 4176 y Fv(\))824 b(\(2.57\))515 4354 y(while)1356 4420 y Fq(@)p 1310 4457 140 4 v 1310 4533 a(@)5 b(\025)1407 4545 y Fk(\015)1518 4420 y Fq(@)p 1470 4457 145 4 v 1470 4533 a(@)g(\025)1567 4545 y Fk(\013)1624 4476 y Fq(p)p Fv(\(0)p Fq(;)14 b Fv(0\))22 b(=)h Fp(\000)p Fq(\014)2077 4442 y Fm(2)2128 4397 y Fj(X)2169 4572 y Fk(x)2262 4476 y Fq(\027)5 b Fv(\()p Fq(J)2394 4442 y Fk(\013)2386 4497 y Fm(0)2442 4476 y Fq(J)2496 4442 y Fk(\015)2488 4497 y(x)2538 4476 y Fv(\))p Fq(:)574 b Fv(\(2.58\))515 4689 y(Conclusion,)1352 4771 y Fq(@)p 1298 4808 158 4 v 1298 4884 a(@)5 b(E)1413 4860 y Fk(\015)1465 4827 y Fq(\026)p Fv(\()p Fq(J)1601 4793 y Fk(\013)1593 4848 y Fm(0)1649 4827 y Fv(\)\()p Fq(E)29 b Fv(=)22 b(0\))h(=)g Fq(\014)2140 4748 y Fj(X)2181 4923 y Fk(x)2274 4827 y Fq(\027)5 b Fv(\()p Fq(J)2406 4793 y Fk(\013)2398 4848 y Fm(0)2454 4827 y Fq(J)2508 4793 y Fk(\015)2500 4848 y(x)2550 4827 y Fv(\))p Fq(;)562 b Fv(\(2.59\))515 5044 y(and)28 b(the)i(c)n(hange)d(in)h(relativ)n(e)e(en)n(trop)n(y)i Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))h(\(see)g (\(2.52\)\))g(from)e(Prop)r(osition)e(2.1)j(is)515 5144 y(in)f(quadratic)e(appro)n(ximation)e(for)k(small)d Fq(E)33 b Fv(giv)n(en)26 b(b)n(y)1263 5327 y Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))e(=)f(2)p Fq(\014)1744 5293 y Fm(2)1795 5248 y Fj(X)1804 5423 y Fk(\013;\015)1929 5327 y Fq(E)1995 5293 y Fk(\013)2042 5327 y Fq(E)2108 5293 y Fk(\015)2165 5248 y Fj(X)2206 5423 y Fk(x)2298 5327 y Fq(\027)5 b Fv(\()p Fq(J)2430 5293 y Fk(\013)2422 5348 y Fm(0)2478 5327 y Fq(J)2532 5293 y Fk(\015)2524 5348 y(x)2575 5327 y Fv(\))p Fq(:)537 b Fv(\(2.60\))1905 5650 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fv(Equation)17 b(\(2.59\))i(is)f(a)h(standard)f (Green-Kub)r(o)h(relation)d(while)i(\(2.60\))g(expresses)g(the)i(rel-) 515 623 y(ativ)n(e)i(en)n(trop)n(y)h(densit)n(y)f Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))j(\(or)e(c)n(hange)g (in)g(free)h(energy\))f(in)g(terms)f(of)i(the)g(curren)n(t-)515 722 y(curren)n(t)34 b(correlations)e(\(with)j(the)h(ob)n(vious)d (analogues)g(of)j(Onsager)d(symmetries\).)58 b(In)515 822 y(conclusion,)25 b(w)n(e)i(ha)n(v)n(e)f(iden)n(ti\014ed)g(a)h(\(mo) r(del-dep)r(enden)n(t\))g(con)n(tin)n(uous)f(function)1598 993 y(_)1574 1014 y Fq(S)5 b Fv(\()p Fq(\033)s Fv(\))24 b Fp(\021)1856 935 y Fj(X)1894 1109 y Fk(\013)1990 1014 y Fq(E)2056 979 y Fk(\013)2103 1014 y Fq(J)2157 979 y Fk(\013)2149 1034 y Fm(0)2204 1014 y Fv(\()p Fq(\033)s Fv(\))849 b(\(2.61\))515 1278 y(with)27 b Fq(\031)777 1257 y Fv(_)754 1278 y Fq(S)h Fv(=)23 b Fp(\000)1008 1257 y Fv(_)986 1278 y Fq(S)t(;)14 b(\026)p Fv(\()1183 1257 y(_)1160 1278 y Fq(S)5 b Fv(\))23 b Fq(>)g Fv(0)p Fq(;)14 b(\027)5 b Fv(\()1539 1257 y(_)1516 1278 y Fq(S)t Fv(\))24 b(=)e(0)28 b(and)f(symmetric)d(resp)r(onse)j(matrix)1284 1440 y Fq(@)p 1230 1477 158 4 v 1230 1553 a(@)5 b(E)1345 1529 y Fk(\015)1397 1496 y Fq(\026)p Fv(\()1546 1440 y Fq(@)p 1489 1477 163 4 v 1489 1553 a(@)g(E)1604 1529 y Fk(\013)1685 1475 y Fv(_)1661 1496 y Fq(S)g Fv(\)\()p Fq(E)29 b Fv(=)23 b(0\))f(=)h Fq(\014)2208 1417 y Fj(X)2249 1592 y Fk(x)2342 1496 y Fq(\027)5 b Fv(\()p Fq(J)2474 1462 y Fk(\013)2466 1517 y Fm(0)2522 1496 y Fq(J)2576 1462 y Fk(\015)2568 1517 y(x)2618 1496 y Fv(\))p Fq(:)494 b Fv(\(2.62\))515 1741 y(Symmetries)24 b(in)i(higher)g(order)g(terms)g (can)h(b)r(e)h(obtained)e(b)n(y)h(taking)f(higher)f(deriv)-5 b(ativ)n(es)515 1841 y(of)27 b(the)h(generating)d(form)n(ula)g (\(2.32\).)515 1941 y(The)32 b(notation)1049 1920 y(_)1026 1941 y Fq(S)37 b Fv(should)30 b(not)i(b)r(e)h(read)e(as)g(a)g (time-deriv)-5 b(ativ)n(e)27 b(\(c)n(hange)k(of)h(en)n(trop)n(y)f(in) 515 2040 y(time\).)68 b(More)37 b(appropriate)f(will)f(b)r(e)k(to)f (regard)e Fq(\026)p Fv(\()2263 2019 y(_)2239 2040 y Fq(S)6 b Fv(\))p Fq(=)p Fv(2)37 b(as)h(the)h(limit)c(\()p Fq(S)3008 2052 y Fk(f)3076 2040 y Fp(\000)26 b Fq(S)3218 2052 y Fk(i)3245 2040 y Fv(\))p Fq(=T)515 2140 y Fv(as)35 b(time)g Fq(T)47 b Fv(go)r(es)36 b(to)g(in\014nit)n(y)e(of)j(the)f(total)f(c)n (hange)g(of)i(en)n(trop)n(y)e Fq(S)2726 2152 y Fk(f)2793 2140 y Fp(\000)24 b Fq(S)2933 2152 y Fk(i)2996 2140 y Fv(in)36 b(a)g(reser-)515 2240 y(v)n(oir)24 b(during)g(the)j (nonequilibrium)21 b(pro)r(cess.)35 b(The)27 b(reserv)n(oir)22 b(is)j(initially)c(in)26 b(equilibrium)515 2339 y(with)g(thermo)r (dynamic)d(en)n(trop)n(y)i Fq(S)1649 2351 y Fk(i)1703 2339 y Fv(and)i(after)f(absorbing)e(the)j(heat)f(dissipated)e(b)n(y)j (the)515 2439 y(nonequilibrium)e(pro)r(cess)j(it)i(reac)n(hes)e(a)h (new)h(equilibrium)c(with)j(en)n(trop)n(y)g Fq(S)2995 2451 y Fk(f)3038 2439 y Fv(.)44 b(W)-7 b(e)30 b(will)515 2538 y(come)c(bac)n(k)h(to)g(this)g(once)g(time)f(has)h(b)r(een)h (explicitly)c(in)n(tro)r(duced)i(\(in)i(Section)e(3\).)515 2812 y Fr(3)134 b(Fluctuation)44 b(theorem)h(for)g(PCA.)515 2994 y Fv(PCA)34 b(\(short)f(for)h(probabilistic)29 b(cellular)h (automata\))j(are)g(discrete)g(time)f(parallel)e(up-)515 3093 y(dating)35 b(sto)r(c)n(hastic)g(dynamics)g(for)h(lattice)f(spin)h (systems.)63 b(They)37 b(are)f(used)h(in)f(man)n(y)515 3193 y(con)n(texts)31 b(but)i(w)n(e)f(see)g(them)g(here)g(as)g(in)n (teresting)d(examples)h(of)i(non-equilibrium)27 b(dy-)515 3292 y(namics.)39 b(W)-7 b(e)29 b(refer)g(to)g([13)o(,)g(18)o(])h(for)e (details)f(and)i(examples)e(and)i(w)n(e)g(restrict)e(ourselv)n(es)515 3392 y(here)33 b(to)h(the)g(essen)n(tials)d(w)n(e)i(need.)56 b(W)-7 b(e)35 b(w)n(ork)d(with)h(time-homogeneous)d(translation-)515 3492 y(in)n(v)-5 b(arian)n(t)36 b(nearest-neigh)n(b)r(or)h(PCA)i(whic)n (h)g(are)g(sp)r(eci\014ed)g(b)n(y)g(giving)e(the)j(single-site)515 3591 y(transition)25 b(probabilities)671 3766 y(0)e Fq(<)f(p)865 3778 y Fk(i;n)953 3766 y Fv(\()p Fq(a)p Fp(j)p Fq(\033)s Fv(\))i(=)f Fq(p)1288 3778 y Fk(i)1316 3766 y Fv(\()p Fq(a)p Fp(j)p Fq(\033)s Fv(\()p Fq(j;)14 b(n)19 b Fp(\000)f Fv(1\))p Fq(;)c Fp(j)p Fq(j)23 b Fp(\000)18 b Fq(i)p Fp(j)23 b(\024)g Fv(1\))f Fq(<)h Fv(1)p Fq(;)14 b(a)22 b Fp(2)i Fq(S;)14 b(\033)26 b Fp(2)e Fv(\012)2865 3778 y Fk(d)p Fm(+1)2987 3766 y Fq(:)157 b Fv(\(3.63\))515 3942 y(This)21 b(de\014nes)i(a)f(Mark)n(o)n(v)e(pro)r(cess)i(\()p Fq(X)1714 3954 y Fk(n)1759 3942 y Fv(\))1791 3954 y Fk(n)p Fm(=0)p Fk(;)p Fm(1)p Fk(:::)2055 3942 y Fv(on)h(\012)2226 3954 y Fk(d)2287 3942 y Fv(for)f(whic)n(h)f(for)i(all)d(\014nite)i Fq(V)42 b Fp(\032)p 3272 3944 45 4 v 3272 3887 43 4 v 3272 3942 3 3 v 3273 3940 V 3274 3938 V 3275 3936 V 3276 3935 V 3277 3933 V 3278 3931 V 3279 3929 V 3280 3927 V 3281 3925 V 3282 3923 V 3283 3921 V 3284 3919 V 3285 3917 V 3285 3916 V 3286 3914 V 3287 3912 V 3288 3910 V 3289 3908 V 3290 3906 V 3291 3904 V 3292 3902 V 3293 3901 V 3294 3899 V 3295 3897 V 3296 3895 V 3297 3893 V 3298 3891 V 3299 3889 V 3299 3887 V 3287 3942 V 3288 3940 V 3289 3938 V 3290 3936 V 3291 3935 V 3292 3933 V 3293 3931 V 3294 3929 V 3295 3927 V 3296 3925 V 3297 3923 V 3298 3921 V 3299 3919 V 3299 3917 V 