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b([10,)34 b(11],)h(and)f(w)m (e)i(ha)m(v)m(e)g(sho)m(wn)f(ho)m(w)h(our)e(approac)m(h)h(quite)236 3441 y(naturally)24 b(giv)m(es)i(rise)f(to)g(a)g(\015uctuation)g (theorem)h(and)f(resulting)g(Green-Kub)s(o)g(t)m(yp)s(e)h(relations.) 236 3562 y(W)-8 b(e)32 b(will)d(also)i(not)g(discuss)i(these)g(p)s(oin) m(ts)e(here)h(an)m(y)g(further)g(\(except)h(for)e(the)h(op)s(en)g (questions)236 3682 y(in)f(Section)g(4\).)43 b(F)-8 b(urther)31 b(motiv)-5 b(ations)29 b(ha)m(v)m(e)k(already)e(app)s(eared)h(in)e([8]) h(and)h(seem)g(to)f(go)g(bac)m(k)236 3802 y(so)i(far)f(as)h(Kirc)m (ho\013)7 b('s)32 b(curren)m(t)i(la)m(w,)e([9].)44 b(One)33 b(p)s(oin)m(t)f(is)g(that)g(under)i(the)f(natural)e(assumption)236 3923 y(\(2.7\),)h(the)h(mean)f(en)m(trop)m(y)i(pro)s(duction)e(can)h(b) s(e)g(directly)f(expressed)j(as)e(the)g(bilinear)d(form)1340 4169 y Fh(M)10 b(E)c(P)14 b Fj(\()p Fh(k)s(;)j(\032)p Fj(\))28 b(=)1964 4102 y(1)p 1964 4146 49 4 v 1964 4238 a(2)2040 4075 y Fb(X)2200 4169 y Fh(J)2254 4184 y Fn(ab)2326 4169 y Fj(\()p Fh(\032)p Fj(\))p Fh(A)2525 4184 y Fn(ab)2597 4169 y Fj(\()p Fh(\032)p Fj(\))854 b(\(2.24\))236 4402 y(where,)34 b(assuming)e(\(2.7\),)1339 4604 y Fh(J)1393 4619 y Fn(ab)1465 4604 y Fj(\()p Fh(\032)p Fj(\))c Fg(\021)g Fh(k)s Fj(\()p Fh(a;)17 b(b)p Fj(\))p Fh(\032)p Fj(\()p Fh(a)p Fj(\))23 b Fg(\000)g Fh(k)s Fj(\()p Fh(b;)17 b(a)p Fj(\))p Fh(\032)p Fj(\()p Fh(b)p Fj(\))236 4806 y(can)41 b(b)s(e)h(iden)m(ti\014ed)f(with)f(the)i(thermo)s(dynamic)d(\015ux)j(b) s(et)m(w)m(een)h(the)e(states)h Fh(a)f Fj(and)g Fh(b)p Fj(,)j(and)d(is)236 4927 y(linear)31 b(in)h Fh(\032)p Fj(,)h(while)1526 5178 y Fh(A)1599 5193 y Fn(ab)1671 5178 y Fj(\()p Fh(\032)p Fj(\))28 b Fg(\021)g Fj(log)2083 5111 y Fh(\032)p Fj(\()p Fh(a)p Fj(\))p Fh(k)s Fj(\()p Fh(a;)17 b(b)p Fj(\))p 2083 5155 444 4 v 2088 5246 a Fh(\032)p Fj(\()p Fh(b)p Fj(\))p Fh(k)s Fj(\()p Fh(b;)g(a)p Fj(\))236 5430 y(is)33 b(the)h(thermo)s(dynamic)e(force)i(b)m(y)g(whic) m(h)g(the)g(system)g(is)f(driv)m(en)h(from)e(equilibrium.)44 b(This)33 b(is)236 5551 y(most)21 b(transparan)m(t)h(in)f(case)i(w)m(e) g(ha)m(v)m(e)g(the)f(form)f Fh(k)s Fj(\()p Fh(a;)c(b)p Fj(\))p Fh(=k)s Fj(\()p Fh(b;)g(a)p Fj(\))28 b(=)f(exp)18 b Fg(\000)p Fj([)p Fh(V)k Fj(\()p Fh(b)p Fj(\))h Fg(\000)f Fh(V)g Fj(\()p Fh(a)p Fj(\)])g(+)g Fh(E)6 b Fj(\010)3806 5566 y Fn(ab)2007 5800 y Fj(8)p eop %%Page: 9 9 9 8 bop 236 548 a Fj(\(whic)m(h)36 b(is)f(called)f(lo)s(cal)f(detailed) h(balance)h(in)f([13]\))h(where)h Fh(E)42 b Fj(is)34 b(some)h(amplitude)f(\(external)236 668 y(\014eld\))f(and)h(\010)747 683 y Fn(ab)848 668 y Fj(=)29 b Fg(\000)p Fj(\010)1100 683 y Fn(ba)1206 668 y Fj(is)k(related)g(to)g(the)h(microscopic)e (curren)m(t)j(in)m(v)m(olving)d(states)i Fh(a)g Fj(and)f Fh(b)p Fj(.)236 789 y(Similar)24 b(discussions)29 b(ha)m(v)m(e)g (already)e(app)s(eared)h(in)f([6,)h(7)o(,)g(8,)g(13)o(],)h(e.g.)42 b(ab)s(out)27 b(the)h(relation)e(with)236 909 y(the)32 b(time-deriv)-5 b(ativ)m(e)30 b(of)h(the)h(relativ)m(e)f(en)m(trop)m(y) i(of)e(a)g(time-ev)m(olv)m(ed)g(measure)h(with)f(resp)s(ect)i(to)236 1029 y(the)g(stationary)f(state.)236 1150 y(Most)38 b(imp)s(ortan)m(t)c (to)j(us)g(ho)m(w)m(ev)m(er)i(seem)f(the)f(thermo)s(dynamic)e (analogues)h(of)h(the)g(iden)m(tities)236 1270 y(\(2.21\))24 b(and)i(\(2.22\).)40 b(It)25 b(app)s(ears)g(that)g(the)h(mean)e(en)m (trop)m(y)j(pro)s(duction)d(is)h(the)g(relativ)m(e)f(en)m(trop)m(y)236 1391 y(b)s(et)m(w)m(een)37 b(the)d(forw)m(ard)h(and)f(the)h(bac)m(kw)m (ard)h(steady)g(state.)49 b(That)34 b(these)i(states)f(are)g (di\013eren)m(t)236 1511 y(is)40 b(a)h(consequence)i(of)e(main)m (taining)c(a)j(curren)m(t)i(in)e(the)h(system.)69 b(The)41 b(system)h(is)e(then)h(able)236 1631 y(to)36 b(p)s(erform)g(w)m(ork)h (and)f(the)h(en)m(trop)m(y)h(pro)s(duction)e(measures)h(the)g(rate)f (at)g(whic)m(h)h(en)m(trop)m(y)h(is)236 1752 y(pro)s(duced)c(in)d(the)i (reserv)m(oirs.)236 2083 y Fq(3)161 b(Thermo)t(dynamic)58 b(limit:)84 b(in)l(teracting)58 b(particle)h(sys-)478 2266 y(tems)236 2514 y Fi(3.1)136 b(Dynamics)236 2698 y Fj(T)-8 b(o)32 b(eac)m(h)g(site)g Fh(x)c Fg(2)p 969 2700 53 4 v 969 2634 51 4 v 969 2698 3 3 v 970 2696 V 971 2694 V 972 2691 V 973 2689 V 975 2687 V 976 2685 V 977 2683 V 978 2681 V 979 2679 V 980 2676 V 981 2674 V 982 2672 V 983 2670 V 984 2668 V 986 2665 V 987 2663 V 988 2661 V 989 2659 V 990 2657 V 991 2654 V 992 2652 V 993 2650 V 994 2648 V 995 2646 V 997 2643 V 998 2641 V 999 2639 V 1000 2637 V 1001 2635 V 987 2698 V 988 2696 V 989 2694 V 990 2691 V 991 2689 V 992 2687 V 993 2685 V 994 2683 V 995 2681 V 997 2679 V 998 2676 V 999 2674 V 1000 2672 V 1001 2670 V 1002 2668 V 1003 2665 V 1004 2663 V 1005 2661 V 1006 2659 V 1007 2657 V 1009 2654 V 1010 2652 V 1011 2650 V 1012 2648 V 1013 2646 V 1014 2643 V 1015 2641 V 1016 2639 V 1017 2637 V 1018 2635 V 967 2646 4 14 v 969 2641 4 9 v 970 2639 4 7 v 971 2637 4 5 v 972 2635 4 3 v 1020 2698 4 14 v 1019 2698 4 9 v 1017 2698 4 7 v 1016 2698 4 5 v 1016 2698 4 3 v 1022 2662 a Fn(d)1062 2698 y Fj(,)k(w)m(e)h(asso)s(ciate)e(a)h(v)-5 b(ariable)29 b Fh(\033)t Fj(\()p Fh(x)p Fj(\))j(taking)f(v)-5 b(alues)32 b(in)f(a)g(\014nite)g(set)i Fh(S)6 b Fj(.)43 b(The)236 2843 y(con\014guration)32 b(space)h(is)f(denoted)i(b)m(y)f (\012)28 b Fg(\021)h Fh(S)p 1979 2809 53 4 v 1979 2743 51 4 v 1979 2807 3 3 v 1980 2805 V 1981 2803 V 1982 2800 V 1984 2798 V 1985 2796 V 1986 2794 V 1987 2792 V 1988 2790 V 1989 2787 V 1990 2785 V 1991 2783 V 1992 2781 V 1993 2779 V 1995 2776 V 1996 2774 V 1997 2772 V 1998 2770 V 1999 2768 V 2000 2765 V 2001 2763 V 2002 2761 V 2003 2759 V 2004 2757 V 2006 2754 V 2007 2752 V 2008 2750 V 2009 2748 V 2010 2746 V 2011 2743 V 1997 2807 V 1998 2805 V 1999 2803 V 2000 2800 V 2001 2798 V 2002 2796 V 2003 2794 V 2004 2792 V 2006 2790 V 2007 2787 V 2008 2785 V 2009 2783 V 2010 2781 V 2011 2779 V 2012 2776 V 2013 2774 V 2014 2772 V 2015 2770 V 2016 2768 V 2018 2765 V 2019 2763 V 2020 2761 V 2021 2759 V 2022 2757 V 2023 2754 V 2024 2752 V 2025 2750 V 2026 2748 V 2027 2746 V 2029 2743 V 1978 2754 4 14 v 1979 2750 4 9 v 1980 2748 4 7 v 1981 2746 4 5 v 1982 2743 4 3 v 2030 2807 4 14 v 2029 2807 4 9 v 2028 2807 4 7 v 2026 2807 4 5 v 2026 2807 4 3 v 2032 2766 a Fd(d)2072 2843 y Fj(.)44 b(W)-8 b(e)32 b(ha)m(v)m(e)i(an)f(in)m(v)m(olution)e Fh(\031)36 b Fj(on)c(\012)h(obtained)236 2964 y(from)g(an)g(in)m(v)m (olution)f Fh(i)i Fj(on)f Fh(S)40 b Fj(with)33 b Fh(\031)t(\033)t Fj(\()p Fh(x)p Fj(\))d Fg(\021)g Fh(i)p Fj(\()p Fh(\033)t Fj(\()p Fh(x)p Fj(\)\).)47 b(Lo)s(cal)32 b(v)m(ersions)j(can)f(b)s(e)f (obtained)g(via)236 3084 y Fh(\031)291 3099 y Fo(\003)345 3084 y Fh(\033)t Fj(\()p Fh(x)p Fj(\))28 b Fg(\021)g Fh(i)p Fj(\()p Fh(\033)t Fj(\()p Fh(x)p Fj(\)\))33 b(for)f Fh(x)c Fg(2)g Fj(\003)p Fh(;)17 b(\031)1493 3099 y Fo(\003)1546 3084 y Fh(\033)t Fj(\()p Fh(x)p Fj(\))28 b Fg(\021)g Fh(\033)t Fj(\()p Fh(x)p Fj(\))33 b(for)f Fh(x)c Fg(2)g Fj(\003)2486 3048 y Fn(c)2548 3084 y Fg(\021)p 2669 3086 53 4 v 2669 3020 51 4 v 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V 3752 3528 V 3753 3526 V 3754 3524 V 3756 3522 V 3757 3519 V 3758 3517 V 3759 3515 V 3760 3513 V 3761 3511 V 3762 3508 V 3763 3506 V 3764 3504 V 3765 3502 V 3751 3566 V 