Content-Type: multipart/mixed; boundary="-------------9912140614928" This is a multi-part message in MIME format. ---------------9912140614928 Content-Type: text/plain; name="99-477.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-477.comments" Related papers by the same authors can be found via url:http://tfdec1.fys.kuleuven.ac.be/~christ/ ---------------9912140614928 Content-Type: text/plain; name="99-477.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-477.keywords" entropy production, nonequilibrium Gibbs states ---------------9912140614928 Content-Type: application/postscript; name="brus.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="brus.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.66 (MiKTeX 1.07) Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: c:\users\christ\irrev\brus.dvi %%CreationDate: Tue Dec 14 12:42:22 1999 %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 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(giv)n(e)h(rise)g(to)h(a)h(space-time)d(steady)i(state)h(whic)n(h)e(is) h(in)n(v)-5 b(ari-)515 1519 y(an)n(t)36 b(under)h(time-rev)n(ersal.)59 b(Imp)r(osing)34 b(gradien)n(ts)g(add)j(ho)n(w)n(ev)n(er)d(time-rev)n (ersal)e(sym-)515 1719 y(metry)k(breaking)f(terms)h(to)i(the)g(action)e (functional)f(\(generating)h(curren)n(ts\))g(and)i(the)515 1918 y(time-rev)n(ersed)23 b(state)28 b(is)e(no)h(longer)e(equal)h(to)h (the)h(original)23 b(one.)36 b(F)-7 b(or)27 b(almost)e(all)g(space-)515 2117 y(time)31 b(con\014gurations,)h(the)i(mean)d(en)n(trop)n(y)h(pro)r (duction)g(\(that)i(w)n(e)e(will)f(de\014ne)i(b)r(elo)n(w\))515 2316 y(can)d(b)r(e)i(though)n(t)f(of)g(as)f(measuring)e(the)k(distance) d(\(in)i(the)g(sense)g(of)g(relativ)n(e)d(en)n(trop)n(y\))515 2516 y(b)r(et)n(w)n(een)34 b(the)g(original)c(space-time)i(state)i(and) g(the)g(time-rev)n(ersed)d(space-time)g(state.)515 2715 y(The)c(\015uctuations)f(of)h(the)g(en)n(trop)n(y)f(pro)r(duction)f (\(as)i(a)g(random)e(v)-5 b(ariable\))24 b(around)i(that)515 2914 y(v)-5 b(alue)23 b(are)h(negligibly)c(small)i(in)i(the)h (spatio-temp)r(oral)c(size)i(of)i(the)g(observ)-5 b(ation)23 b(windo)n(w)515 3113 y(and)34 b(they)g(satisfy)f(the)i(same)d(symmetry) g(as)i(in)f(the)i(Galla)n(v)n(otti-Cohen)29 b(theorem,)34 b(see)515 3313 y([2)o(,)28 b(3,)f(5,)h(7)o(,)g(8)o(,)g(11)o(].)515 3512 y(In)j(the)h(presen)n(t)f(pap)r(er)g(w)n(e)g(are)f(mostly)f (concerned)i(with)g(the)h(relation)c(b)r(et)n(w)n(een)k(p)r(osi-)515 3711 y(tivit)n(y)c(of)i(en)n(trop)n(y)e(pro)r(duction)h(and)h(detailed) e(balance.)42 b(This)29 b(relation)e(b)r(ecomes)i(non-)515 3910 y(trivial)d(only)j(when)h(dealing)d(with)i(systems)g(in)g(the)h (thermo)r(dynamic)d(limit.)40 b(W)-7 b(e)31 b(pro)n(v)n(e)515 4110 y(the)d(equiv)-5 b(alence)26 b(b)r(et)n(w)n(een)i(zero)f(en)n (trop)n(y)g(pro)r(duction)g(and)h(detailed)e(balance)h(only)g(for)515 4309 y(spin\015ip)f(pro)r(cesses.)38 b(W)-7 b(e)28 b(wish)g(to)g (restrict)e(us)j(here)e(to)i(that)f(mathematically)23 b(simplest)515 4508 y(case)g(while)g(it)h(app)r(ears)f(eviden)n(t)h (that)h(the)g(pro)r(of)e(can)i(b)r(e)g(extended)f(to)h(for)f(example)e (dif-)515 4707 y(fusiv)n(e)c(systems)g(whic)n(h)h(ph)n(ysically)-7 b(,)17 b(for)j(the)g(purp)r(ose)f(of)g(studies)g(of)h(en)n(trop)n(y)e (pro)r(duction,)515 4907 y(are)26 b(more)g(in)n(teresting)f(b)n(y)i (the)h(presence)f(of)h(conserv)n(ed)e(quan)n(tities.)1926 5650 y(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fo(2)134 b(En)l(trop)l(y)46 b(pro)t(duction)515 838 y Fv(2.1)112 b(Heuristics)515 1091 y Fp(Supp)r(ose)28 b(w)n(e)f(ha)n(v)n(e)f(obtained)h(b)n(y)g(some)g(pro)r(cedure)f(a)i (probabilit)n(y)23 b(distribution)i Fn(P)3217 1060 y Fk(J)3291 1091 y Fp(on)515 1290 y(the)30 b(\(microscopic\))c(tra)5 b(jectories)27 b(in)i(phase)g(space)h(of)f(our)g(system)g(whic)n(h)g (is)g(consisten)n(t)515 1489 y(with)35 b(the)i(v)-5 b(alues)34 b(of)i(space-time)e(macroscopic)e(v)-5 b(ariables)32 b Fn(J)8 b Fp(.)63 b(This)35 b(distribution)e Fn(P)3333 1459 y Fk(J)515 1688 y Fp(represen)n(ts)22 b(the)i(plausibilities)17 b(that)24 b(w)n(e)f(attac)n(h)g(within)f(a)h(mo)r(del)f(to)h(certain)f (time-series)515 1888 y(of)32 b(microscopic)d(states)j(giv)n(en)f (certain)g(macroscopic)e(information.)48 b(That)33 b(time-series)515 2087 y(is)25 b(pro)r(duced)h(b)n(y)g(the)h(dynamics)d(but)j(our)f (probabilities)21 b(are)26 b(certainly)d(not)k(`created)e(b)n(y)515 2286 y(nature')i(or)g(b)n(y)g(some)f(`truly)h(random)e(or)i(c)n (haotic')f(dynamics.)515 2485 y(One)21 b(w)n(a)n(y)g(to)h(construct)f Fn(P)1367 2455 y Fk(J)1435 2485 y Fp(is)g(to)g(start)h(from)e(an)i (equilibrium)17 b(dynamics)i(and)j(to)g(add)f(to)515 2685 y(the)29 b(basic)e(equations)f(driving)g(terms)h(and)i(reserv)n (oirs)24 b(to)29 b(generate)e(curren)n(ts.)39 b(Starting)515 2884 y(from)33 b(suc)n(h)i(a)f(non-equilibrium)c(dynamics)i(one)j (constructs)f(the)h(pathspace)f(measure)515 3083 y Fn(P)580 3053 y Fk(J)654 3083 y Fp(with)28 b(resp)r(ect)f(to)h(some)f(natural)f (stationary)g(measure.)36 b(Macroscopic)25 b(information)515 3282 y(is)k(here)i(con)n(tained)e(in)h(the)h(c)n(hoice)e(of)i(the)g (amplitude)d(b)n(y)j(whic)n(h)e(the)j(non-equilibrium)515 3482 y(condition)c(is)i(main)n(tained.)43 b(This)30 b(amplitude)e(ma)n (y)h(for)i(example)d(consist)h(of)i(the)g(v)-5 b(alue)515 3681 y(of)35 b(the)g(temp)r(erature)e(gradien)n(t)g(b)r(et)n(w)n(een)i (t)n(w)n(o)f(reserv)n(oirs)c(with)35 b(whic)n(h)e(the)j(system)d(is)515 3880 y(brough)n(t)28 b(in)n(to)f(con)n(tact)i(and)f(in)h(this)f(case)g (it)g(is)g(conjugate)g(to)h(the)h(heat)e(curren)n(t.)41 b(Or)28 b(it)515 4079 y(ma)n(y)21 b(b)r(e)j(the)g(v)-5 b(alue)22 b(of)h(an)g(external)e(\(electric\))h(\014eld)g(\(measuring)e (the)k(gradien)n(t)d(in)i(c)n(hem-)515 4279 y(ical)g(or)i(electric)e(p) r(oten)n(tial)g(at)j(the)g(ends)f(of)h(the)g(system\))e(in)h(whic)n(h)g (case)f(it)h(is)g(conjugate)515 4478 y(to)i(the)h(particle)d(or)i(c)n (harge)f(curren)n(t.)515 4677 y(A)e(second)f(w)n(a)n(y)f(to)i(obtain)e Fn(P)1446 4647 y Fk(J)1517 4677 y Fp(is)g(ph)n(ysically)e(more)i (pragmatic.)32 b(There)23 b(one)h(starts)f(with)515 4876 y(the)g(c)n(hoice)e(of)h(a)h(reference)e(pathspace)h(measure)f Fn(P)2159 4846 y Fj(0)2219 4876 y Fp(whic)n(h)g(is)h(supp)r(osed)h(to)f (decrib)r(e)g(the)515 5076 y(unp)r(erturb)r(ed)34 b(equilibrium)28 b(steady)33 b(state)h(via)e(a)h(time-rev)n(ersal)28 b(in)n(v)-5 b(arian)n(t)31 b(space-time)515 5275 y(action)g(functional)f Fn(A)1225 5245 y Fj(0)1295 5275 y Fp(\()p Fn(P)1392 5245 y Fj(0)1461 5275 y Fi(')h Fp(exp\()p Fi(\000)p Fn(A)1843 5245 y Fj(0)1880 5275 y Fp(\)\).)52 b(It)33 b(could)e(corresp)r(ond)g (to)i(a)f(classical)c(\(or)1926 5650 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fp(quan)n(tum\))24 b(KMS)h(state)g(whic)n(h)f(is)h (time-rev)n(ersal)20 b(in)n(v)-5 b(arian)n(t,)22 b(see)j([1])g(for)g(a) f(quan)n(tum)h(ex-)515 722 y(ample.)41 b(Macroscopic)27 b(information)f(is)j(con)n(tained)f(in)h(a)h(c)n(hoice)e(of)h(v)-5 b(alues)29 b(for)g(curren)n(ts)515 922 y Fn(J)41 b Fp(and,)35 b(in)e(the)h(spirit)d(of)j(the)g(maxim)n(um)29 b(en)n(trop)n(y)j (principle)f(for)i(equilibrium)28 b(states,)515 1121 y(one)d(simply)d(exp)r(ects)j Fn(P)1286 1091 y Fk(J)1356 1121 y Fi(\030)d Fp(exp)14 b Fi(\000)p Fp([)p Fn(A)1734 1091 y Fj(0)1785 1121 y Fp(+)g Fn(E)k Fi(\001)c Fn(J)8 b Fp(])25 b(as)g(the)h(space-time)c(Gibbs)j(state)g(corre-)515 1320 y(sp)r(onding)i(to)h(the)g(new)h(action)e Fn(A)1598 1290 y Fk(J)1668 1320 y Fp(=)d Fn(A)1819 1290 y Fj(0)1876 1320 y Fp(+)18 b Fn(E)24 b Fi(\001)19 b Fn(J)8 b Fp(.)39 b(The)28 b(pro)r(duct)g Fn(E)c Fi(\001)19 b Fn(J)37 b Fp(is)27 b(resp)r(onsible)515 1519 y(for)f(breaking)e(time-rev)n(ersal) d(in)n(v)-5 b(ariance)23 b(and)k(p)r(ossibly)d(includes)g(a)j(sum)e(o)n (v)n(er)g(di\013eren)n(t)515 1719 y(t)n(yp)r(es)i(of)h(curren)n(ts)e (as)h(w)n(ell)f(as)h(a)g(space-time)e(in)n(tegration.)515 1918 y(Whatev)n(er)i(the)h(case,)g(w)n(e)f(tak)n(e)g(it)h(as)f(part)h (of)g(our)f(Gibbsian)f(h)n(yp)r(othesis)g(that)j(w)n(e)e(ha)n(v)n(e)515 2117 y(found)32 b(the)h(distribution)c Fn(P)1422 2087 y Fk(J)1500 2117 y Fp(whic)n(h,)j(for)g(the)g(purp)r(ose)g(at)g(hand,)h (correctly)d(describ)r(es)515 2316 y(the)22 b(plausibilit)n(y)17 b(of)22 b(microscopic)17 b(tra)5 b(jectories)19 b(and)j(for)f(whic)n(h) g(it)g(is)g(p)r(ossible)f(to)i(iden)n(tify)515 