3300 3916 V 3301 3914 V 3302 3912 V 3303 3910 V 3304 3908 V 3305 3906 V 3306 3904 V 3307 3902 V 3308 3901 V 3309 3899 V 3310 3897 V 3311 3895 V 3312 3893 V 3313 3891 V 3313 3889 V 3314 3887 V 3271 3897 4 12 v 3272 3893 4 8 v 3273 3891 4 6 v 3274 3889 4 4 v 3274 3887 4 2 v 3316 3942 4 12 v 3314 3942 4 8 v 3313 3942 4 6 v 3312 3942 4 4 v 3312 3942 4 2 v 3317 3910 a Fk(d)3356 3942 y Fv(,)666 4136 y(Prob)27 b([)p Fq(X)963 4148 y Fk(n)1008 4136 y Fv(\()p Fq(i)p Fv(\))c(=)f Fq(a)1255 4148 y Fk(i)1283 4136 y Fq(;)14 b Fp(8)p Fq(i)22 b Fp(2)h Fq(V)c Fp(j)p Fq(X)1655 4148 y Fk(n)p Fl(\000)p Fm(1)1785 4136 y Fv(])k(=)1927 4058 y Fj(Y)1919 4236 y Fk(i)p Fl(2)p Fk(V)2054 4136 y Fq(p)2096 4148 y Fk(i)2124 4136 y Fv(\()p Fq(a)2200 4148 y Fk(i)2228 4136 y Fp(j)p Fq(X)2320 4148 y Fk(n)p Fl(\000)p Fm(1)2450 4136 y Fv(\()p Fq(j)5 b Fv(\))p Fq(;)14 b Fp(j)p Fq(i)k Fp(\000)g Fq(j)5 b Fp(j)23 b(\024)g Fv(1\))p Fq(;)14 b(a)3071 4148 y Fk(i)3121 4136 y Fp(2)23 b Fq(S)3167 4319 y Fv(\(3.64\))515 4419 y(with)31 b(some)f(giv)n(en)g(initial)f (con\014guration)g Fq(X)1962 4431 y Fm(0)2029 4419 y Fv(=)h Fq(\030)k Fp(2)d Fv(\012)2340 4431 y Fk(d)2378 4419 y Fv(.)50 b(Notice)31 b(that)h(w)n(e)f(ha)n(v)n(e)g(k)n(ept)515 4518 y(the)26 b(notation)e Fq(\033)30 b Fv(for)25 b(a)g(general)f (con\014guration)f(on)i(the)i(space-time)c(lattice.)34 b(Remem)n(b)r(er)515 4618 y(that)28 b Fq(x)d Fv(=)g(\()p Fq(i;)14 b(n)p Fv(\))24 b Fp(2)p 1155 4620 45 4 v 1155 4563 43 4 v 1155 4618 3 3 v 1156 4616 V 1157 4614 V 1157 4612 V 1158 4611 V 1159 4609 V 1160 4607 V 1161 4605 V 1162 4603 V 1163 4601 V 1164 4599 V 1165 4597 V 1166 4595 V 1167 4594 V 1168 4592 V 1169 4590 V 1170 4588 V 1171 4586 V 1171 4584 V 1172 4582 V 1173 4580 V 1174 4579 V 1175 4577 V 1176 4575 V 1177 4573 V 1178 4571 V 1179 4569 V 1180 4567 V 1181 4565 V 1182 4564 V 1170 4618 V 1171 4616 V 1172 4614 V 1172 4612 V 1173 4611 V 1174 4609 V 1175 4607 V 1176 4605 V 1177 4603 V 1178 4601 V 1179 4599 V 1180 4597 V 1181 4595 V 1182 4594 V 1183 4592 V 1184 4590 V 1185 4588 V 1186 4586 V 1186 4584 V 1187 4582 V 1188 4580 V 1189 4579 V 1190 4577 V 1191 4575 V 1192 4573 V 1193 4571 V 1194 4569 V 1195 4567 V 1196 4565 V 1197 4564 V 1153 4573 4 12 v 1154 4569 4 8 v 1155 4567 4 6 v 1156 4565 4 4 v 1157 4564 4 2 v 1198 4618 4 12 v 1197 4618 4 8 v 1196 4618 4 6 v 1194 4618 4 4 v 1194 4618 4 2 v 1200 4586 a Fk(d)p Fm(+1)1351 4618 y Fv(stands)k(for)g(a)g(space-time)e(p)r(oin)n(t)h (with)h(time-co)r(ordinate)d Fq(n)j Fv(at)515 4717 y(site)e Fq(i)d Fp(2)p 811 4719 45 4 v 811 4663 43 4 v 811 4717 3 3 v 812 4715 V 813 4713 V 814 4711 V 815 4710 V 816 4708 V 817 4706 V 818 4704 V 819 4702 V 820 4701 V 821 4699 V 822 4697 V 823 4695 V 823 4693 V 824 4691 V 825 4689 V 826 4688 V 827 4686 V 828 4684 V 829 4682 V 830 4680 V 831 4678 V 832 4676 V 833 4674 V 834 4673 V 835 4671 V 836 4669 V 837 4667 V 837 4665 V 838 4663 V 826 4717 V 827 4715 V 828 4713 V 829 4711 V 830 4710 V 831 4708 V 832 4706 V 833 4704 V 834 4702 V 835 4701 V 836 4699 V 837 4697 V 837 4695 V 838 4693 V 839 4691 V 840 4689 V 841 4688 V 842 4686 V 843 4684 V 844 4682 V 845 4680 V 846 4678 V 847 4676 V 848 4674 V 849 4673 V 850 4671 V 851 4669 V 851 4667 V 852 4665 V 853 4663 V 810 4673 4 12 v 811 4669 4 8 v 812 4667 4 6 v 813 4665 4 4 v 813 4663 4 2 v 854 4717 4 12 v 853 4717 4 8 v 852 4717 4 6 v 851 4717 4 4 v 851 4717 4 2 v 856 4686 a Fk(d)895 4717 y Fv(.)515 4817 y(The)32 b Fq(\031)737 4829 y Fm(\003)818 4817 y Fv(are)f(restricted)f(to)i(time-rev)n(ersal)27 b(transformations)h(and)j(the)i(v)n(olumes)c(\003)i(are)515 4917 y(to)c(gro)n(w)f(\014rst)h(in)g(the)h(time-direction)c(\(for)j(a)g (\014xed)h(spatial)d(windo)n(w\).)515 5148 y Fy(3.1)112 b(Steady)38 b(state)g(\015uctuation)g(theorem.)515 5301 y Fv(If)c(w)n(e)f(tak)n(e)g(a)g(translation-in)n(v)-5 b(aria)o(n)n(t)28 b(stationary)j(state)j Fq(\032)f Fv(of)h(a)f(PCA)h (as)f(ab)r(o)n(v)n(e,)h(then)515 5401 y(its)h(Mark)n(o)n(v)f(extension) g(de\014nes)i(a)g(translation-in)n(v)-5 b(ari)o(an)n(t)30 b(Gibbs)35 b(measure)f Fq(\026)i Fv(for)g(the)1905 5650 y(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fv(\(formal\))25 b(Hamiltonian)1229 704 y Fq(H)7 b Fv(\()p Fq(\033)s Fv(\))24 b(=)f Fp(\000)1610 625 y Fj(X)1627 802 y Fk(i;n)1743 704 y Fv(ln)13 b Fq(p)1868 716 y Fk(i;n)1956 704 y Fv(\()p Fq(\033)s Fv(\()p Fq(i;)h(n)p Fv(\))p Fp(j)p Fq(\033)s Fv(\()p Fp(\001)p Fq(;)g(n)19 b Fp(\000)g Fv(1\)\))p Fq(:)502 b Fv(\(3.65\))515 957 y(W)-7 b(e)41 b(refer)g(to)f([13,)h(18)o(])g(for)g(a)f(precise)g(form)n (ulation.)73 b Fq(\026)41 b Fv(describ)r(es)f(the)h(distribution)515 1057 y(of)33 b(the)h(space-time)d(con\014gurations)f(in)j(the)h(steady) f(state)g(and)g(its)g(restriction)e(to)i(an)n(y)515 1156 y(spatial)24 b(la)n(y)n(er)g(is)i(equal)g(to)h(the)g(stationary)d (state)j Fq(\032)g Fv(w)n(e)g(started)f(from.)36 b(T)-7 b(o)26 b(c)n(haracterize)515 1256 y Fq(\032)p Fv(,)44 b(one)c(m)n(ust)f(study)i(the)g(pro)5 b(jection)39 b(of)h Fq(\026)h Fv(to)f(a)g(la)n(y)n(er)e(\(see)j([20)o(])f(for)g(a)h(v)-5 b(ariational)515 1356 y(c)n(haracterization)23 b(of)k(suc)n(h)h(a)f (pro)5 b(jection\).)515 1455 y(Since)21 b Fq(\026)g Fv(is)g(Gibbsian)f (w)n(e)h(can)g(try)h(applying)d(the)j(theory)f(of)g(the)h(previous)e (section.)33 b(Most)515 1555 y(in)n(teresting)26 b(is)h(to)i(consider)d (a)i(sequence)g(of)h(rectangular)c(b)r(o)n(xes)j(\003)2674 1567 y Fk(L;N)2827 1555 y Fp(\021)c(f)p Fq(x)h Fv(=)f(\()p Fq(i;)14 b(n)p Fv(\))25 b Fp(2)p 529 1657 45 4 v 529 1600 43 4 v 529 1655 3 3 v 530 1653 V 530 1651 V 531 1649 V 532 1648 V 533 1646 V 534 1644 V 535 1642 V 536 1640 V 537 1638 V 538 1636 V 539 1634 V 540 1632 V 541 1630 V 542 1628 V 543 1627 V 544 1625 V 544 1623 V 545 1621 V 546 1619 V 547 1617 V 548 1615 V 549 1613 V 550 1612 V 551 1610 V 552 1608 V 553 1606 V 554 1604 V 555 1602 V 556 1600 V 544 1655 V 544 1653 V 545 1651 V 546 1649 V 547 1648 V 548 1646 V 549 1644 V 550 1642 V 551 1640 V 552 1638 V 553 1636 V 554 1634 V 555 1632 V 556 1630 V 557 1628 V 558 1627 V 558 1625 V 559 1623 V 560 1621 V 561 1619 V 562 1617 V 563 1615 V 564 1613 V 565 1612 V 566 1610 V 567 1608 V 568 1606 V 569 1604 V 570 1602 V 571 1600 V 527 1610 4 12 v 528 1606 4 8 v 529 1604 4 6 v 530 1602 4 4 v 531 1600 4 2 v 572 1655 4 12 v 571 1655 4 8 v 570 1655 4 6 v 568 1655 4 4 v 568 1655 4 2 v 573 1623 a Fk(d)p Fm(+1)719 1655 y Fv(=)p 821 1657 45 4 v 821 1600 43 4 v 821 1655 3 3 v 822 1653 V 823 1651 V 824 1649 V 824 1648 V 825 1646 V 826 1644 V 827 1642 V 828 1640 V 829 1638 V 830 1636 V 831 1634 V 832 1632 V 833 1630 V 834 1628 V 835 1627 V 836 1625 V 837 1623 V 838 1621 V 838 1619 V 839 1617 V 840 1615 V 841 1613 V 842 1612 V 843 1610 V 844 1608 V 845 1606 V 846 1604 V 847 1602 V 848 1600 V 836 1655 V 837 1653 V 838 1651 V 838 1649 V 839 1648 V 840 1646 V 841 1644 V 842 1642 V 843 1640 V 844 1638 V 845 1636 V 846 1634 V 847 1632 V 848 1630 V 849 1628 V 850 1627 V 851 1625 V 852 1623 V 852 