3752 3564 V 3753 3562 V 3754 3559 V 3756 3557 V 3757 3555 V 3758 3553 V 3759 3550 V 3760 3548 V 3761 3546 V 3762 3544 V 3763 3541 V 3764 3539 V 3765 3537 V 3767 3535 V 3768 3533 V 3769 3530 V 3770 3528 V 3771 3526 V 3772 3524 V 3773 3522 V 3774 3519 V 3775 3517 V 3776 3515 V 3778 3513 V 3779 3511 V 3780 3508 V 3781 3506 V 3782 3504 V 3783 3502 V 3732 3513 4 14 v 3733 3508 4 9 v 3735 3506 4 7 v 3736 3504 4 5 v 3736 3502 4 3 v 3785 3566 4 14 v 3783 3566 4 9 v 3782 3566 4 7 v 3781 3566 4 5 v 3780 3566 4 3 v 3786 3529 a Fn(d)236 3686 y Fj(suc)m(h)47 b(that)f(for)f Fh(\033)n(;)17 b(\021)54 b Fg(2)c Fj(\012)c(agreeing)f(on)g(\003,)k Fh(f)11 b Fj(\()p Fh(\033)t Fj(\))49 b(=)h Fh(f)11 b Fj(\()p Fh(\021)t Fj(\).)82 b(A)45 b(quite)h(general)f(in)m(teracting)236 3806 y(particle)28 b(system)i(can)f(no)m(w)g(b)s(e)g(de\014ned)i(b)m(y) e(in)m(tro)s(ducing)f(p)s(ositiv)m(e,)h(translation)e(in)m(v)-5 b(arian)m(t)28 b(and)236 3927 y(lo)s(cal)g(rates)i Fh(k)s Fj(\()p Fh(\033)n(;)17 b(\021)937 3942 y Fo(\003)991 3927 y Fh(\033)1046 3942 y Fo(\003)1095 3923 y Fd(c)1131 3927 y Fj(\))30 b(b)m(y)h(whic)m(h)g Fh(\033)h Fg(2)c Fj(\012)i(is)g(c)m(hanged)h(in)m(to)e(the)h(con\014guration)f Fh(\021)3356 3942 y Fo(\003)3410 3927 y Fh(\033)3465 3942 y Fo(\003)3514 3923 y Fd(c)3580 3927 y Fj(whic)m(h)236 4047 y(equals)j Fh(\033)j Fj(on)d(\003)824 4011 y Fn(c)890 4047 y Fj(and)f(equals)h Fh(\021)g Fg(2)c Fj(\012)k(on)f(\003,)g(where) i(\003)e(runs)i(o)m(v)m(er)f(some)f(sp)s(eci\014ed)i(set)f(of)f (\014nite)236 4167 y(subsets)g(of)p 694 4169 53 4 v 694 4103 51 4 v 694 4167 3 3 v 695 4165 V 696 4163 V 697 4160 V 698 4158 V 699 4156 V 700 4154 V 702 4152 V 703 4150 V 704 4148 V 705 4145 V 706 4143 V 707 4141 V 708 4139 V 709 4137 V 710 4135 V 711 4132 V 713 4130 V 714 4128 V 715 4126 V 716 4124 V 717 4121 V 718 4119 V 719 4117 V 720 4115 V 721 4113 V 722 4110 V 723 4108 V 725 4106 V 726 4104 V 711 4167 V 713 4165 V 714 4163 V 715 4160 V 716 4158 V 717 4156 V 718 4154 V 719 4152 V 720 4150 V 721 4148 V 722 4145 V 723 4143 V 725 4141 V 726 4139 V 727 4137 V 728 4135 V 729 4132 V 730 4130 V 731 4128 V 732 4126 V 733 4124 V 734 4121 V 736 4119 V 737 4117 V 738 4115 V 739 4113 V 740 4110 V 741 4108 V 742 4106 V 743 4104 V 692 4115 4 14 v 694 4110 4 9 v 695 4108 4 7 v 696 4106 4 5 v 697 4104 4 3 v 745 4167 4 14 v 744 4167 4 9 v 742 4167 4 7 v 741 4167 4 5 v 740 4167 4 3 v 747 4131 a Fn(d)787 4167 y Fj(.)42 b(T)-8 b(o)29 b(b)s(e)h(somewhat)f(more)f(sp)s(eci\014c)i(w)m(e)g(\014x)g(a)f (\014nite)f(cub)s(e)i(\003)3047 4182 y Fo(0)3114 4167 y Fg(\032)p 3235 4169 53 4 v 3235 4103 51 4 v 3235 4167 3 3 v 3236 4165 V 3237 4163 V 3238 4160 V 3240 4158 V 3241 4156 V 3242 4154 V 3243 4152 V 3244 4150 V 3245 4148 V 3246 4145 V 3247 4143 V 3248 4141 V 3249 4139 V 3250 4137 V 3252 4135 V 3253 4132 V 3254 4130 V 3255 4128 V 3256 4126 V 3257 4124 V 3258 4121 V 3259 4119 V 3260 4117 V 3261 4115 V 3263 4113 V 3264 4110 V 3265 4108 V 3266 4106 V 3267 4104 V 3253 4167 V 3254 4165 V 3255 4163 V 3256 4160 V 3257 4158 V 3258 4156 V 3259 4154 V 3260 4152 V 3261 4150 V 3263 4148 V 3264 4145 V 3265 4143 V 3266 4141 V 3267 4139 V 3268 4137 V 3269 4135 V 3270 4132 V 3271 4130 V 3272 4128 V 3274 4126 V 3275 4124 V 3276 4121 V 3277 4119 V 3278 4117 V 3279 4115 V 3280 4113 V 3281 4110 V 3282 4108 V 3283 4106 V 3285 4104 V 3233 4115 4 14 v 3235 4110 4 9 v 3236 4108 4 7 v 3237 4106 4 5 v 3238 4104 4 3 v 3286 4167 4 14 v 3285 4167 4 9 v 3284 4167 4 7 v 3282 4167 4 5 v 3282 4167 4 3 v 3288 4131 a Fn(d)3328 4167 y Fj(,)g(con)m(taining)236 4288 y(the)40 b(origin)e(and)i(de\014ne)h Fg(P)1250 4303 y Fo(0)1329 4288 y Fj(to)f(b)s(e)g(a)f(set)i(of)e(transformations)f Fh(U)2738 4303 y Fo(0)2818 4288 y Fj(a\013ecting)h(only)g(the)h(spins) 236 4408 y(inside)32 b(\003)581 4423 y Fo(0)620 4408 y Fj(,)h(i.e.,)f(for)g Fh(U)1079 4423 y Fo(0)1147 4408 y Fg(2)c(P)1310 4423 y Fo(0)1349 4408 y Fj(,)33 b(and)g(for)f(ev)m(ery) i Fh(\033)e Fg(2)c Fj(\012:)377 4602 y(i.)48 b Fh(U)546 4617 y Fo(0)586 4602 y Fh(\033)31 b Fg(2)e Fj(\012,)k(and)f(\()p Fh(U)1190 4617 y Fo(0)1230 4602 y Fh(\033)t Fj(\))16 b(\()p Fh(y)t Fj(\))27 b(=)h Fh(\033)t Fj(\()p Fh(y)t Fj(\),)j(for)h Fh(y)f Fg(2)d Fj(\003)2237 4566 y Fn(c)2237 4627 y Fo(0)2276 4602 y Fj(,)350 4802 y(ii.)47 b(for)32 b(ev)m(ery)j Fh(U)953 4817 y Fo(0)1020 4802 y Fg(2)28 b(P)1183 4817 y Fo(0)1223 4802 y Fj(,)k Fh(U)1358 4761 y Ff(\000)p Fo(1)1348 4827 y(0)1481 4802 y Fg(2)c(P)1644 4817 y Fo(0)1684 4802 y Fj(.)236 4996 y(Moreo)m(v)m(er,)36 b(it)d(is)g(assumed)i(\(with)e(some)g(abuse)i(of)f(notation\))e(that)h Fh(\031)t Fg(P)2945 5011 y Fo(0)2985 4996 y Fh(\031)h Fj(=)29 b Fg(P)3248 5011 y Fo(0)3288 4996 y Fj(.)47 b(W)-8 b(e)34 b(further)236 5116 y(denote)g(\003)619 5131 y Fn(x)690 5116 y Fg(\021)28 b(f)p Fh(y)d Fj(+)d Fh(x)28 b Fj(:)g Fh(y)j Fg(2)d Fj(\003)1395 5131 y Fo(0)1434 5116 y Fg(g)k Fj(and)h Fh(U)1772 5131 y Fn(x)1844 5116 y Fg(\021)28 b Fh(\034)1991 5131 y Fn(x)2035 5116 y Fh(U)2101 5131 y Fo(0)2141 5116 y Fh(\034)2183 5131 y Ff(\000)p Fn(x)2282 5116 y Fj(.)236 5237 y(The)33 b(dynamics)f(is)f(generated)i (via)e(lo)s(cal)f(translation)g(in)m(v)-5 b(arian)m(t)31 b(rates)h Fh(c)p Fj(\()p Fh(U)3083 5252 y Fn(x)3127 5237 y Fh(;)17 b(\033)t Fj(\))32 b(for)f(the)h(tran-)236 5357 y(sition)f Fh(\033)h Fg(!)44 b Fh(U)799 5372 y Fn(x)843 5357 y Fh(\033)t Fj(.)g(W)-8 b(e)32 b(assume)i(the)f(follo)m(wing:)377 5551 y(i.)48 b(P)m(ositivit)m(y:)43 b Fh(c)p Fj(\()p Fh(U)1105 5566 y Fo(0)1144 5551 y Fh(;)17 b(\033)t Fj(\))28 b(=)f(0)33 b(when)g Fh(U)1818 5566 y Fo(0)1858 5551 y Fh(\033)f Fj(=)27 b Fh(\033)37 b Fj(and)c(if)e(not,)h Fh(c)p Fj(\()p Fh(U)2765 5566 y Fo(0)2805 5551 y Fh(;)17 b(\033)t Fj(\))27 b Fg(\025)i Fh(\017)j Fj(for)g(some)h Fh(\017)28 b(>)g Fj(0,)2007 5800 y(9)p eop %%Page: 10 10 10 9 bop 350 548 a Fj(ii.)47 b(Finite)34 b(range:)49 b(there)36 b(is)f(a)g(\014nite)1776 523 y(\026)1767 548 y(\003)d Fg(\032)p 1993 550 53 4 v 1993 484 51 4 v 1993 548 3 3 v 1994 546 V 1995 544 V 1996 541 V 1997 539 V 1998 537 V 1999 535 V 2000 533 V 2001 530 V 2002 528 V 2004 526 V 2005 524 V 2006 522 V 2007 519 V 2008 517 V 2009 515 V 2010 513 V 2011 511 V 2012 508 V 2013 506 V 2015 504 V 2016 502 V 2017 500 V 2018 497 V 2019 495 V 2020 493 V 2021 491 V 2022 489 V 2023 487 V 2024 484 V 2010 548 V 2011 546 V 2012 544 V 2013 541 V 2015 539 V 2016 537 V 2017 535 V 2018 533 V 2019 530 V 2020 528 V 2021 526 V 2022 524 V 2023 522 V 2024 519 V 2026 517 V 2027 515 V 2028 513 V 2029 511 V 2030 508 V 2031 506 V 2032 504 V 2033 502 V 2034 500 V 2035 497 V 2036 495 V 2038 493 V 2039 491 V 2040 489 V 2041 487 V 2042 484 V 1991 495 4 14 v 1992 491 4 9 v 1994 489 4 7 v 1995 486 4 5 v 1995 484 4 3 v 2044 548 4 14 v 2042 548 4 9 v 2041 548 4 7 v 2040 548 4 5 v 2039 548 4 3 v 2045 512 a Fn(d)2121 548 y Fj(suc)m(h)37 b(that)e(for)g(all)e Fh(\033)n(;)17 b(\021)36 b Fg(2)d Fj(\012,)k(and)e Fh(U)3520 563 y Fo(0)3592 548 y Fg(2)e(P)3760 563 y Fo(0)3800 548 y Fj(:)480 668 y Fh(c)p Fj(\()p Fh(U)626 683 y Fo(0)666 668 y Fh(;)17 b(\033)t Fj(\))27 b(=)h Fh(c)p Fj(\()p Fh(U)1084 683 y Fo(0)1123 668 y Fh(;)17 b(\033)1229 672 y Fo(\026)1222 689 y(\003)1275 668 y Fh(\021)1330 672 y Fo(\026)1323 689 y(\003)1372 670 y Fd(c)1409 668 y Fj(\),)323 871 y(iii.)46 b(T)-8 b(ranslation)31 b(in)m(v)-5 b(ariance:)43 b(for)32 b(all)e Fh(x)e Fg(2)p 1974 873 53 4 v 1974 807 51 4 v 1974 871 3 3 v 1976 869 V 1977 867 V 1978 864 V 1979 862 V 1980 860 V 1981 858 V 1982 856 V 1983 854 V 1984 852 V 1985 849 V 1986 847 V 1988 845 V 1989 843 V 1990 841 V 1991 838 V 1992 836 V 1993 834 V 1994 832 V 1995 830 V 1996 828 V 1997 825 V 1999 823 V 2000 821 V 2001 819 V 2002 817 V 2003 814 V 2004 812 V 2005 810 V 2006 808 V 1992 871 V 1993 869 V 1994 867 V 1995 864 V 1996 862 V 1997 860 V 1999 858 V 2000 856 V 2001 854 V 2002 852 V 2003 849 V 2004 847 V 2005 845 V 2006 843 V 2007 841 V 2008 838 V 2010 836 V 2011 834 V 2012 832 V 2013 830 V 2014 828 V 2015 825 V 2016 823 V 2017 821 V 2018 819 V 2019 817 V 2020 814 V 2022 812 V 2023 810 V 2024 808 V 1973 819 4 14 v 1974 814 4 9 v 1975 812 4 7 v 1977 810 4 5 v 1977 808 4 3 v 2025 871 4 14 v 2024 871 4 9 v 2023 871 4 7 v 2021 871 4 5 v 2021 871 4 3 v 2027 835 a Fn(d)2068 871 y Fj(,)k Fh(U)2193 886 y Fn(x)2265 871 y Fg(2)c(P)2428 886 y Fn(x)2472 871 y Fj(,)33 b Fh(\033)f Fg(2)c Fj(\012:)44 b Fh(c)p Fj(\()p Fh(U)3000 886 y Fn(x)3044 871 y Fh(;)17 b(\033)t Fj(\))27 b(=)g Fh(c)p Fj(\()p Fh(U)3461 886 y Fo(0)3501 871 y Fh(;)17 b(\034)3587 886 y Ff(\000)p Fn(x)3686 871 y Fh(\033)t Fj(\),)326 1074 y(iv.)48 b Fh(U)546 1089 y Fo(0)586 1074 y Fh(\033)34 b Fg(6)p Fj(=)c Fh(U)857 1038 y Ff(0)847 1099 y Fo(0)886 1074 y Fh(\033)38 b Fj(for)c(all)d Fh(\033)j Fg(2)d Fj(\012)p Fh(;)17 b(U)1632 1089 y Fo(0)1702 1074 y Fg(6)p Fj(=)29 b Fh(U)1883 1038 y Ff(0)1873 1099 y Fo(0)1943 1074 y Fg(2)h(P)2108 1089 y Fo(0)2182 1074 y Fj(for)j(whic)m(h)i Fh(U)2679 1089 y Fo(0)2718 1074 y Fh(\033)f Fg(6)p Fj(=)c Fh(\033)38 b Fj(and)c Fh(U)3273 1038 y Ff(0)3263 1099 y Fo(0)3303 1074 y Fh(\033)g Fg(6)p Fj(=)c Fh(\033)t Fj(.)47 b(\(this)480 1195 y(is)32 b(for)g(con)m(v)m (enience)j(only)-8 b(.\))236 1397 y(The)31 b(generator)g Fh(L)f Fj(corresp)s(onding)g(to)g(the)h(giv)m(en)f(rates)h(is)f(no)m(w) h(de\014ned)h(on)e(lo)s(cal)e(functions)i Fh(f)236 1518 y Fj(as)1099 1736 y Fh(Lf)11 b Fj(\()p Fh(\033)t Fj(\))28 b Fg(\021)1518 1642 y Fb(X)1492 1883 y Fn(x)p Ff(2)p 1599 1885 53 4 v 1599 1819 51 4 v 1599 1883 3 3 v 1600 1881 V 1602 1879 V 1603 1876 V 1604 1874 V 1605 1872 V 1606 1870 V 1607 1868 V 1608 1866 V 1609 1864 V 1610 1861 V 1611 1859 V 1613 1857 V 1614 1855 V 1615 1853 V 1616 1850 V 1617 1848 V 1618 1846 V 1619 1844 V 1620 1842 V 1621 1839 V 1622 1837 V 1623 1835 V 1625 1833 V 1626 1831 V 1627 1828 V 1628 1826 V 1629 1824 V 1630 1822 V 1631 1820 V 1617 1883 V 1618 1881 V 1619 1879 V 1620 1876 V 1621 1874 V 1622 1872 V 1623 1870 V 1625 1868 V 1626 1866 V 1627 1864 V 1628 1861 V 1629 1859 V 1630 1857 V 1631 1855 V 1632 1853 V 1633 1850 V 1634 1848 V 1636 1846 V 1637 1844 V 1638 1842 V 1639 1839 V 1640 1837 V 1641 1835 V 1642 1833 V 1643 1831 V 1644 1828 V 1645 1826 V 1647 1824 V 1648 1822 V 1649 1820 V 1598 1831 4 14 v 1599 1826 4 9 v 1600 1824 4 7 v 1602 1822 4 5 v 1602 1820 4 3 v 1650 1883 4 14 v 1649 1883 4 9 v 1648 1883 4 7 v 1646 1883 4 5 v 1646 1883 4 3 v 1652 1842 a Fd(d)1760 1642 y Fb(X)1721 1853 y Fn(U)1769 1861 y Fd(x)1808 1853 y Ff(2P)1904 1861 y Fd(x)1960 1736 y Fh(c)p Fj(\()p Fh(U)2106 1751 y Fn(x)2150 1736 y Fh(;)17 b(\033)t Fj(\)[)p Fh(f)11 b Fj(\()p Fh(U)2481 1751 y Fn(x)2525 1736 y Fh(\033)t Fj(\))22 b Fg(\000)g Fh(f)11 b Fj(\()p Fh(\033)t Fj(\)])613 b(\(3.25\))236 2079 y(W)-8 b(e)33 b(refer)g(to)f([4])h(for)f(further)h(details)e(on)i (constructing)f(the)h(in\014nite)f(v)m(olume)g(pro)s(cess.)236 2368 y Fi(3.2)136 b(Mean)44 b(En)l(trop)l(y)i(pro)t(duction)236 2552 y Fj(F)-8 b(rom)31 b(no)m(w)i(on)g(w)m(e)g(use)h(the)f(sets)h (\003)1570 2567 y Fn(n)1644 2552 y Fj(=)28 b([)p Fg(\000)p Fh(n;)17 b(n)p Fj(])2039 2516 y Fn(d)2102 2552 y Fg(\\)p 2206 2554 53 4 v 2206 2488 51 4 v 2206 2552 3 3 v 2207 2550 V 2209 2548 V 2210 2545 V 2211 2543 V 2212 2541 V 2213 2539 V 2214 2537 V 2215 2535 V 2216 2533 V 2217 2530 V 2218 2528 V 2220 2526 V 2221 2524 V 2222 2522 V 2223 2519 V 2224 2517 V 2225 2515 V 2226 2513 V 2227 2511 V 2228 2508 V 2229 2506 V 2231 2504 V 2232 2502 V 2233 2500 V 2234 2497 V 2235 2495 V 2236 2493 V 2237 2491 V 2238 2489 V 2224 2552 V 2225 2550 V 2226 2548 V 2227 2545 V 2228 2543 V 2229 2541 V 2231 2539 V 2232 2537 V 2233 2535 V 2234 2533 V 2235 2530 V 2236 2528 V 2237 2526 V 2238 2524 V 2239 2522 V 2240 2519 V 2241 2517 V 2243 2515 V 2244 2513 V 2245 2511 V 2246 2508 V 2247 2506 V 2248 2504 V 2249 2502 V 2250 2500 V 2251 2497 V 2252 2495 V 2254 2493 V 2255 2491 V 2256 2489 V 2205 2500 4 14 v 2206 2495 4 9 v 2207 2493 4 7 v 2209 2491 4 5 v 2209 2489 4 3 v 2257 2552 4 14 v 2256 2552 4 9 v 2255 2552 4 7 v 2253 2552 4 5 v 2253 2552 4 3 v 2259 2516 a Fn(d)2332 2552 y Fj(for)32 b Fh(n)h Fj(large)e(and)i(their) f(corresp)s(onding)236 2673 y(subsets)43 b(\003)650 2637 y Fn(])650 2697 y(n)696 2673 y Fj(,)g(meaning)c(the)h(\(p)s(ossibly)g (empt)m(y\))h(maximal)c(subset)42 b(of)e(\003)3004 2688 y Fn(n)3051 2673 y Fj(,)i(suc)m(h)g(that)e(for)g(all)236 2793 y Fh(x)c Fg(2)g Fj(\003)497 2757 y Fn(])497 2818 y(n)544 2793 y Fj(,)j(and)e Fh(U)870 2808 y Fn(x)950 2793 y Fg(2)f(P)1121 2808 y Fn(x)1165 2793 y Fj(,)j Fh(c)p Fj(\()p Fh(U)1377 2808 y Fn(x)1421 2793 y Fh(;)17 b(\033)t Fj(\))37 b(dep)s(ends)h(only)f(on)g(co)s(ordinates)g(inside)g(\003,)h (and)g(\003)3540 2808 y Fn(x)3619 2793 y Fg(\032)e Fj(\003.)236 2913 y(Consider)d(no)m(w)g(the)g(Mark)m(o)m(v)h(c)m(hain)f(\(as)g(in)e (section)i(2\))f(on)h(\012)2549 2928 y Fo(\003)2598 2936 y Fd(n)2672 2913 y Fg(\021)c Fh(S)2844 2877 y Fo(\003)2893 2885 y Fd(n)2972 2913 y Fj(with)j(generator)1090 3150 y Fh(L)1156 3165 y Fo(\003)1209 3150 y Fh(f)11 b Fj(\()p Fh(\033)t Fj(\))28 b Fg(\021)1553 3055 y Fb(X)1536 3288 y Fn(x)p Ff(2)p Fo(\003)1672 3256 y Fd(])1672 3299 y(n)1770 3055 y Fb(X)1731 3266 y Fn(U)1779 3274 y Fd(x)1817 3266 y Ff(2P)1913 3274 y Fd(x)1969 3150 y Fh(c)p Fj(\()p Fh(U)2115 3165 y Fn(x)2159 3150 y Fh(;)17 b(\033)t Fj(\)[)p Fh(f)11 b Fj(\()p Fh(U)2490 3165 y Fn(x)2534 3150 y Fh(\033)t Fj(\))22 b Fg(\000)g Fh(f)11 b Fj(\()p Fh(\033)t Fj(\)])236 3485 y(Just)33 b(as)g(in)f(section)h(2.3,)f(w)m(e)i(can)e(compute)h (the)g(corresp)s(onding)g(relativ)m(e)e(Hamiltonian)779 3765 y Fh(R)853 3780 y Fn(T)6 b(;n;\031)1029 3765 y Fj(\()p Fh(!)t Fj(\))83 b(=)1429 3671 y Fb(X)1411 3904 y Fn(x)p Ff(2)p Fo(\003)1547 3871 y Fd(])1547 3914 y(n)1645 3671 y Fb(X)1606 3882 y Fn(U)1654 3890 y Fd(x)1693 3882 y Ff(2P)1789 3890 y Fd(x)1845 3630 y Fb(Z)1945 3656 y Fn(T)1900 3855 y Ff(\000)p Fn(T)2027 3765 y Fj(log)2353 3698 y Fh(c)p Fj(\()p Fh(U)2499 3713 y Fn(x)2543 3698 y Fh(;)17 b(!)t Fj(\()p Fh(s)2736 3662 y Ff(\000)2794 3698 y Fj(\)\))p 2179 3742 865 4 v 2179 3834 a Fh(c)p Fj(\()p Fh(\031)t(U)2394 3805 y Ff(\000)p Fo(1)2384 3858 y Fn(x)2489 3834 y Fh(\031)t(;)g(\031)t (U)2717 3849 y Fn(x)2761 3834 y Fh(!)t Fj(\()p Fh(s)2910 3805 y Ff(\000)2968 3834 y Fj(\)\))3054 3765 y Fh(dN)3193 3724 y Fn(U)3241 3732 y Fd(x)3183 3790 y Fn(s)1253 4112 y Fj(+)1411 3977 y Fb(Z)1511 4003 y Fn(T)1467 4202 y Ff(\000)p Fn(T)1577 4112 y Fj([)p Fh(c)p Fj(\()p Fh(U)1750 4127 y Fn(x)1794 4112 y Fh(;)g(\031)t(!)t Fj(\()p Fh(s)p Fj(\)\))k Fg(\000)h Fh(c)p Fj(\()p Fh(U)2388 4127 y Fn(x)2432 4112 y Fh(;)17 b(!)t Fj(\()p Fh(s)p Fj(\)\)])p Fh(ds)752 b Fj(\(3.26\))236 4404 y(where)24 b Fh(N)596 4361 y Fn(U)644 4369 y Fd(x)586 4426 y Fn(t)715 4404 y Fg(\021)821 4329 y Fb(P)926 4433 y Ff(\000)p Fn(T)10 b Ff(\024)p Fn(s)p Ff(\024)p Fn(t)1220 4404 y Fh(I)25 b Fj(\()p Fh(!)t Fj(\()p Fh(s)p Fj(\))h(=)i Fh(U)1709 4419 y Fn(x)1753 4404 y Fj(\()p Fh(!)t Fj(\()p Fh(s)1940 4368 y Ff(\000)1998 4404 y Fj(\))g Fg(6)p Fj(=)f Fh(!)t Fj(\()p Fh(s)2316 4368 y Ff(\000)2375 4404 y Fj(\)\))22 b(is)h(the)g(n)m(um)m(b)s(er)h (of)e(times)g(the)i(trans-)236 4524 y(formation)35 b Fh(U)754 4539 y Fn(x)835 4524 y Fj(app)s(eared)i(in)f(the)h (realization)e Fh(!)40 b Fj(up)d(to)g(time)e Fh(t)p Fj(.)57 b(\(3.26\))36 b(m)m(ust)h(b)s(e)g(compared)236 4645 y(with)46 b(\(2.6\))f(but)h(w)m(e)h(ha)m(v)m(e)g(omitted)d(putting)h(additional)e (subscripts)k(to)e(the)h Fh(\031)k Fj(or)45 b(the)i Fh(U)3783 4660 y Fn(x)236 4765 y Fj(meaning)31 b(that)i(they)g(are)g(restricted)g (here)g(to)g(w)m(ork)g(on)f(elemen)m(ts)h Fh(!)t Fj(\()p Fh(s)p Fj(\))27 b Fg(2)h Fj(\012)3135 4780 y Fo(\003)3189 4765 y Fj(.)236 4885 y(The)34 b(mean)e(en)m(trop)m(y)i(pro)s(duction)e (for)g(the)h(in)m(teracting)e(particle)g(system)j(is)e(no)m(w)h (de\014ned)h(as)1154 5143 y(MEP)1376 5158 y Fn(\031)1423 5143 y Fj(\()p Fg(P)1530 5158 y Fo(0)1570 5143 y Fh(;)17 b(\032)p Fj(\))28 b Fg(\021)70 b Fj(lim)1835 5206 y Fn(n;T)10 b Ff("1)2201 5076 y Fj(1)p 2081 5120 290 4 v 2081 5212 a(2)p Fg(j)p Fj(\003)2226 5227 y Fn(n)2272 5212 y Fg(j)p Fh(T)2380 5143 y Fj(I)-16 b(E)2465 5101 y Fn(n;T)2465 5168 y(\032)2583 5143 y Fj(\()p Fh(R)2695 5158 y Fn(T)6 b(;n;\031)2871 5143 y Fj(\))668 b(\(3.27\))236 5430 y(By)36 b(I)-16 b(E)477 5389 y Fn(n;T)477 5455 y(\032)595 5430 y Fj(,)36 b(w)m(e)h(mean)e(the)h(exp)s(ectation)g(with)f(resp)s(ect)i (to)e(the)h(path)g(space)g(measure,)h(started)236 5551 y(from)g(the)i(stationary)e(state)i Fh(\032)p Fj(,)h(restricted)f(to)f (tra)5 b(jectories)38 b(within)f Fh(S)2935 5515 y Fo(\003)2984 5523 y Fd(n)3031 5551 y Fj(.)60 b(The)39 b(\014rst)g(part)f(of)1983 5800 y(10)p eop %%Page: 11 11 11 10 bop 236 548 a Fj(the)32 b(next)h(theorem)e(then)i(pro)m(v)m(es)g (the)f(existence)h(of)e(the)i(limit)28 b(\(3.27\).)42 b(A)32 b(second)h(prop)s(ert)m(y)f(of)236 668 y(\(3.27\))k(that)g (requires)h(a)g(pro)s(of)e(is)h(its)g(p)s(ositivit)m(y)-8 b(.)54 b(In)37 b(order)f(to)g(complete)g(this)g(task,)j(w)m(e)e(will) 236 789 y(here)d(apply)f(a)f(similar)e(argumen)m(t)j(as)g(used)h(in)e (\(2.22\))g(for)h(the)g(case)h(of)e(\014nite)h(Mark)m(o)m(v)h(c)m (hains.)236 909 y(Denote)41 b(b)m(y)g Fh(\026)781 924 y Fn(\016)859 909 y Fj(the)g(join)m(t)e(distribution)g(of)h(\()p Fh(X)2048 924 y Fo(0)2087 909 y Fh(;)17 b(X)2212 924 y Fn(\016)2250 909 y Fj(\),)42 b(where)g Fh(X)2728 924 y Fo(0)2807 909 y Fj(is)e(distributed)g(according)236 1029 y(to)46 b Fh(\032)p Fj(.)83 b(Let)45 b Fh(\027)764 1044 y Fn(\016)848 1029 y Fj(denote)i(the)f(join)m(t)e(probabilit)m(y)g (distribution)g(of)h(\()p Fh(\031)t(X)2956 1044 y Fn(\016)2994 1029 y Fh(;)17 b(\031)t(X)3178 1044 y Fo(0)3217 1029 y Fj(\))45 b(where)i Fh(X)3676 1044 y Fo(0)3761 1029 y Fj(is)236 1174 y(distributed)41 b(according)f(to)g Fh(\032\031)t Fj(.)69 b Fh(\026)1582 1189 y Fn(\016)1660 1174 y Fj(and)41 b Fh(\027)1906 1189 y Fn(\016)1985 1174 y Fj(are)g(measures)h(on)e(\()p Fh(S)34 b Fg(\002)28 b Fh(S)6 b Fj(\))p 3088 1140 53 4 v 3088 1074 51 4 v 3088 1138 3 3 v 3089 1136 V 3090 1134 V 3091 1132 V 3092 1129 V 3093 1127 V 3094 1125 V 3095 1123 V 3096 1121 V 3098 1118 V 3099 1116 V 3100 1114 V 3101 1112 V 3102 1110 V 3103 1107 V 3104 1105 V 3105 1103 V 3106 1101 V 3107 1099 V 3109 1097 V 3110 1094 V 3111 1092 V 3112 1090 V 3113 1088 V 3114 1086 V 3115 1083 V 3116 1081 V 3117 1079 V 3118 1077 V 3120 1075 V 3105 1138 V 3106 1136 V 3107 1134 V 3109 1132 V 3110 1129 V 3111 1127 V 3112 1125 V 3113 1123 V 3114 1121 V 3115 1118 V 3116 1116 V 3117 1114 V 3118 1112 V 3120 1110 V 3121 1107 V 3122 1105 V 3123 1103 V 3124 1101 V 3125 1099 V 3126 1097 V 3127 1094 V 3128 1092 V 3129 1090 V 3130 1088 V 3132 1086 V 3133 1083 V 3134 1081 V 3135 1079 V 3136 1077 V 3137 1075 V 3086 1086 4 14 v 3087 1081 4 9 v 3089 1079 4 7 v 3090 1077 4 5 v 3090 1075 4 3 v 3139 1138 4 14 v 3137 1138 4 9 v 3136 1138 4 7 v 3135 1138 4 5 v 3134 1138 4 3 v 3140 1097 a Fd(d)3181 1174 y Fj(.)68 b(Consider)41 b(the)236 1295 y(relativ)m(e)32 b(en)m(trop)m(y)i Fh(S)998 1310 y Fn(n)1078 1295 y Fj(of)e(the)h(\014nite)f(v)m(olume)g (\003)2010 1310 y Fn(n)2057 1295 y Fj(-)g(restrictions)g(of)g Fh(\026)2797 1310 y Fn(\016)2834 1295 y Fh(;)17 b(\027)2926 1310 y Fn(\016)2964 1295 y Fj(.)236 1569 y Fh(S)296 1584 y Fn(n)343 1569 y Fj(\()p Fh(\026)440 1584 y Fn(\016)478 1569 y Fg(j)p Fh(\027)554 1584 y Fn(\016)591 1569 y Fj(\))83 b Fg(\021)939 1474 y Fb(X)873 1691 y Fn(\033)913 1702 y Fe(\003)954 1710 y Fd(n)1001 1691 y Fn(;\033)1063 1668 y Fa(0)1061 1714 y Fe(\003)1102 1722 y Fd(n)1166 1569 y Fh(\026)1225 1584 y Fn(\016)1262 1569 y Fj(\()p Fh(\033)1355 1584 y Fo(\003)1404 1592 y Fd(n)1451 1569 y Fh(;)17 b(\033)1554 1528 y Ff(0)1550 1594 y Fo(\003)1599 1602 y Fd(n)1646 1569 y Fj(\))g(log)1853 1497 y Fh(\026)1912 1512 y Fn(\016)1950 1497 y Fj(\()p Fh(\033)2043 1512 y Fo(\003)2092 1520 y Fd(n)2139 1497 y Fh(;)g(\033)2242 1461 y Ff(0)2238 1522 y Fo(\003)2287 1530 y Fd(n)2333 1497 y Fj(\))p 1853 1546 519 4 v 1858 1637 a Fh(\027)1906 1652 y Fn(\016)1945 1637 y Fj(\()p Fh(\033)2038 1652 y Fo(\003)2087 1660 y Fd(n)2133 1637 y Fh(;)g(\033)2236 1603 y Ff(0)2232 1664 y Fo(\003)2281 1672 y Fd(n)2328 1637 y Fj(\))3698 1569 y(\(3.28\))713 1913 y(=)939 1818 y Fb(X)873 2035 y Fn(\033)913 2046 y Fe(\003)954 2054 y Fd(n)1001 2035 y Fn(;\033)1063 2012 y Fa(0)1061 2058 y Fe(\003)1102 2066 y Fd(n)1166 1913 y Fc(P)1225 1928 y Fn(\032)1267 1913 y Fj([)p Fh(X)1375 1928 y Fo(0)p Fn(;)p Fo(\003)1479 1936 y Fd(n)1553 1913 y Fj(=)28 b Fh(\033)1712 1928 y Fo(\003)1761 1936 y Fd(n)1808 1913 y Fh(;)17 b(X)1933 1928 y Fn(\016)o(;)p Fo(\003)2032 1936 y Fd(n)2105 1913 y Fj(=)28 b Fh(\033)2268 1872 y Ff(0)2264 1937 y Fo(\003)2313 1945 y Fd(n)2360 1913 y Fj(])17 b(log)2636 1840 y Fc(P)2695 1855 y Fn(\032)2738 1840 y Fj([)p Fh(X)2846 1855 y Fo(0)p Fn(;)p Fo(\003)2950 1863 y Fd(n)3024 1840 y Fj(=)27 b Fh(\033)3182 1855 y Fo(\003)3231 1863 y Fd(n)3278 1840 y Fh(;)17 b(X)3403 1855 y Fn(\016)o(;)p Fo(\003)3502 1863 y Fd(n)3576 1840 y Fj(=)27 b Fh(\033)3738 1804 y Ff(0)3734 1866 y Fo(\003)3783 1874 y Fd(n)3830 1840 y Fj(])p 2556 1890 1382 4 v 2556 1981 a Fc(P)2615 1996 y Fn(\032\031)2700 1981 y Fj([)p Fh(X)2808 1996 y Fo(0)p Fn(;)p Fo(\003)2912 2004 y Fd(n)2986 1981 y Fj(=)h Fh(\031)t(\033)3208 1947 y Ff(0)3204 2008 y Fo(\003)3253 2016 y Fd(n)3300 1981 y Fh(;)17 b(X)3425 1996 y Fn(\016)o(;)p Fo(\003)3524 2004 y Fd(n)3597 1981 y Fj(=)28 b Fh(\031)t(\033)3815 1996 y Fo(\003)3864 2004 y Fd(n)3911 1981 y Fj(])236 2387 y Fp(Theorem)37 b(3.1)49 b Fm(F)-7 b(or)34 b(a)h(stationary)g(and) f(tr)-5 b(anslation)35 b(invariant)f(me)-5 b(asur)g(e)34 b Fh(\032)p Fm(,)372 2591 y(i.)835 2841 y(MEP)1055 2856 y Fn(\031)1102 2841 y Fj(\()p Fg(P)1209 2856 y Fo(0)1249 2841 y Fh(;)17 b(\032)p Fj(\))83 b(=)1657 2746 y Fb(X)1622 2958 y Fn(U)1670 2967 y Fe(0)1705 2958 y Ff(2P)1801 2967 y Fe(0)1852 2841 y Fj(\()1890 2705 y Fb(Z)2007 2841 y Fh(\032)p Fj(\()p Fh(d\033)t Fj(\))17 b Fh(c)p Fj(\()p Fh(U)2406 2856 y Fo(0)2445 2841 y Fh(;)g(\033)t 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Fn(\016)2953 3685 y Fj(\))586 b(\(3.30\))236 4022 y Fp(Pro)s(of:)75 b Fm(i.)91 b Fj(Start)48 b(b)m(y)h(noting)f(that)g Fh(N)1816 3980 y Fn(U)1864 3988 y Fd(x)1806 4045 y Fn(t)1940 4022 y Fg(\000)2050 3942 y Fb(R)2117 3968 y Fn(t)2097 4057 y Ff(\000)p Fn(T)2224 4022 y Fh(c)p Fj(\()p Fh(U)2370 4037 y Fn(x)2414 4022 y Fh(;)17 b(!)2519 4037 y Fn(s)2555 4022 y Fj(\))p Fh(ds)48 b Fj(is)g(a)g(martingale.)88 b(By)49 b(the)236 4143 y(stationarit)m(y)31 b(of)i Fh(\032)p Fj(,)f(the)h(limit)d Fh(T)41 b Fg(!)27 b(1)32 b Fj(can)h(b)s(e)g(tak)m (en)h(immediately)464 4412 y(MEP)686 4427 y Fn(\031)733 4412 y Fj(\()p Fg(P)840 4427 y Fo(0)880 4412 y Fh(;)17 b(\032)p Fj(\))83 b(=)89 b(lim)1254 4475 y Fn(n)p Ff("1)1489 4345 y Fj(1)p 1429 4389 171 4 v 1429 4481 a Fg(j)p Fj(\003)1525 4496 y Fn(n)1571 4481 y Fg(j)1643 4318 y Fb(X)1625 4551 y Fn(x)p Ff(2)p Fo(\003)1761 4518 y Fd(])1761 4561 y(n)1859 4318 y Fb(X)1820 4529 y Fn(U)1868 4537 y Fd(x)1907 4529 y Ff(2P)2003 4537 y Fd(x)2059 4277 y Fb(Z)2175 4412 y Fh(\032)p Fj(\()p Fh(d\033)t Fj(\))p Fh(c)p Fj(\()p Fh(U)2557 4427 y Fn(x)2601 4412 y Fh(;)17 b(\033)t Fj(\))g(log)3055 4345 y Fh(c)p Fj(\()p Fh(U)3201 4360 y Fn(x)3245 4345 y Fh(;)g(\031)t(\033)t Fj(\))p 2911 4389 679 4 v 2911 4481 a Fh(c)p Fj(\()p Fh(\031)t(U)3126 4452 y Ff(\000)p Fo(1)3116 4505 y Fn(x)3221 4481 y Fh(\031)t(;)g(\031)t(U)3449 4496 y Fn(x)3492 4481 y Fh(\033)t Fj(\))1095 4740 y(+)1254 4604 y Fb(Z)1370 4740 y Fh(\032)p Fj(\()p Fh(d\033)t Fj(\)[)p Fh(c)p Fj(\()p Fh(U)1779 4755 y Fn(x)1823 4740 y Fh(;)g(\031)t(\033)t Fj(\))22 b Fg(\000)g Fh(c)p Fj(\()p Fh(U)2290 4755 y Fn(x)2334 4740 y Fh(;)17 b(\033)t Fj(\)])1095 5009 y(=)89 b(lim)1254 5071 y Fn(n)p Ff("1)1429 4941 y Fg(j)p Fj(\003)1525 4905 y Fn(])1525 4966 y(n)1571 4941 y Fg(j)p 1429 4986 171 4 v 1429 5077 a(j)p Fj(\003)1525 5092 y Fn(n)1571 5077 y Fg(j)1660 4914 y Fb(X)1625 5126 y Fn(U)1673 5135 y Fe(0)1708 5126 y Ff(2P)1804 5135 y Fe(0)1855 4873 y Fb(Z)1972 5009 y Fh(\032)p Fj(\()p Fh(d\033)t Fj(\))p Fh(c)p Fj(\()p Fh(U)2354 5024 y Fo(0)2393 5009 y Fh(;)17 b(\033)t Fj(\))g(log)2847 4941 y Fh(c)p Fj(\()p Fh(U)2993 4956 y Fo(0)3033 4941 y Fh(;)g(\031)t(\033)t Fj(\))p 2703 4986 674 4 v 2703 5082 a Fh(c)p Fj(\()p Fh(\031)t(U)2918 5041 y Ff(\000)p Fo(1)2908 5106 y(0)3013 5082 y Fh(\031)t(;)g(\031)t(U)3241 5097 y Fo(0)3280 5082 y Fh(\033)t Fj(\))1095 5314 y(+)1254 5178 y Fb(Z)1370 5314 y Fh(\032)p Fj(\()p Fh(d\033)t Fj(\)[)p Fh(c)p Fj(\()p Fh(U)1779 5329 y Fo(0)1818 5314 y Fh(;)g(\031)t(\033)t Fj(\))22 b Fg(\000)h Fh(c)p Fj(\()p Fh(U)2286 5329 y Fo(0)2325 5314 y Fh(;)17 b(\033)t Fj(\)])1983 5800 y(11)p eop %%Page: 12 12 12 11 bop 236 548 a Fj(By)40 b(translation)d(in)m(v)-5 b(ariance)38 b(of)g Fh(\032)h Fj(together)h(with)e(the)i(fact)e Fg(j)p Fj(\003)2658 563 y Fn(n)2731 548 y Fg(n)26 b Fj(\003)2875 512 y Fn(])2875 573 y(n)2922 548 y Fg(j)38 b Fj(=)h(0\()p Fh(n)3248 512 y Fn(d)p Ff(\000)p Fo(1)3378 548 y Fj(\),)i(the)e (\014rst)236 668 y(part)33 b(of)f(the)h(theorem)f(follo)m(ws.)236 789 y Fm(ii.)43 b Fj(W)-8 b(e)33 b(ha)m(v)m(e)478 988 y Fh(S)538 1003 y Fn(n)585 988 y Fj(\()p Fh(\026)682 1003 y Fn(\016)720 988 y Fg(j)p Fh(\027)796 1003 y Fn(\016)833 988 y Fj(\))319 1196 y(=)83 b Fh(\016)t Fg(f)575 1101 y Fb(X)583 1310 y Fn(\033)623 1321 y Fe(\003)664 1329 y Fd(n)753 1101 y Fb(X)735 1328 y Fn(x)p Ff(2)829 1312 y Fo(~)822 1328 y(\003)871 1336 y Fd(n)969 1101 y Fb(X)930 1312 y Fn(U)978 1320 y Fd(x)1017 1312 y Ff(2P)1113 1320 y Fd(x)1169 1055 y Fb(\024)1221 1060 y(Z)1338 1196 y Fh(I)8 b Fj(\()p Fh(\030)1470 1211 y Fo(\003)1519 1219 y Fd(n)1592 1196 y Fj(=)28 b Fh(\033)1751 1211 y Fo(\003)1800 1219 y Fd(n)1847 1196 y Fj(\))p Fh(c)p Fj(\()p Fh(U)2031 1211 y Fn(x)2075 1196 y Fh(;)17 b(\030)5 b Fj(\))17 b(log)2602 1046 y Fb(R)2685 1126 y Fh(I)8 b Fj(\()p Fh(\030)2817 1141 y Fo(\003)2866 1149 y Fd(n)2940 1126 y Fj(=)27 b Fh(\033)3098 1141 y Fo(\003)3147 1149 y Fd(n)3194 1126 y Fj(\))p Fh(c)p Fj(\()p Fh(U)3378 1141 y Fn(x)3422 1126 y Fh(;)17 b(\030)5 b Fj(\))p Fh(\032)p Fj(\()p Fh(d\030)g 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y Fd(n)2562 1515 y Fj(=)28 b Fh(\033)2721 1530 y Fo(\003)2770 1538 y Fd(n)2817 1515 y Fj(\))p Fg(g)21 b Fj(+)h Fh(O)s Fj(\()p Fh(\016)3187 1474 y Fo(2)3226 1515 y Fj(\))236 1845 y(where)527 1820 y(~)518 1845 y(\003)586 1860 y Fn(n)660 1845 y Fg(\021)29 b(f)p Fh(x)e Fg(2)p 1008 1847 53 4 v 1008 1781 51 4 v 1008 1845 3 3 v 1010 1843 V 1011 1841 V 1012 1838 V 1013 1836 V 1014 1834 V 1015 1832 V 1016 1830 V 1017 1827 V 1018 1825 V 1019 1823 V 1021 1821 V 1022 1819 V 1023 1816 V 1024 1814 V 1025 1812 V 1026 1810 V 1027 1808 V 1028 1806 V 1029 1803 V 1030 1801 V 1032 1799 V 1033 1797 V 1034 1795 V 1035 1792 V 1036 1790 V 1037 1788 V 1038 1786 V 1039 1784 V 1040 1781 V 1026 1845 V 1027 1843 V 1028 1841 V 1029 1838 V 1030 1836 V 1032 1834 V 1033 1832 V 1034 1830 V 1035 1827 V 1036 1825 V 1037 1823 V 1038 1821 V 1039 1819 V 1040 1816 V 1041 1814 V 1042 1812 V 1044 1810 V 1045 1808 V 1046 1806 V 1047 1803 V 1048 1801 V 1049 1799 V 1050 1797 V 1051 1795 V 1052 1792 V 1053 1790 V 1055 1788 V 1056 1786 V 1057 1784 V 1058 1781 V 1007 1792 4 14 v 1008 1788 4 9 v 1009 1786 4 7 v 1011 1784 4 5 v 1011 1781 4 3 v 1059 1845 4 14 v 1058 1845 4 9 v 1057 1845 4 7 v 1056 1845 4 5 v 1055 1845 4 3 v 1061 1809 a Fn(d)1129 1845 y Fj(:)h(\003)1252 1860 y Fn(x)1318 1845 y Fg(\\)22 b Fj(\003)1474 1860 y Fn(n)1549 1845 y Fg(6)p Fj(=)27 b Fg(;g)p Fj(.)43 b(W)-8 b(e)33 b(rewrite)288 2099 y(lim)304 2162 y Fn(\016)r Ff(#)p Fo(0)450 2032 y Fj(1)p 450 2076 49 4 v 451 2168 a Fh(\016)509 2099 y(S)569 2114 y Fn(n)616 2099 y Fj(\()p Fh(\026)713 2114 y Fn(\016)750 2099 y Fg(j)p Fh(\027)826 2114 y Fn(\016)864 2099 y Fj(\))83 b(=)1144 2005 y Fb(X)1152 2214 y Fn(\033)1192 2225 y Fe(\003)1233 2233 y Fd(n)1322 2005 y Fb(X)1305 2238 y Fn(x)p Ff(2)p Fo(\003)1441 2205 y Fd(])1441 2248 y(n)1538 2005 y Fb(X)1499 2216 y Fn(U)1547 2224 y Fd(x)1586 2216 y Ff(2P)1682 2224 y Fd(x)1738 2099 y Fh(\032)p Fj(\()p Fh(\033)1881 2114 y Fo(\003)1930 2122 y Fd(n)1977 2099 y Fj(\))p Fh(c)p Fj(\()p Fh(U)2161 2114 y Fn(x)2205 2099 y Fh(;)17 b(\033)2304 2114 y Fo(\003)2353 2122 y Fd(n)2400 2099 y Fj(\))g(log)2836 2032 y Fh(c)p Fj(\()p Fh(U)2982 2047 y Fn(x)3026 2032 y Fh(;)g(\033)3125 2047 y Fo(\003)3174 2055 y Fd(n)3221 2032 y Fj(\))p Fh(\032)p Fj(\()p Fh(\033)3402 2047 y Fo(\003)3451 2055 y Fd(n)3498 2032 y Fj(\))p 2607 2076 1158 4 v 2607 2168 a Fh(c)p Fj(\()p Fh(\031)t(U)2822 2139 y Ff(\000)p Fo(1)2812 2192 y Fn(x)2917 2168 y Fh(\031)t(;)g(\031)t(U)3145 2183 y Fn(x)3189 2168 y Fh(\033)3244 2183 y Fo(\003)3293 2191 y Fd(n)3340 2168 y Fj(\))p Fh(\032)p Fj(\()p Fh(U)3532 2183 y Fn(x)3576 2168 y Fh(\033)3631 2183 y Fo(\003)3680 2191 y Fd(n)3727 2168 y Fj(\))985 2427 y(+)1144 2332 y Fb(X)1152 2542 y Fn(\033)1192 2553 y Fe(\003)1233 2561 y Fd(n)1322 2332 y Fb(X)1305 2566 y Fn(x)p Ff(2)p Fo(\003)1441 2533 y Fd(])1441 2576 y(n)1538 2332 y Fb(X)1499 2544 y Fn(U)1547 2552 y Fd(x)1586 2544 y Ff(2P)1682 2552 y Fd(x)1738 2292 y Fb(Z)1854 2427 y Fh(\032)p Fj(\()p Fh(d\033)2048 2442 y Fo(\003)2097 2450 y Fd(n)2144 2427 y Fj(\)\()p Fh(c)p Fj(\()p Fh(U)2366 2442 y Fn(x)2410 2427 y Fh(;)g(\031)t(\033) 2568 2442 y Fo(\003)2617 2450 y Fd(n)2664 2427 y Fj(\))22 b Fg(\000)g Fh(c)p Fj(\()p Fh(U)2969 2442 y Fn(x)3013 2427 y Fh(;)17 b(\033)3112 2442 y Fo(\003)3161 2450 y Fd(n)3208 2427 y Fj(\)\))985 2694 y(+)83 b Fh(o)p Fj(\()p Fg(j)p Fj(\003)1325 2709 y Fn(n)1371 2694 y Fg(j)p Fj(\))236 2893 y(Comparing)31 b(this)h(expression)i(with)f(that)f(of)g(\(3.29\),) g(w)m(e)h(still)e(ha)m(v)m(e)j(to)e(pro)m(v)m(e)i(that)832 3148 y(lim)808 3207 y Fn(n)p Ff(!1)1079 3080 y Fj(1)p 1018 3125 171 4 v 1018 3216 a Fg(j)p Fj(\003)1114 3231 y Fn(n)1161 3216 y Fg(j)1215 3053 y Fb(X)1223 3262 y Fn(\033)1263 3273 y Fe(\003)1304 3281 y Fd(n)1393 3053 y Fb(X)1375 3286 y Fn(x)p Ff(2)p Fo(\003)1511 3254 y Fd(])1511 3297 y(n)1609 3053 y Fb(X)1570 3264 y Fn(U)1618 3272 y Fd(x)1657 3264 y Ff(2P)1753 3272 y Fd(x)1809 3148 y Fh(\032)p Fj(\()p Fh(\033)1952 3163 y Fo(\003)2001 3171 y Fd(n)2048 3148 y Fj(\))p Fh(c)p Fj(\()p Fh(U)2232 3163 y Fn(x)2276 3148 y Fh(;)17 b(\033)2375 3163 y Fo(\003)2424 3171 y Fd(n)2471 3148 y Fj(\))g(log)2733 3080 y Fh(\032)p Fj(\()p Fh(\033)2876 3095 y Fo(\003)2925 3103 y Fd(n)2972 3080 y Fj(\))p 2678 3125 388 4 v 2678 3216 a Fh(\032)p Fj(\()p Fh(U)2832 3231 y Fn(x)2876 3216 y Fh(\033)2931 3231 y Fo(\003)2980 3239 y Fd(n)3027 3216 y Fj(\))3103 3148 y(=)27 b(0)322 b(\(3.31\))236 3469 y(Put)30 b Fh(F)487 3484 y Fo(\003)536 3492 y Fd(n)583 3469 y Fj(\()p Fh(\033)t Fj(\))d Fg(\021)h(\000)17 b Fj(log)g Fh(\032)p Fj(\()p Fh(\033)1230 3484 y Fo(\003)1279 3492 y Fd(n)1326 3469 y Fj(\);)30 b(this)f(function)g(dep)s(ends)i(only)d(on)h(spins)h (inside)e(\003)3286 3484 y Fn(n)3333 3469 y Fj(.)42 b(\(3.31\))28 b(can)236 3589 y(b)s(e)33 b(rewritten)g(as)849 3812 y(lim)825 3872 y Fn(n)p Ff(!1)1096 3744 y Fj(1)p 1035 3789 171 4 v 1035 3880 a Fg(j)p Fj(\003)1131 3895 y Fn(n)1178 3880 y Fg(j)1232 3676 y Fb(Z)1348 3812 y Fh(\032)p Fj(\()p Fh(d\033)t Fj(\))1618 3717 y Fb(X)1601 3950 y Fn(x)p Ff(2)p Fo(\003)1737 3918 y Fd(])1737 3961 y(n)1835 3717 y Fb(X)1796 3929 y Fn(U)1844 3937 y Fd(x)1882 3929 y Ff(2P)1978 3937 y Fd(x)2034 3812 y Fh(c)p Fj(\()p Fh(U)2180 3827 y Fn(x)2224 3812 y Fh(;)17 b(\033)t Fj(\)[)p Fh(F)2455 3827 y Fo(\003)2504 3835 y Fd(n)2551 3812 y Fj(\()p Fh(U)2655 3827 y Fn(x)2699 3812 y Fh(\033)t Fj(\))22 b Fg(\000)g Fh(F)2980 3827 y Fo(\003)3029 3835 y Fd(n)3076 3812 y Fj(\()p Fh(\033)t Fj(\)])339 b(\(3.32\))236 4133 y(By)33 b(stationarit)m(y)f(of)g Fh(\032)g Fj(\(3.32\))g(equals)616 4381 y(lim)592 4440 y Fn(n)p Ff(!1)863 4313 y Fj(1)p 803 4358 V 803 4449 a Fg(j)p Fj(\003)899 4464 y Fn(n)945 4449 y Fg(j)999 4245 y Fb(Z)1116 4381 y Fh(\032)p Fj(\()p Fh(d\033)1310 4396 y Fo(\003)1359 4404 y Fd(n)1406 4381 y Fj(\))1493 4286 y Fb(X)1461 4520 y Fn(x)p Ff(2)p Fo(\003)1597 4487 y Fd(])1597 4531 y(n)1638 4473 y(c)1726 4286 y Fb(X)1687 4497 y Fn(U)1735 4505 y Fd(x)1774 4497 y Ff(2P)1870 4505 y Fd(x)1925 4381 y Fh(c)p Fj(\()p Fh(U)2071 4396 y Fn(x)2116 4381 y Fh(;)17 b(\033)2215 4396 y Fo(\003)2264 4404 y Fd(n)2310 4381 y Fj(\)[)p Fh(F)2438 4396 y Fo(\003)2487 4404 y Fd(n)2534 4381 y Fj(\()p Fh(U)2638 4396 y Fn(x)2682 4381 y Fh(\033)t Fj(\))22 b Fg(\000)h Fh(F)2964 4396 y Fo(\003)3013 4404 y Fd(n)3060 4381 y Fj(\()p Fh(\033)t Fj(\)])236 4719 y(The)35 b(only)f Fh(x)d Fg(2)f Fj(\003)904 4683 y Fn(])904 4744 y(n)951 4663 y(c)1020 4719 y Fj(for)j(whic)m(h)i Fh(F)1514 4734 y Fo(\003)1563 4742 y Fd(n)1610 4719 y Fj(\()p Fh(U)1714 4734 y Fn(x)1758 4719 y Fh(\033)t Fj(\))23 b Fg(\000)h Fh(F)2042 4734 y Fo(\003)2091 4742 y Fd(n)2137 4719 y Fj(\()p Fh(\033)t Fj(\))34 b(is)g(non-zero)g(are)g Fh(x)d Fg(2)g Fj(\003)3218 4683 y Fn(])3218 4744 y(n)3265 4663 y(c)3322 4719 y Fg(\\)3445 4694 y Fj(~)3412 4719 y(\003)3480 4734 y Fn(n)3527 4719 y Fj(,)j(hence)236 4839 y(the)f(in)m(tegral)e(is)h(of)g(the)h(order)g Fg(j)1428 4814 y Fj(~)1419 4839 y(\003)1487 4854 y Fn(n)1556 4839 y Fg(n)22 b Fj(\003)1696 4854 y Fn(n)1742 4839 y Fg(j)g Fj(+)g Fg(j)p Fj(\003)1986 4854 y Fn(n)2055 4839 y Fg(n)f Fj(\003)2194 4803 y Fn(])2194 4864 y(n)2241 4839 y Fg(j)28 b Fj(=)f Fh(O)s Fj(\()p Fh(n)2574 4803 y Fn(d)p Ff(\000)p Fo(1)2704 4839 y Fj(\),)33 b(whic)m(h)g(pro)m(v)m(es)h(\(3.31\).)p 3780 4839 48 48 v 236 5246 a Fi(3.3)136 b(Detailed)46 b(balance)236 5430 y Fj(Giv)m(en)f(our)g(con\014gurational)e(time)g (rev)m(ersal)j Fh(\031)t Fj(,)i(w)m(e)e(sa)m(y)g(that)f(the)g(dynamics) g(satis\014es)g(the)236 5551 y(condition)33 b(of)h(\(generalized\))f (detailed)g(balance)h(with)g(resp)s(ect)i(to)d(the)i(Hamiltonian)c Fh(H)41 b Fj(if)33 b(for)1983 5800 y(12)p eop %%Page: 13 13 13 12 bop 236 548 a Fj(all)31 b Fh(x)d Fg(2)p 565 550 53 4 v 565 484 51 4 v 565 548 3 3 v 566 546 V 567 544 V 568 541 V 569 539 V 570 537 V 571 535 V 572 533 V 574 530 V 575 528 V 576 526 V 577 524 V 578 522 V 579 519 V 580 517 V 581 515 V 582 513 V 583 511 V 585 508 V 586 506 V 587 504 V 588 502 V 589 500 V 590 497 V 591 495 V 592 493 V 593 491 V 594 489 V 596 487 V 597 484 V 582 548 V 583 546 V 585 544 V 586 541 V 587 539 V 588 537 V 589 535 V 590 533 V 591 530 V 592 528 V 593 526 V 594 524 V 596 522 V 597 519 V 598 517 V 599 515 V 600 513 V 601 511 V 602 508 V 603 506 V 604 504 V 605 502 V 607 500 V 608 497 V 609 495 V 610 493 V 611 491 V 612 489 V 613 487 V 614 484 V 563 495 4 14 v 564 491 4 9 v 566 489 4 7 v 567 486 4 5 v 568 484 4 3 v 616 548 4 14 v 615 548 4 9 v 613 548 4 7 v 612 548 4 5 v 611 548 4 3 v 617 512 a Fn(d)658 548 y Fh(;)17 b(\033)31 b Fg(2)d Fj(\012)p Fh(;)17 b(U)1062 563 y Fn(x)1134 548 y Fg(2)28 b(P)1297 563 y Fn(x)1342 548 y Fj(,)950 741 y Fh(c)p Fj(\()p Fh(U)1096 756 y Fn(x)1140 741 y Fh(;)17 b(\033)t Fj(\))27 b(=)h Fh(c)p Fj(\()p Fh(\031)t(U)1627 700 y Ff(\000)p Fo(1)1617 766 y Fn(x)1721 741 y Fh(\031)t(;)17 b(\031)t(U)1949 756 y Fn(x)1993 741 y Fh(\033)t Fj(\))g(exp)q(\()p Fg(\000)p Fh(H)8 b Fj(\()p Fh(U)2564 756 y Fn(x)2608 741 y Fh(\033)t Fj(\))22 b(+)g Fh(H)8 b Fj(\()p Fh(\033)t Fj(\)\))p Fh(:)463 b Fj(\(3.33\))236 934 y(This)34 b(is)e(the)i (analogue)e(of)h(\(2.17\).)44 b(The)35 b(energy)f(di\013erence)g(in)e (\(3.33\))h(should)g(b)s(e)g(in)m(terpreted)236 1055 y(in)f(terms)h(of)f(an)g(absolutely)g(con)m(v)m(ergen)m(t)j(sum)d(of)g (p)s(oten)m(tials:)742 1260 y Fh(H)8 b Fj(\()p Fh(\033)924 1275 y Fo(\003)977 1260 y Fh(\021)1025 1275 y Fo(\003)1074 1256 y Fd(c)1110 1260 y Fj(\))22 b Fg(\000)h Fh(H)8 b Fj(\()p Fh(\030)1440 1275 y Fo(\003)1492 1260 y Fh(\021)1540 1275 y Fo(\003)1589 1256 y Fd(c)1625 1260 y Fj(\))28 b(=)1842 1165 y Fb(X)1794 1381 y Fn(A)p Ff(\\)p Fo(\003)p Ff(6)p Fo(=)p Ff(;)2050 1260 y Fj(\()p Fh(V)21 b Fj(\()p Fh(A;)c(\033)2376 1275 y Fo(\003)2430 1260 y Fh(\021)2478 1275 y Fo(\003)2527 1256 y Fd(c)2563 1260 y Fj(\))22 b Fg(\000)h Fh(V)e Fj(\()p Fh(A;)c(\030)2999 1275 y Fo(\003)3052 1260 y Fh(\021)3100 1275 y Fo(\003)3149 1256 y Fd(c)3185 1260 y Fj(\)\))f Fh(;)273 b Fj(\(3.34\))236 1567 y(where)36 b(\()p Fh(V)22 b Fj(\()p Fh(A;)17 b Fg(\001)p Fj(\))31 b(:)g Fh(S)1013 1531 y Fn(A)1101 1567 y Fg(!)g Fj(\()p Fg(\0001)p Fh(;)17 b Fj(+)p Fg(1)p Fj(\))p Fh(;)g(A)34 b Fj(\014nite)h(subsets)i(of)p 2577 1569 53 4 v 2577 1503 51 4 v 2577 1567 3 3 v 2578 1565 V 2579 1563 V 2580 1560 V 2581 1558 V 2582 1556 V 2583 1554 V 2585 1552 V 2586 1550 V 2587 1548 V 2588 1545 V 2589 1543 V 2590 1541 V 2591 1539 V 2592 1537 V 2593 1534 V 2594 1532 V 2596 1530 V 2597 1528 V 2598 1526 V 2599 1523 V 2600 1521 V 2601 1519 V 2602 1517 V 2603 1515 V 2604 1512 V 2605 1510 V 2606 1508 V 2608 1506 V 2609 1504 V 2594 1567 V 2596 1565 V 2597 1563 V 2598 1560 V 2599 1558 V 2600 1556 V 2601 1554 V 2602 1552 V 2603 1550 V 2604 1548 V 2605 1545 V 2607 1543 V 2608 1541 V 2609 1539 V 2610 1537 V 2611 1534 V 2612 1532 V 2613 1530 V 2614 1528 V 2615 1526 V 2616 1523 V 2617 1521 V 2619 1519 V 2620 1517 V 2621 1515 V 2622 1512 V 2623 1510 V 2624 1508 V 2625 1506 V 2626 1504 V 2575 1515 4 14 v 2577 1510 4 9 v 2578 1508 4 7 v 2579 1506 4 5 v 2580 1504 4 3 v 2628 1567 4 14 v 2627 1567 4 9 v 2625 1567 4 7 v 2624 1567 4 5 v 2624 1567 4 3 v 2630 1531 a Fn(d)2670 1567 y Fj(\),)e(is)g(a)f(translation)f(in)m(v)-5 b(arian)m(t)236 1688 y(\(uniformly\))30 b(absolutely)i(summable)g(p)s(oten)m(tial:)1503 1804 y Fb(X)1508 2015 y Fn(A)p Ff(3)p Fo(0)1667 1898 y Fj(max)1664 1968 y Fn(\033)r Ff(2)p Fn(S)1800 1949 y Fd(A)1867 1898 y Fg(j)p Fh(V)21 b Fj(\()p Fh(A;)c(\033)t Fj(\))p Fg(j)27 b Fh(<)h Fj(+)p Fg(1)1017 b Fj(\(3.35\))236 2180 y(The)32 b(follo)m(wing)d(theorem)i(states)h(that)f(detailed)f (balance)h(mak)m(es)h(the)g(mean)e(en)m(trop)m(y)j(pro)s(duc-)236 2300 y(tion)f(zero.)236 2480 y Fp(Theorem)37 b(3.2)49 b Fm(Supp)-5 b(ose)35 b(that)i(the)f(tr)-5 b(ansformation)35 b(r)-5 b(ates)36 b(satisfy)g(\(3.33\))e(for)i(the)g(p)-5 b(otential)236 2601 y Fh(V)22 b Fj(\()p Fh(A;)17 b Fg(\001)p Fj(\))p Fm(,)34 b(and)g(that)h Fh(\032)h Fm(and)e Fh(\032\031)39 b Fm(ar)-5 b(e)35 b(tr)-5 b(anslation)34 b(invariant)g(stationary)h (distributions.)45 b(Then,)1299 2794 y Fj(MEP)1521 2809 y Fn(\031)1568 2794 y Fj(\()p Fg(P)1675 2809 y Fo(0)1714 2794 y Fh(;)17 b(\032)p Fj(\))28 b(=)g(MEP)2200 2809 y Fn(\031)2247 2794 y Fj(\()p Fg(P)2354 2809 y Fo(0)2394 2794 y Fh(;)17 b(\032\031)t Fj(\))27 b(=)h(0)236 2987 y Fp(Pro)s(of:)42 b Fj(Let)30 b(us)h(\014rst)f(assume)h(that)f Fh(\032)d Fj(=)h Fh(\032\031)t Fj(.)43 b(Using)29 b(translation)f(in)m (v)-5 b(ariance)30 b(of)f Fh(\032)p Fj(,)i(w)m(e)g(obtain)236 3108 y(from)h(subsituting)g(\(3.33\))f(in)h(\(3.29\))282 3349 y(MEP)q(\()p Fh(\032)p Fj(\))c(=)f Fg(\000)907 3282 y Fj(1)p 848 3326 167 4 v 848 3418 a Fg(j)p Fj(\003)944 3433 y Fo(n)987 3418 y Fg(j)1072 3255 y Fb(X)1058 3466 y Fo(x)p Ff(2)p Fo(\003)1191 3474 y Fe(n)1301 3255 y Fb(X)1262 3466 y Fo(U)1315 3474 y Fe(x)1351 3466 y Ff(2P)1447 3474 y Fe(x)1517 3214 y Fb(Z)1633 3349 y Fh(\032)p Fj(\(d)p Fh(\033)1830 3364 y Fo(\003)1879 3372 y Fe(n)1922 3349 y Fj(\))17 b(c\(U)2131 3364 y Fo(x)2173 3349 y Fh(;)g(\033)2272 3364 y Fo(\003)2321 3372 y Fe(n)2363 3349 y Fj(\))2499 3255 y Fb(X)2434 3470 y Fo(A)p Ff(\\)p Fo(\003)2583 3478 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Fo(\003)2568 4134 y Fj(\()p Fh(U)2672 4149 y Fn(x)2716 4134 y Fh(\033)t Fj(\))22 b Fg(\000)h Fh(H)3016 4149 y Fo(\003)3069 4134 y Fj(\()p Fh(\033)t Fj(\)])346 b(\(3.37\))236 4425 y(for)30 b Fh(H)464 4440 y Fo(\003)513 4448 y Fd(n)560 4425 y Fj(\()p Fh(\033)t Fj(\))d Fg(\021)827 4351 y Fb(P)932 4454 y Fn(A)p Ff(\032)p Fo(\003)1089 4462 y Fd(n)1152 4425 y Fh(V)22 b Fj(\()p Fh(A;)17 b(\033)t Fj(\).)42 b(Since,)32 b(b)m(y)f(the)g(stationarit)m(y)e(of)h Fh(\032)p Fj(,)h(\(3.37\))f(w)m(ould)g(b)s(e)h(zero)g(if)236 4546 y(summed)i(o)m(v)m(er)g(all)e Fh(x)p Fj(,)i(w)m(e)g(see)h(that)e(its)g (absolute)h(v)-5 b(alue)32 b(is)g(b)s(ounded)h(b)m(y)874 4721 y(2)17 b(sup)1087 4744 y Fn(\033)1150 4721 y Fh(c)p Fj(\()p Fh(U)1296 4736 y Fo(0)1336 4721 y Fh(;)g(\033)t Fj(\))p 874 4765 602 4 v 1090 4857 a Fg(j)p Fj(\003)1186 4872 y Fn(n)1233 4857 y Fg(j)1520 4694 y Fb(X)1503 4905 y Fn(x)p Ff(2)p Fo(\003)1639 4886 y Fd(c)1639 4922 y(n)1938 4694 y Fb(X)1714 4909 y Fn(U)1762 4917 y Fd(x)1801 4909 y Ff(2P)1897 4917 y Fd(x)1936 4909 y Fn(;)33 b Fo(\003)2038 4917 y Fd(x)2076 4909 y Ff(\\)p Fo(\003)2172 4917 y Fd(n)2215 4909 y Ff(6)p Fo(=)p Ff(;)2405 4694 y Fb(X)2338 4909 y Fn(A)p Ff(\\)p Fo(\003)2487 4917 y Fd(x)2526 4909 y Ff(6)p Fo(=)p Ff(;)2649 4788 y Fj(sup)2659 4867 y Fn(\033)2699 4878 y Fe(\003)2740 4886 y Fd(n)2813 4788 y Fg(j)p Fh(V)21 b Fj(\()p Fh(A;)c(\033)t Fj(\))p Fg(j)378 b Fj(\(3.38\))236 5099 y(Since)40 b(from)f(\(3.35\),)1055 5025 y Fb(P)1160 5128 y Fn(A)p Ff(\\)p Fo(\003)1309 5136 y Fd(x)1347 5128 y Ff(6)p Fo(=)p Ff(;)1458 5099 y Fj(sup)1605 5123 y Fn(\033)1669 5099 y Fg(j)p Fh(V)21 b Fj(\()p Fh(A;)c(\033)t Fj(\))p Fg(j)40 b Fh(<)g(C)7 b Fg(j)p Fj(\003)2384 5114 y Fo(0)2423 5099 y Fg(j)39 b Fj(and)i Fg(jf)p Fh(U)2832 5114 y Fn(x)2916 5099 y Fg(2)g(P)3092 5114 y Fn(x)3136 5099 y Fg(gj)f(\024)h(j)p Fh(S)6 b Fg(j)3494 5063 y Ff(j)p Fo(\003)3563 5072 y Fe(0)3596 5063 y Ff(j)3620 5099 y Fj(!,)42 b(w)m(e)236 5220 y(can)33 b(estimate)f(the)h(\014rst)g(term)f(\(3.37\))g(b)m(y)1364 5457 y(C')1532 5390 y(1)p 1472 5435 171 4 v 1472 5526 a Fg(j)p Fj(\003)1568 5541 y Fn(n)1614 5526 y Fg(j)1652 5457 y(jf)p Fh(x)27 b Fg(2)h Fj(\003)1974 5416 y Fn(c)1974 5482 y(n)2049 5457 y Fj(:)f(\003)2171 5472 y Fn(n)2240 5457 y Fg(\\)c Fj(\003)2397 5472 y Fn(x)2468 5457 y Fg(6)p Fj(=)28 b Fg(;gj)877 b Fj(\(3.39\))1983 5800 y(13)p eop %%Page: 14 14 14 13 bop 236 548 a Fj(whic)m(h)33 b(con)m(v)m(erges)i(to)d(zero)h(as)g (\003)27 b Fg(")p 1586 550 53 4 v 1586 484 51 4 v 1586 548 3 3 v 1587 546 V 1588 544 V 1589 541 V 1590 539 V 1591 537 V 1592 535 V 1594 533 V 1595 530 V 1596 528 V 1597 526 V 1598 524 V 1599 522 V 1600 519 V 1601 517 V 1602 515 V 1603 513 V 1604 511 V 1606 508 V 1607 506 V 1608 504 V 1609 502 V 1610 500 V 1611 497 V 1612 495 V 1613 493 V 1614 491 V 1615 489 V 1617 487 V 1618 484 V 1603 548 V 1605 546 V 1606 544 V 1607 541 V 1608 539 V 1609 537 V 1610 535 V 1611 533 V 1612 530 V 1613 528 V 1614 526 V 1615 524 V 1617 522 V 1618 519 V 1619 517 V 1620 515 V 1621 513 V 1622 511 V 1623 508 V 1624 506 V 1625 504 V 1626 502 V 1628 500 V 1629 497 V 1630 495 V 1631 493 V 1632 491 V 1633 489 V 1634 487 V 1635 484 V 1584 495 4 14 v 1586 491 4 9 v 1587 489 4 7 v 1588 486 4 5 v 1589 484 4 3 v 1637 548 4 14 v 1636 548 4 9 v 1634 548 4 7 v 1633 548 4 5 v 1632 548 4 3 v 1639 512 a Fn(d)1679 548 y Fj(.)236 668 y(A)33 b(similar)c(analysis)j(can)h(b)s (e)g(made)f(for)g(the)h(second)h(part,)e(that)h(is)302 910 y Fg(\000)450 843 y Fj(1)p 389 887 171 4 v 389 979 a Fg(j)p Fj(\003)485 994 y Fn(n)532 979 y Fg(j)620 816 y Fb(X)602 1027 y Fn(x)p Ff(2)p Fo(\003)738 1035 y Fd(n)853 816 y Fb(X)814 1027 y Fn(U)862 1035 y Fd(x)901 1027 y Ff(2P)997 1035 y Fd(x)1069 775 y Fb(Z)1185 910 y Fh(\032)p Fj(\()p Fh(d\033)1379 925 y Fo(\003)1428 933 y Fd(n)1475 910 y Fj(\))17 b Fh(c)p Fj(\()p Fh(U)1676 925 y Fn(x)1720 910 y Fh(;)g(\033)1819 925 y Fo(\003)1868 933 y Fd(n)1915 910 y Fj(\))2220 816 y Fb(X)1986 1031 y Fn(A)p Ff(\\)p Fo(\003)2135 1039 y Fd(x)2174 1031 y Ff(6)p Fo(=)p Ff(;)p Fn(;)32 b(A)p Ff(\\)p Fo(\003)2465 1012 y Fd(c)2465 1048 y(n)2507 1031 y Ff(6)p Fo(=)p Ff(;)2614 910 y Fj([)p Fh(V)22 b Fj(\()p Fh(A;)17 b(U)2941 925 y Fn(x)3001 910 y Fh(\033)3056 925 y Fo(\003)3105 933 y Fd(n)3152 910 y Fj(\))22 b Fg(\000)h Fh(V)e Fj(\()p Fh(A;)c(\033)3600 925 y Fo(\003)3649 933 y Fd(n)3696 910 y Fj(\)])236 1220 y(This)33 b(con)m(v)m(erges)i(to)d(zero)h(b)m(y)g(\(3.35\))f(and)h (translation)e(in)m(v)-5 b(ariance)31 b(of)h Fh(V)22 b Fj(.)p 3780 1220 48 48 v 236 1340 a(When)29 b Fh(\032)f Fg(6)p Fj(=)g Fh(\032\031)t Fj(,)h(w)m(e)f(apply)g(the)g(previous)h(to) e(the)h(measure)h Fh(\032)2515 1304 y Ff(0)2566 1340 y Fj(=)e(1)p Fh(=)p Fj(2\()p Fh(\032)13 b Fj(+)g Fh(\032\031)t Fj(\))27 b(whic)m(h)h(is)g(clearly)236 1460 y Fh(\031)t Fj(-in)m(v)-5 b(arian)m(t.)56 b(Use)38 b(then)f(that)g(MEP)q(\()p Fh(\032)1733 1424 y Ff(0)1757 1460 y Fj(\))e(=)h(0)h(implies)d(MEP)q (\()p Fh(\032)p Fj(\))i(=)f(MEP)r(\()p Fh(\032\031)t Fj(\))g(=)g(0)i(b)s(ecause)236 1581 y(MEP)29 b Fg(\025)f Fj(0)k(\(b)m(y)i(\(3.30\)\))e(and)g(b)s(ecause)i(MEP)g(is)e(a\016ne)h (b)m(y)h(de\014nition)d(\(3.27\).)236 1910 y Fq(4)161 b(Additional)55 b(remarks)e(and)h(questions)236 2129 y Fj(The)27 b(sp)s(eci\014c)g(en)m(terprise)g(that)f(w)m(as)h(started)f (in)f(the)i(presen)m(t)g(pap)s(er)g(concerns)g(a)f(mathematical)236 2250 y(study)37 b(of)f(en)m(trop)m(y)h(pro)s(duction)e(in)g(spatially)f (\(in\014nitely\))g(extended)k(systems.)55 b(W)-8 b(e)36 b(list)f(here)236 2370 y(some)g(of)f(the)h(man)m(y)g(additional)d (questions)j(concerning)g(general)f(asp)s(ects)i(of)f(the)g(theory)-8 b(.)50 b(Of)236 2491 y(course,)34 b(sp)s(eci\014c)f(mo)s(dels)f(will)e (allo)m(w)h(man)m(y)h(additional)e(\(sp)s(eci\014c\))j(questions.)382 2676 y Fg(\017)48 b Fj(Is)32 b(it)e(correct)h(that)g(for)f(the)i(in)m (teracting)e(particle)f(systems)k(that)d(ha)m(v)m(e)j(app)s(eared)e(in) f(the)480 2796 y(presen)m(t)39 b(pap)s(er,)g(v)-5 b(anishing)36 b(mean)h(en)m(trop)m(y)h(pro)s(duction)f(implies)e(that)i(the)h(pro)s (cess)g(is)480 2917 y(rev)m(ersible?)44 b(More)33 b(sp)s(eci\014cally) -8 b(,)32 b(can)h(one)f(sho)m(w)i(that)660 3160 y Fh(M)10 b(E)c(P)905 3175 y Fn(\031)952 3160 y Fj(\()p Fg(P)1059 3175 y Fo(0)1099 3160 y Fh(;)17 b(\032)p Fj(\))28 b(=)f(0)h Fg(\))f Fh(c)p Fj(\()p Fh(U)1712 3175 y Fo(0)1752 3160 y Fh(;)17 b(\033)t Fj(\))27 b(=)g Fh(c)p Fj(\()p Fh(\031)t(U)2238 3118 y Ff(\000)p Fo(1)2228 3184 y(0)2333 3160 y Fh(\031)t(;)17 b(\031)t(U)2561 3175 y Fo(0)2600 3160 y Fh(\033)t Fj(\))2707 3092 y Fh(d\032)23 b Fg(\016)e Fh(U)2968 3107 y Fo(0)p 2707 3137 301 4 v 2807 3228 a Fh(d\032)3018 3160 y(;)c(U)3128 3175 y Fo(0)3195 3160 y Fg(2)28 b(P)3358 3175 y Fo(0)3577 3160 y Fj(\(4.40\))480 3403 y(where)38 b(w)m(e)g(assume)f(that)f Fh(\032)25 b Fg(\016)g Fh(U)1686 3418 y Fo(0)1762 3403 y Fj(has)37 b(a)f(densit)m(y)i(with)e(resp)s(ect)i(to)e Fh(\032)p Fj(.)56 b(If)37 b(the)g(set)g Fg(P)3685 3418 y Fo(0)3761 3403 y Fj(is)480 3524 y(su\016cien)m(tly)26 b(ric)m(h)f(\(in)f(the)h(sense)i(that)e(an)m(y)g(con\014guration)f(can) i(b)s(e)f(c)m(hanged)h(in)e(an)m(y)i(other)480 3644 y(con\014guration) 36 b(that)g(lo)s(cally)e(di\013ers)i(b)m(y)h(applications)e(of)h(the)h Fh(U)2936 3659 y Fn(x)2980 3644 y Fj(\),)g(that)f(w)m(ould)h(imply)480 3765 y(that)e(zero)h(mean)f(en)m(trop)m(y)h(pro)s(duction)f(can)g(only) g(o)s(ccur)h(if)e Fh(\032)h Fj(is)g(a)g Fh(\031)t Fj(-rev)m(ersible)g (Gibbs)480 3885 y(measure.)67 b(Additionally)-8 b(,)40 b(do)s(es)g(this)g(`no)h(curren)m(t)g(without)f(heat')h(result)f(fail)e (in)i(case)480 4005 y(the)28 b(dynamics)f(or)g(the)h(in)m(v)m(olution)e Fh(\031)31 b Fj(is)c(no)g(longer)g(\(quasi-\)lo)s(cal?)38 b(A)28 b(partial)d(result)i(has)480 4126 y(b)s(een)33 b(w)m(ork)m(ed)i(out)d(in)g([3])g(for)g(the)h(case)h(of)e(spin\015ip)g (dynamics.)382 4323 y Fg(\017)48 b Fj(Is)34 b(there)g(a)f(thermo)s (dynamic)f(analogue)g(of)h(the)g(circulation)f(form)m(ulae)f(that)i(ha) m(v)m(e)i(b)s(een)480 4443 y(dev)m(elop)s(ed)f(in)d([6,)i(7,)f(8])h (for)f(the)h(mean)f(en)m(trop)m(y)i(pro)s(duction?)382 4641 y Fg(\017)48 b Fj(Can)36 b(one)g(extend)h(the)f(lo)s(cal)d (\015uctuation)i(theorems)h(for)f(the)h(en)m(trop)m(y)g(pro)s(duction)f (ob-)480 4761 y(tained)24 b(in)f([1])g(for)h(probabilistic)d(cellular)h (automata)g(to)i(the)g(case)h(of)e(in)m(teracting)g(particle)480 4882 y(systems,)34 b(including)d(those)i(that)g(ha)m(v)m(e)g(a)g (conserv)-5 b(ation)32 b(la)m(w?)236 5211 y Fq(References)285 5430 y Fj([1])49 b(Maes)40 b(C.,)i Fm(Fluctuation)f(the)-5 b(or)g(em)41 b(as)g(a)g(Gibbs)g(pr)-5 b(op)g(erty)9 b Fj(,)41 b(J.)f(Stat.)f(Ph)m(ys.)i Fp(95)p Fj(,)h(367-392)437 5551 y(\(1999\))1983 5800 y(14)p eop %%Page: 15 15 15 14 bop 285 548 a Fj([2])49 b(Maes)30 b(C.,)h(Redig)e(F.,)h(V)-8 b(an)30 b(Mo\013aert)f(A.,)i Fm(On)h(the)g(de\014nition)f(of)h(entr)-5 b(opy)32 b(pr)-5 b(o)g(duction)32 b(via)437 668 y(examples)f Fj(J.)i(Math.)f(Ph)m(ys.)j(\(Marc)m(h)e(2000\).)285 872 y([3])49 b(Maes)33 b(C.,)g(Redig)f(F.,)h Fm(Positivity)i(of)f(entr)-5 b(opy)35 b(pr)-5 b(o)g(duction)32 b Fj(T)-8 b(o)33 b(app)s(ear)f(in)g (J.)h(Stat.)f(Ph)m(ys.)285 1075 y([4])49 b(Liggett)35 b(T.)j(M.,)g Fm(Inter)-5 b(acting)39 b(p)-5 b(article)38 b(systems)f Fj(Springer-V)-8 b(erlag,)36 b(New)i(Y)-8 b(ork,)38 b(Heidel-)437 1196 y(b)s(erg,)32 b(Berlin.)g(\(1985\))285 1399 y([5])49 b(Kipnis)41 b(C.,)47 b(Landim)41 b(C.,)46 b Fm(Sc)-5 b(aling)43 b(limits)h(of)g(inter)-5 b(acting)44 b(p)-5 b(article)44 b(systems)f Fj(Springer-)437 1519 y(V)-8 b(erlag,)31 b(New)j(Y)-8 b(ork,)33 b(Heidelb)s(erg,)e(Berlin.)h (\(1999\))285 1723 y([6])49 b(Qian)37 b(M.)h(P)-8 b(.,)40 b(Qian)d(M.,)j(Qian)d(C.,)j Fm(Cir)-5 b(culations)39 b(of)g(markov)g(chains)g(with)h(c)-5 b(ontinuous)437 1843 y(time)29 b(and)g(pr)-5 b(ob)g(ability)29 b(interpr)-5 b(etation)30 b(of)f(some)g(determinants)d Fj(Sci.)g(Sinica)f Fp(27)p Fj(,)k(470-481.)437 1963 y(\(1984\))285 2167 y([7])49 b(Qian)36 b(M.)j(P)-8 b(.,)39 b(Qian)e(M.,)i Fm(The)g(entr)-5 b(opy)40 b(pr)-5 b(o)g(duction)39 b(and)g(r)-5 b(eversibility)39 b(of)h(Markov)f(pr)-5 b(o-)437 2287 y(c)g(esses)31 b Fj(Pro)s(ceedings)i(of)g(the)g(\014rst)g(w)m(orld)f (congress)i(Bernoulli)c(so)s(c.)j(1988,)f(307-316)285 2491 y([8])49 b(Sc)m(hnak)m(en)m(b)s(erg)43 b(J.,)h Fm(Network)f(the)-5 b(ory)43 b(of)f(b)-5 b(ehavior)42 b(of)g(master)h(e)-5 b(quation)42 b(systems)f Fj(Rev.)437 2611 y(Mo)s(d.)32 b(Ph)m(ys.)j Fp(48)p Fj(,)e(4,)f(571-585.)f(\(1976\))285 2814 y([9])49 b(Kirc)m(hho\013)32 b(G.,)g(P)m(oggendor\013s)h(Ann.)h (Ph)m(ys.)g(Chem.,)f Fp(72)p Fj(,)g(495.)f(\(1847\))236 3018 y([10])49 b(Galla)m(v)m(otti)24 b(G.,)j(Cohen)h(E.)f(G.)f(D.,)i Fm(Dynamic)-5 b(al)28 b(ensembles)g(in)h(stationary)h(states)d Fj(J.)g(Stat.)437 3138 y(Ph)m(ys.)34 b Fp(80)p Fj(,)f(931-970.)e (\(1995\))236 3342 y([11])49 b(Galla)m(v)m(otti)32 b(G.,)k Fm(Chaotic)h(hyp)-5 b(othesis,)38 b(Onsager)f(r)-5 b(e)g(cipr)g(o)g (city)37 b(and)g(the)h(\015uctuation-)g(dis-)437 3462 y(sip)-5 b(ation)34 b(the)-5 b(or)g(em)32 b Fj(J.)h(Stat.)f(Ph)m(ys.)j Fp(84)p Fj(,)d(899-926.)f(\(1996\))236 3665 y([12])49 b(De)43 b(Gro)s(ot)e(S.)j(R.,)h(Mazur)f(P)-8 b(.,)46 b Fm(Non-)e(e)-5 b(quilibrium)44 b(thermo)-5 b(dynamics)42 b Fj(North-)h(Holland)437 3786 y(publishing)31 b(compan)m(y)-8 b(,)33 b(Amsterdam)f(\(1962\))236 3989 y([13])49 b(Leb)s(o)m(witz)34 b(J.)g(L.,)g(Sp)s(ohn)g(H.,)h Fm(A)h(Gal)5 b(lavotti-Cohen)35 b(typ)-5 b(e)37 b(symmetry)f(in)g(the)g(lar)-5 b(ge)36 b(devi-)437 4110 y(ation)e(functional)h(for)f(sto)-5 b(chastic)35 b(dynamics)c Fj(J.)i(Stat.)f(Ph)m(ys.)j Fp(95)p Fj(,)d(333-365.)f(\(1999\))1983 5800 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0009121021689--