2516 y(an)27 b(action)f(functional)g Fn(A)1331 2485 y Fk(J)1405 2516 y Fp(on)h(space-time.)515 2715 y(In)i(fact,)g(for)g(the)g(de\014nition) f(of)h(en)n(trop)n(y)e(pro)r(duction,)h(not)h(all)e(details)g(of)i(the) g(distribu-)515 2914 y(tion)23 b Fn(P)747 2884 y Fk(J)818 2914 y Fp(m)n(ust)h(b)r(e)h(tak)n(en)f(in)g(consideration.)33 b(The)24 b(main)f(p)r(oin)n(t)h(is)g(that)g(w)n(e)h(wish)e(to)i(giv)n (e)515 3113 y(meaning)g(to)i(the)h(quan)n(tit)n(y)1100 3339 y(_)1077 3360 y Fn(S)5 b Fp(\()p Fn(!)s Fp(\))23 b(=)g(log)1514 3304 y Fn(P)1579 3274 y Fk(J)1625 3304 y Fp(\()p Fn(!)s Fp(\))p 1494 3341 272 4 v 1494 3417 a Fn(P)1559 3393 y Fk(J)1605 3417 y Fp(\()p Fn(\022)r(!)s Fp(\))1798 3360 y(=)g Fn(A)1948 3326 y Fk(J)1995 3360 y Fp(\()p Fn(\022)r(!)s Fp(\))c Fi(\000)f Fn(A)2319 3326 y Fk(J)2365 3360 y Fp(\()p Fn(!)s Fp(\))24 b(=)e(2)p Fn(E)h Fi(\001)c Fn(J)399 b Fp(\(2.1\))515 3607 y(where)40 b Fn(\022)j Fp(is)d(the)h(time-rev)n(ersal)36 b(op)r(eration)j(on)h (space-time)f(tra)5 b(jectories)38 b Fn(!)s Fp(,)44 b(de\014ned)515 3807 y(from)23 b(\()p Fn(\022)r(!)s Fp(\)\()p Fn(t)p Fp(\))h(=)e Fn(\031)s Fp(\()p Fn(!)s Fp(\()p Fi(\000)p Fn(t)p Fp(\)\),)k(where)e Fn(\031)j Fp(is)d(an)g(in)n(v)n(olution)c (\(also)i(sometimes)f(called,)i(time-)515 4006 y(rev)n(ersal\))j(on)j (phase)f(space,)h(and)g(where)g(w)n(e)g(used)g(the)g(an)n(tisymmetry)d (of)j(the)g(curren)n(ts)515 4205 y(under)g(time-rev)n(ersal:)36 b Fn(J)8 b Fp(\()p Fn(\022)r(!)s Fp(\))27 b(=)g Fi(\000)p Fn(J)8 b Fp(\()p Fn(!)s Fp(\).)43 b(The)29 b(relation)e(\(2.1\))j (`de\014nes')f(what)h(w)n(e)f(call)515 4404 y(the)e(en)n(trop)n(y)f (pro)r(duction)f(for)h(a)h(microscopic)22 b(tra)5 b(jectory)25 b Fn(!)30 b Fp(when)d(the)g(relativ)n(e)d(w)n(eigh)n(t)515 4604 y Fn(P)580 4574 y Fk(J)626 4604 y Fp(\()p Fn(!)s Fp(\))p Fn(=P)852 4574 y Fk(J)898 4604 y Fp(\()p Fn(\022)r(!)s Fp(\))35 b(is)e(w)n(ell-de\014ned.)54 b(This)33 b(last)g(condition)f(m) n(ust)h(allo)n(w)e(us)j(to)g(compare)515 4803 y(the)j(plausibilit)n(y) 31 b(of)36 b(ev)n(ery)f(tra)5 b(jectory)35 b(with)h(its)f(time-rev)n (ersed)e(motion.)61 b(It)37 b(is)e(clear)515 5002 y(that)e(one)f(needs) h(to)g(ha)n(v)n(e)e(that)i Fn(P)1642 4972 y Fk(J)1689 5002 y Fp(\()p Fn(\022)r(!)s Fp(\))f(=)f(0)i(implies)c Fn(P)2404 4972 y Fk(J)2450 5002 y Fp(\()p Fn(!)s Fp(\))j(=)f(0.)52 b(In)33 b(other)f(w)n(ords,)515 5201 y(time-rev)n(ersed)17 b(tra)5 b(jectories)19 b(m)n(ust)h(remain)f(p)r(ossible)g(tra)5 b(jectories)19 b(for)h(the)i(system)e(\(with)515 5401 y(p)r(erhaps)g(a)g(m)n(uc)n(h)f(smaller)d(or)k(a)g(m)n(uc)n(h)f(larger) e(plausibilit)n(y\))f(at)k(least)f(for)h(the)h(o)n(v)n(erwhelm-)1926 5650 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fp(ing)20 b(ma)5 b(jorit)n(y)18 b(of)j(tra)5 b(jectories)19 b(as)h(measured)g(b)n(y)h Fn(P)2132 493 y Fk(J)2178 523 y Fp(.)35 b(W)-7 b(e)22 b(call)d(this)h(prop)r(ert)n(y) h(dynamical)515 722 y(rev)n(ersibilit)n(y)13 b(\(as)19 b(in)g([12)o(]\))g(but)h(it)f(should)f(not)h(b)r(e)g(confused)h(with)e (microscopic)d(rev)n(ersibil-)515 922 y(it)n(y)26 b(\(whic)n(h)h(is)g (more)e(or)i(less)f(the)i(same)e(as)h(detailed)f(balance\).)515 1121 y(Clearly)-7 b(,)24 b(and)k(on)f(the)h(same)e(formal)f(lev)n(el,)g (if)i(w)n(e)h(tak)n(e)f(the)h(exp)r(ectation)e(of)3007 1100 y(_)2984 1121 y Fn(S)32 b Fp(o)n(v)n(er)26 b Fn(P)3310 1091 y Fk(J)3356 1121 y Fp(,)515 1320 y(that)f(is)e(in)h(the)h (non-equilibrium)19 b(steady)24 b(state)h(\(NESS\),)g(then)g(the)g (mean)f(en)n(trop)n(y)f(pro-)515 1519 y(duction)30 b(is)g(obtained)g (and,)h(b)n(y)g(conca)n(vit)n(y)d(of)j(the)h(logarithmic)25 b(function,)32 b(it)e(is)g(clearly)515 1719 y(non-negativ)n(e:)1342 1918 y Fn(P)1407 1884 y Fk(J)1453 1918 y Fp(\()1509 1897 y(_)1485 1918 y Fn(S)5 b Fp(\))23 b(=)g Fi(h)p Fp(log)1868 1862 y Fn(P)1933 1832 y Fk(J)1979 1862 y Fp(\()p Fn(!)s Fp(\))p 1847 1899 272 4 v 1847 1975 a Fn(P)1912 1951 y Fk(J)1959 1975 y Fp(\()p Fn(\022)r(!)s Fp(\))2129 1918 y Fi(i)2161 1946 y Fp(NESS)2400 1918 y Fi(\025)f Fp(0)p Fn(:)656 b Fp(\(2.2\))515 2167 y(with)27 b(equalit)n(y)e(only)h(if)h (the)h(pathspace)f(action)f Fn(A)2120 2137 y Fk(J)2194 2167 y Fp(is)h(time-rev)n(ersal)22 b(in)n(v)-5 b(arian)n(t.)515 2516 y Fv(2.2)112 b(De\014nition)515 2768 y Fp(The)29 b(ab)r(o)n(v)n(e)e(discussion)g(and)h(de\014nition)g(\(2.1\))g(ha)n(v)n (e)g(remained)e(mathematically)e(naiv)n(e)515 2968 y(and)36 b(ph)n(ysically)d(v)-5 b(ague.)63 b(The)36 b(main)f(mathematical)d (question)j(is)g(here)h(to)h(dev)n(elop)d(a)515 3167 y(su\016cien)n(tly)16 b(general)h(theory)g(of)i(space-time)d(Gibbs)i (measures.)31 b(F)-7 b(or)18 b(simplicit)n(y)c(ho)n(w)n(ev)n(er,)515 3366 y(w)n(e)22 b(merely)f(rep)r(eat)h(here)h(the)g(de\014nition)e(of)i (en)n(trop)n(y)f(pro)r(duction)g(for)g(a)h(Mark)n(o)n(v)d(pro)r(cess) 515 3565 y(with)j(a)g(\014nite)g(state)h(space.)35 b(F)-7 b(or)23 b(more)e(complicated)g(mo)r(dels)g(the)j(story)f(is)f(v)n(ery)h (similar.)515 3765 y(Let)30 b Fn(X)37 b Fp(b)r(e)30 b(a)g(\014nite)g (set,)h(the)f(phase)g(space,)g(and)g(let)f Fn(K)36 b Fp(b)r(e)31 b(a)e(\014nite)h(set)g(for)g(the)g(v)-5 b(alues)515 3964 y(of)30 b(parameters.)42 b(F)-7 b(or)30 b(ev)n(ery)f(parameter)e (v)-5 b(alue)30 b Fn(a)d Fi(2)h Fn(K)36 b Fp(there)30 b(is)f(a)h(stationary)d(Mark)n(o)n(v)515 4163 y(pro)r(cess)18 b Fn(x)843 4175 y Fk(t)872 4163 y Fn(;)c(t)23 b Fi(2)h Fp(I)-14 b(R)p Fn(;)19 b Fp(with)g(transition)e(rates)h Fn(p)1949 4175 y Fk(a)1989 4163 y Fp(\()p Fn(x;)c(y)s Fp(\))20 b(for)e(jumps)h Fn(x)24 b Fi(!)f Fn(y)f Fp(and)d(a)g (stationary)515 4362 y(densit)n(y)26 b Fn(\032)844 4374 y Fk(a)911 4362 y Fp(on)h Fn(X)7 b Fp(.)36 b(W)-7 b(e)28 b(are)e(giv)n(en)g(time-rev)n(ersal)c(transformations)h Fn(\031)2798 4374 y Fk(X)2889 4362 y Fp(on)k Fn(X)34 b Fp(and)27 b Fn(\031)3315 4374 y Fk(K)515 4562 y Fp(on)20 b Fn(K)6 b Fp(,)22 b(i.e.)33 b(an)21 b(in)n(v)n(olution)c(lea)n(ving)g (in)n(v)-5 b(arian)n(t)18 b(the)j(coun)n(ting)e(measure.)32 b(F)-7 b(or)20 b(a)h(tra)5 b(jectory)515 4761 y Fn(!)34 b Fp(=)c(\()p Fn(x)775 4773 y Fj(1)813 4761 y Fn(;)14 b(x)897 4773 y Fj(2)935 4761 y Fn(;)g(:)g(:)g(:)g(;)g(x)1167 4773 y Fk(n)1212 4761 y Fp(\))33 b(in)f Fn(X)39 b Fp(and)32 b(a)g(v)-5 b(alue)32 b Fn(a)f Fi(2)g Fn(K)38 b Fp(for)32 b(the)h(parameters,)e(the)i(en)n(trop)n(y)515 4960 y(pro)r(duction)26 b(corresp)r(onding)f(to)i(the)h(time-rev)n(ersal)23 b Fn(\031)31 b Fp(is)1293 5221 y(_)1269 5242 y Fn(S)1320 5254 y Fk(a)1360 5242 y Fp(\()p Fn(!)s Fp(\))24 b(=)e(ln)1683 5186 y Fn(p)1725 5198 y Fk(a)1765 5186 y Fp(\()p Fn(x)1844 5198 y Fj(1)1882 5186 y Fn(;)14 b(x)1966 5198 y Fj(2)2004 5186 y Fp(\))g Fn(:)g(:)g(:)g(p)2203 5198 y Fk(a)2243 5186 y Fp(\()p Fn(x)2322 5198 y Fk(n)p Fh(\000)p Fj(1)2453 5186 y Fn(;)g(x)2537 5198 y Fk(n)2582 5186 y Fp(\))p 1683 5223 931 4 v 1690 5299 a(~)-49 b Fn(p)1725 5311 y Fk(a)1765 5299 y Fp(\()p Fn(x)1844 5311 y Fk(n)1890 5299 y Fn(;)14 b(x)1974 5311 y Fk(n)p Fh(\000)p Fj(1)2105 5299 y Fp(\))g Fn(:)g(:)g(:)21 b Fp(~)-49 b Fn(p)2304 5311 y Fk(a)2344 5299 y Fp(\()p Fn(x)2423 5311 y Fj(2)2460 5299 y Fn(;)14 b(x)2544 5311 y Fj(1)2582 5299 y Fp(\))1926 5650 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fp(where)25 b(~)-49 b Fn(p)788 535 y Fk(a)828 523 y Fp(\()p Fn(u;)14 b(v)s Fp(\))23 b(=)g Fn(p)1173 538 y Fk(\031)1212 546 y Fg(K)1266 538 y Fj(\()p Fk(a)p Fj(\))1358 523 y Fp(\()p Fn(\031)1437 535 y Fk(X)1500 523 y Fp(\()p Fn(u)p Fp(\))p Fn(;)14 b(\031)1696 535 y Fk(X)1760 523 y Fp(\()p Fn(v)s Fp(\)\).)35 b(The)18 b(mean)g(en)n(trop)n(y)f(pro)r(duction)g(\(MEP\))i(in)515 722 y(the)25 b(stationary)d(state)j Fn(\032)1291 734 y Fk(a)1356 722 y Fp(is)f(the)h(exp)r(ectation)f(v)-5 b(alue)23 b(of)2348 701 y(_)2325 722 y Fn(S)2376 734 y Fk(a)2416 722 y Fp(\()p Fn(!)s Fp(\))i(in)f(the)i(Mark)n(o)n(v)c(pro) r(cess.)515 922 y(It)28 b(equals)1167 1121 y(MEP)1356 1133 y Fk(\031)1401 1121 y Fp(\()p Fn(\032)1476 1133 y Fk(a)1517 1121 y Fp(\))23 b(=)1698 1042 y Ff(X)1660 1220 y Fk(x;y)r Fh(2)p Fk(X)1870 1121 y Fn(\032)1913 1133 y Fk(a)1953 1121 y Fp(\()p Fn(x)p Fp(\))p Fn(p)2106 1133 y Fk(a)2147 1121 y Fp(\()p Fn(x;)14 b(y)s Fp(\))g(ln)2447 1065 y Fn(p)2489 1077 y Fk(a)2529 1065 y Fp(\()p Fn(x;)g(y)s Fp(\))p 2447 1102 275 4 v 2454 1178 a(~)-49 b Fn(p)2489 1190 y Fk(a)2529 1178 y Fp(\()p Fn(y)s(;)14 b(x)p Fp(\))2731 1121 y Fn(:)454 b Fp(\(2.3\))515 1363 y(A)39 