1621 V 853 1619 V 854 1617 V 855 1615 V 856 1613 V 857 1612 V 858 1610 V 859 1608 V 860 1606 V 861 1604 V 862 1602 V 863 1600 V 819 1610 4 12 v 820 1606 4 8 v 821 1604 4 6 v 822 1602 4 4 v 823 1600 4 2 v 864 1655 4 12 v 863 1655 4 8 v 862 1655 4 6 v 861 1655 4 4 v 860 1655 4 2 v 866 1623 a Fk(d)921 1655 y Fp(\002)p 1016 1657 45 4 v 1016 1600 43 4 v 1016 1655 3 3 v 1017 1653 V 1018 1651 V 1019 1649 V 1020 1648 V 1021 1646 V 1021 1644 V 1022 1642 V 1023 1640 V 1024 1638 V 1025 1636 V 1026 1634 V 1027 1632 V 1028 1630 V 1029 1628 V 1030 1627 V 1031 1625 V 1032 1623 V 1033 1621 V 1034 1619 V 1035 1617 V 1035 1615 V 1036 1613 V 1037 1612 V 1038 1610 V 1039 1608 V 1040 1606 V 1041 1604 V 1042 1602 V 1043 1600 V 1031 1655 V 1032 1653 V 1033 1651 V 1034 1649 V 1035 1648 V 1036 1646 V 1036 1644 V 1037 1642 V 1038 1640 V 1039 1638 V 1040 1636 V 1041 1634 V 1042 1632 V 1043 1630 V 1044 1628 V 1045 1627 V 1046 1625 V 1047 1623 V 1048 1621 V 1049 1619 V 1050 1617 V 1050 1615 V 1051 1613 V 1052 1612 V 1053 1610 V 1054 1608 V 1055 1606 V 1056 1604 V 1057 1602 V 1058 1600 V 1014 1610 4 12 v 1015 1606 4 8 v 1016 1604 4 6 v 1018 1602 4 4 v 1018 1600 4 2 v 1059 1655 4 12 v 1058 1655 4 8 v 1057 1655 4 6 v 1056 1655 4 4 v 1055 1655 4 2 v 98 w Fv(:)e Fp(j)p Fq(i)p Fp(j)g(\024)f Fq(L;)14 b Fp(j)p Fq(n)p Fp(j)23 b(\024)f Fq(N)9 b Fp(g)p Fv(.)36 b(The)27 b(idea)e(is)h(that)h(w)n(e)f(wish)g(to)g(k)n(eep)h(the)g(spatial)515 1754 y(size)k Fq(L)g Fv(m)n(uc)n(h)h(smaller)c(than)k(the)h (time-extension)c Fq(N)39 b(>>)30 b(L)p Fv(.)51 b(As)32 b(transformation)c(w)n(e)515 1854 y(tak)n(e)f Fq(\031)742 1866 y Fm(\003)787 1874 y Fh(L;N)927 1854 y Fp(\021)22 b Fq(\031)1061 1866 y Fk(L;N)1217 1854 y Fv(corresp)r(onding)j(to)j(a)f (time-rev)n(ersal:)1244 2018 y Fq(\031)1291 2030 y Fk(L;N)1420 2018 y Fq(\033)s Fv(\()p Fq(i;)14 b(n)p Fv(\))23 b(=)g Fq(\033)s Fv(\()p Fq(i;)14 b Fp(\000)p Fq(n)p Fv(\))p Fq(;)g Fp(j)p Fq(n)p Fp(j)23 b(\024)f Fq(N)t(;)14 b Fp(j)p Fq(i)p Fp(j)23 b(\024)g Fq(L)517 b Fv(\(3.66\))515 2183 y(and)27 b Fq(\031)723 2195 y Fk(L;N)852 2183 y Fq(\033)s Fv(\()p Fq(i;)14 b(n)p Fv(\))23 b(=)g Fq(\033)s Fv(\()p Fq(i;)14 b(n)p Fv(\))28 b(whenev)n(er)e(\()p Fq(i;)14 b(n)p Fv(\))33 b Fq(=)-52 b Fp(2)24 b Fv(\003)2155 2195 y Fk(L;N)2283 2183 y Fv(.)639 2282 y(De\014ne)k(the)g(curren)n(t)760 2447 y Fq(J)806 2459 y Fk(i;n)895 2447 y Fv(\()p Fq(\033)s Fv(\))c Fp(\021)e Fv(ln)13 b Fq(p)1245 2459 y Fk(i)1273 2447 y Fv(\()p Fq(\033)s Fv(\()p Fq(i;)h(n)p Fv(\))p Fp(j)p Fq(\033)s Fv(\()p Fp(\001)p Fq(;)g(n)19 b Fp(\000)f Fv(1\)\))h Fp(\000)f Fv(ln)13 b Fq(p)2185 2459 y Fk(i)2212 2447 y Fv(\()p Fq(\033)s Fv(\()p Fq(i;)h(n)19 b Fp(\000)f Fv(1\))p Fp(j)p Fq(\033)s Fv(\()p Fp(\001)p Fq(;)c(n)p Fv(\)\))p Fq(:)247 b Fv(\(3.67\))515 2611 y(Notice)33 b(that)h(in)g(con)n(trast)f(with)g(the)i(previous)d(section,)j(w)n(e)e (do)h(not)g(sp)r(ecify)g(here)g(the)515 2711 y(unp)r(erturb)r(ed)e (state)g(\(but)h(one)e(can)h(alw)n(a)n(ys)d(tak)n(e)i(some)g (homogeneous)e(pro)r(duct)j(mea-)515 2811 y(sure\))d(and)h(w)n(e)g(tak) n(e)f Fq(m)e Fv(=)g(1)f(=)h(2)p Fq(E)34 b Fv(for)c(simplicit)n(y)-7 b(.)39 b Fq(J)2272 2823 y Fk(i;n)2391 2811 y Fv(is)28 b(a)i(lo)r(cal)d(function)j(and)g(it)f(is)515 2910 y(the)f(space-time)d (translate)g(of)j Fq(J)1556 2922 y Fm(0)1593 2910 y Fv(.)37 b(In)28 b(the)g(same)e(w)n(a)n(y)g(as)h(in)g(\(2.24\),)g(w)n(e)g (de\014ne)1398 3075 y Fq(R)1461 3087 y Fk(L;N)1589 3075 y Fv(\()p Fq(\033)s Fv(\))d Fp(\021)f Fq(H)7 b Fv(\()p Fq(\031)1970 3087 y Fk(L;N)2098 3075 y Fq(\033)s Fv(\))19 b Fp(\000)f Fq(H)7 b Fv(\()p Fq(\033)s Fv(\))p Fq(:)672 b Fv(\(3.68\))515 3239 y(Starting)19 b(from)g(\(3.65\))h Fq(R)1316 3251 y Fk(L;N)1465 3239 y Fv(can)h(b)r(e)g(written)f(out)g (as)g(a)h(\014nite)f(sum)g(but)h(most)e(imp)r(ortan)n(t)515 3339 y(is)26 b(that)1117 3499 y Fq(R)1180 3511 y Fk(L;N)1309 3499 y Fv(\()p Fq(\033)s Fv(\))e(=)1606 3395 y Fk(N)6 b Fl(\000)p Fm(1)1618 3420 y Fj(X)1534 3598 y Fk(n)p Fm(=)p Fl(\000)p Fk(N)g Fm(+1)1898 3420 y Fj(X)1835 3602 y Fl(j)p Fk(i)p Fl(j\024)p Fk(L)p Fl(\000)p Fm(1)2094 3499 y Fq(J)2140 3511 y Fk(i;n)2228 3499 y Fv(\()p Fq(\033)s Fv(\))20 b(+)e Fq(G)2510 3511 y Fk(L;N)2638 3499 y Fv(\()p Fq(\033)s Fv(\))p Fq(;)392 b Fv(\(3.69\))515 3730 y(where)733 3894 y Fp(k)p Fq(G)840 3906 y Fk(L;N)968 3894 y Fp(k)23 b(\024)f Fq(c)p Fv(\(2)p Fq(N)27 b Fv(+)18 b(1\)\(2)p Fq(L)g Fv(+)g(1\))1787 3860 y Fk(d)p Fl(\000)p Fm(1)1929 3894 y Fv(+)g Fq(c)2048 3860 y Fl(0)2071 3894 y Fv(\(2)p Fq(L)g Fv(+)g(1\))2377 3860 y Fk(d)2438 3894 y Fp(\024)23 b Fq(c)p Fv(\()p Fq(d)p Fv(\))p Fq(N)9 b(L)2802 3860 y Fk(d)p Fl(\000)p Fm(1)2926 3894 y Fq(;)218 b Fv(\(3.70\))515 4059 y(with)23 b(a)g(constan)n(t)f Fq(c)p Fv(\()p Fq(d)p Fv(\))i(dep)r(ending)f(on)g(the)h(dimension)c Fq(d)k Fv(and)f(on)g(the)h(transition)d(proba-)515 4159 y(bilities)g (\(3.63\).)35 b(W)-7 b(e)25 b(are)f(therefore)g(in)g(a)g(p)r(osition)f (to)h(rep)r(eat)h(the)g(\015uctuation)f(Theorem)515 4258 y(2.1)i(in)h(that)h(con)n(text.)515 4423 y Fs(Theorem)i(3.1)41 b Fe(T)-6 b(ake)30 b Fq(L)23 b Fv(=)f Fq(L)p Fv(\()p Fq(N)9 b Fv(\))23 b Fp(\024)g Fq(N)38 b Fe(gr)l(owing)31 b(to)f(in\014nity)f(as)h Fq(N)i Fp(")23 b(1)p Fe(.)38 b(The)31 b(limit)1063 4626 y Fq(e)p Fv(\()p Fq(\025)p Fv(\))24 b Fp(\021)e(\000)14 b Fv(lim)1432 4680 y Fk(N)1638 4570 y Fv(1)p 1543 4607 233 4 v 1543 4683 a Fp(j)p Fv(\003)1624 4695 y Fk(L;N)1752 4683 y Fp(j)1799 4626 y Fv(ln)f Fq(\026)p Fv(\(exp[)p Fp(\000)p Fq(\025)2374 4547 y Fj(X)2241 4725 y Fk(x)p Fl(2)p Fm(\003)2369 4733 y Fh(L)p Ff(\000)p Fg(1)p Fh(;N)5 b Ff(\000)p Fg(1)2641 4626 y Fq(J)2687 4638 y Fk(i;n)2775 4626 y Fv(]\))337 b(\(3.71\))515 4881 y Fe(exists)29 b(for)i(al)t(l)f(r)l(e)l(al)h Fq(\025)f Fe(and)1657 4981 y Fq(e)p Fv(\()p Fq(\025)p Fv(\))24 b(=)e Fq(e)p Fv(\(1)c Fp(\000)g Fq(\025)p Fv(\))p Fq(:)931 b Fv(\(3.72\))515 5119 y Fe(Mor)l(e)l(over,)31 b(for)g(\014xe)l(d)e Fq(L)p Fe(,)1149 5318 y Fq(e)1188 5330 y Fk(L;N)1315 5318 y Fv(\()p Fq(\025)p Fv(\))24 b Fp(\021)f(\000)1631 5262 y Fv(1)p 1614 5299 76 4 v 1614 5375 a Fq(N)1713 5318 y Fv(ln)13 b Fq(\026)p Fv(\(exp[)p Fp(\000)p Fq(\025)2288 5239 y Fj(X)2155 5417 y Fk(x)p Fl(2)p Fm(\003)2283 5425 y Fh(L)p Ff(\000)p Fg(1)p Fh(;N)5 b Ff(\000)p Fg(1)2555 5318 y Fq(J)2601 5330 y Fk(i;n)2690 5318 y Fv(]\))422 b(\(3.73\))1905 5650 y(15)p eop %%Page: 16 16 16 15 bop 515 523 a Fe(\(which,)32 b(gener)l(al)t(ly,)f(is)f(of)h(or)l (der)f Fq(L)1646 493 y Fk(d)1685 523 y Fe(\))f(satis\014es)1262 699 y Fp(j)p Fq(e)1324 711 y Fk(L;N)1452 699 y Fv(\()p Fq(\025)p Fv(\))19 b Fp(\000)f Fq(e)1705 711 y Fk(L;N)1833 699 y Fv(\(1)g Fp(\000)g Fq(\025)p Fv(\))p Fp(j)24 b(\024)f Fq(c)p Fv(\()p Fq(\025;)14 b(d)p Fv(\))p Fq(L)2508 665 y Fk(d)p Fl(\000)p Fm(1)3167 699 y Fv(\(3.74\))515 875 y Fe(uniformly)30 b(in)g Fq(N)i Fp(")23 b(1)p Fe(.)515 1098 y Fs(Pro)s(of:)46 b Fv(The)32 b(pro)r(of)g(is)f(a)h(cop)n(y)g(of)g (the)h(pro)r(of)f(of)g(Theorem)f(2.1.)50 b(As)33 b(b)r(efore,)g(w)n(e)f (ha)n(v)n(e)515 1198 y(automatically)-7 b(,)22 b(from)27 b(the)h(Gibbs)e(prop)r(ert)n(y)h(\(as)g(in)g(\(1.14\)\),)g(that)1184 1374 y Fq(\026)p Fv(\(exp)q([)p Fp(\000)p Fq(\025R)1593 1386 y Fk(L;N)1721 1374 y Fv(]\))c(=)g Fq(\026)p Fv(\(exp[)p Fp(\000)p Fv(\(1)18 b Fp(\000)g Fq(\025)p Fv(\))p Fq(R)2502 1386 y Fk(L;N)2631 1374 y Fv(]\))p Fq(:)458 b Fv(\(3.75\))515 1550 y(W)-7 b(e)28 b(no)n(w)f(substitute)g(\(3.69\))g(and)g(use)h(the)g (estimate)d(\(3.70\))i(to)h(p)r(erform)e(the)i(limits.)p 3268 1550 48 48 v 515 1697 a Fs(Remark)20 b(1:)32 b Fv(One)18 b(ma)n(y)e(w)n(onder)i(ab)r(out)g(the)h(existence)e(of)h(the)h(limit)c Fq(e)2743 1709 y Fk(L)2792 1697 y Fv(\()p Fq(\025)p Fv(\))24 b Fp(\021)f Fv(lim)3131 1709 y Fk(N)3208 1697 y Fq(e)3247 1709 y Fk(L;N)3375 1697 y Fv(\()p Fq(\025)p Fv(\))515 1796 y(for)f(\014xed)h Fq(L)p Fv(.)34 b(This)22 b(is)g(certainly)e(exp) r(ected)j(when)f(the)h(steady-state)f Fq(\026)h Fv(is)e(a)i(high)e (temp)r(er-)515 1896 y(ature)31 b(Gibbs)g(state.)48 b(In)32 b(that)g(case,)g(the)g(limit)c(lim)2188 1908 y Fk(L)2252 1896 y Fq(e)2291 1908 y Fk(L)2340 1896 y Fv(\()p Fq(\025)p Fv(\))p Fq(=L)2551 1866 y Fk(d)2620 1896 y Fv(=)h(lim)2829 1908 y Fk(L)2893 1896 y Fq(e)2932 1908 y Fk(L)2981 1896 y Fv(\(1)21 b Fp(\000)g Fq(\025)p Fv(\))p Fq(=L)3341 1866 y Fk(d)515 1996 y Fv(satis\014es)26 b(\(3.72\).)515 2095 y Fs(Remark)c(2:)33 b Fv(Some)19 b(quite)g(similar)c(results)k(w)n (ere)g(discussed)g(already)f(in)h([8].)34 b(There)20 b(ho)n(w-)515 2195 y(ev)n(er)33 b(the)h(dynamics)e(w)n(as)h (deterministic)e(\(w)n(eakly)h(coupled)h(strongly)f(c)n(haotic)g (maps\).)515 2295 y(There)e(again,)g(the)i(metho)r(ds)e(of)h([2,)g(3,)g (22)o(,)g(15)o(])g(can)g(reduce)g(the)g(problem)e(to)i(a)g(higher)515 2394 y(dimensional)i(sym)n(b)r(olic)h(dynamics)h(and)j(the)g(metho)r 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Fv(\()p Fp(\001)p Fq(;)h Fp(\000)p Fv(1\)\))p Fq(:)398 b Fv(\(3.76\))515 3629 y(On)19 b(the)i(other)e (hand,)i(the)f(free)g(energy)f(densit)n(y)f(v)-5 b(anishes)19 b(iden)n(tically)c(for)20 b(PCA)f(\(b)r(ecause)515 3728 y(of)27 b(the)h(normalization)22 b(in)27 b(\(3.65\),)g(see)g([13)o(,)h (18)o(]\),)g(so)f(that,)h(from)e(\(2.54\),)996 3904 y Fq(P)12 b Fv(\()p Fq(U)d Fv(\))23 b(=)g(0)p Fq(;)14 b(P)e Fv(\()p Fq(U)d Fv(\))18 b Fp(\000)g Fq(F)12 b Fv(\()p Fq(U;)i(\031)s(\026)p Fv(\))23 b(=)g Fp(\000)p Fq(s)p Fv(\()p Fq(\026)p Fv(\))c(+)f Fq(\014)t(\031)s(\026)p Fv(\(\010)2674 3916 y Fm(0)2712 3904 y Fv(\()p Fq(U)9 b Fv(\)\))p Fq(:)270 b Fv(\(3.77\))515 4080 y(Hence,)28 b(still)e(in)i(the)g(notation)f(of)h(the)h(previous)d(section,)h (whenev)n(er)h Fq(P)12 b Fv(\()p Fq(U)d Fv(\))24 b(=)g(0)k(\(whic)n(h) 515 4180 y(is)e(v)n(eri\014ed)g(for)h(PCA\),)1360 4356 y Fp(\000)p Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))c(=)g Fq(s)p Fv(\()p Fq(\026)p Fv(\))c Fp(\000)f Fq(\014)t(\031)s(\026)p Fv(\(\010)2310 4368 y Fm(0)2348 4356 y Fv(\()p Fq(U)9 b Fv(\)\))p Fq(:)634 b Fv(\(3.78\))515 4532 y(\(This)25 b(form)n(ula)d(is)j(not)g(correct)f(when)i(w)n(e)f (replace)f(in)h(it)g Fq(\026)h Fv(b)n(y)f Fq(\031)s(\026)p Fv(.\))37 b(That)25 b(is)g(in)n(teresting)515 4632 y(b)r(ecause)i(w)n (e)h(found)h(that)f(no)n(w)f Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))e Fq(>)f Fv(0)j(is)g(min)n(us)g(the)h(sp) r(eci\014c)f(en)n(trop)n(y)g(rate)g Fq(s)p Fv(\()p Fq(\026)p Fv(\))515 4731 y(mo)r(dulo)i(a)j(term)e(whic)n(h)h(is)f(linear)f(in)i Fq(\026)p Fv(.)49 b(W)-7 b(riting)30 b(this)h(out)h(in)f(our)f(presen)n (t)i(notation,)515 4831 y(this)27 b(is)f(nothing)h(else)f(than)627 5019 y Fp(\000)p Fq(\026)p Fv(\()p Fq(J)820 5031 y Fm(0)857 5019 y Fv(\))e(=)e Fp(\000)p Fq(\032)p Fv(\()1142 4940 y Fj(X)1140 5118 y Fk(a)p Fl(2)p Fk(S)1279 5019 y Fq(p)1321 5031 y Fm(0)1358 5019 y Fv(\()p Fq(a)p Fp(j)p Fq(\033)s Fv(\()p Fp(\001)p Fq(;)14 b Fp(\000)p Fv(1\)\))g(ln)f Fq(p)1909 5031 y Fm(0)1946 5019 y Fv(\()p Fq(a)p Fp(j)p Fq(\033)s Fv(\()p Fp(\001)p Fq(;)h Fp(\000)p Fv(1\)\))19 b Fp(\000)f Fq(\026)p Fv(\()p Fp(\000)c Fv(ln)f Fq(p)2746 5031 y Fm(0)2783 5019 y Fv(\()p Fq(\033)s Fv(\(0\))p Fp(j)p Fq(\033)s Fv(\()p Fp(\001)p Fq(;)h Fv(1\)\))p Fq(:)3167 5201 y Fv(\(3.79\))515 5301 y(The)30 b(\014rst)h(term)f(to)g (the)h(righ)n(t)e(is)h(the)h(sp)r(eci\014c)f(en)n(trop)n(y)f(rate)h (\(3.76\))g(\(alw)n(a)n(ys)e(p)r(ositiv)n(e\))515 5401 y(and)f(the)h(second)f(term)g(\(linear)e(in)i Fq(\026)p Fv(\))h(subtracts)e(from)h(this)g(exactly)f(so)g(m)n(uc)n(h)h(that)h (the)1905 5650 y(16)p eop %%Page: 17 17 17 16 bop 515 523 a Fv(net-result)32 b(to)h(the)h(left)f(v)-5 b(anishes)32 b(in)g(the)i(case)e(of)i(time-rev)n(ersal)28 b(symmetry)j(\(detailed)515 623 y(balance\).)i(Of)23 b(course,)f(as)g(in)g(Prop)r(osition)d(2.1)j(w)n(e)g(ha)n(v)n(e)f(an)i (equalit)n(y)d(b)r(et)n(w)n(een)i(the)h(a)n(v)n(er-)515 722 y(aged)j(curren)n(t)h(in)g(the)h(steady)f(state)h Fq(\026)g Fv(and)f(the)h(relativ)n(e)c(en)n(trop)n(y)j Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))h(\(remem)n(b)r (er)515 822 y(that)k(w)n(e)g(to)r(ok)f Fq(m)g Fv(=)e(1)h(=)g(2)p Fq(E)5 b Fv(!\).)51 b(W)-7 b(e)32 b(can)g(therefore)f(conclude)f(that)j (indeed)e Fq(\026)p Fv(\()p Fq(J)3203 834 y Fm(0)3241 822 y Fv(\))h(or)515 922 y Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))22 b(m)n(ust)f(b)r(e)h(regarded)d(as)i(the)h (\(p)r(ositiv)n(e\))d(en)n(trop)n(y)i(pro)r(duction)f(b)n(y)h(our)f (dynamics.)515 1021 y(The)29 b(curren)n(t)g(asso)r(ciated)d(to)k(the)f (breaking)e(of)j(time-rev)n(ersal)24 b(symmetry)i(giv)n(es)i(rise)f(to) 515 1121 y(nothing)k(else)h(than)h(the)g(lo)r(cal)d(\(in)i (space-time\))e(en)n(trop)n(y)i(pro)r(duction)f(whose)h(\015uctua-)515 1220 y(tions)22 b(w)n(e)g(ha)n(v)n(e)g(in)n(v)n(estigated)e(in)j (Theorems)e(2.1)h(and)h(3.1.)35 b(The)23 b(p)r(oin)n(ts)f(made)g(in)g (Section)515 1320 y(2.4)27 b(related)g(to)h(the)h(Green-Kub)r(o)e(form) n(ula)f(remain)g(unaltered)h(and)h(w)n(e)g(do)g(not)g(rep)r(eat)515 1420 y(them)f(here.)515 1519 y(Y)-7 b(et,)27 b(to)e(obtain)g(a)g(ph)n (ysically)d(inspiring)g(picture,)k(w)n(e)f(should)g(connect)g(the)i(ab) r(o)n(v)n(e)d(anal-)515 1619 y(ysis)d(to)h(measurable)d(quan)n(tities.) 