b(more)f(general)f(set-up) i(with)f(v)-5 b(arious)37 b(examples)g(also)g(of)i(calculations)c(in)k (sp)r(eci\014c)515 1562 y(mo)r(dels)24 b(can)h(b)r(e)i(found)f(in)f ([7,)h(8)o(];)h(w)n(e)f(just)g(tak)n(e)g(\(2.3\))f(as)h(our)f (de\014nition)f(in)h(the)i(presen)n(t)515 1762 y(simplest)e(case.)515 2108 y Fv(2.3)112 b(P)m(ositivit)m(y)515 2361 y Fp(F)-7 b(or)31 b(the)h(mean)e(en)n(trop)n(y)h(pro)r(duction)f(w)n(e)h(are)g (in)n(terested)f(in)h(conditions)f(under)h(whic)n(h)515 2560 y(it)c(is)g(p)r(ositiv)n(e)f(or)h(when)h(it)g(is)f(zero.)37 b(F)-7 b(or)27 b(systems)g(with)g(a)h(\014nite)g(phase)f(space,)h(w)n (e)f(ha)n(v)n(e)515 2760 y(the)h(follo)n(wing)23 b(easy)k(prop)r (osition:)515 3016 y Fq(Prop)s(osition)j(2.1)40 b Fe(The)31 b(me)l(an)f(entr)l(opy)g(pr)l(o)l(duction)g(de\014ne)l(d)g(in)g (\(2.3\))h(satis\014es)1696 3287 y(MEP)1883 3299 y Fk(\031)1928 3287 y Fp(\()p Fn(\032)2003 3299 y Fk(a)2043 3287 y Fp(\))24 b Fi(\025)e Fp(0)980 b(\(2.4\))515 3558 y Fe(with)30 b(e)l(quality)g(if)h(and)f(only)h(if)1244 3829 y Fn(p)1286 3841 y Fk(a)1326 3829 y Fp(\()p Fn(x;)14 b(y)s Fp(\))p Fn(\032)1561 3841 y Fk(a)1601 3829 y Fp(\()p Fn(x)p Fp(\))24 b(=)30 b(~)-49 b Fn(p)1866 3841 y Fk(a)1906 3829 y Fp(\()p Fn(y)s(;)14 b(x)p Fp(\))p Fn(\032)2141 3841 y Fk(a)2182 3829 y Fp(\()p Fn(y)s Fp(\))p Fn(;)g(x;)g(y)26 b Fi(2)d Fn(X)r(:)558 b Fp(\(2.5\))515 4099 y(The)37 b(condition)e(\(2.5\))i(is) f(sometimes)e(called)h(generalized)f(detailed)h(balance.)64 b(It)38 b(will)515 4299 y(reapp)r(ear)26 b(as)h(the)h(equalit)n(y)d(in) i(\(4.24\).)515 4721 y Fo(3)134 b(En)l(trop)l(y)46 b(pro)t(duction)d (for)i(spin\015ip)e(pro)t(cesses)515 5002 y Fp(Spin\015ip)29 b(dynamics)e(are)i(Mark)n(o)n(v)f(\(jump\))i(pro)r(cesses)f(for)g (Ising)f(spin)h(con\014gurations)f Fn(\033)515 5201 y Fp(on)i(an)g(in\014nite)g(lattice)f(whereb)n(y)g(the)j(elemen)n(tary)27 b(transitions)h(are)i(spin)f(\015ips)h Fn(\033)i Fi(!)c Fn(\033)3351 5171 y Fk(i)515 5401 y Fp(go)n(v)n(erned)23 b(b)n(y)h(\(spin\015ip\))g(rates)g Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))26 b(at)f(the)g(sites)f Fn(i)h Fp(of)g(the)g(lattice.)34 b(F)-7 b(or)24 b(simplicit)n(y)c(w)n(e)1926 5650 y(7)p eop %%Page: 8 8 8 7 bop 515 523 a Fp(tak)n(e)27 b(the)h Fn(d)p Fi(\000)p Fp(dimensional)c(cubic)j(lattice)p 1889 525 45 4 v 1889 469 43 4 v 1889 523 3 3 v 1890 521 V 1890 519 V 1891 517 V 1892 516 V 1893 514 V 1894 512 V 1895 510 V 1896 508 V 1897 506 V 1898 504 V 1899 502 V 1900 501 V 1901 499 V 1902 497 V 1903 495 V 1904 493 V 1904 491 V 1905 489 V 1906 488 V 1907 486 V 1908 484 V 1909 482 V 1910 480 V 1911 478 V 1912 476 V 1913 474 V 1914 473 V 1915 471 V 1916 469 V 1904 523 V 1904 521 V 1905 519 V 1906 517 V 1907 516 V 1908 514 V 1909 512 V 1910 510 V 1911 508 V 1912 506 V 1913 504 V 1914 502 V 1915 501 V 1916 499 V 1917 497 V 1918 495 V 1918 493 V 1919 491 V 1920 489 V 1921 488 V 1922 486 V 1923 484 V 1924 482 V 1925 480 V 1926 478 V 1927 476 V 1928 474 V 1929 473 V 1930 471 V 1931 469 V 1887 478 4 12 v 1888 474 4 8 v 1889 473 4 6 v 1890 471 4 4 v 1891 469 4 2 v 1932 523 4 12 v 1931 523 4 8 v 1929 523 4 6 v 1928 523 4 4 v 1928 523 4 2 v 1933 491 a Fk(d)2000 523 y Fp(to)h(the)g(sites)f(of)h(whic)n(h)f (w)n(e)g(ha)n(v)n(e)g(assigned)515 722 y(Ising)j(spins)h Fn(\033)s Fp(\()p Fn(i)p Fp(\))g(=)f Fi(\006)p Fp(1)p Fn(;)14 b(i)29 b Fi(2)p 1504 724 45 4 v 1504 668 43 4 v 1504 722 3 3 v 1505 720 V 1506 718 V 1507 716 V 1508 715 V 1509 713 V 1510 711 V 1511 709 V 1512 707 V 1512 705 V 1513 704 V 1514 702 V 1515 700 V 1516 698 V 1517 696 V 1518 694 V 1519 692 V 1520 691 V 1521 689 V 1522 687 V 1523 685 V 1524 683 V 1525 681 V 1526 679 V 1526 677 V 1527 676 V 1528 674 V 1529 672 V 1530 670 V 1531 668 V 1519 722 V 1520 720 V 1521 718 V 1522 716 V 1523 715 V 1524 713 V 1525 711 V 1526 709 V 1526 707 V 1527 705 V 1528 704 V 1529 702 V 1530 700 V 1531 698 V 1532 696 V 1533 694 V 1534 692 V 1535 691 V 1536 689 V 1537 687 V 1538 685 V 1539 683 V 1540 681 V 1540 679 V 1541 677 V 1542 676 V 1543 674 V 1544 672 V 1545 670 V 1546 668 V 1502 677 4 12 v 1504 674 4 8 v 1505 672 4 6 v 1506 670 4 4 v 1506 668 4 2 v 1547 722 4 12 v 1546 722 4 8 v 1545 722 4 6 v 1544 722 4 4 v 1544 722 4 2 v 1549 691 a Fk(d)1588 722 y Fp(.)50 b(The)32 b(dynamics)e(is)h(de\014ned)h(via)e (rates)i Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\):)46 b(they)515 922 y(giv)n(e)37 b(the)j(probabilit)n(y)35 b(p)r(er)k(unit)g(time)f (that)i(the)f(spin)g(at)g(site)f Fn(i)h Fp(will)e(\015ip)h(when)i(the) 515 1121 y(curren)n(t)20 b(con\014guration)f(is)i Fn(\033)s Fp(.)35 b(After)22 b(\015ipping)e(at)h(lattice)f(site)h Fn(i)g Fp(the)h(new)f(con\014guration)e(is)515 1320 y(denoted)h(b)n(y)g Fn(\033)979 1290 y Fk(i)1007 1320 y Fp(.)35 b(W)-7 b(e)21 b(mak)n(e)d(the)j(standard)e(assumptions)f(that)j(the)f(rates)g(are)f (translation)515 1519 y(in)n(v)-5 b(arian)n(t,)34 b(\014nite)i(range)e (and)h(p)r(ositiv)n(e.)59 b(This)35 b(means)f(that)i(the)g(pro)r(cess)e (is)h(spatially)515 1719 y(homogeneous,)24 b(that)j(the)g(spin\015ip)e (rates)h Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))27 b(only)e(dep)r(end)i (on)g(the)g(con\014guration)d Fn(\033)515 1918 y Fp(restricted)h(to)h (a)g(\014nite)g(neigh)n(b)r(orho)r(o)r(d)e(of)j(the)g(site)e Fn(i)h Fp(and)h(that)f(there)h(exists)e(a)h(constan)n(t)515 2117 y Fn(\016)g(>)c Fp(0)k(so)g(that)h Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))23 b Fi(\025)g Fn(\016)s Fp(.)36 b(W)-7 b(e)27 b(refer)f(to)g([6)o(])h(for)e(a)h(mathematically)21 b(precise)k(de\014nition)515 2316 y(and)i(for)g(more)f(details.)639 2516 y(T)-7 b(o)23 b(start)f(w)n(e)h(recall)d(the)k(notion)d(of)i(a)g (p)r(oten)n(tial)e(for)h(lattice)g(spin)g(systems.)34 b(A)23 b(p)r(oten-)515 2715 y(tial)j(is)h(a)h(family)d(of)j(lo)r(cal)e (functions)i Fn(V)1770 2727 y Fk(A)1852 2715 y Fp(on)g(the)h(spin)e (con\014gurations)f Fn(\033)31 b Fp(parametrized)515 2914 y(b)n(y)c(the)h(\014nite)f(subsets)h Fn(A)f Fp(of)p 1471 2916 45 4 v 1471 2860 43 4 v 1471 2914 3 3 v 1472 2912 V 1473 2910 V 1474 2908 V 1475 2907 V 1476 2905 V 1476 2903 V 1477 2901 V 1478 2899 V 1479 2897 V 1480 2895 V 1481 2894 V 1482 2892 V 1483 2890 V 1484 2888 V 1485 2886 V 1486 2884 V 1487 2882 V 1488 2880 V 1489 2879 V 1490 2877 V 1490 2875 V 1491 2873 V 1492 2871 V 1493 2869 V 1494 2867 V 1495 2866 V 1496 2864 V 1497 2862 V 1498 2860 V 1486 2914 V 1487 2912 V 1488 2910 V 1489 2908 V 1490 2907 V 1490 2905 V 1491 2903 V 1492 2901 V 1493 2899 V 1494 2897 V 1495 2895 V 1496 2894 V 1497 2892 V 1498 2890 V 1499 2888 V 1500 2886 V 1501 2884 V 1502 2882 V 1503 2880 V 1504 2879 V 1504 2877 V 1505 2875 V 1506 2873 V 1507 2871 V 1508 2869 V 1509 2867 V 1510 2866 V 1511 2864 V 1512 2862 V 1513 2860 V 1469 2869 4 12 v 1470 2866 4 8 v 1471 2864 4 6 v 1473 2862 4 4 v 1473 2860 4 2 v 1514 2914 4 12 v 1513 2914 4 8 v 1512 2914 4 6 v 1511 2914 4 4 v 1510 2914 4 2 v 1516 2882 a Fk(d)1582 2914 y Fp(with)1501 3196 y Fn(V)1549 3208 y Fk(A)1604 3196 y Fp(\()p Fn(\033)s Fp(\))d(=)e Fn(V)1877 3208 y Fk(A)1932 3196 y Fp(\()p Fn(\033)s Fp(\()p Fn(i)p Fp(\))p Fn(;)14 b(i)23 b Fi(2)h Fn(A)p Fp(\))p Fn(;)515 3479 y Fp(and)j(suc)n(h)g(that)1406 3599 y Ff(X)1407 3777 y Fk(A)p Fh(3)p Fk(i)1540 3678 y Fp(sup)1582 3744 y Fk(\033)1678 3678 y Fi(j)p Fn(V)1749 3690 y Fk(A)1804 3678 y Fp(\()p Fn(\033)s Fp(\))p Fi(j)d Fn(<)f Fp(+)p Fi(1)p Fn(;)14 b(i)22 b Fi(2)p 2381 3680 45 4 v 2381 3623 43 4 v 2381 3678 3 3 v 2382 3676 V 2383 3674 V 2384 3672 V 2385 3671 V 2386 3669 V 2387 3667 V 2388 3665 V 2389 3663 V 2390 3661 V 2391 3659 V 2391 3657 V 2392 3655 V 2393 3654 V 2394 3652 V 2395 3650 V 2396 3648 V 2397 3646 V 2398 3644 V 2399 3642 V 2400 3641 V 2401 3639 V 2402 3637 V 2403 3635 V 2404 3633 V 2405 3631 V 2405 3629 V 2406 3627 V 2407 3626 V 2408 3624 V 2396 3678 V 2397 3676 V 2398 3674 V 2399 3672 V 2400 3671 V 2401 3669 V 2402 3667 V 2403 3665 V 2404 3663 V 2405 3661 V 2405 3659 V 2406 3657 V 2407 3655 V 2408 3654 V 2409 3652 V 2410 3650 V 2411 3648 V 2412 3646 V 2413 3644 V 2414 3642 V 2415 3641 V 2416 3639 V 2417 3637 V 2418 3635 V 2419 3633 V 2420 3631 V 2420 3629 V 2421 3627 V 2422 3626 V 2423 3624 V 2380 3633 4 12 v 2381 3629 4 8 v 2382 3627 4 6 v 2383 3626 4 4 v 2383 3624 4 2 v 2424 3678 4 12 v 2423 3678 4 8 v 2422 3678 4 6 v 2421 3678 4 4 v 2421 3678 4 2 v 2426 3644 a Fk(d)2465 3678 y Fn(:)720 b Fp(\(3.6\))515 3927 y(W)-7 b(e)29 b(sa)n(y)f(that)i (the)f(spin\015ip)f(dynamics)e(is)i(a)h(sto)r(c)n(hastic)e(Ising)h(mo)r (del)f(for)i(the)g(p)r(oten)n(tial)515 4126 y(\()p Fn(V)595 4138 y Fk(A)650 4126 y Fp(\),)f(or)e(shorter,)h(satis\014es)e(the)j (condition)e(of)h(detailed)f(balance,)g(when)1272 4352 y Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))p 1258 4389 245 4 v 1258 4465 a Fn(c)p Fp(\()p Fn(i;)g(\033)1442 4441 y Fk(i)1470 4465 y Fp(\))1535 4408 y(=)23 b(exp\()p Fi(\000)1861 4330 y Ff(X)1861 4508 y Fk(A)p Fh(3)p Fk(i)1980 4408 y Fp([)p Fn(V)2051 4420 y Fk(A)2106 4408 y Fp(\()p Fn(\033)2188 4374 y Fk(i)2216 4408 y Fp(\))c Fi(\000)f Fn(V)2398 4420 y Fk(A)2452 4408 y Fp(\()p Fn(\033)s Fp(\)]\))p Fn(:)564 b Fp(\(3.7\))515 4691 y(A)27 b(standard)e(example)g(is)g(the)i (Glaub)r(er)e(dynamics)g(for)h(the)h(Ising)e(mo)r(del)g(in)g(whic)n(h)h (case)515 4890 y Fn(V)563 4902 y Fk(A)617 4890 y Fp(\()p Fn(\033)s Fp(\))34 b(=)d(0)i(except)g(when)h Fn(A)e Fp(=)g Fi(f)p Fn(i;)14 b(j)5 b Fi(g)32 b Fp(is)g(a)h(pair)e(of)i(nearest)f (neigh)n(b)r(or)g(sites)g Fn(i;)14 b(j)37 b Fp(and)515 5089 y(then)22 b Fn(V)746 5101 y Fk(A)824 5089 y Fp(=)h Fi(\000)p Fn(\014)t(J)8 b(\033)s Fp(\()p Fn(i)p Fp(\))p Fn(\033)s Fp(\()p Fn(j)d Fp(\))23 b(for)f(some)f(in)n(v)n(erse)e(temp)r (erature)i Fn(\014)27 b Fp(and)22 b(coupling)d(co)r(e\016cien)n(t)515 5289 y Fn(J)8 b Fp(.)1926 5650 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fp(Supp)r(ose)27 b(no)n(w)g(that)g(w)n(e)g(ha)n(v)n (e)f(a)h(translation-in)n(v)-5 b(aria)o(n)n(t)22 b(stationary)i (probabilit)n(y)g(distri-)515 722 y(bution)31 b Fn(\032)i Fp(for)e(our)h(spin\015ip)f(dynamics.)48 b(Stationarit)n(y)29 b(of)k(a)e(probabilit)n(y)e(measure)h Fn(\032)i Fp(is)515 922 y(expressed)26 b(b)n(y)h(the)h(condition)1254 1091 y Ff(Z)1351 1204 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))1588 1125 y Ff(X)1565 1324 y Fk(i)p Fh(2)p 1650 1326 45 4 v 1650 1270 43 4 v 1650 1324 3 3 v 1651 1322 V 1652 1320 V 1653 1318 V 1654 1317 V 1655 1315 V 1656 1313 V 1657 1311 V 1658 1309 V 1659 1307 V 1660 1305 V 1661 1303 V 1661 1302 V 1662 1300 V 1663 1298 V 1664 1296 V 1665 1294 V 1666 1292 V 1667 1290 V 1668 1288 V 1669 1287 V 1670 1285 V 1671 1283 V 1672 1281 V 1673 1279 V 1674 1277 V 1675 1275 V 1675 1274 V 1676 1272 V 1677 1270 V 1665 1324 V 1666 1322 V 1667 1320 V 1668 1318 V 1669 1317 V 1670 1315 V 1671 1313 V 1672 1311 V 1673 1309 V 1674 1307 V 1675 1305 V 1675 1303 V 1676 1302 V 1677 1300 V 1678 1298 V 1679 1296 V 1680 1294 V 1681 1292 V 1682 1290 V 1683 1288 V 1684 1287 V 1685 1285 V 1686 1283 V 1687 1281 V 1688 1279 V 1689 1277 V 1689 1275 V 1690 1274 V 1691 1272 V 1692 1270 V 1649 1279 4 12 v 1650 1275 4 8 v 1651 1273 4 6 v 1652 1272 4 4 v 1652 1270 4 2 v 1693 1324 4 12 v 1692 1324 4 8 v 1691 1324 4 6 v 1690 1324 4 4 v 1690 1324 4 2 v 1695 1292 a Fg(d)1744 1204 y Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\)[)p Fn(f)9 b Fp(\()p Fn(\033)2115 1170 y Fk(i)2143 1204 y Fp(\))19 b Fi(\000)f Fn(f)9 b Fp(\()p Fn(\033)s Fp(\)])24 b(=)e(0)p Fn(;)568 b Fp(\(3.8\))515 1486 y(for)33 b(all)e(lo)r(cal)g(functions)i Fn(f)43 b Fp(on)33 b(the)h(spin)f(con\014gurations.)52 b(F)-7 b(rom)33 b Fn(\032)g Fp(w)n(e)h(can)f(de\014ne)h(its)515 1685 y(lo)r(cal)25 b(spin\015ip)h(transform)f Fn(\032)1426 1655 y Fk(i)1454 1685 y Fn(;)14 b(i)22 b Fi(2)p 1635 1687 45 4 v 1635 1631 43 4 v 1635 1685 3 3 v 1636 1683 V 1636 1681 V 1637 1679 V 1638 1678 V 1639 1676 V 1640 1674 V 1641 1672 V 1642 1670 V 1643 1669 V 1644 1667 V 1645 1665 V 1646 1663 V 1647 1661 V 1648 1659 V 1649 1657 V 1650 1655 V 1650 1654 V 1651 1652 V 1652 1650 V 1653 1648 V 1654 1646 V 1655 1644 V 1656 1642 V 1657 1641 V 1658 1639 V 1659 1637 V 1660 1635 V 1661 1633 V 1662 1631 V 1650 1685 V 1650 1683 V 1651 1681 V 1652 1679 V 1653 1678 V 1654 1676 V 1655 1674 V 1656 1672 V 1657 1670 V 1658 1669 V 1659 1667 V 1660 1665 V 1661 1663 V 1662 1661 V 1663 1659 V 1664 1657 V 1664 1655 V 1665 1654 V 1666 1652 V 1667 1650 V 1668 1648 V 1669 1646 V 1670 1644 V 1671 1642 V 1672 1641 V 1673 1639 V 1674 1637 V 1675 1635 V 1676 1633 V 1677 1631 V 1633 1641 4 12 v 1634 1637 4 8 v 1635 1635 4 6 v 1636 1633 4 4 v 1637 1631 4 2 v 1678 1685 4 12 v 1677 1685 4 8 v 1676 1685 4 6 v 1674 1685 4 4 v 1674 1685 4 2 v 1679 1654 a Fk(d)1718 1685 y Fp(,)28 b(via)1390 1855 y Ff(Z)1487 1968 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(f)9 b Fp(\()p Fn(\033)1819 1933 y Fk(i)1848 1968 y Fp(\))23 b(=)1991 1855 y Ff(Z)2088 1968 y Fn(\032)2131 1933 y Fk(i)2158 1968 y Fp(\()p Fn(d\033)s Fp(\))p Fn(f)9 b Fp(\()p Fn(\033)s Fp(\))p Fn(;)515 2250 y Fp(for)30 b(all)f(lo)r(cal)g (functions)i Fn(f)9 b Fp(.)47 b(W)-7 b(e)31 b(sa)n(y)g(that)g Fn(\032)1966 2220 y Fk(i)2025 2250 y Fp(is)f(absolutely)e(con)n(tin)n (uous)i(with)g(resp)r(ect)515 2449 y(to)d Fn(\032)h Fp(if)f(there)g (exists)g(an)g Fn(L)1377 2419 y Fj(1)1414 2449 y Fp(\()p Fn(d\032)p Fp(\))p Fi(\000)h Fp(function)f Fn(r)2019 2461 y Fk(i)2047 2449 y Fn(;)2084 2382 y Ff(R)2153 2449 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fi(j)p Fn(r)2413 2461 y Fk(i)2442 2449 y Fp(\()p Fn(\033)s Fp(\))p Fi(j)d Fn(<)f Fp(+)p Fi(1)p Fp(,)k(for)g(whic)n(h)1292 2618 y Ff(Z)1389 2731 y Fn(\032)1432 2697 y Fk(i)1459 2731 y Fp(\()p Fn(d\033)s Fp(\))p Fn(f)9 b Fp(\()p Fn(\033)s Fp(\))25 b(=)1893 2618 y Ff(Z)1990 2731 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(r)2227 2743 y Fk(i)2256 2731 y Fp(\()p Fn(d\033)s Fp(\))p Fn(f)9 b Fp(\()p Fn(\033)s Fp(\))p Fn(;)515 3014 y Fp(for)27 b(all)e(lo)r(cal)g Fn(f)9 b Fp(.)37 b(W)-7 b(e)28 b(sa)n(y)e(that)i Fn(\032)g Fp(is)e(a)h(Gibbs)g(distribution)e(for)i(the)h(p)r(oten)n(tial)d(\()p Fn(V)3146 3026 y Fk(A)3201 3014 y Fp(\))j(if)908 3183 y Ff(Z)1005 3296 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(f)9 b Fp(\()p Fn(\033)1337 3262 y Fk(i)1366 3296 y Fp(\))23 b(=)1509 3183 y Ff(Z)1606 3296 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(f)9 b Fp(\()p Fn(\033)s Fp(\))14 b(exp)q(\()p Fi(\000)2223 3217 y Ff(X)2224 3395 y Fk(A)p Fh(3)p Fk(i)2343 3296 y Fp([)p Fn(V)2414 3308 y Fk(A)2469 3296 y Fp(\()p Fn(\033)2551 3262 y Fk(i)2579 3296 y Fp(\))19 b Fi(\000)f Fn(V)2761 3308 y Fk(A)2815 3296 y Fp(\()p Fn(\033)s Fp(\)]\))224 b(\(3.9\))515 3578 y(for)27 b(all)e Fn(i)e Fi(2)p 901 3580 45 4 v 901 3524 43 4 v 901 3578 3 3 v 902 3576 V 903 3574 V 904 3572 V 905 3571 V 906 3569 V 907 3567 V 908 3565 V 908 3563 V 909 3561 V 910 3560 V 911 3558 V 912 3556 V 913 3554 V 914 3552 V 915 3550 V 916 3548 V 917 3547 V 918 3545 V 919 3543 V 920 3541 V 921 3539 V 922 3537 V 922 3535 V 923 3533 V 924 3532 V 925 3530 V 926 3528 V 927 3526 V 928 3524 V 916 3578 V 917 3576 V 918 3574 V 919 3572 V 920 3571 V 921 3569 V 922 3567 V 923 3565 V 923 3563 V 924 3561 V 925 3560 V 926 3558 V 927 3556 V 928 3554 V 929 3552 V 930 3550 V 931 3548 V 932 3547 V 933 3545 V 934 3543 V 935 3541 V 936 3539 V 937 3537 V 937 3535 V 938 3533 V 939 3532 V 940 3530 V 941 3528 V 942 3526 V 943 3524 V 899 3533 4 12 v 900 3530 4 8 v 902 3528 4 6 v 903 3526 4 4 v 903 3524 4 2 v 944 3578 4 12 v 943 3578 4 8 v 942 3578 4 6 v 941 3578 4 4 v 940 3578 4 2 v 946 3546 a Fk(d)1012 3578 y Fp(and)28 b(for)f(all)e(lo)r(cal)g(functions)i Fn(f)9 b Fp(.)37 b(In)27 b(this)g(case)g(w)n(e)g(ha)n(v)n(e)1110 3861 y Fn(r)1147 3873 y Fk(i)1175 3861 y Fp(\()p Fn(\033)s Fp(\))d(=)1411 3804 y Fn(d\032)1497 3774 y Fk(i)p 1411 3841 114 4 v 1425 3917 a Fn(d\032)1535 3861 y Fp(\()p Fn(\033)s Fp(\))g(=)e(exp\()p Fi(\000)1998 3782 y Ff(X)1999 3960 y Fk(A)p Fh(3)p Fk(i)2118 3861 y Fp([)p Fn(V)2189 3873 y Fk(A)2243 3861 y Fp(\()p Fn(\033)2325 3826 y Fk(i)2354 3861 y Fp(\))d Fi(\000)f Fn(V)2536 3873 y Fk(A)2590 3861 y Fp(\()p Fn(\033)s Fp(\)]\))p Fn(:)639 4143 y Fp(In)26 b([8)o(])g(w)n(e)f(obtained)f(the)i(follo)n(wing)21 b(expression)i(for) h(the)i(mean)e(en)n(trop)n(y)g(pro)r(duction)515 4342 y(\(MEP\))j(in)g(the)h(stationary)d(state)i Fn(\032)p Fp(:)1290 4624 y(MEP)o(\()p Fn(\032)p Fp(\))d(=)1697 4511 y Ff(Z)1794 4624 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))g(ln)2350 4568 y Fn(c)p Fp(\(0)p Fn(;)g(\033)s Fp(\))p 2332 4605 267 4 v 2332 4681 a Fn(c)p Fp(\(0)p Fn(;)g(\033)2529 4657 y Fj(0)2566 4681 y Fp(\))2608 4624 y Fn(:)536 b Fp(\(3.10\))515 4907 y(Here)33 b(w)n(e)g(used)g(the)h(standard)e (time-rev)n(ersal)c Fn(\022)r Fp(\()p Fn(\033)2161 4919 y Fk(t)2192 4907 y Fp(\))k(=)h Fn(\033)2401 4919 y Fh(\000)p Fk(t)2482 4907 y Fp(,)i(with)e Fn(\031)k Fp(the)c(iden)n(tit)n(y)-7 b(.)53 b(It)515 5106 y(is)30 b(easy)h(to)g(see)g(that)h(when)f(the)h (spin\015ip)e(dynamics)f(satis\014es)h(detailed)f(balance,)i(as)g(in) 515 5305 y(\(3.7\),)c(then)h(MEP\()p Fn(\032)p Fp(\))23 b(=)g(0)k(for)g(all)e(translation)g(in)n(v)-5 b(arian)n(t)24 b(stationary)h(distributions)f Fn(\032)p Fp(.)1926 5650 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fq(Prop)s(osition)30 b(3.1)40 b Fe(Supp)l(ose)24 b(that)f(the)h(spin\015ip)g(r)l(ates)f(satisfy)i(\(3.7\))f(for)h(some)e (p)l(otential)515 722 y Fp(\()p Fn(V)595 734 y Fk(A)650 722 y Fp(\))36 b Fe(and)h(that)f Fn(\032)g Fe(is)g(a)h(tr)l(anslation)f (invariant)h(stationary)g(distribution)g(as)f(in)g(\(3.8\).)515 922 y(Then,)1727 1121 y(MEP)q Fp(\()p Fn(\032)p Fp(\))23 b(=)g(0)p Fn(:)515 1386 y Fq(Pro)s(of:)71 b Fp(Using)20 b(translation)e(in)n(v)-5 b(ariance,)19 b(w)n(e)h(can)h(substitute)g (\(3.7\))f(in)g(\(3.10\))h(to)f(obtain)515 1586 y(that)960 1785 y(MEP)o(\()p Fn(\032)p Fp(\))k(=)e Fi(\000)1473 1729 y Fp(1)p 1442 1766 104 4 v 1442 1842 a Fi(j)p Fp(\003)p Fi(j)1569 1706 y Ff(X)1572 1884 y Fk(i)p Fh(2)p Fj(\003)1703 1672 y Ff(Z)1800 1785 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))2230 1706 y Ff(X)2232 1884 y Fk(A)p Fh(3)p Fk(i)2351 1785 y Fp([)p Fn(V)2422 1797 y Fk(A)2477 1785 y Fp(\()p Fn(\033)2559 1751 y Fk(i)2587 1785 y Fp(\))19 b Fi(\000)f Fn(V)2769 1797 y Fk(A)2823 1785 y Fp(\()p Fn(\033)s Fp(\)])515 2034 y(for)25 b(all)f(\014nite)h(\003)e Fi(\032)p 1146 2036 45 4 v 1146 1980 43 4 v 1146 2034 3 3 v 1147 2032 V 1148 2030 V 1149 2028 V 1150 2027 V 1151 2025 V 1152 2023 V 1153 2021 V 1154 2019 V 1155 2017 V 1156 2015 V 1156 2013 V 1157 2012 V 1158 2010 V 1159 2008 V 1160 2006 V 1161 2004 V 1162 2002 V 1163 2000 V 1164 1999 V 1165 1997 V 1166 1995 V 1167 1993 V 1168 1991 V 1169 1989 V 1170 1987 V 1170 1985 V 1171 1984 V 1172 1982 V 1173 1980 V 1161 2034 V 1162 2032 V 1163 2030 V 1164 2028 V 1165 2027 V 1166 2025 V 1167 2023 V 1168 2021 V 1169 2019 V 1170 2017 V 1170 2015 V 1171 2013 V 1172 2012 V 1173 2010 V 1174 2008 V 1175 2006 V 1176 2004 V 1177 2002 V 1178 2000 V 1179 1999 V 1180 1997 V 1181 1995 V 1182 1993 V 1183 1991 V 1184 1989 V 1184 1987 V 1185 1985 V 1186 1984 V 1187 1982 V 1188 1980 V 1145 1989 4 12 v 1146 1985 4 8 v 1147 1984 4 6 v 1148 1982 4 4 v 1148 1980 4 2 v 1189 2034 4 12 v 1188 2034 4 8 v 1187 2034 4 6 v 1186 2034 4 4 v 1186 2034 4 2 v 1191 2002 a Fk(d)1230 2034 y Fp(.)36 b(W)-7 b(e)26 b(no)n(w)g(split)e(the)i (sum)f(o)n(v)n(er)f Fn(A)f Fi(3)h Fn(i)h Fp(in)g(t)n(w)n(o)g(parts:)36 b Fn(A)23 b Fi(\032)g Fp(\003)i(and)515 2233 y Fn(A)18 b Fi(\\)h Fp(\003)727 2203 y Fk(c)784 2233 y Fi(6)p Fp(=)j Fi(;)p Fp(.)37 b(F)-7 b(or)27 b(the)h(\014rst)f(part)g(w)n(e)h(get)1201 2516 y Fi(\000)1306 2459 y Fp(1)p 1276 2496 104 4 v 1276 2573 a Fi(j)p Fp(\003)p Fi(j)1403 2437 y Ff(X)1406 2615 y Fk(i)p Fh(2)p Fj(\003)1536 2403 y Ff(Z)1633 2516 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\)[)p Fn(H)2141 2528 y Fj(\003)2192 2516 y Fp(\()p Fn(\033)2274 2481 y Fk(i)2303 2516 y Fp(\))19 b Fi(\000)f Fn(H)2506 2528 y Fj(\003)2555 2516 y Fp(\()p Fn(\033)s Fp(\)])515 2798 y(with)31 b Fn(H)777 2810 y Fj(\003)827 2798 y Fp(\()p Fn(\033)s Fp(\))g(=)1067 2736 y Ff(P)1155 2823 y Fk(A)p Fh(\032)p Fj(\003)1320 2798 y Fn(V)1368 2810 y Fk(A)1423 2798 y Fp(\()p Fn(\033)s Fp(\))i(only)d(dep)r(ending)i(on)g(the)g(spin)f(con\014guration)e Fn(\033)36 b Fp(in)31 b(\003.)515 2997 y(Therefore,)26 b(and)i(b)n(y)f(the)h(stationarit)n(y)c(\(3.8\))j(of)h Fn(\032)p Fp(,)g(this)e(\014rst)i(part)f(is)f(equal)g(to)1167 3223 y(1)p 1136 3260 V 1136 3336 a Fi(j)p Fp(\003)p Fi(j)1263 3166 y Ff(Z)1360 3279 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))1588 3200 y Ff(X)1574 3379 y Fk(i)p Fh(2)p Fj(\003)1687 3362 y Fg(c)1734 3279 y Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\)[)p Fn(H)2042 3291 y Fj(\003)2092 3279 y Fp(\()p Fn(\033)2174 3245 y Fk(i)2202 3279 y Fp(\))19 b Fi(\000)f Fn(H)2405 3291 y Fj(\003)2455 3279 y Fp(\()p Fn(\033)s Fp(\)])24 b(=)e(0)p Fn(:)515 3562 y Fp(F)-7 b(or)27 b(the)h(second)f (part)g(w)n(e)g(get)1001 3844 y Fi(\000)1107 3788 y Fp(1)p 1076 3825 V 1076 3901 a Fi(j)p Fp(\003)p Fi(j)1203 3765 y Ff(X)1206 3943 y Fk(i)p Fh(2)p Fj(\003)1337 3731 y Ff(Z)1434 3844 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))2002 3765 y Ff(X)1864 3947 y Fk(A)p Fh(3)p Fk(i;A)p Fh(\\)p Fj(\003)2142 3930 y Fg(c)2174 3947 y Fh(6)p Fj(=)p Fh(;)2259 3844 y Fp([)p Fn(V)2330 3856 y Fk(A)2385 3844 y Fp(\()p Fn(\033)2467 3810 y Fk(i)2495 3844 y Fp(\))19 b Fi(\000)f Fn(V)2677 3856 y Fk(A)2731 3844 y Fp(\()p Fn(\033)s Fp(\)])p Fn(:)515 4126 y Fp(But)k(since)f(the)i(p)r(oten)n(tial)d(\()p Fn(V)1435 4138 y Fk(A)1490 4126 y Fp(\))i(is)f(b)n(y)h(de\014nition)f (uniformly)e(and)j(absolutely)d(summable)515 4325 y(b)n(y)34 b(3.6,)h(it)e(easily)f(follo)n(ws)f(that)k(this)e(second)h(part)f(go)r (es)h(to)g(zero)f(as)g(\003)h Fi(")p 2992 4327 45 4 v 2992 4271 43 4 v 2992 4325 3 3 v 2993 4323 V 2993 4321 V 2994 4319 V 2995 4318 V 2996 4316 V 2997 4314 V 2998 4312 V 2999 4310 V 3000 4309 V 3001 4307 V 3002 4305 V 3003 4303 V 3004 4301 V 3005 4299 V 3006 4297 V 3007 4296 V 3007 4294 V 3008 4292 V 3009 4290 V 3010 4288 V 3011 4286 V 3012 4284 V 3013 4282 V 3014 4281 V 3015 4279 V 3016 4277 V 3017 4275 V 3018 4273 V 3019 4271 V 3007 4325 V 3008 4323 V 3008 4321 V 3009 4319 V 3010 4318 V 3011 4316 V 3012 4314 V 3013 4312 V 3014 4310 V 3015 4309 V 3016 4307 V 3017 4305 V 3018 4303 V 3019 4301 V 3020 4299 V 3021 4297 V 3022 4296 V 3022 4294 V 3023 4292 V 3024 4290 V 3025 4288 V 3026 4286 V 3027 4284 V 3028 4282 V 3029 4281 V 3030 4279 V 3031 4277 V 3032 4275 V 3033 4273 V 3034 4271 V 2990 4281 4 12 v 2991 4277 4 8 v 2992 4275 4 6 v 2993 4273 4 4 v 2994 4271 4 2 v 3035 4325 4 12 v 3034 4325 4 8 v 3033 4325 4 6 v 3031 4325 4 4 v 3031 4325 4 2 v 3036 4294 a Fk(d)3109 4325 y Fp(along)e(a)515 4525 y(sequence)27 b(of)g(cub)r(es.)p 1177 4525 48 48 v 515 4724 a Fq(Remark:)35 b Fp(When)28 b(the)g(p)r(oten)n(tial)e Fn(V)1693 4736 y Fk(A)1775 4724 y Fp(satis\014es)1509 4927 y Ff(X)1505 5105 y Fk(A)p Fh(3)p Fj(0)1647 5006 y Fi(j)p Fn(A)p Fi(j)14 b Fp(sup)1811 5072 y Fk(\033)1908 5006 y Fi(j)p Fn(V)1979 5018 y Fk(A)2034 5006 y Fp(\()p Fn(\033)s Fp(\))p Fi(j)24 b Fn(<)e Fi(1)p Fn(;)779 b Fp(\(3.11\))515 5289 y(Prop)r(osition)24 b(3.1)i(can)i(also) d(b)r(e)j(obtained)e(as)h(an)h(application)c(of)j(Corollary)c(4.2)k(in) g([4].)1905 5650 y(10)p eop %%Page: 11 11 11 10 bop 639 523 a Fp(W)-7 b(e)26 b(see)g(from)e(the)i(previous)d (Prop)r(osition)f(that)k(the)g(condition)e(of)h(detailed)f(balance)515 722 y(\(3.7\))f(implies)c(that)24 b(the)f(MEP)g(is)f(zero)g(for)h(all)e (translation)f(in)n(v)-5 b(arian)n(t)20 b(stationary)g(states.)515 922 y(W)-7 b(e)31 b(are)f(no)n(w)h(going)e(to)i(sho)n(w)f(a)h(sort)f (of)h(in)n(v)n(erse:)42 b(if)31 b(there)g(is)f(a)g(`nice')h(stationary) d(dis-)515 1121 y(tribution)d Fn(\032)i Fp(suc)n(h)g(that)g(MEP\()p Fn(\032)p Fp(\))c(=)f(0,)27 b(then)g(the)h(spin\015ip)d(dynamics)g (satis\014es)g(detailed)515 1320 y(balance.)35 b(In)28 b(particular,)c Fn(\032)k Fp(is)e(then)i(necessarily)c(a)j(rev)n (ersible)d(Gibbs)j(measure!)515 1602 y Fq(Prop)s(osition)j(3.2)40 b Fe(Supp)l(ose)35 b Fp(sup)1640 1623 y Fk(\033)1698 1602 y Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))31 b(=)f Fn(M)38 b(<)30 b Fi(1)p Fe(.)51 b(Supp)l(ose)34 b Fn(\032)g Fe(is)g(a)g(tr)l(anslation)515 1802 y(invariant)c(pr)l(ob)l(ability)i (me)l(asur)l(e)e(such)g(that)1964 1765 y Fk(d\032)2033 1739 y Fd(0)p 1964 1782 102 4 v 1980 1830 a Fk(d\032)2105 1802 y Fe(exists)g(and)g(is)g(b)l(ounde)l(d)g(fr)l(om)g(b)l(elow:)1724 2028 y Fn(d\032)1810 1998 y Fj(0)p 1724 2065 124 4 v 1743 2141 a Fn(d\032)1881 2084 y Fi(\025)22 b Fn(c)h(>)g Fp(0)p Fn(:)987 b Fp(\(3.12\))515 2366 y Fe(Then)30 b(we)g(have)h(MEP)p Fp(\()p Fn(\032)p Fp(\))25 b(=)d(0)30 b Fe(i\013)i Fp(\026)-44 b Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))23 b(=)g Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))p Fe(,)30 b Fn(\032)p Fe(-a.s.,)h(wher)l(e)1505 2648 y Fp(\026)-44 b Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))23 b(=)1853 2592 y Fn(d\032)1939 2562 y Fj(0)p 1853 2629 V 1871 2705 a Fn(d\032)1986 2648 y Fp(\()p Fn(\033)s Fp(\))p Fn(c)p Fp(\(0)p Fn(;)14 b(\033)2297 2614 y Fj(0)2335 2648 y Fp(\))p Fn(:)777 b Fp(\(3.13\))515 2931 y Fq(Pro)s(of:)52 b Fp(Let)36 b Fi(F)1043 2943 y Fj(\003)1128 2931 y Fp(denote)f(the)i Fn(\033)s Fp(-\014eld)e (generated)f(b)n(y)h Fn(\033)s Fp(\()p Fn(x)p Fp(\))p Fn(;)14 b(x)39 b Fi(2)d Fp(\003.)61 b(Denote)36 b(b)n(y)f Fn(\032)3306 2943 y Fj(\003)3356 2931 y Fp(,)515 3130 y(resp.)h Fn(\032)766 3100 y Fj(0)766 3153 y(\003)843 3130 y Fp(the)28 b(restriction)d(of)i Fn(\032)p Fp(,)h(resp.)36 b Fn(\032)1823 3100 y Fj(0)1888 3130 y Fp(to)28 b(the)f Fn(\033)s Fp(-\014eld)g Fi(F)2450 3142 y Fj(\003)2500 3130 y Fp(.)37 b(W)-7 b(e)28 b(ob)n(viously)c(ha)n(v)n(e)1580 3356 y Fn(d\032)1666 3326 y Fj(0)1666 3379 y(\003)p 1580 3393 136 4 v 1580 3469 a Fn(d\032)1666 3481 y Fj(\003)1749 3412 y Fp(=)f(I)-14 b(E)1923 3295 y Ff(\024)1977 3356 y Fn(d\032)2063 3326 y Fj(0)p 1977 3393 124 4 v 1996 3469 a Fn(d\032)2110 3412 y Fi(jF)2193 3424 y Fj(\003)2243 3295 y Ff(\025)2300 3412 y Fn(:)844 b Fp(\(3.14\))515 3694 y(Since)739 3657 y Fk(d\032)808 3632 y Fd(0)p 739 3675 102 4 v 755 3723 a Fk(d\032)873 3694 y Fi(2)24 b Fn(L)1009 3664 y Fj(1)1045 3694 y Fp(\()p Fn(d\032)p Fp(\),)j(w)n(e)d(conclude)g(from)g(the)h(martingale)c(con)n(v)n (ergence)h(theorem)i(that)1666 3977 y(lim)1636 4051 y Fj(\003)p Fh(")p 1732 4053 45 4 v 1732 3997 43 4 v 1732 4051 3 3 v 1733 4049 V 1734 4047 V 1734 4045 V 1735 4044 V 1736 4042 V 1737 4040 V 1738 4038 V 1739 4036 V 1740 4034 V 1741 4033 V 1742 4031 V 1743 4029 V 1744 4027 V 1745 4025 V 1746 4023 V 1747 4021 V 1748 4019 V 1748 4018 V 1749 4016 V 1750 4014 V 1751 4012 V 1752 4010 V 1753 4008 V 1754 4006 V 1755 4005 V 1756 4003 V 1757 4001 V 1758 3999 V 1759 3997 V 1747 4051 V 1748 4049 V 1748 4047 V 1749 4045 V 1750 4044 V 1751 4042 V 1752 4040 V 1753 4038 V 1754 4036 V 1755 4034 V 1756 4033 V 1757 4031 V 1758 4029 V 1759 4027 V 1760 4025 V 1761 4023 V 1762 4021 V 1762 4019 V 1763 4018 V 1764 4016 V 1765 4014 V 1766 4012 V 1767 4010 V 1768 4008 V 1769 4006 V 1770 4005 V 1771 4003 V 1772 4001 V 1773 3999 V 1774 3997 V 1730 4006 4 12 v 1731 4003 4 8 v 1732 4001 4 6 v 1733 3999 4 4 v 1734 3997 4 2 v 1775 4051 4 12 v 1774 4051 4 8 v 1773 4051 4 6 v 1771 4051 4 4 v 1771 4051 4 2 v 1776 4019 a Fg(d)1835 3921 y Fn(d\032)1921 3890 y Fj(0)1921 3943 y(\003)p 1835 3958 136 4 v 1835 4034 a Fn(d\032)1921 4046 y Fj(\003)2004 3977 y Fp(=)2101 3921 y Fn(d\032)2187 3890 y Fj(0)p 2101 3958 124 4 v 2120 4034 a Fn(d\032)2235 3977 y(;)909 b Fp(\(3.15\))515 4259 y(in)37 b Fn(L)679 4229 y Fj(1)716 4259 y Fp(\()p Fn(d\032)p Fp(\).)70 b(Denote)38 b(b)n(y)g Fn(\027)1421 4271 y Fj(\003)1509 4259 y Fp(the)g(Bernoulli)d(measure)h(with)i Fn(\027)5 b Fp(\()p Fn(\033)s Fp(\(0\))42 b(=)e(+1\))g(=)g(1)p Fn(=)p Fp(2,)515 4458 y(restricted)26 b(to)h Fi(F)1046 4470 y Fj(\003)1095 4458 y Fp(.)37 b(Then,)28 b(b)n(y)f(stationarit)n (y)e(of)i Fn(\032)h Fp(w)n(e)f(ha)n(v)n(e,)f(with)i Fn(f)2691 4470 y Fj(\003)2763 4458 y Fp(=)2860 4421 y Fk(d\032)2929 4429 y Fd(\003)p 2860 4439 111 4 v 2861 4487 a Fk(d\027)2929 4495 y Fd(\003)2981 4458 y Fp(,)1036 4741 y(0)82 b(=)1308 4662 y Ff(X)1311 4840 y Fk(i)p Fh(2)p Fj(\003)1442 4628 y Ff(Z)1538 4741 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))29 b Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\)[ln)g Fn(f)2131 4753 y Fj(\003)2180 4741 y Fp(\()p Fn(\033)2262 4706 y Fk(i)2290 4741 y Fp(\))19 b Fi(\000)f Fp(ln)13 b Fn(f)2548 4753 y Fj(\003)2597 4741 y Fp(\()p Fn(\033)s Fp(\)])1160 5010 y(=)1308 4932 y Ff(X)1311 5110 y Fk(i)p Fh(2)p Fj(\003)1442 4897 y Ff(Z)1538 5010 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))29 b Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))g(ln)2090 4954 y Fn(d\032)2176 4924 y Fk(i)2176 4977 y Fj(\003)p 2090 4991 136 4 v 2090 5067 a Fn(d\032)2176 5079 y Fj(\003)2236 5010 y Fp(\()p Fn(\033)s Fp(\))1160 5280 y(=)1308 5201 y Ff(X)1311 5380 y Fk(i)p Fh(2)p Fj(\003)1442 5167 y Ff(Z)1538 5280 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))29 b Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))g(ln)2090 5224 y Fn(d\032)2176 5194 y Fk(i)p 2090 5261 114 4 v 2104 5337 a Fn(d\032)1905 5650 y Fp(11)p eop %%Page: 12 12 12 11 bop 1160 565 a Fp(+)1308 486 y Ff(X)1311 664 y Fk(i)p Fh(2)p Fj(\003)1442 452 y Ff(Z)1538 565 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))1968 448 y Ff(\024)2014 565 y Fp(ln)2107 509 y Fn(d\032)2193 478 y Fk(i)2193 531 y Fj(\003)p 2107 546 136 4 v 2107 622 a Fn(d\032)2193 634 y Fj(\003)2252 565 y Fp(\()p Fn(\033)s Fp(\))20 b Fi(\000)e Fp(ln)2562 509 y Fn(d\032)2648 478 y Fk(i)p 2562 546 114 4 v 2576 622 a Fn(d\032)2685 565 y Fp(\()p Fn(\033)s Fp(\))2799 448 y Ff(\025)1160 834 y Fp(=)83 b Fi(j)p Fp(\003)p Fi(j)1426 721 y Ff(Z)1522 834 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))29 b Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))g(ln)2087 777 y Fn(d\032)2173 747 y Fj(0)p 2087 815 124 4 v 2106 891 a Fn(d\032)2221 834 y Fp(\()p Fn(\033)s Fp(\))1160 1058 y(+)1308 979 y Ff(X)1311 1157 y Fk(i)p Fh(2)p Fj(\003)1442 945 y Ff(Z)1538 1058 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))29 b Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))p Fn(F)2048 1024 y Fk(i)2036 1078 y Fj(\003)2087 1058 y Fp(\()p Fn(\033)s Fp(\))p Fn(:)943 b Fp(\(3.16\))515 1340 y(Here)27 b(w)n(e)g (abbreviated)1425 1622 y Fn(F)1490 1588 y Fk(i)1478 1643 y Fj(\003)1528 1622 y Fp(\()p Fn(\033)s Fp(\))d(=)1753 1505 y Ff(\022)1814 1622 y Fp(ln)1907 1566 y Fn(d\032)1993 1536 y Fk(i)1993 1589 y Fj(\003)p 1907 1603 136 4 v 1907 1679 a Fn(d\032)1993 1691 y Fj(\003)2071 1622 y Fi(\000)18 b Fp(ln)2247 1566 y Fn(d\032)2333 1536 y Fk(i)p 2247 1603 114 4 v 2261 1679 a Fn(d\032)2371 1505 y Ff(\023)2446 1622 y Fn(;)698 b Fp(\(3.17\))515 1905 y(and)24 b(used)g(the)g (translation)d(in)n(v)-5 b(ariance)21 b(of)j Fn(\032)g Fp(in)f(the)i(last)e(step.)36 b(F)-7 b(rom)22 b(\(3.16\))i(w)n(e)f (obtain:)799 2067 y Ff(\014)799 2116 y(\014)799 2166 y(\014)799 2216 y(\014)826 2074 y(Z)923 2187 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))29 b Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))g(ln)1488 2131 y Fn(d\032)1574 2101 y Fj(0)p 1488 2168 124 4 v 1507 2244 a Fn(d\032)1621 2187 y Fp(\()p Fn(\033)s Fp(\))1735 2067 y Ff(\014)1735 2116 y(\014)1735 2166 y(\014)1735 2216 y(\014)1847 2187 y Fi(\024)2036 2131 y Fp(1)p 2005 2168 104 4 v 2005 2244 a Fi(j)p Fp(\003)p Fi(j)2132 2108 y Ff(X)2135 2286 y Fk(i)p Fh(2)p Fj(\003)2266 2067 y Ff(\014)2266 2116 y(\014)2266 2166 y(\014)2266 2216 y(\014)2294 2074 y(Z)2390 2187 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))29 b Fn(c)p Fp(\()p Fn(i;)14 b(\033)s Fp(\))p Fn(F)2900 2153 y Fk(i)2888 2207 y Fj(\003)2939 2187 y Fp(\()p Fn(\033)s Fp(\))3053 2067 y Ff(\014)3053 2116 y(\014)3053 2166 y(\014)3053 2216 y(\014)1847 2445 y Fi(\024)83 b Fn(M)2125 2389 y Fp(1)p 2094 2426 V 2094 2502 a Fi(j)p Fp(\003)p Fi(j)2222 2366 y Ff(X)2225 2544 y Fk(i)p Fh(2)p Fj(\003)2356 2332 y Ff(Z)2452 2445 y Fn(d\032)28 b Fi(j)p Fn(F)2654 2411 y Fj(0)2642 2466 y(\003)p Fh(\000)p Fk(i)2768 2445 y Fi(j)p Fn(;)353 b Fp(\(3.18\))515 2727 y(where)37 b(in)h(the)g(last)f (step)h(w)n(e)g(used)g(translation)d(in)n(v)-5 b(ariance)35 b(of)j Fn(\032)g Fp(once)f(again.)67 b(No)n(w)515 2927 y(from)26 b(the)i(elemen)n(tary)c(inequalit)n(y)g Fi(j)14 b Fp(ln)f Fn(a)18 b Fi(\000)g Fp(ln)13 b Fn(b)p Fi(j)23 b(\024)g(j)p Fp(\()p Fn(a)18 b Fi(\000)g Fn(b)p Fp(\))p Fi(j)p Fn(=a)g Fi(^)h Fn(b)27 b Fp(w)n(e)g(can)g(deduce)h(the)515 3126 y(follo)n(wing:)34 b(if)28 b Fn(f)1017 3138 y Fk(n)1091 3126 y Fp(con)n(v)n(erges)e(to)i Fn(f)37 b Fp(in)28 b Fn(L)1799 3096 y Fj(1)1836 3126 y Fp(\()p Fn(d\032)p Fp(\))h(and)g Fn(f)2219 3138 y Fk(n)2263 3126 y Fn(;)14 b(f)38 b Fp(are)27 b(b)r(ounded)i(from)e(b)r(elo)n(w)g(b)n(y)515 3325 y(some)i(constan)n(t)h Fn(c)e(>)g Fp(0,)j(then)g(ln)13 b Fn(f)1633 3337 y Fk(n)1709 3325 y Fp(con)n(v)n(erges)28 b(to)i(ln)13 b Fn(f)40 b Fp(in)30 b Fn(L)2509 3295 y Fj(1)2545 3325 y Fp(\()p Fn(d\032)p Fp(\).)47 b(This)30 b(fact)h(implies)515 3524 y(that)d(for)f(giv)n(en)e Fn(\017)e(>)g Fp(0,)k(w)n(e)g(can)h(c)n(ho)r(ose)e(\001)d Fi(\032)p 2007 3526 45 4 v 2007 3470 43 4 v 2007 3524 3 3 v 2008 3522 V 2008 3520 V 2009 3518 V 2010 3517 V 2011 3515 V 2012 3513 V 2013 3511 V 2014 3509 V 2015 3507 V 2016 3506 V 2017 3504 V 2018 3502 V 2019 3500 V 2020 3498 V 2021 3496 V 2022 3494 V 2022 3493 V 2023 3491 V 2024 3489 V 2025 3487 V 2026 3485 V 2027 3483 V 2028 3481 V 2029 3479 V 2030 3478 V 2031 3476 V 2032 3474 V 2033 3472 V 2034 3470 V 2022 3524 V 2023 3522 V 2023 3520 V 2024 3518 V 2025 3517 V 2026 3515 V 2027 3513 V 2028 3511 V 2029 3509 V 2030 3507 V 2031 3506 V 2032 3504 V 2033 3502 V 2034 3500 V 2035 3498 V 2036 3496 V 2037 3494 V 2037 3493 V 2038 3491 V 2039 3489 V 2040 3487 V 2041 3485 V 2042 3483 V 2043 3481 V 2044 3479 V 2045 3478 V 2046 3476 V 2047 3474 V 2048 3472 V 2049 3470 V 2005 3479 4 12 v 2006 3476 4 8 v 2007 3474 4 6 v 2008 3472 4 4 v 2009 3470 4 2 v 2050 3524 4 12 v 2049 3524 4 8 v 2048 3524 4 6 v 2046 3524 4 4 v 2046 3524 4 2 v 2051 3493 a Fk(d)2118 3524 y Fp(suc)n(h)k(that)h(for)f(all)e(\001)2796 3494 y Fh(0)2843 3524 y Fi(\033)e Fp(\001)1634 3694 y Ff(Z)1731 3807 y Fn(d\032)p Fi(j)p Fn(F)1905 3772 y Fj(0)1893 3827 y(\001)1948 3810 y Fc(0)1975 3807 y Fi(j)g(\024)2167 3750 y Fn(\017)p 2118 3787 132 4 v 2118 3864 a Fp(2)p Fn(M)3167 3807 y Fp(\(3.19\))515 4089 y(Next)28 b(w)n(e)f(can)g(c)n(ho) r(ose)f(\003)d Fi(\032)p 1438 4091 45 4 v 1438 4034 43 4 v 1438 4089 3 3 v 1439 4087 V 1440 4085 V 1440 4083 V 1441 4082 V 1442 4080 V 1443 4078 V 1444 4076 V 1445 4074 V 1446 4072 V 1447 4070 V 1448 4068 V 1449 4066 V 1450 4065 V 1451 4063 V 1452 4061 V 1453 4059 V 1454 4057 V 1454 4055 V 1455 4053 V 1456 4052 V 1457 4050 V 1458 4048 V 1459 4046 V 1460 4044 V 1461 4042 V 1462 4040 V 1463 4038 V 1464 4037 V 1465 4035 V 1453 4089 V 1454 4087 V 1454 4085 V 1455 4083 V 1456 4082 V 1457 4080 V 1458 4078 V 1459 4076 V 1460 4074 V 1461 4072 V 1462 4070 V 1463 4068 V 1464 4066 V 1465 4065 V 1466 4063 V 1467 4061 V 1468 4059 V 1468 4057 V 1469 4055 V 1470 4053 V 1471 4052 V 1472 4050 V 1473 4048 V 1474 4046 V 1475 4044 V 1476 4042 V 1477 4040 V 1478 4038 V 1479 4037 V 1480 4035 V 1436 4044 4 12 v 1437 4040 4 8 v 1438 4038 4 6 v 1439 4037 4 4 v 1440 4035 4 2 v 1481 4089 4 12 v 1480 4089 4 8 v 1479 4089 4 6 v 1477 4089 4 4 v 1477 4089 4 2 v 1482 4057 a Fk(d)1549 4089 y Fp(suc)n(h)k(that)1223 4315 y Fi(jf)p Fn(i)22 b Fp(:)h(\001)c(+)f Fn(i)23 b Fi(6\032)f Fp(\003)p Fi(gj)p 1223 4352 595 4 v 1468 4428 a(j)p Fp(\003)p Fi(j)1851 4371 y(\024)2293 4315 y Fn(\017)p 1948 4352 723 4 v 1948 4428 a Fp(2)p Fn(M)g Fp(sup)2218 4448 y Fj(\003)2282 4428 y Fi(k)p Fn(F)2389 4399 y Fj(0)2377 4452 y(\003)2426 4428 y Fi(k)2468 4443 y Fk(L)2514 4426 y Fd(1)2546 4443 y Fj(\()p Fk(d\032)p Fj(\))3167 4371 y Fp(\(3.20\))515 4653 y(and)27 b(w)n(e)g(obtain)g (from)f(\(3.18\):)1147 4815 y Ff(\014)1147 4865 y(\014)1147 4915 y(\014)1147 4965 y(\014)1175 4823 y(Z)1272 4936 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))i Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))g(ln)1837 4880 y Fn(d\032)1923 4849 y Fj(0)p 1837 4917 124 4 v 1855 4993 a Fn(d\032)1970 4936 y Fp(\()p Fn(\033)s Fp(\))2084 4815 y Ff(\014)2084 4865 y(\014)2084 4915 y(\014)2084 4965 y(\014)1000 5160 y Fi(\024)1188 5104 y Fp(1)p 1157 5141 104 4 v 1157 5217 a Fi(j)p Fp(\003)p Fi(j)1405 5081 y Ff(X)1285 5259 y Fk(i)p Fh(2)p Fj(\003)p Fk(;)p Fj(\001+)p Fk(i)p Fh(\032)p Fj(\003)1672 5104 y Fn(\017)p 1668 5141 42 4 v 1668 5217 a Fp(2)1738 5160 y(+)1831 5104 y Fi(jf)p Fn(i)22 b Fp(:)h(\001)c(+)f Fn(i)23 b Fi(6\032)f Fp(\003)p Fi(gj)p 1831 5141 595 4 v 2076 5217 a(j)p Fp(\003)p Fi(j)2449 5160 y Fp(sup)2489 5230 y Fj(\003)2588 5160 y Fi(k)p Fn(F)2695 5126 y Fj(0)2683 5180 y(\003)2733 5160 y Fi(k)2775 5175 y Fk(L)2821 5158 y Fd(1)2852 5175 y Fj(\()p Fk(d\032)p Fj(\))1000 5384 y Fi(\024)82 b Fn(\017:)1963 b Fp(\(3.21\))1905 5650 y(12)p eop %%Page: 13 13 13 12 bop 515 523 a Fp(No)n(w)38 b(start)h(from)f(the)h(expression)e (\(3.10\))h(for)g(the)i(en)n(trop)n(y)e(pro)r(duction.)70 b(A)39 b(simple)515 722 y(computation)25 b(sho)n(ws)h(that)568 1005 y(MEP)o(\()p Fn(\032)p Fp(\))d(=)985 948 y(1)p 985 985 42 4 v 985 1062 a(2)1050 892 y Ff(Z)1147 1005 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\)\()p Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))20 b Fi(\000)g Fp(\026)-44 b Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\)\))g(ln)2080 948 y Fn(c)p Fp(\(0)p Fn(;)g(\033)s Fp(\))p 2080 985 230 4 v 2082 1062 a(\026)-44 b Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))2337 1005 y(+)2421 892 y Ff(Z)2517 1005 y Fn(\032)p Fp(\()p Fn(d\033)s Fp(\))p Fn(c)p Fp(\(0)p Fn(;)g(\033)s Fp(\))g(ln)3055 948 y Fn(d\032)3141 918 y Fj(0)p 3055 985 124 4 v 3073 1062 a Fn(d\032)3188 1005 y Fp(\()p Fn(\033)s Fp(\))p Fn(:)3167 1204 y Fp(\(3.22\))515 1403 y(But)31 b(\(3.21\))e(sho)n(ws)h(that)g(the)h(second)f(term)g(in)f (\(3.22\))h(is)g(equal)f(to)h(zero,)g(and)g(the)h(\014rst)515 1602 y(term)26 b(is)h(zero)f(if)h(and)h(only)e(if)h Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))23 b(=)i(\026)-44 b Fn(c)p Fp(\(0)p Fn(;)14 b(\033)s Fp(\))28 b Fn(\032)p Fp(-a.s.)p 2293 1602 48 48 v 515 2026 a Fo(4)134 b(En)l(trop)l(y)46 b(pro)t(duction)d(and)i(information)515 2308 y Fp(W)-7 b(e)28 b(restrict)d(ourselv)n(es)g(again)g(to)j(a)f(simple)e(example.) 34 b(Supp)r(ose)28 b(that)g Fn(X)33 b Fp(is)27 b(a)g(\014nite)g(set)515 2507 y(and)i(consider)e(a)i(Mark)n(o)n(v)e(pro)r(cess)i(on)g Fn(X)d Fi(\002)19 b(f\000)p Fp(1)p Fn(;)14 b Fp(+1)p Fi(g)26 b Fp(with)j(transition)e(rates)h Fn(p)3146 2519 y Fk(a)3186 2507 y Fp(\()p Fn(x;)14 b(y)s Fp(\))515 2706 y(for)32 b(the)g(c)n(hange)g(\()p Fn(x;)14 b(a)p Fp(\))32 b Fi(!)f Fp(\()p Fn(y)s(;)14 b(a)p Fp(\))32 b(and)h(transition)c(rates) j Fn(q)2429 2718 y Fk(x)2471 2706 y Fp(\()p Fn(a;)14 b(b)p Fp(\))31 b(=)f(1)j(for)e(the)i(c)n(hange)515 2906 y(\()p Fn(x;)14 b(a)p Fp(\))39 b Fi(!)f Fp(\()p Fn(x;)14 b(b)p Fp(\).)65 b(No)37 b(other)f(transitions)e(are)i(allo)n(w)n(ed.)61 b(W)-7 b(e)37 b(th)n(us)g(ha)n(v)n(e)f(the)h(Master)515 3105 y(Equation)560 3331 y Fn(@)5 b(P)662 3343 y Fk(t)691 3331 y Fp(\()p Fn(x;)14 b(a)p Fp(\))p 560 3368 325 4 v 682 3444 a Fn(@)5 b(t)917 3387 y Fp(=)1014 3308 y Ff(X)1004 3486 y Fk(y)r Fh(2)p Fk(X)1144 3387 y Fp([)p Fn(p)1209 3399 y Fk(a)1249 3387 y Fp(\()p Fn(y)s(;)14 b(x)p Fp(\))p Fn(P)1494 3399 y Fk(t)1524 3387 y Fp(\()p Fn(y)s(;)g(a)p Fp(\))k Fi(\000)g Fn(p)1856 3399 y Fk(a)1896 3387 y Fp(\()p Fn(x;)c(y)s Fp(\))p Fn(P)2141 3399 y Fk(t)2172 3387 y Fp(\()p Fn(x;)g(a)p Fp(\)])19 b(+)2511 3308 y Ff(X)2489 3487 y Fk(b)p Fj(=)p Fh(\006)p Fj(1)2654 3387 y Fp([)p Fn(P)2730 3399 y Fk(t)2760 3387 y Fp(\()p Fn(x;)14 b(b)p Fp(\))19 b Fi(\000)f Fn(P)3099 3399 y Fk(t)3128 3387 y Fp(\()p Fn(x;)c(a)p Fp(\)])3167 3586 y(\(4.23\))515 3786 y(for)27 b(a)g(probabilit)n(y)d(distribution)h Fn(P)1642 3798 y Fk(t)1699 3786 y Fp(on)j Fn(X)d Fi(\002)18 b(f\000)p Fp(1)p Fn(;)c Fp(+1)p Fi(g)p Fp(.)34 b(Assuming)26 b(that)i Fn(p)2988 3798 y Fk(a)3056 3786 y Fp(is)e(double)515 3985 y(sto)r(c)n(hastic:)1041 4188 y Ff(X)1083 4363 y Fk(y)1175 4267 y Fn(p)1217 4279 y Fk(a)1257 4267 y Fp(\()p Fn(x;)14 b(y)s Fp(\))24 b(=)e(1)h(=)1712 4188 y Ff(X)1754 4363 y Fk(y)1846 4267 y Fn(p)1888 4279 y Fk(a)1928 4267 y Fp(\()p Fn(y)s(;)14 b(x)p Fp(\))p Fn(;)42 b Fp(for)27 b Fn(x)d Fi(2)f Fn(X)r(;)14 b(a)23 b Fp(=)f Fi(\006)p Fp(1)p Fn(;)515 4549 y Fp(the)j(distribution)d Fn(\032)k Fp(de\014ned)f(b)n(y)g Fn(\032)p Fp(\()p Fn(x;)14 b(a)p Fp(\))24 b(=)e(1)p Fn(=)p Fp(\(2)p Fi(j)p Fn(X)7 b Fi(j)p Fp(\))p Fn(;)14 b(x)22 b Fi(2)i Fn(X)r(;)14 b(a)22 b Fp(=)h Fi(\006)p Fp(1)h(is)g(stationary)f(for)515 4749 y(the)28 b(Mark)n(o)n(v)d(pro)r(cess)h(\(4.23\).)515 4948 y(Assume)g(furthermore)g(that)1387 5230 y Fn(p)1429 5242 y Fk(a)1469 5230 y Fp(\()p Fn(x;)14 b(y)s Fp(\))23 b(=)g Fn(p)1814 5242 y Fh(\000)p Fk(a)1906 5230 y Fp(\()p Fn(y)s(;)14 b(x)p Fp(\))24 b Fi(6)p Fp(=)e Fn(p)2251 5242 y Fk(a)2291 5230 y Fp(\()p Fn(y)s(;)14 b(x)p Fp(\))p Fn(:)661 b Fp(\(4.24\))1905 5650 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fp(W)-7 b(e)30 b(no)n(w)e(de\014ne)i(t)n(w)n(o)f (mean)f(en)n(trop)n(y)g(pro)r(ductions)g(that)i(di\013er)e(only)g(b)n (y)i(the)f(c)n(hoice)f(of)515 722 y(time-rev)n(ersal)22 b(transformation.)33 b(F)-7 b(or)27 b(the)h(\014rst)g(c)n(hoice)d(w)n (e)j(de\014ne)1568 1004 y Fn(\022)r Fp(\()p Fn(x)1688 1016 y Fk(t)1718 1004 y Fn(;)14 b(a)1799 1016 y Fk(t)1828 1004 y Fp(\))23 b(=)g(\()p Fn(x)2050 1016 y Fh(\000)p Fk(t)2132 1004 y Fn(;)14 b(a)2213 1016 y Fh(\000)p Fk(t)2294 1004 y Fp(\))515 1285 y(and)27 b(the)h(corresp)r(onding)d(mean)h(en)n (trop)n(y)h(pro)r(duction)f(is)932 1567 y(MEP)o(\()p Fn(\032)p Fp(\))d(=)1410 1511 y(1)p 1349 1548 163 4 v 1349 1624 a(2)p Fi(j)p Fn(X)7 b Fi(j)1574 1488 y Ff(X)1536 1666 y Fk(x;y)r Fh(2)p Fk(X)1733 1567 y Fp([)p Fn(p)1798 1579 y Fh(\000)p Fj(1)1887 1567 y Fp(\()p Fn(x;)14 b(y)s Fp(\))19 b Fi(\000)f Fn(p)2223 1579 y Fj(+1)2311 1567 y Fp(\()p Fn(x;)c(y)s Fp(\)])g(ln)2633 1511 y Fn(p)2675 1523 y Fh(\000)p Fj(1)2764 1511 y Fp(\()p Fn(x;)g(y)s Fp(\))p 2633 1548 324 4 v 2633 1624 a Fn(p)2675 1636 y Fj(+1)2764 1624 y Fp(\()p Fn(x;)g(y)s Fp(\))2967 1567 y Fn(;)515 1849 y Fp(whic)n(h)22 b(is)h(clearly)d(strictly)h(p)r (ositiv)n(e)h(as)h(a)g(consequence)f(of)i(the)g(assumption)d Fn(p)3007 1861 y Fh(\000)p Fk(a)3099 1849 y Fp(\()p Fn(x;)14 b(y)s Fp(\))23 b Fi(6)p Fp(=)515 2048 y Fn(p)557 2060 y Fk(a)597 2048 y Fp(\()p Fn(x;)14 b(y)s Fp(\))37 b(in)f(\(4.24\).)64 b(F)-7 b(or)36 b(the)i(second)e(c)n(hoice,)h(w)n(e)g(de\014ne)g(the)g (in)n(v)n(olution)c Fn(\031)s Fp(\()p Fn(x;)14 b(a)p Fp(\))39 b(=)515 2247 y(\()p Fn(x;)14 b Fi(\000)p Fn(a)p Fp(\))28 b(on)f Fn(X)e Fi(\002)18 b(f\000)p Fp(1)p Fn(;)c Fp(+1)p Fi(g)25 b Fp(and)i(the)h(corresp)r(onding)d(time-rev)n(ersal) 1246 2529 y Fn(\022)1287 2494 y Fh(0)1311 2529 y Fp(\()p Fn(x)1390 2541 y Fk(t)1420 2529 y Fn(;)14 b(a)1501 2541 y Fk(t)1530 2529 y Fp(\))23 b(=)g(\()p Fn(x)1752 2541 y Fh(\000)p Fk(t)1834 2529 y Fn(;)14 b Fi(\000)p Fn(a)1980 2541 y Fh(\000)p Fk(t)2060 2529 y Fp(\))23 b(=)g Fn(\022)e Fi(\016)d Fn(\031)s Fp(\()p Fn(x)2452 2541 y Fk(t)2482 2529 y Fn(;)c(a)2563 2541 y Fk(t)2592 2529 y Fp(\))p Fn(:)515 2810 y Fp(In)33 b(that)g(case,)h(b)n(y)f(\(4.24\),)h(the)f (corresp)r(onding)d(mean)i(en)n(trop)n(y)g(pro)r(duction)g(MEP)3246 2822 y Fk(\031)3323 2810 y Fp(is)515 3010 y(easily)20 b(found)j(to)f(b)r(e)h(zero!)34 b(As)23 b(an)g(ob)n(vious)d (conclusion:)31 b(the)24 b(mean)d(en)n(trop)n(y)g(pro)r(duction)515 3209 y(in)i(a)g(stationary)e(state)j(dep)r(ends)g(on)f(the)h(c)n(hoice) e(of)i(time-rev)n(ersal)18 b(transformation)i(used)515 3408 y(to)29 b(de\014ne)i(it.)43 b(More)29 b(ph)n(ysically)c(relev)-5 b(an)n(t)28 b(ho)n(w)n(ev)n(er)g(is)h(that)h(this)f(example)f(reminds)f (us)515 3607 y(of)32 b(the)g(signi\014can)n(t)d(di\013erence)h(b)r(et)n (w)n(een)i(en)n(trop)n(y)f(and)g(functions)h(directly)d(de\014ned)j(on) 515 3807 y(the)e(microscopic)c(phase)j(space.)44 b(In)30 b(con)n(trast)f(to)h(e.g.)44 b(the)31 b(energy)-7 b(,)29 b(whic)n(h)g(is)g(obtained)515 4006 y(b)n(y)g(ev)-5 b(aluating)26 b(the)k(microscopic)25 b(phase)j(p)r(oin)n(t,)h(a)g(de\014nition)f(of)h (en)n(trop)n(y)f(should)g(start)515 4205 y(b)n(y)c(sp)r(ecifying)e(the) j(macroscopic)c(or)i(thermo)r(dynamic)e(v)-5 b(ariables)22 b(whic)n(h)h(determine)g(the)515 4404 y(macroscopic)j(state.)46 b(In)31 b(other)f(w)n(ords,)g(what)h(are)e(the)i(con)n(trol)e (parameters)f(and)i(what)515 4604 y(v)-5 b(ariables)23 b(are)i(considered)f(part)i(of)g(the)h(dynamics.)34 b(In)27 b(the)f(previous)e(example,)h(w)n(e)g(can)515 4803 y(think)32 b(of)h(particles)d(p)r(erforming)g(a)i(random)f(w)n(alk)g(\(on)h(the)i (set)e Fn(X)7 b Fp(\))33 b(with)f(a)g(lo)r(cal)f(bias)515 5002 y(de\014ned)k(in)f(terms)g(of)h(a)g(spin)f(v)-5 b(ariable)32 b Fn(a)j Fp(=)g Fi(\006)p Fp(1.)58 b(In)35 b(the)h(\014rst)f(de\014nition)e(of)i(the)g(en-)515 5201 y(trop)n(y)27 b(pro)r(duction)h(ab)r(o)n(v)n(e,)f(this)h(bias)f(is)h (considered)f(as)h(an)g Fe(external)h Fp(\014eld)e(that)i(breaks)515 5401 y(detailed)22 b(balance.)34 b(In)24 b(the)h(second)e(de\014nition) g(ho)n(w)n(ev)n(er,)g(this)g(\014eld)h(app)r(ears)f(as)g(another)1905 5650 y(14)p eop %%Page: 15 15 15 14 bop 515 523 a Fp(dynamical)27 b(v)-5 b(ariable)28 b(whic)n(h,)j(under)g(time-rev)n(ersal,)c(m)n(ust)j(b)r(e)h (re\015ected.)47 b(In)32 b(this)e(case)515 722 y(microscopic)21 b(rev)n(ersibilit)n(y)f(is)25 b(restored)f(\(the)i(\014eld)f(has)g(b)r (ecome)g(part)g(of)h(the)g(|)g(no)n(w)f(|)515 922 y(equilibrium)d (dynamics\))k(and)h(the)h(mean)f(en)n(trop)n(y)f(pro)r(duction)g(is)h (zero.)515 1121 y(Other,)i(ph)n(ysically)c(more)i(realistic,)e (examples)i(of)i(this)f(same)g(phenomenon)f(can)i(easily)515 1320 y(b)r(e)39 b(giv)n(en.)67 b(In)39 b(this)f(w)n(a)n(y)-7 b(,)40 b(one)e(can)g(argue,)i(p)r(ositivit)n(y)35 b(of)j(en)n(trop)n(y) g(pro)r(duction)f(can-)515 1519 y(not)32 b(b)r(e)h(reduced)g(to)f(a)h (dynamical)c(prop)r(ert)n(y)i(alone)g(\(as)h(is)g(done)g(in)g([9,)h(10) o(]\).)52 b(Just)33 b(as)515 1719 y(for)c(the)i(equilibrium)25 b(en)n(trop)n(y:)41 b(its)29 b(de\014nition)g(and)g(its)h(increase)e (for)h(isolated)f(systems)515 1918 y(dep)r(ends)39 b(\(also\))f(on)h (the)g(c)n(hoice)e(of)i(v)-5 b(ariables)36 b(re\015ecting)h(the)j (observ)-5 b(ational)35 b(set-up.)515 2117 y(More)27 b(generally)-7 b(,)24 b(one)k(w)n(ould)e(sa)n(y)h(that)h(questions)f (on)g(the)h(p)r(ositivit)n(y)d(of)j(transp)r(ort)f(co-)515 2316 y(e\016cien)n(ts)34 b(cannot)g(b)r(e)i(answ)n(ered)d(without)i(sp) r(ecifying)e(exp)r(erimen)n(tal)f(conditions)g(and)515 2516 y(con)n(trol;)i(referring)d(to)j(microscopic)29 b(c)n(haos)j(or)h(ergo)r(dic-theoretic)d(considerations)f(can)515 2715 y(nev)n(er)22 b(pro)n(vide)f(a)i(complete)e(explanation.)33 b(Remark)21 b(nev)n(ertheless)h(that)h(the)h(traditional)515 2914 y(de\014nition)g(of)j(transp)r(ort)e(co)r(e\016cien)n(ts)g(\(as)g (used)i(within)d(linear)g(resp)r(onse)h(theory\))h(relies)515 3113 y(on)g(a)h(go)r(o)r(d)f(space-time)e(deca)n(y)h(of)i(curren)n(t)f (correlations)c(whic)n(h)k(is)g(b)r(oth)h(a)f(static)g(and)g(a)515 3313 y(dynamic)f(issue.)515 3737 y Fo(References)563 4001 y Fr([1])42 b(An)n(tal,)f(T.,)h(R\023)-38 b(acz,)41 b(Z.)d(and)f(Sasv\023)-38 b(ari,)41 b(L.)d(\(1997\))h(Nonequilibrium)d (steady)i(state)g(in)g(a)685 4184 y(quan)n(tum)30 b(system:)47 b(one-dimensional)32 b(transv)n(erse)h(Ising)g(mo)r(del)f(with)g (energy)g(curren)n(t,)685 4367 y Fb(Ph)n(ys.)26 b(Rev.)f(Lett.)h Fs(78)p Fr(,)g(167.)563 4616 y([2])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 4798 y(dissipation)c (theorem,)e Fb(J.)i(Stat.)f(Ph)n(ys.)f Fs(84)p Fr(,)h(899-926.)563 5048 y([3])42 b(Galla)n(v)n(otti,)i(G.)39 b(and)f(Cohen,)43 b(E.G.D.)d(\(1995\))g(Dynamical)e(ensem)n(bles)g(in)g(stationary)685 5230 y(states,)27 b Fb(J.)g(Stat.)e(Ph)n(ys.)h Fs(80)p Fr(,)g(931{970.)1905 5650 y Fp(15)p eop %%Page: 16 16 16 15 bop 563 523 a Fr([4])42 b(K)r(\177)-41 b(unsc)n(h,)42 b(H.)37 b(\(1984\))j(Non)d(rev)n(ersible)i(stationary)f(measures)g(for) h(in\014nite)e(in)n(teracting)685 706 y(particle)27 b(systems,)e Fb(Z.)h(W)-6 b(ahrsc)n(h.)26 b(V)-6 b(erw.)25 b(Gebiete)i Fs(66)p Fr(,)f(407.)563 955 y([5])42 b(Leb)r(o)n(witz,)26 b(J.L.)g(and)e(Sp)r(ohn,)g(H.)g(\(1999\))i(A)e(Galla)n(v)n(otti-Cohen)i (T)n(yp)r(e)e(Symmetry)d(in)j(the)685 1137 y(Large)g(Deviation)g(F)-6 b(unctional)23 b(for)h(Sto)r(c)n(hastic)g(Dynamics,)f Fb(J.)g(Stat.)h(Ph)n(ys.)f Fs(95)p Fr(,)h(333{365.)563 1386 y([6])42 b(Liggett,)30 b(T.M.)f(\(1985\))g Fa(Inter)l(acting)j (Particle)e(Systems)p Fr(,)g(Springer,)f(Berlin,)g(Heidelb)r(erg,)685 1569 y(New)d(Y)-6 b(ork.)563 1818 y([7])42 b(Maes,)33 b(C.)e(\(1999\))g(The)f(Fluctuation)h(Theorem)e(as)i(a)f(Gibbs)g(Prop)r (ert)n(y)-6 b(,)31 b Fb(J.)g(Stat.)f(Ph)n(ys.)685 2001 y Fs(95)p Fr(,)d(367{392.)563 2250 y([8])42 b(Maes,)35 b(C.,)g(Redig,)e(F.)g(and)e(V)-6 b(an)31 b(Mo\013aert,)k(A.)c(\(1999\)) i(On)e(the)h(de\014nition)f(of)i(en)n(trop)n(y)685 2433 y(pro)r(duction,)26 b(via)g(examples,)g(to)f(app)r(ear)i(in)e Fb(J.)h(Math.)h(Ph)n(ys)e Fr(\(Marc)n(h)h(2000)h(issue\).)563 2682 y([9])42 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 2864 y(mec)n(hanics,)d Fb(J.)g(Stat.)g(Ph)n(ys.)g Fs(85)p Fr(,)g(1{25.)524 3113 y([10])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 3296 y Fb(Comm)n(un.)24 b(Math.)j(Ph)n(ys.)e Fs(189)p Fr(,)i(365{371.)524 3545 y([11])43 b(Ruelle,)26 b(D.)f(\(1999\))h(Smo)r(oth)e(dynamics)g(and)h(new)g(theoretical)h (ideas)g(in)f(nonequilibrium)685 3728 y(statistical)j(mec)n(hanics,)e Fb(J.)g(Stat.)g(Ph)n(ys.)f Fs(95)p Fr(,)i(393{468.)524 3977 y([12])43 b(T)-6 b(olman,)49 b(R.C.)44 b(\(1938\))h Fa(The)g(principles)f(of)g(statistic)l(al)i(me)l(chanics)p Fr(,)k(Oxford)43 b(at)h(the)685 4159 y(Clarendon)27 b(Press,)g(p.)e (102.)1905 5650 y Fp(16)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9912140614928--