33 b(The)22 b(\(second)g(part)g(of)h(the\))g(second)e(la)n(w)g(of)i (thermo)r(dy-)515 1719 y(namics)i(connects)i(the)h(thermo)r(dynamic)d (en)n(trop)n(y)h(of)i(an)f(initial)d(and)j(\014nal)g(equilibrium)515 1818 y(state)h(after)f(some)g(thermo)r(dynamically)22 b(irrev)n(ersible)h(pro)r(cess)k(has)g(tak)n(en)g(place.)37 b(In)28 b(an)515 1918 y(adiabatic)d(non-quasi-stationary)f(pro)r(cess)j (the)i(en)n(trop)n(y)e(can)h(only)f(increase:)37 b Fq(S)3121 1930 y Fk(f)3188 1918 y Fq(>)24 b(S)3328 1930 y Fk(i)3356 1918 y Fv(.)515 2017 y(If)29 b(w)n(e)e(no)n(w)h(w)n(ere)g(to)g(rerun)f (the)i(pro)r(cess)e(in)h(the)h(opp)r(osite)e(direction,)f(simply)f(b)n (y)j(\(ther-)515 2117 y(mo)r(dynamically\))20 b(in)n(v)n(erting)i(all)h (the)i(curren)n(ts)g(\(b)n(y)g(c)n(hanging)e(the)i(sign)f(of)i(all)d (gradien)n(ts)515 2217 y(of)32 b(the)h(in)n(tensiv)n(e)d(v)-5 b(ariables\),)31 b(again)f(the)j(en)n(trop)n(y)e(w)n(ould)g(increase)f (and)j(b)n(y)f(the)h(same)515 2316 y(amoun)n(t)d(as)i(b)r(efore)f(\()p Fq(s)p Fv(\()p Fq(\026)p Fv(\))h(=)e Fq(s)p Fv(\()p Fq(\031)s(\026)p Fv(\)\))j(and)f(w)n(e)f(w)n(ould)g(reac)n(h)g(a)g(new)h(equilibrium)c (with)515 2416 y(en)n(trop)n(y)k(equal)g(to)h Fq(S)1206 2428 y Fk(i)1256 2416 y Fv(+)22 b(2\()p Fq(S)1468 2428 y Fk(f)1533 2416 y Fp(\000)g Fq(S)1671 2428 y Fk(i)1698 2416 y Fv(\).)55 b(While)32 b(w)n(e)g(lac)n(k)g(at)h(this)g(p)r(oin)n (t)f(a)h(more)f(formal)515 2516 y(understanding,)h(w)n(e)g(b)r(eliev)n (e)e(that)j(our)e(en)n(trop)n(y)g(pro)r(duction)g(exactly)g(measures)f (that)515 2615 y(di\013erence:)44 b([)p Fq(S)1001 2627 y Fk(f)1065 2615 y Fp(\000)21 b Fq(S)1202 2627 y Fk(i)1229 2615 y Fv(])h Fp(\000)e Fv([)p Fq(S)1433 2627 y Fk(i)1482 2615 y Fp(\000)h Fq(S)1619 2627 y Fk(f)1662 2615 y Fv(])30 b(=)f(2\()p Fq(S)1934 2627 y Fk(f)1998 2615 y Fp(\000)21 b Fq(S)2135 2627 y Fk(i)2163 2615 y Fv(\))30 b(=)g Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))g Fq(>)g Fv(0.)49 b(More)31 b(generally)515 2715 y(and)h(dep)r(ending)f(on)h (the)g(ph)n(ysical)d(realization)f(of)k(the)g(pro)r(cess,)g(these)g (considerations)515 2814 y(m)n(ust)d(apply)g(to)g(the)i(relev)-5 b(an)n(t)28 b(thermo)r(dynamic)f(p)r(oten)n(tial)h(and)i(`en)n(trop)n (y)e(pro)r(duction')515 2914 y(m)n(ust)e(for)i(example)d(b)r(e)j (replaced)d(b)n(y)j(`w)n(ork)e(done')h(or)g(`free)g(energy)g(pro)r (duction.')515 3014 y(W)-7 b(e)23 b(will)d(further)j(illustrate)c(this) j(b)n(y)h(an)f(example)e(in)i(the)i(follo)n(wing)18 b(subsection)k(but) h(it)f(is)515 3113 y(in)n(teresting)h(to)j(remark)d(already)g(that)k Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))f(repro)r (duces,)f(via)g(the)h(formal)d(analogies)515 3213 y(on)g(the)g(lev)n (el)e(of)i(the)h(v)-5 b(ariational)18 b(principle)j(\(b)r(oth)i(for)g (Gibbs)g(and)g(for)f(SRB)i(states\),)g(the)515 3313 y(en)n(trop)n(y)31 b(pro)r(duction)g(in)g(the)i(con)n(text)e(of)i(the)f(theory)g(of)g (dynamical)c(systems.)50 b(There)515 3412 y(w)n(e)32 b(ha)n(v)n(e)g(that)h(the)g(en)n(trop)n(y)f(pro)r(duction)f(is)h(giv)n (en)f(b)n(y)i(\(1.3\))f(with)h Fq(\032)p Fv(\()2835 3391 y(_)2812 3412 y Fq(S)t Fv(\))h(equal)d(to)i(the)515 3512 y(sum)26 b(of)i(the)g(p)r(ositiv)n(e)d(Ly)n(apuno)n(v)h(exp)r(onen)n (ts)h(with)h(resp)r(ect)f(to)g Fq(\036)2633 3482 y Fl(\000)p Fm(1)2751 3512 y Fv(min)n(us)e(the)j(sum)f(of)515 3611 y(p)r(ositiv)n(e)e(Ly)n(apuno)n(v)h(exp)r(onen)n(ts)h(with)g(resp)r (ect)g(to)h Fq(\036)p Fv(.)37 b(If)28 b Fq(\032)g Fv(is)e(singular)f (with)i(resp)r(ect)g(to)515 3711 y Fq(d\030)37 b Fv(and)c(has)f(no)h(v) -5 b(anishing)30 b(Ly)n(apuno)n(v)h(exp)r(onen)n(t,)k(then)e Fq(\032)p Fv(\()2513 3690 y(_)2490 3711 y Fq(S)5 b Fv(\))32 b Fq(>)f Fv(0,)j(see)f([23)o(].)53 b(In)33 b(our)515 3811 y(case,)27 b Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))c(=)g Fq(\014)t(\031)s(\026)p Fv(\(\010)1344 3823 y Fm(0)1382 3811 y Fv(\()p Fq(U)9 b Fv(\)\))19 b Fp(\000)f Fq(\014)t(\026)p Fv(\(\010)1839 3823 y Fm(0)1877 3811 y Fv(\()p Fq(U)9 b Fv(\)\))24 b(=)f Fq(P)12 b Fv(\()p Fq(U)d Fv(\))18 b Fp(\000)g Fq(F)12 b Fv(\()p Fq(U;)i(\031)s(\026)p Fv(\))23 b Fq(>)g Fv(0.)515 4037 y Fy(3.3)112 b(Illustration.)515 4190 y Fv(W)-7 b(e)26 b(tak)n(e)g(here)g(a)g(closer)e(lo)r(ok)h(at)h (the)h(curren)n(t)e(\(3.67\))h(for)f(Mark)n(o)n(v)g(c)n(hains.)34 b(The)27 b(spatial)515 4290 y(degree)f(of)i(freedom)e Fq(i)d Fp(2)p 1328 4292 45 4 v 1328 4235 43 4 v 1328 4290 3 3 v 1329 4288 V 1330 4286 V 1331 4284 V 1332 4283 V 1333 4281 V 1333 4279 V 1334 4277 V 1335 4275 V 1336 4273 V 1337 4271 V 1338 4269 V 1339 4267 V 1340 4265 V 1341 4263 V 1342 4262 V 1343 4260 V 1344 4258 V 1345 4256 V 1346 4254 V 1347 4252 V 1348 4250 V 1348 4249 V 1349 4247 V 1350 4245 V 1351 4243 V 1352 4241 V 1353 4239 V 1354 4237 V 1355 4235 V 1343 4290 V 1344 4288 V 1345 4286 V 1346 4284 V 1347 4283 V 1348 4281 V 1348 4279 V 1349 4277 V 1350 4275 V 1351 4273 V 1352 4271 V 1353 4269 V 1354 4267 V 1355 4265 V 1356 4263 V 1357 4262 V 1358 4260 V 1359 4258 V 1360 4256 V 1361 4254 V 1362 4252 V 1362 4250 V 1363 4249 V 1364 4247 V 1365 4245 V 1366 4243 V 1367 4241 V 1368 4239 V 1369 4237 V 1370 4235 V 1326 4245 4 12 v 1327 4241 4 8 v 1328 4239 4 6 v 1330 4237 4 4 v 1330 4235 4 2 v 1371 4290 4 12 v 1370 4290 4 8 v 1369 4290 4 6 v 1368 4290 4 4 v 1367 4290 4 2 v 1373 4258 a Fk(d)1439 4290 y Fv(has)k(no)n(w)g(disapp)r (eared)f(and)h(w)n(e)g(m)n(ust)g(study)1053 4435 y Fq(J)1099 4447 y Fk(n)1144 4435 y Fv(\()p Fq(\033)s Fv(\))d Fp(\021)f Fv(ln)13 b Fq(p)p Fv(\()p Fq(\033)s Fv(\()p Fq(n)p Fv(\))p Fp(j)p Fq(\033)s Fv(\()p Fq(n)19 b Fp(\000)g Fv(1\)\))f Fp(\000)g Fv(ln)13 b Fq(p)p Fv(\()p Fq(\033)s Fv(\()p Fq(n)19 b Fp(\000)f Fv(1\))p Fp(j)p Fq(\033)s Fv(\()p Fq(n)p Fv(\)\))327 b(\(3.80\))515 4581 y(for)27 b Fq(\033)f Fp(2)e Fv(\012)854 4593 y Fm(1)919 4581 y Fv(and)j(transition)e (probabilities)1202 4727 y(Prob)h([)p Fq(X)1498 4739 y Fk(n)1566 4727 y Fv(=)d Fq(a)p Fp(j)p Fq(X)1790 4739 y Fk(n)p Fl(\000)p Fm(1)1943 4727 y Fv(=)g Fq(b)p Fv(])f(=)h Fq(p)p Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))p Fq(;)14 b(a;)g(b)22 b Fp(2)i Fq(S)452 b Fv(\(3.81\))515 4873 y(for)37 b(the)h(stationary)d Fq(S)5 b Fv(-v)-5 b(alued)36 b(Mark)n(o)n(v)f(c)n(hain)h Fq(X)2174 4885 y Fk(n)2219 4873 y Fv(.)67 b(The)37 b(steady)g(state)h Fq(\026)f Fv(is)g(no)n(w)f(a)515 4972 y(homogeneous)29 b(one-dimensional)d(Gibbs)31 b(measure)e(and)j(its)e(single-time)e (restriction)g(is)515 5072 y(the)g(stationary)d(measure)g Fq(\032)j Fv(on)f Fq(S)h Fv(:)1687 5010 y Fj(P)1775 5097 y Fk(b)1822 5072 y Fq(p)p Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))p Fq(\032)p Fv(\()p Fq(b)p Fv(\))23 b(=)g Fq(\032)p Fv(\()p Fq(a)p Fv(\))p Fq(;)14 b(a)23 b Fp(2)g Fq(S)5 b Fv(.)639 5172 y(The)28 b(steady)f(state)g(exp)r(ectation)g(of)g(the)h (curren)n(t)f(\(3.80\))g(is)1113 5334 y Fq(\026)p Fv(\()p Fq(J)8 b Fv(\))24 b(=)1393 5255 y Fj(X)1438 5434 y Fk(b)1526 5334 y Fq(\032)p Fv(\()p Fq(b)p Fv(\))1683 5255 y Fj(X)1725 5429 y Fk(a)1817 5334 y Fq(p)p Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\)[ln)13 b Fq(p)p Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))18 b Fp(\000)g Fv(ln)13 b Fq(p)p Fv(\()p Fq(b)p Fp(j)p Fq(a)p Fv(\)])p Fq(:)387 b Fv(\(3.82\))1905 5650 y(17)p eop %%Page: 18 18 18 17 bop 515 523 a Fv(No)n(w)37 b(use)g(that)h(the)g(transition)d (probabilities)e Fq(q)s Fv(\()p Fp(\001j\001)p Fv(\))38 b(for)f(the)h(rev)n(ersed)e(c)n(hain)g(\()p Fq(Y)3229 535 y Fk(n)3314 523 y Fp(\021)515 623 y Fq(X)584 635 y Fl(\000)p Fk(n)681 623 y Fv(\))713 635 y Fk(n)782 623 y Fv(\(with)23 b(distribution)d Fq(\031)s(\026)k Fv(but)g(with)f(the)h (same)e(stationary)e(measure)i Fq(\032)p Fv(\))h(are)g(giv)n(en)515 722 y(b)n(y)1074 865 y Fq(q)s Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))g Fp(\021)50 b Fv(Prob)27 b([)p Fq(X)1716 877 y Fk(n)1784 865 y Fv(=)22 b Fq(a)p Fp(j)p Fq(X)2007 877 y Fk(n)p Fm(+1)2160 865 y Fv(=)g Fq(b)p Fv(])h(=)g Fq(p)p Fv(\()p Fq(b)p Fp(j)p Fq(a)p Fv(\))2636 809 y Fq(\032)p Fv(\()p Fq(a)p Fv(\))p 2636 846 152 4 v 2640 922 a Fq(\032)p Fv(\()p Fq(b)p Fv(\))2797 865 y Fq(:)347 b Fv(\(3.83\))515 1063 y(Since)726 1001 y Fj(P)814 1088 y Fk(b)861 1063 y Fq(\032)p Fv(\()p Fq(b)p Fv(\))1018 1001 y Fj(P)1105 1088 y Fk(a)1159 1063 y Fq(p)p Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))14 b(ln)f Fq(\032)p Fv(\()p Fq(b)p Fv(\))23 b(=)1719 1001 y Fj(P)1807 1088 y Fk(b)1854 1063 y Fq(\032)p Fv(\()p Fq(b)p Fv(\))14 b(ln)f Fq(\032)p Fv(\()p Fq(b)p Fv(\))23 b(=)2348 1001 y Fj(P)2435 1088 y Fk(b)2482 1063 y Fq(\032)p Fv(\()p Fq(b)p Fv(\))2639 1001 y Fj(P)2727 1088 y Fk(a)2781 1063 y Fq(p)p Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))14 b(ln)f Fq(\032)p Fv(\()p Fq(a)p Fv(\),)23 b(w)n(e)515 1163 y(can)32 b(substitute)h(\(3.83\))e(in)n(to)h(\(3.82\)) g(\()p Fq(q)s Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))h(for)f Fq(p)p Fv(\()p Fq(b)p Fp(j)p Fq(a)p Fv(\)\))h(with)f(no)h(extra)f(cost) g(and)g(w)n(e)515 1263 y(obtain)1626 1362 y Fq(\026)p Fv(\()p Fq(J)1754 1374 y Fm(0)1792 1362 y Fv(\))23 b(=)g Fq(\032)p Fv(\()p Fq(S)5 b Fv(\()p Fq(p)p Fp(j)p Fq(q)s Fv(\)\))900 b(\(3.84\))515 1512 y(where)1370 1638 y Fq(S)5 b Fv(\()p Fq(p)p Fp(j)p Fq(q)s Fv(\))23 b(=)1706 1559 y Fj(X)1747 1734 y Fk(a)1839 1638 y Fq(p)p Fv(\()p Fq(a)p Fp(j\001)p Fv(\))14 b(ln)2142 1582 y Fq(p)p Fv(\()p Fq(a)p Fp(j\001)p Fv(\))p 2142 1619 197 4 v 2143 1695 a Fq(q)s Fv(\()p Fq(a)p Fp(j\001)p Fv(\))2372 1638 y Fp(\025)22 b Fv(0)p Fq(:)643 b Fv(\(3.85\))515 1858 y(is)25 b(the)h(relativ)n(e)d (en)n(trop)n(y)h(b)r(et)n(w)n(een)i(the)g(forw)n(ard)e(and)i(the)h(bac) n(kw)n(ard)c(transition)h(proba-)515 1957 y(bilities.)33 b(\(3.85\))25 b(is)h(zero)f(only)g(if)h(the)h(Mark)n(o)n(v)d(c)n(hain)h (is)g(time-rev)n(ersible)c(\(in)26 b(whic)n(h)f(case)515 2057 y Fq(\026)i Fv(=)f Fq(\031)s(\026)p Fv(\).)45 b(Then,)31 b(\(3.83\))e(for)g Fq(q)s Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))e(=)g Fq(p)p Fv(\()p Fq(a)p Fp(j)p Fq(b)p Fv(\))j(b)r(ecomes)e (the)j(detailed)c(balance)i(condi-)515 2157 y(tion.)50 b(Relation)29 b(\(3.84\))j(is)f(nothing)g(but)i(\(2.52\))e(sp)r (eci\014ed)h(to)g(the)g(con)n(text)g(of)g(Mark)n(o)n(v)515 2256 y(c)n(hains.)639 2356 y(A)23 b(second)e(less)g(trivial)e(and)j(ph) n(ysically)c(in)n(teresting)i(illustration)d(can)22 b(b)r(e)h(tak)n(en) e(from)515 2456 y(a)28 b(mo)r(del)f(of)h(hopping)f(conductivit)n(y)-7 b(.)38 b(It)29 b(is)e(a)h(bulk)g(driv)n(en)f(di\013usiv)n(e)g(lattice)f (gas)i(where)515 2555 y(c)n(harged)d(particles,)e(sub)5 b(ject)27 b(to)f(an)g(on-site)f(exclusion,)f(hop)j(on)f(a)g(ring)e(in)i (the)h(presence)515 2655 y(of)33 b(an)f(electric)f(\014eld.)52 b(The)33 b(con\014guration)d(space)j(is)e(\012)2315 2667 y Fm(1)2385 2655 y Fv(=)g Fp(f)p Fv(0)p Fq(;)14 b Fv(1)p Fp(g)2686 2625 y Fl(T)2774 2655 y Fv(with)33 b Fq(\030)t Fv(\()p Fq(i)p Fv(\))f(=)f(0)i(or)515 2754 y(1)g(dep)r(ending)f(on)i (whether)f(the)h(site)e Fq(i)g Fp(2)h(T)55 b Fv(is)32 b(empt)n(y)h(or)f(o)r(ccupied.)53 b(W)-7 b(e)34 b(tak)n(e)f(for)g Fp(T)515 2854 y Fv(the)h(set)h Fp(f)p Fv(1)p Fq(;)14 b(:)g(:)g(:)e(;)i(`)p Fp(g)34 b Fv(with)f(p)r(erio)r(dic)g(b)r(oundary) g(conditions.)55 b(T)-7 b(o)33 b(eac)n(h)h(b)r(ond)g(\()p Fq(i;)14 b(i)23 b Fv(+)f(1\))515 2954 y(in)k(the)i(ring)e(and)h(indep)r (enden)n(tly)f(of)i(all)d(the)j(rest)f(there)g(is)f(asso)r(ciated)f(a)i (P)n(oisson)e(clo)r(c)n(k)515 3053 y(\(with)i(rate)g(1\).)36 b(If)28 b(the)g(clo)r(c)n(k)d(rings)h(and)h Fq(\030)t Fv(\()p Fq(i)p Fv(\))d(=)e(1)p Fq(;)14 b(\030)t Fv(\()p Fq(i)k Fv(+)f(1\))23 b(=)g(0)k(then)h(the)f(particle)e(at)i Fq(i)515 3153 y Fv(jumps)j(to)g Fq(i)20 b Fv(+)g(1)29 b(with)h(probabilit)n(y)c Fq(p)p Fv(.)45 b(If)31 b(on)f(the)h(other)e (hand,)i Fq(\030)t Fv(\()p Fq(i)p Fv(\))d(=)f(0)p Fq(;)14 b(\030)t Fv(\()p Fq(i)20 b Fv(+)g(1\))27 b(=)g(1)515 3253 y(the)i(particle)d(jumps)j(to)g Fq(i)g Fv(with)f(probabilit)n(y)d Fq(q)s Fv(.)41 b(Therefore,)28 b(the)i(`probabilit)n(y)25 b(p)r(er)j(unit)515 3352 y(time')c(to)i(mak)n(e)e(the)i(transition)e (from)g Fq(\030)30 b Fv(to)c Fq(\030)1959 3322 y Fk(i;i)p Fm(+1)2140 3352 y Fv(\(in)g(whic)n(h)e(the)j(o)r(ccupations)d(of)i Fq(i)f Fv(and)515 3452 y Fq(i)18 b Fv(+)g(1)27 b(are)g(in)n(terc)n (hanged\))e(is)i(giv)n(en)e(b)n(y)j(the)g(exc)n(hange)e(rate)964 3635 y Fq(c)p Fv(\()p Fq(i;)14 b(i)k Fv(+)g(1)p Fq(;)c(\030)t Fv(\))23 b(=)f Fq(p\030)t Fv(\()p Fq(i)p Fv(\)\(1)d Fp(\000)f Fq(\030)t Fv(\()p Fq(i)g Fv(+)g(1\)\))h(+)f Fq(q)s(\030)t Fv(\()p Fq(i)g Fv(+)g(1\)\(1)g Fp(\000)g Fq(\030)t Fv(\()p Fq(i)p Fv(\)\))p Fq(:)238 b Fv(\(3.86\))515 3817 y(and)27 b(should)f(b)r(e)i(though)n(t)f(of)g(as)f(a)h(con)n(tin)n(uous)e(time)h (analogue)f(of)i(\(3.63\).)36 b(It)28 b(is)e(natural)515 3917 y(to)g(call)e Fq(E)k Fv(=)22 b(ln)13 b Fq(p=q)29 b Fv(the)d(electric)e(\014eld.)36 b(This)25 b(mo)r(del)f(is)h(called)f (the)j(asymmetric)22 b(simple)515 4016 y(exclusion)j(pro)r(cess)j(and)g (it)f(is)h(also)e(considered)g(in)i([19)o(].)39 b(Strictly)26 b(sp)r(eaking,)h(it)h(is)f(not)h(a)515 4116 y(PCA)j(but)h(a)f(con)n (tin)n(uous)e(time)h(pro)r(cess)h(with)f(sequen)n(tial)f(up)r(dating.) 48 b(Ho)n(w)n(ev)n(er,)30 b(since)515 4216 y(it)24 b(is)g(a)g(jump)h (pro)r(cess,)f(the)h(c)n(hange)f(with)g(resp)r(ect)h(to)f(the)i(PCA)e (of)h(ab)r(o)n(v)n(e)e(just)j(amoun)n(ts)515 4315 y(to)h(randomizing)d (the)j(time)g(b)r(et)n(w)n(een)g(successiv)n(e)e(transitions.)515 4415 y(Eac)n(h)i(uniform)f(pro)r(duct)i(measure)e Fq(\032)i Fv(is)f(time-in)n(v)-5 b(arian)n(t)23 b(for)k(this)g(pro)r(cess)g(and)h (w)n(e)g(con-)515 4515 y(sider)j(the)i(steady)g(state)f Fq(\026)h Fv(starting)e(in)h(this)g(in)n(v)-5 b(arian)n(t)30 b(state.)53 b(If)33 b(w)n(e)f(no)n(w)h(consider)d(a)515 4614 y(realization)25 b Fq(\033)33 b Fv(of)c(the)h(pro)r(cess)e(in)h (whic)n(h)f(at)h(a)g(certain)f(time,)g(when)i(the)g(con\014guration)515 4714 y(is)c Fq(\030)h Fp(2)c Fv(\012)799 4726 y Fm(1)837 4714 y Fv(,)k(a)g(particle)e(hops)h(from)g(site)g Fq(i)h Fv(to)g Fq(i)18 b Fv(+)f(1,)27 b(then)h(the)f(time-rev)n(ersed)d(tra)5 b(jectory)515 4813 y(sho)n(ws)27 b(a)h(particle)d(jumping)i(from)g Fq(i)18 b Fv(+)h(1)27 b(to)h Fq(i)p Fv(.)39 b(The)28 b(con)n(tribution)d(of)k(this)e(ev)n(en)n(t)h(to)g(the)515 4913 y(en)n(trop)n(y)e(pro)r(duction)g(is)h(therefore)556 5096 y(ln)13 b Fq(c)p Fv(\()p Fq(i;)h(i)r Fv(+)r(1)p Fq(;)g(\030)t Fv(\))r Fp(\000)r Fv(ln)d Fq(c)p Fv(\()p Fq(i;)j(i)r Fv(+)r(1)p Fq(;)g(\030)1523 5061 y Fk(i;i)p Fm(+1)1676 5096 y Fv(\))24 b(=)e Fq(E)5 b Fv([)p Fq(\030)t Fv(\()p Fq(i)p Fv(\)\(1)r Fp(\000)r Fq(\030)t Fv(\()p Fq(i)r Fv(+)r(1\)\))r Fp(\000)r Fq(\030)t Fv(\()p Fq(i)r Fv(+)r(1\)\(1)r Fp(\000)r Fq(\030)t Fv(\()p Fq(i)p Fv(\)\)])p Fq(:)40 b Fv(\(3.87\))515 5278 y(This)25 b(form)n(ula)d(is)j(the)h(con) n(tin)n(uous)e(time)g(analogue)g(of)h(\(3.80\))g(or)g(\(3.67\))g(\(but) i(w)n(e)f(do)f(not)515 5378 y(tak)n(e)32 b Fq(E)k Fv(=)31 b(1)p Fq(=)p Fv(2)g(here\))h(with)g Fq(\030)37 b Fv(the)c (con\014guration)d(righ)n(t)h(b)r(efore)h(the)h(jump)g(and)f Fq(\030)3224 5348 y Fk(i;i)p Fm(+1)1905 5650 y Fv(18)p eop %%Page: 19 19 19 18 bop 515 523 a Fv(the)26 b(con\014guration)e(righ)n(t)g(after)i (the)g(jump)g(in)g(the)g(tra)5 b(jectory)25 b Fq(\033)s Fv(.)36 b(Of)27 b(course,)e(this)g(jump)515 623 y(in)f Fq(\033)k Fv(itself)c(happ)r(ens)h(with)f(a)h(rate)f Fq(c)p Fv(\()p Fq(i;)14 b(i)f Fv(+)g(1)p Fq(;)h(\030)t Fv(\).)35 b(W)-7 b(e)26 b(see)e(therefore)g(that)i(the)f(deriv)-5 b(ativ)n(e)515 722 y(of)27 b(\(3.87\))g(with)g(resp)r(ect)h(to)f Fq(E)33 b Fv(has)27 b(exp)r(ectation)515 905 y Fq(\026)p Fv(\()p Fq(J)643 917 y Fk(i;t)716 905 y Fv(\))c(=)g Fq(\032)p Fv(\()p Fq(c)p Fv(\()p Fq(i;)14 b(i)h Fv(+)g(1)p Fq(;)f(\030)t Fv(\)[)p Fq(\030)t Fv(\()p Fq(i)p Fv(\)\(1)i Fp(\000)f Fq(\030)t Fv(\()p Fq(i)g Fv(+)h(1\)\))f Fp(\000)h Fq(\030)t Fv(\()p Fq(i)f Fv(+)h(1\)\(1)f Fp(\000)g Fq(\030)t Fv(\()p Fq(i)p Fv(\)\)]\))24 b(=)f(\()p Fq(p)15 b Fp(\000)h Fq(q)s Fv(\))p Fq(u)p Fv(\(1)f Fp(\000)g Fq(u)p Fv(\))3167 1005 y(\(3.88\))515 1104 y(for)34 b Fq(u)h Fp(\021)g Fq(\032)p Fv(\()p Fq(\030)t Fv(\()p Fq(i)p Fv(\)\))g(the)h(densit)n(y)-7 b(.)57 b(\(3.88\))34 b(is)g(indeed)g(the)h(curren)n(t)f(as)h(it)f(app)r (ears)g(in)g(the)515 1204 y(h)n(ydro)r(dynamic)f(equation,)k(here)f (the)h(Burgers)e(equation,)i(through)f(whic)n(h)f(a)h(densit)n(y)515 1303 y(pro\014le)28 b(ev)n(olv)n(es.)42 b(The)30 b(\015uctuations)g(of) g(the)g(particle)e(curren)n(t)h(satisfy)g(\(2.32\))h(or)f(\(3.72\))515 1403 y(\(with)e Fq(E)h Fv(=)23 b(1)p Fq(=)p Fv(2\),)j(see)i(also)d([19) o(].)37 b(The)28 b(en)n(trop)n(y)e(pro)r(duction)h(\(as)g(in)g (\(3.84\)-\(3.85\)\))f(is)658 1577 y(1)p 658 1614 42 4 v 658 1691 a(2)710 1634 y Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))e(=)e Fq(\032)p Fv(\()p Fq(c)p Fv(\()p Fq(i;)14 b(i)k Fv(+)g(1)p Fq(;)c(\030)t Fv(\))g(ln)1772 1577 y Fq(c)p Fv(\()p Fq(i;)g(i)j Fv(+)h(1)p Fq(;)c(\030)t Fv(\))p 1694 1614 570 4 v 1694 1691 a Fq(c)p Fv(\()p Fq(i;)g(i)k Fv(+)g(1)p Fq(;)c(\030)2077 1667 y Fk(i;i)p Fm(+1)2231 1691 y Fv(\))2274 1634 y(\))23 b(=)g Fq(E)5 b Fv(\()p Fq(p)18 b Fp(\000)g Fq(q)s Fv(\))p Fq(u)p Fv(\(1)g Fp(\000)g Fq(u)p Fv(\))134 b(\(3.89\))515 1860 y(whic)n(h)36 b(is)f(the)j(\014eld)e(times)f(the)i(curren)n(t)f (and)h(is)f(left)g(in)n(v)-5 b(arian)n(t)34 b(b)n(y)j(c)n(hanging)d Fq(E)42 b Fv(in)n(to)515 1960 y Fp(\000)p Fq(E)5 b Fv(.)57 b(If,)36 b(to)f(b)r(e)f(sp)r(eci\014c,)i(w)n(e)e(tak)n(e)g Fq(p)g Fv(=)g(1)p Fq(=)p Fv(\(1)22 b(+)g Fq(e)2186 1930 y Fl(\000)p Fk(E)2294 1960 y Fv(\))34 b(=)g(1)23 b Fp(\000)f Fq(q)s Fv(,)37 b(then,)f(in)e(quadratic)515 2060 y(appro)n(ximation,) 1529 2137 y(1)p 1529 2175 42 4 v 1529 2251 a(2)1581 2194 y Fq(s)p Fv(\()p Fq(\026)p Fp(j)p Fq(\031)s(\026)p Fv(\))24 b(=)e Fq(u)p Fv(\(1)c Fp(\000)g Fq(u)p Fv(\))p Fq(E)2337 2159 y Fm(2)3167 2194 y Fv(\(3.90\))515 2366 y(whic)n(h)36 b(is)g(the)h(dissipated)e(heat)i(through)f(a)h(conductor)f(in)h(an)f (electric)f(\014eld)h Fq(E)43 b Fv(with)515 2466 y(Ohmic)34 b(conductivit)n(y)f Fq(u)p Fv(\(1)24 b Fp(\000)f Fq(u)p Fv(\))37 b(=)g Fq(\026)p Fv(\()p Fq(J)1865 2436 y Fm(2)1857 2487 y(0)1902 2466 y Fv(\)\()p Fq(E)43 b Fv(=)36 b(0\))h(=)g Fq(\032)p Fv(\()p Fq(c)p Fv(\(0)p Fq(;)14 b Fv(1)p Fq(;)g(\030)t Fv(\)[)p Fq(\030)t Fv(\(0\)\(1)23 b Fp(\000)h Fq(\030)t Fv(\(1\)\))g Fp(\000)515 2566 y Fq(\030)t Fv(\(1\)\(1)g Fp(\000)g Fq(\030)t Fv(\(0\)\)])1049 2535 y Fm(2)1087 2566 y Fv(\))36 b(giv)n(en)f(b)n(y)h(the)h(v)-5 b(ariance)34 b(of)i(the)h(curren)n(t.)62 b(This)35 b(mo)r(del)g(\(together)515 2665 y(with)30 b(the)h(mo)r(dels)e(discussed)g(in)h([19)o(]\))h (illustrates)c(that)k(the)g(metho)r(ds)f(exp)r(osed)h(in)f(the)515 2765 y(presen)n(t)h(pap)r(er)g(are)g(not)g(restricted)f(to)i(just)g (PCA.)g(W)-7 b(e)32 b(ha)n(v)n(e)e(restricted)g(us)i(here)f(to)g(a)515 2864 y(somewhat)23 b(informal)f(treatmen)n(t)i(of)h(the)h(asp)r(ects)f (concerning)e(the)i(en)n(trop)n(y)f(pro)r(duction)515 2964 y(in)g(the)h(mo)r(del)e(as)h(it)g(will)e(b)r(e)j(included)f(in)g (a)g(future)h(publication)d(dealing)g(with)i(the)i(lo)r(cal)515 3064 y(\015uctuation)g(theorem,)h([21)o(].)515 3338 y Fr(4)134 b(Concluding)43 b(remark.)515 3520 y Fv(It)c(do)r(es)g(not)h (seem)e(unreasonable)e(that)k(Gibbs')e(v)-5 b(ariational)35 b(principle)h(determining)515 3620 y(the)27 b(conditions)d(of)i (equilibrium)c(can)k(b)r(e)h(generalized)c(to)j(certain)f (nonequilibrium)d(con-)515 3719 y(ditions.)57 b(In)35 b(this)g(note)g(w)n(e)f(ha)n(v)n(e)g(sho)n(wn)h(that)g(describing)d (the)k(steady)e(state)h(via)f(the)515 3819 y(standard)23 b(metho)r(ds)g(of)h(the)g(Gibbs)f(formalism)c(leads)k(directly)e(to)j (the)g(\015uctuation)g(theo-)515 3919 y(rem.)34 b(This)23 b(is)g(true)h(close)e(or)h(far)g(from)g(equilibrium)c(b)r(ecause)k(it)h (follo)n(ws)c(quite)k(generally)515 4018 y(from)g(the)i(de\014ning)f (Gibbs)g(prop)r(ert)n(y)f(itself.)35 b(F)-7 b(rom)24 b(this)h(`Gibbsian')f(p)r(oin)n(t)h(of)h(view,)f(ap-)515 4118 y(plying)e(the)i(lo)r(cal)e(\015uctuation)h(theorem)f(to)i(v)-5 b(arious)23 b(sp)r(eci\014c)h(mo)r(dels)f(is)h(to)h(add)g(sp)r (eci\014c)515 4218 y(observ)-5 b(able)25 b(consequences)h(to)i(the)g (studies)e(of)i(E.T.)f(Ja)n(ynes,)f([14].)515 4492 y Fr(References)563 4666 y Ft([1])42 b(Bonetto,)28 b(F.,)g(Galla)n(v)n (otti,)h(G.)e(and)f(Garrido,)i(P)-6 b(.)27 b(\(1997\))h(Chaotic)g (principle:)37 b(an)26 b(exp)r(eri-)685 4757 y(men)n(tal)f(test,)h Fd(Ph)n(ysica)g(D)g Fu(105)p Ft(,)g(226.)563 4915 y([2])42 b(Bricmon)n(t,)22 b(J.)f(and)g(Kupiainen,)g(A.)g(\(1997\))h(In\014nite) e(dimensional)h(SRB)f(measures,)i Fd(Ph)n(ys-)685 5006 y(ica)27 b(D)e Fu(103)p Ft(,)i(18{33.)563 5164 y([3])42 b(Bricmon)n(t,)22 b(J.)f(and)f(Kupiainen,)h(A.)f(\(1996\))i(High)e (temp)r(erature)f(expansions)i(and)f(dynam-)685 5255 y(ical)27 b(systems,)f Fd(Comm.)e(Math.)j(Ph)n(ys.)e Fu(178)p Ft(,)i(703{732.)1905 5650 y Fv(19)p eop %%Page: 20 20 20 19 bop 563 523 a Ft([4])42 b(v)l(an)30 b(En)n(ter,)i(A.C.D.,)h(F)-6 b(ern\023)-38 b(andez,)32 b(R.)e(and)h(Sok)l(al)f(A.D.)h(\(1993\))h (Regularit)n(y)e(prop)r(erties)685 614 y(and)37 b(pathologies)i(of)f(p) r(osition-space)h(renormalization)f(transformations:)58 b(scop)r(e)38 b(and)685 706 y(limitations)27 b(of)f(Gibbsian)g(theory) -6 b(,)26 b Fd(J.)g(Stat.)g(Ph)n(ys.)f Fu(72)p Ft(,)i(879{1167.)563 863 y([5])42 b(Ev)l(ans,)31 b(D.J.,)h(Cohen,)f(E.G.D)g(and)e(Morriss,)k (G.P)-6 b(.)30 b(\(1993\))h(Probabilit)n(y)g(of)f(second)g(la)n(w)685 955 y(violations)e(in)d(steady)h(\015o)n(ws,)g Fd(Ph)n(ys.)g(Rev.)f (Lett.)h Fu(71)p Ft(,)g(2401{2404.)563 1112 y([6])42 b(Galla)n(v)n(otti,)35 b(G.)d(\(1996\))g(Chaotic)h(h)n(yp)r(othesis:)45 b(Onsager)32 b(recipro)r(cit)n(y)g(and)f(\015uctuation-)685 1204 y(dissipation)c(theorem,)e Fd(J.)i(Stat.)f(Ph)n(ys.)f Fu(84)p Ft(,)h(899-926.)563 1362 y([7])42 b(Galla)n(v)n(otti,)34 b(G.)c(\(1996\))i(Extension)e(of)h(Onsager's)h(recipro)r(cit)n(y)f(to)f (large)i(\014elds)e(and)g(the)685 1453 y(c)n(haotic)d(h)n(yp)r (othesis,)f Fd(Ph)n(ys.)f(Rev.)g(Lett.)h Fu(77)p Ft(,)g(4334{4337.)563 1611 y([8])42 b(Galla)n(v)n(otti,)28 b(G.)e(\(1998\))h(A)e(lo)r(cal)i (\015uctuation)f(theorem.)f(Preprin)n(t.)563 1768 y([9])42 b(Galla)n(v)n(otti,)27 b(G.)e(\(1998\))h(Chaotic)g(dynamics,)e (\015uctuations,)i(nonequilibrium)d(ensem)n(bles,)685 1860 y Fd(Chaos)k Fu(8)p Ft(,)f(384{392.)524 2017 y([10])43 b(Galla)n(v)n(otti,)24 b(G.)d(and)f(Cohen,)j(E.G.D.)e(\(1995\))h (Dynamical)e(ensem)n(bles)h(in)f(nonequilibrium)685 2109 y(statistical)28 b(mec)n(hanics,)e Fd(Ph)n(ys.)f(Rev.)g(Lett.)h Fu(74)p Ft(,)g(2694{2697.)524 2267 y([11])43 b(Galla)n(v)n(otti,)h(G.) 39 b(and)f(Cohen,)43 b(E.G.D.)d(\(1995\))g(Dynamical)e(ensem)n(bles)g (in)g(stationary)685 2358 y(states,)27 b Fd(J.)g(Stat.)e(Ph)n(ys.)h Fu(80)p Ft(,)g(931{970.)524 2516 y([12])43 b(Georgii,)27 b(H.-O.)c(\(1988\))j Fc(Gibbs)g(me)l(asur)l(es)i(and)e(phase)i(tr)l (ansitions)p Ft(,)e(de)e(Gruyter,)g(Berlin)h Fb(\001)685 2607 y Ft(New)h(Y)-6 b(ork.)524 2765 y([13])43 b(Goldstein,)e(S.,)f (Kuik,)f(R.,)g(Leb)r(o)n(witz,)i(J.L.)d(and)e(Maes,)41 b(C.)c(\(1989\))h(F)-6 b(rom)36 b(PCA's)i(to)685 2856 y(Equilibrium)25 b(Systems)g(and)g(Bac)n(k,)h Fd(Comm.)f(Math.)h(Ph)n (ys.)g Fu(125)p Ft(,)g(71{79.)524 3014 y([14])43 b(Ja)n(ynes,)i(E.T.)c (\(1989\))g(Clearing)h(up)d(Mysteries;)49 b(the)40 b(Original)h(Goal,)k (in:)63 b(Pro)r(ceed-)685 3105 y(ings)41 b(of)f(the)f(8'th)h(In)n (ternational)f(W)-6 b(orkshop)39 b(in)h(Maxim)n(um)d(En)n(trop)n(y)i (and)g(Ba)n(y)n(esian)685 3196 y(Metho)r(ds,)g(Cam)n(bridge,)e (England,)h(August)d(1{5,)k(1988;)i(J.)36 b(Skilling,)i(Editor;)j(Klu)n (w)n(er)685 3288 y(Academic)57 b(Publishers,)66 b(Dordrec)n(h)n(t,)e (Holland.)58 b(See)f(also)i(in)e Fc(Pap)l(ers)h(on)e(Pr)l(ob)l(a-)685 3379 y(bility,)61 b(Statistics,)i(and)55 b(Statistic)l(al)h(Physics)p Ft(,)64 b(D.)55 b(Reidel)g(Publishing)h(Co.,)64 b(Dor-)685 3470 y(drec)n(h)n(t,)47 b(Holland,)i(R.D.)42 b(Rosenkran)n(tz,)48 b(Editor.)c(Reprin)n(ts)e(of)i(13)g(pap)r(ers.)g(See)f(also)685 3562 y(h)n(ttp://ba)n(y)n(es.wustl.edu/etj/no)r(de1.h)n(tml.)524 3719 y([15])g(Miaoh)n(uang)f(Jiang)g(and)f(P)n(esin,)k(Y.B.)d(\(1997\)) g(Equilibrium)e(Measures)i(for)g(Coupled)685 3811 y(Map)34 b(Lattices:)51 b(Existence,)36 b(Uniqueness)d(and)g(Finite-Dimensional) h(Appro)n(ximations.)685 3902 y(Preprin)n(t.)524 4060 y([16])43 b(Kurc)n(han,)38 b(J.)f(\(1998\))g(Fluctuation)g(theorem)e (for)i(sto)r(c)n(hastic)g(dynamics,)h Fd(J.)e(Ph)n(ys.)g(A:)685 4151 y(Math.)27 b(Gen.)f Fu(31)p Ft(,)g(3719{3729.)524 4309 y([17])43 b(Lanford)33 b(I)r(I)r(I,)g(O.E.)g(\(1973\))g(En)n(trop) n(y)f(and)g(equilibrium)f(states)j(in)e(classical)j(statistical)685 4400 y(mec)n(hanics,)25 b(in)e Fd(Statistical)j(Mec)n(hanics)f(and)e (Mathematical)i(Problems)f(\(Batelle)i(Seattle)685 4491 y(Rencon)n(tres)g(1971\))p Ft(,)j(Lecture)d(Notes)g(in)h(Ph)n(ysics)f (No.)h(20)g(\(Springer-V)-6 b(erlag,)27 b(Berlin\),)g(1{)685 4583 y(113.)33 b(Comets,)g(F.)e(\(1986\))i(Grandes)e(d)n(\023)-36 b(eviations)32 b(p)r(our)f(des)g(c)n(hamps)f(de)h(Gibbs)g(sur)p 3277 4585 45 4 v 3277 4528 43 4 v 3277 4583 3 3 v 3278 4581 V 3279 4579 V 3279 4577 V 3280 4576 V 3281 4574 V 3282 4572 V 3283 4570 V 3284 4568 V 3285 4566 V 3286 4564 V 3287 4562 V 3288 4560 V 3289 4559 V 3290 4557 V 3291 4555 V 3292 4553 V 3293 4551 V 3293 4549 V 3294 4547 V 3295 4545 V 3296 4544 V 3297 4542 V 3298 4540 V 3299 4538 V 3300 4536 V 3301 4534 V 3302 4532 V 3303 4531 V 3304 4529 V 3292 4583 V 3293 4581 V 3293 4579 V 3294 4577 V 3295 4576 V 3296 4574 V 3297 4572 V 3298 4570 V 3299 4568 V 3300 4566 V 3301 4564 V 3302 4562 V 3303 4560 V 3304 4559 V 3305 4557 V 3306 4555 V 3307 4553 V 3307 4551 V 3308 4549 V 3309 4547 V 3310 4545 V 3311 4544 V 3312 4542 V 3313 4540 V 3314 4538 V 3315 4536 V 3316 4534 V 3317 4532 V 3318 4531 V 3319 4529 V 3275 4538 4 12 v 3276 4534 4 8 v 3277 4532 4 6 v 3278 4531 4 4 v 3279 4529 4 2 v 3320 4583 4 12 v 3319 4583 4 8 v 3318 4583 4 6 v 3316 4583 4 4 v 3316 4583 4 2 v 3322 4551 a Fa(d)3358 4583 y Ft(,)685 4674 y Fd(C.R.)26 b(Acad.)f(Sci.)h(P)n(aris)g(I)f Fu(303)p Ft(,)h(511{513.)i(Olla,)f(S.)e (\(1988\))h(Large)g(deviations)g(for)g(Gibbs)685 4765 y(random)f(\014elds,)h Fd(Prob.)g(Th.)h(Rel.)e(Fields)i Fu(77)p Ft(,)f(343{357.)524 4923 y([18])43 b(Leb)r(o)n(witz,)29 b(J.L.,)f(Maes,)g(C.)f(and)g(Sp)r(eer,)g(E.R.)g(\(1990\))h(Statistical) g(mec)n(hanics)e(of)h(proba-)685 5015 y(bilistic)g(cellular)g (automata,)f Fd(J.)h(Stat.)e(Ph)n(ys.)h Fu(59)p Ft(,)g(117{170.)524 5172 y([19])43 b(Leb)r(o)n(witz,)24 b(J.L.)e(and)f(Sp)r(ohn,)h(H.)f (\(1998\))h(The)g(Galla)n(v)n(otti-Cohen)g(Fluctuation)g(Theorem)685 5264 y(for)27 b(Sto)r(c)n(hastic)f(Dynamics,)f(Rutgers)h(Univ)n(ersit)n (y)e(preprin)n(t.)1905 5650 y Fv(20)p eop %%Page: 21 21 21 20 bop 524 523 a Ft([20])43 b(Maes,)c(C.,)e(Redig,)g(F.)e(and)f(V)-6 b(an)34 b(Mo\013aert,)k(A.)c(\(1998\))i(The)f(restriction)g(of)g(the)g (Ising)685 614 y(mo)r(del)26 b(to)g(a)g(la)n(y)n(er.)g(Preprin)n(t.)524 772 y([21])43 b(Maes,)28 b(C.,)e(Redig,)g(F.)g(and)g(V)-6 b(an)25 b(Mo\013aert,)h(A.)g(\(1998\))h(W)-6 b(ork)25 b(in)h(progress.)524 930 y([22])43 b(P)n(esin,)32 b(Y.B.)f(and)f (Sinai,)i(Y.G.)e(\(1991\))h(Space-time)e(c)n(haos)i(in)f(c)n(hains)g (of)h(w)n(eakly)f(in)n(ter-)685 1021 y(acting)d(h)n(yp)r(erb)r(olic)e (mappings,)h Fd(Adv.)e(So)n(v.)i(Math.)g Fu(3)p Ft(,)g(165{198.)524 1179 y([23])43 b(Ruelle,)32 b(D.)e(\(1996\))h(P)n(ositivit)n(y)g(of)g (en)n(trop)n(y)e(pro)r(duction)h(in)g(nonequilibrium)f(statistical)685 1270 y(mec)n(hanics,)d Fd(J.)g(Stat.)g(Ph)n(ys.)g Fu(85)p Ft(,)g(1{25.)524 1428 y([24])43 b(Ruelle,)37 b(D.)d(\(1997\))i(En)n (trop)n(y)d(pro)r(duction)i(in)f(nonequilibrium)f(statistical)j(mec)n (hanics,)685 1519 y Fd(Comm.)25 b(Math.)h(Ph)n(ys.)g Fu(189)p Ft(,)h(365{371.)524 1677 y([25])43 b(Ruelle,)35 b(D.)e(\(1978\))g(Sensitiv)n(e)g(dep)r(endence)e(on)i(initial)h (conditions)f(and)f(turbulen)n(t)f(b)r(e-)685 1768 y(ha)n(vior)25 b(of)g(dynamical)f(systems,)g Fd(Annals)g(of)h(the)f(New)g(Y)-6 b(ork)24 b(Academ)n(y)e(of)j(Sciences)g Fu(356)p Ft(,)685 1860 y(408{416.)524 2017 y([26])43 b(Ruelle,)26 b(D.)f(\(1998\))h(Smo)r (oth)e(dynamics)g(and)h(new)g(theoretical)h(ideas)g(in)f (nonequilibrium)685 2109 y(statistical)20 b(mec)n(hanics.)d(Rutgers)h (Univ)n(ersit)n(y)e(Lecture)i(Notes,)h(Octob)r(er-No)n(v)n(em)n(b)r(er) d(1997-)685 2200 y(98)27 b(\(unpublished\).)524 2358 y([27])43 b(Simon,)20 b(B.)g(\(1993\))h Fc(The)h(Statistic)l(al)h(Me)l (chanics)g(of)e(L)l(attic)l(e)j(Gases)p Ft(,)e(V)-6 b(olume)18 b(1,)j(Princeton)685 2449 y(Univ)n(ersit)n(y)k(Press,)i(Princeton.)1905 5650 y Fv(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF