Content-Type: multipart/mixed; boundary="-------------0106011008466" This is a multi-part message in MIME format. ---------------0106011008466 Content-Type: text/plain; name="01-204.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-204.comments" Christian.Maes@fys.kuleuven.ac.be, f.h.j.redig@TUE.nl, Michel.Verschuere@fys.kuleuven.ac.be ---------------0106011008466 Content-Type: text/plain; name="01-204.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-204.keywords" entropy production, fluctuation theorem, nonequilibrium steady states, space-time Gibbs measures. ---------------0106011008466 Content-Type: application/postscript; name="Minlos5.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Minlos5.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: Minlos5.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 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b(will)e(call)g(it)h(en)m(trop)m(y)i (pro-)236 4226 y(duction.)64 b(This)40 b(name)e(already)h(exists)i(for) e(a)g(ph)m(ysical)g(quan)m(tit)m(y)h(that)f(app)s(ears)h(in)f(close)g (to)236 4347 y(equilibrium)44 b(thermo)s(dynamics,)50 b(and)e(indeed)f(w)m(e)h(b)s(eliev)m(e)f(that)g(our)g(c)m(hoice)h(of)f (w)m(ords)h(re-)236 4467 y(\015ects)39 b(a)f(generalization.)57 b(The)39 b(basic)f(idea)g(is)f(that)h(nonequilibrium)d(steady)k(states) g(are)f(not)236 4588 y(time-rev)m(ersal)30 b(in)m(v)-5 b(arian)m(t)30 b(and)h(that)f(the)i(mean)e(en)m(trop)m(y)j(pro)s (duction)d(should)h(giv)m(e)g(a)f(measure)236 4708 y(of)35 b(discriminating)d(b)s(et)m(w)m(een)38 b(the)e(original)c(space-time)j (distribution)f(and)i(its)f(time-rev)m(ersal.)236 4828 y(That)c(is)e(the)i(relativ)m(e)f(space-time)f(en)m(trop)m(y)j(densit)m (y)-8 b(.)43 b(F)-8 b(or)30 b(the)g(v)-5 b(ariable)29 b(en)m(trop)m(y)i(pro)s(duction,)236 4949 y(w)m(e)e(m)m(ust)f(lo)s(ok)f 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949 y Fj(1.2)136 b(Example)236 1134 y Fp(W)-8 b(e)43 b(sk)m(etc)m(h)i(here)f(the)f(nature)g(of)f(a)h (lo)s(cal)d(v)m(ersus)45 b(global)40 b(\015uctuation)j(theorem)f(via)g (a)g(sim-)236 1254 y(ple)g(mo)s(del.)69 b(W)-8 b(e)43 b(ha)m(v)m(e)g(in)e(mind)g(a)g(\(1)28 b(+)h(1\)-dimensional)38 b(Ising)k(spin)g(system)h(with)e(formal)236 1375 y(Hamiltonian)1167 1541 y Fn(H)8 b Fp(\()p Fn(\033)t Fp(\))27 b(=)1521 1447 y Ff(X)1551 1656 y Fg(x;t)1682 1541 y Fn(\033)1737 1556 y Fg(t)1767 1541 y Fp(\()p Fn(x)p Fp(\)[)p Fn(\033)1980 1556 y Fg(t)p Fi(+1)2101 1541 y Fp(\()p Fn(x)p Fp(\))22 b(+)g Fn(b\033)2448 1556 y Fg(t)p Fi(+1)2569 1541 y Fp(\()p Fn(x)g Fp(+)g(1\)])730 b(\(1.8\))236 1808 y(where)42 b(w)m(e)g(think)e Fn(x)i Fo(2)p 1164 1810 53 4 v 1164 1744 51 4 v 1164 1808 3 3 v 1165 1806 V 1166 1804 V 1167 1801 V 1169 1799 V 1170 1797 V 1171 1795 V 1172 1793 V 1173 1790 V 1174 1788 V 1175 1786 V 1176 1784 V 1177 1782 V 1178 1779 V 1179 1777 V 1181 1775 V 1182 1773 V 1183 1771 V 1184 1769 V 1185 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Fi(=)p Fh(\000)p Fg(T)1670 2159 y(n)1619 2189 y Ff(X)1595 2398 y Fg(x)p Fi(=)p Fh(\000)p Fg(n)1804 2283 y Fn(\033)1859 2298 y Fg(t)1889 2283 y Fp(\()p Fn(x)p Fp(\)[)p Fn(\033)2102 2298 y Fg(t)p Fh(\000)p Fi(1)2223 2283 y Fp(\()p Fn(x)22 b Fp(+)g(1\))g Fo(\000)h Fn(\033)2700 2298 y Fg(t)p Fi(+1)2820 2283 y Fp(\()p Fn(x)f Fp(+)g(1\)])236 2541 y(of)34 b(the)g(spins)g(in)f(a)h (space-time)f(windo)m(w)i(parametrized)e(b)m(y)i Fn(n;)17 b(T)43 b(>)30 b Fp(0.)47 b(W)-8 b(e)35 b(are)e(in)m(terested)i(in)236 2661 y(its)d(\015uctuations)h(under)g(the)g(probabilit)m(y)e(la)m(ws) 382 2819 y Fo(\017)48 b Fn(P)543 2834 y Fg(n)590 2819 y Fp(,)28 b(the)f(Gibbs)g(measure)g(on)g Fo(f\000)p Fp(1)p Fn(;)17 b Fp(1)p Fo(g)1908 2782 y Fh(f\000)p Fg(n;:::)9 b(;n)p Fi(+1)p Fh(g\002)p 2394 2784 53 4 v 2394 2718 51 4 v 2394 2782 3 3 v 2395 2780 V 2396 2778 V 2397 2775 V 2398 2773 V 2399 2771 V 2401 2769 V 2402 2767 V 2403 2765 V 2404 2763 V 2405 2760 V 2406 2758 V 2407 2756 V 2408 2754 V 2409 2752 V 2410 2749 V 2412 2747 V 2413 2745 V 2414 2743 V 2415 2741 V 2416 2738 V 2417 2736 V 2418 2734 V 2419 2732 V 2420 2730 V 2421 2727 V 2422 2725 V 2424 2723 V 2425 2721 V 2426 2719 V 2412 2782 V 2413 2780 V 2414 2778 V 2415 2775 V 2416 2773 V 2417 2771 V 2418 2769 V 2419 2767 V 2420 2765 V 2421 2763 V 2423 2760 V 2424 2758 V 2425 2756 V 2426 2754 V 2427 2752 V 2428 2749 V 2429 2747 V 2430 2745 V 2431 2743 V 2432 2741 V 2433 2738 V 2435 2736 V 2436 2734 V 2437 2732 V 2438 2730 V 2439 2727 V 2440 2725 V 2441 2723 V 2442 2721 V 2443 2719 V 2392 2730 4 14 v 2394 2725 4 9 v 2395 2723 4 7 v 2396 2721 4 5 v 2397 2719 4 3 v 2445 2782 4 14 v 2444 2782 4 9 v 2442 2782 4 7 v 2441 2782 4 5 v 2441 2782 4 3 v 2478 2819 a Fp(with)26 b(resp)s(ect)i(to)f (the)g(Hamiltonian)1165 3061 y Fn(H)1246 3076 y Fg(n)1292 3061 y Fp(\()p Fn(\033)t Fp(\))h(=)1559 2966 y Ff(X)1618 3176 y Fg(t)1794 2936 y(n)1743 2966 y Ff(X)1719 3176 y Fg(x)p Fi(=)p Fh(\000)p Fg(n)1928 3061 y Fn(\033)1983 3076 y Fg(t)2013 3061 y Fp(\()p Fn(x)p Fp(\)[)p Fn(\033)2226 3076 y Fg(t)p Fi(+1)2347 3061 y Fp(\()p Fn(x)p Fp(\))22 b(+)g Fn(b\033)2694 3076 y Fg(t)p Fi(+1)2815 3061 y Fp(\()p Fn(x)g Fp(+)g(1\)])480 3316 y(and)382 3504 y Fo(\017)48 b Fn(P)14 b Fp(,)32 b(an)m(y)g(in\014nite)f(v)m(olume)g(Gibbs)h (measure)g(on)g Fo(f\000)p Fp(1)p Fn(;)17 b Fp(1)p Fo(g)p 2600 3470 53 4 v 2600 3404 51 4 v 2600 3468 3 3 v 2601 3466 V 2602 3464 V 2603 3462 V 2604 3459 V 2605 3457 V 2607 3455 V 2608 3453 V 2609 3451 V 2610 3448 V 2611 3446 V 2612 3444 V 2613 3442 V 2614 3440 V 2615 3437 V 2616 3435 V 2618 3433 V 2619 3431 V 2620 3429 V 2621 3426 V 2622 3424 V 2623 3422 V 2624 3420 V 2625 3418 V 2626 3415 V 2627 3413 V 2628 3411 V 2630 3409 V 2631 3407 V 2632 3404 V 2618 3468 V 2619 3466 V 2620 3464 V 2621 3462 V 2622 3459 V 2623 3457 V 2624 3455 V 2625 3453 V 2626 3451 V 2627 3448 V 2629 3446 V 2630 3444 V 2631 3442 V 2632 3440 V 2633 3437 V 2634 3435 V 2635 3433 V 2636 3431 V 2637 3429 V 2638 3426 V 2639 3424 V 2641 3422 V 2642 3420 V 2643 3418 V 2644 3415 V 2645 3413 V 2646 3411 V 2647 3409 V 2648 3407 V 2649 3404 V 2598 3415 4 14 v 2600 3411 4 9 v 2601 3409 4 7 v 2602 3407 4 5 v 2603 3404 4 3 v 2651 3468 4 14 v 2650 3468 4 9 v 2648 3468 4 7 v 2647 3468 4 5 v 2647 3468 4 3 v 2653 3427 a Fe(2)2723 3504 y Fp(for)32 b(the)g(Hamiltonian)c(\(1.8\).) 236 3662 y(In)33 b(b)s(oth)f(cases)i(w)m(e)f(tak)m(e)g(the)g(coun)m (ting)f(measure)h(as)f(reference)i(and)f(set)g(the)g(in)m(v)m(erse)g (temp)s(er-)236 3782 y(ature)g Fn(\014)g Fp(=)28 b(1.)236 3903 y(The)35 b(di\013erence)f(is)f(that)g Fn(P)1249 3918 y Fg(n)1329 3903 y Fp(is)g(an)g(Ising)g(mo)s(del)f(on)h(a)h (one-dimensional)c(strip)j(\(\014nite)g(spatial)236 4023 y(v)m(olume)40 b(with)f(in\014nite)g(time-extension\))g(and)h Fn(P)54 b Fp(is)39 b(the)i(corresp)s(onding)f(mo)s(del)e(for)i (in\014nite)236 4143 y(space-time)32 b(v)m(olume.)236 4264 y(W)-8 b(e)37 b(start)f(with)f(the)i(statemen)m(t)g(of)e(a)h (global)e(\015uctuation)h(theorem;)j(that)e(concerns)i(the)e(la)m(w)236 4384 y Fn(P)299 4399 y Fg(n)346 4384 y Fp(.)44 b(A)32 b(simple)f(calculation)f(sho)m(ws)35 b(that)d(for)g(ev)m(ery)i (function)e Fn(g)746 4458 y Ff(Z)862 4594 y Fn(dP)976 4609 y Fg(n)1023 4594 y Fp(\()p Fn(\033)t Fp(\))p Fn(g)t Fp(\()1262 4568 y(\026)1247 4594 y Fn(S)1307 4609 y Fg(n;T)1423 4594 y Fp(\()p Fn(\033)t Fp(\)\))c(=)1727 4458 y Ff(Z)1843 4594 y Fn(dP)1957 4609 y Fg(n)2004 4594 y Fp(\()p Fn(\033)t Fp(\))p Fn(g)t Fp(\()p Fo(\000)2321 4568 y Fp(\026)2305 4594 y Fn(S)2365 4609 y Fg(n;T)2482 4594 y Fp(\()p Fn(\033)t Fp(\)\))p Fn(e)2700 4553 y Fh(\000)2767 4536 y Fi(\026)2755 4553 y Fg(S)2798 4564 y Fd(n;T)2904 4553 y Fi(\()p Fg(\033)r Fi(\)+)p Fg(B)3108 4564 y Fd(n;T)3215 4553 y Fi(\()p Fg(\033)r Fi(\))3626 4594 y Fp(\(1.9\))236 4803 y(with)617 5021 y Fn(B)691 5036 y Fg(n;T)808 5021 y Fp(\()p Fn(\033)t Fp(\))g(=)1125 4896 y Fg(n)1074 4926 y Ff(X)1098 5135 y Fh(\000)p Fg(n)1384 5021 y Fo(f\000)p Fn(b\033)1607 5036 y Fh(\000)p Fg(T)10 b Fh(\000)p Fi(1)1809 5021 y Fp(\()p Fn(x)p Fp(\))p 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1036 y Fg(n;T)2164 1021 y Fp(\()p Fn(\033)t Fp(\))p Fn(=T)j Fp(\))p Fn(e)2502 989 y Fh(\000)2568 972 y Fi(\026)2557 989 y Fg(S)2600 1000 y Fd(n;T)2705 989 y Fi(\()p Fg(\033)r Fi(\))2817 945 y Fo(j)27 b Fp(=)g(0)553 b(\(1.10\))236 1233 y(whic)m(h)33 b(implies)d(the)j(symmetry)g(expressed)j(in)31 b(\(1.7\))h(with)h Fn(N)38 b Fp(=)27 b Fn(T)14 b Fp(.)236 1353 y(No)m(w)37 b(to)e(a)h(lo)s(cal)e(\015uctuation)h(theorem;)j(that)e(concerns)h(the) g(la)m(w)e Fn(P)14 b Fp(.)53 b(A)36 b(similar)d(calculation)236 1474 y(sho)m(ws)h(that)628 1580 y Ff(Z)744 1716 y Fn(dP)14 b Fp(\()p Fn(\033)t Fp(\))p Fn(g)t Fp(\()1111 1691 y(\026)1096 1716 y Fn(S)1156 1731 y Fg(n;T)1272 1716 y Fp(\()p Fn(\033)t Fp(\)\))27 b(=)1576 1580 y Ff(Z)1692 1716 y Fn(dP)14 b Fp(\()p Fn(\033)t Fp(\))p Fn(g)t Fp(\()p Fo(\000)2136 1691 y Fp(\026)2121 1716 y Fn(S)2181 1731 y Fg(n;T)2297 1716 y Fp(\()p Fn(\033)t Fp(\)\))p Fn(e)2515 1675 y Fh(\000)2582 1658 y Fi(\026)2570 1675 y Fg(S)2613 1686 y Fd(n;T)2719 1675 y Fi(\()p Fg(\033)r 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Fi(\))2896 2693 y Fo(j)27 b Fp(=)h(0)473 b(\(1.11\))236 2980 y(This)41 b(is)f(the)h(same)g(symmetry)g(as)f(in)g(\(1.7\))g(but)h(for)f(the)h (lo)s(cal)e(\015uctuations)i(in)e(a)i(spatially)236 3101 y(extended)33 b(system.)44 b(Of)30 b(course,)i(\(1.11\))e(in)m(v)m(olv) m(es)i(limits)27 b(but)k(the)h(basic)e(fact)h(b)s(ehind)f(\(1.11\))g (is)236 3221 y(that)j(there)g(is)f(a)g(lo)s(cal)f(function)h Fn(R)1563 3236 y Fg(n;T)1713 3221 y Fp(with)g Fo(j)p Fn(R)2037 3236 y Fg(n;T)2154 3221 y Fp(\()p Fn(\033)t Fp(\))22 b Fo(\000)2428 3196 y Fp(\026)2411 3221 y Fn(S)2471 3236 y Fg(n;T)2589 3221 y Fp(\()p Fn(\033)t Fp(\))p Fo(j)27 b(\024)h Fn(c)2926 3236 y Fi(1)2965 3221 y Fn(n)23 b Fp(+)f Fn(c)3186 3236 y Fi(2)3225 3221 y Fn(T)46 b Fp(for)32 b(whic)m(h)1051 3356 y Ff(Z)1167 3491 y Fn(dP)14 b Fp(\()p Fn(\033)t Fp(\))p Fn(g)t Fp(\()p Fn(R)1593 3506 y Fg(n;T)1709 3491 y Fp(\))27 b(=)1878 3356 y Ff(Z)1994 3491 y Fn(dP)14 b Fp(\()p Fn(\033)t Fp(\))p Fn(g)t Fp(\()p Fo(\000)p Fn(R)2497 3506 y Fg(n;T)2614 3491 y Fp(\))p Fn(e)2697 3450 y Fh(\000)p Fg(R)2805 3461 y Fd(n;T)2911 3450 y Fi(\()p Fg(\033)r Fi(\))236 3753 y Fp(whic)m(h)39 b(is)f(an)h(exact)g (lo)s(cal)e(\015uctuation)h(symmetry)-8 b(.)62 b(V)-8 b(arious)37 b(things)i(are)f(lac)m(king)g(from)f(this)236 3873 y(example.)68 b(Mathematically)-8 b(,)40 b(things)h(will)e(b)s(e)i (more)f(complicated)f(when)j(the)f Fn(B)3360 3888 y Fg(n;T)3519 3873 y Fp(or)f Fn(F)3709 3888 y Fg(n;T)236 3994 y Fp(are)48 b(not)f(uniformly)e(b)s(ounded)k(or)e(when)h(time)e(is)h(not)h (discrete)g(or)f(when)i(the)e(space-time)236 4114 y(Hamiltonian)c (\(1.8\))j(is)h(not)f(lo)s(cal)f(or)h(con)m(tains)h(hard-core)f(in)m (teractions.)85 b(Ph)m(ysically)-8 b(,)50 b(the)236 4235 y(example)32 b(ab)s(o)m(v)m(e)i(carries)e(no)h(in)m(terpretation)e(of) 2088 4209 y(\026)2071 4235 y Fn(S)2131 4250 y Fg(n;T)2281 4235 y Fp(as)i(en)m(trop)m(y)h(pro)s(duction.)236 4523 y Fj(1.3)136 b(Lo)t(cal)45 b(\015uctuation)g(theorem)236 4708 y Fp(The)33 b(main)c(theme)j(of)f(the)g(presen)m(t)j(pap)s(er)d (is)g(a)g(general)g(strategy)h(to)f(\014nd)h(a)f(lo)s(cal)e (\015uctuation)236 4828 y(theorem)k(for)f(the)i(en)m(trop)m(y)g(pro)s (duction)e(in)g(a)h(nonequilibrium)d(steady)k(state,)g(in)e(the)h(con)m (text)236 4949 y(of)c(sto)s(c)m(hastic)g(in)m(teracting)e(particle)h (systems.)43 b(T)-8 b(o)29 b(get)g(the)g(idea)f(w)m(e)i(presen)m(t)h (the)e(result)f(infor-)236 5069 y(mally)36 b(for)h(a)g(t)m(ypical)g (application.)56 b(The)38 b(details)f(and)g(mathematically)d(precise)39 b(statemen)m(ts)236 5190 y(ab)s(out)i(this)g(mo)s(del)f(are)i(giv)m(en) g(in)e(Section)i(4.)70 b(The)42 b(mo)s(del)e(is)h(a)h(microscopic)e(v)m (ersion)i(of)f(a)236 5310 y(reaction-di\013usion)29 b(system)j(where)h (the)e(reaction)g(consists)h(of)f(the)g(birth)g(and)g(death)h(of)e (parti-)236 5430 y(cles)j(on)g(the)g(sites)h(of)e(a)g(regular)g (lattice)g(and)h(the)g(di\013usion)f(part)g(lets)h(these)h(particles)e (hop)h(to)236 5551 y(nearest)h(neigh)m(b)s(or)e(v)-5 b(acancies)33 b(sub)5 b(ject)34 b(to)e(an)h(external)f(\014eld.)2007 5800 y(6)p eop %%Page: 7 7 7 6 bop 236 548 a Fp(Consider)30 b(the)g(square)h(lattice)p 1420 550 53 4 v 1420 484 51 4 v 1420 548 3 3 v 1421 546 V 1423 544 V 1424 541 V 1425 539 V 1426 537 V 1427 535 V 1428 533 V 1429 530 V 1430 528 V 1431 526 V 1432 524 V 1434 522 V 1435 519 V 1436 517 V 1437 515 V 1438 513 V 1439 511 V 1440 508 V 1441 506 V 1442 504 V 1443 502 V 1445 500 V 1446 497 V 1447 495 V 1448 493 V 1449 491 V 1450 489 V 1451 487 V 1452 484 V 1438 548 V 1439 546 V 1440 544 V 1441 541 V 1442 539 V 1443 537 V 1445 535 V 1446 533 V 1447 530 V 1448 528 V 1449 526 V 1450 524 V 1451 522 V 1452 519 V 1453 517 V 1454 515 V 1455 513 V 1457 511 V 1458 508 V 1459 506 V 1460 504 V 1461 502 V 1462 500 V 1463 497 V 1464 495 V 1465 493 V 1466 491 V 1468 489 V 1469 487 V 1470 484 V 1419 495 4 14 v 1420 491 4 9 v 1421 489 4 7 v 1423 486 4 5 v 1423 484 4 3 v 1471 548 4 14 v 1470 548 4 9 v 1469 548 4 7 v 1467 548 4 5 v 1467 548 4 3 v 1473 512 a Fi(2)1542 548 y Fp(to)e(eac)m(h)i (site)f Fn(i)f Fp(of)h(whic)m(h)g(w)m(e)h(assign)e(a)g(v)-5 b(ariable)28 b Fn(\021)t Fp(\()p Fn(i)p Fp(\))g(=)f(0)p Fn(;)17 b Fp(1,)236 668 y(meaning)27 b(that)h(site)f(is)h(empt)m(y)h (or)e(o)s(ccupied)h(b)m(y)h(a)f(particle.)41 b(The)29 b(con\014guration)e Fn(\021)32 b Fp(can)c(c)m(hange)236 789 y(in)40 b(t)m(w)m(o)h(w)m(a)m(ys:)61 b(\014rst,)42 b(a)f(particle)e(can)h(b)s(e)h(created)g(or)f(destro)m(y)m(ed)i(at)e (lattice)f(site)h Fn(i)p Fp(:)60 b Fn(\021)44 b Fo(!)d Fn(\021)3799 753 y Fg(i)236 909 y Fp(where)30 b Fn(\021)566 873 y Fg(i)622 909 y Fp(is)e(iden)m(tical)e(to)i Fn(\021)k Fp(except)e(that)e(the)h(o)s(ccupation)e(at)h(the)h(site)f Fn(i)g Fp(is)g(\015ipp)s(ed.)42 b(Secondly)-8 b(,)236 1029 y(a)30 b(particle)e(at)h Fn(i)h Fp(can)g(hop)f(to)h(one)f(of)h (the)g(four)f(nearest)h(neigh)m(b)s(or)f(sites)h Fn(j)36 b Fp(under)30 b(the)g(condition)236 1150 y(that)k Fn(j)40 b Fp(is)33 b(empt)m(y:)47 b Fn(\021)33 b Fo(!)d Fn(\021)1230 1114 y Fg(ij)1324 1150 y Fp(where)35 b Fn(\021)1659 1114 y Fg(ij)1753 1150 y Fp(is)f(the)g(new)h(con\014guration)e(obtained)g(b) m(y)i(exc)m(hanging)236 1270 y(the)f(o)s(ccupations)f(at)g(sites)g Fn(i)h Fp(and)f Fn(j)6 b Fp(.)46 b(W)-8 b(e)33 b(mak)m(e)h(a)f (nonequilibrium)d(dynamics)j(b)m(y)h(adding)f(an)236 1391 y(external)j(\014eld)f Fn(E)j(>)32 b Fp(0)j(whic)m(h)h(in)m(tro)s (duces)g(a)f(bias)g(for)f(particle)g(hopping)h(in)f(a)h(certain)g (direc-)236 1511 y(tion.)236 1631 y(In)40 b(form)m(ula,)e(\014rst,)k(a) d(particle)e(is)i(destro)m(y)m(ed)i(or)e(created)h(at)e(an)m(y)i(giv)m (en)f(site)g(at)g(\014xed)h(rates.)236 1752 y(The)34 b(transition)d(from)g(a)h(con\014guration)g Fn(\021)k Fp(to)c(the)h(new)h Fn(\021)2408 1716 y Fg(i)2468 1752 y Fp(tak)m(es)g(place)f(at)f(rate)1389 1946 y Fn(c)p Fp(\()p Fn(i;)17 b(\021)t Fp(\))27 b(=)h Fn(\015)1818 1961 y Fi(+)1876 1946 y Fp(\(1)22 b Fo(\000)h Fn(\021)t Fp(\()p Fn(i)p Fp(\)\))f(+)g Fn(\015)2455 1961 y Fh(\000)2513 1946 y Fn(\021)t Fp(\()p Fn(i)p Fp(\))236 2140 y(where)33 b Fn(\015)568 2155 y Fi(+)658 2140 y Fp(is)f(the)g(rate)g(for)f(the)h (transition)e(0)d Fo(!)h Fp(1)j(and)h Fn(\015)2406 2155 y Fh(\000)2496 2140 y Fp(is)f(the)i(rate)e(for)g(1)d Fo(!)f Fp(0.)43 b(Secondly)-8 b(,)236 2260 y(the)38 b(particles)f(on)g (the)h(lattice)e(undergo)i(a)f(di\013usiv)m(e)g(motion.)57 b(T)-8 b(o)37 b(b)s(e)h(sp)s(eci\014c,)h(w)m(e)g(c)m(ho)s(ose)f(a)236 2380 y(large)k(square)i Fn(V)64 b Fp(cen)m(tered)45 b(around)e(the)g (origin)e(with)h(p)s(erio)s(dic)f(b)s(oundary)j(conditions)e(and)236 2501 y(w)m(e)e(\014rst)e(in)m(tro)s(duce)h(hopping)e(rates)i(o)m(v)m (er)h(a)e(nearest)h(neigh)m(b)s(or)f(pair)f Fo(h)p Fn(ij)6 b Fo(i)38 b Fp(in)g(the)g(horizon)m(tal)236 2621 y(direction,)32 b Fn(i)c Fp(=)f(\()p Fn(i)905 2636 y Fi(1)945 2621 y Fn(;)17 b(i)1022 2636 y Fi(2)1061 2621 y Fp(\))p Fn(;)g(j)34 b Fp(=)27 b(\()p Fn(i)1391 2636 y Fi(1)1453 2621 y Fp(+)22 b(1)p Fn(;)17 b(i)1677 2636 y Fi(2)1716 2621 y Fp(\):)957 2815 y Fn(c)p Fp(\()p Fn(i;)g(j;)g(\021)t Fp(\))27 b(=)h Fn(e)1464 2774 y Fg(E)t(=)p Fi(2)1594 2815 y Fn(\021)t Fp(\()p Fn(i)p Fp(\)\(1)22 b Fo(\000)h Fn(\021)t Fp(\()p Fn(j)6 b Fp(\)\))21 b(+)h Fn(e)2340 2774 y Fh(\000)p Fg(E)t(=)p Fi(2)2525 2815 y Fn(\021)t Fp(\()p Fn(j)6 b Fp(\)\(1)22 b Fo(\000)g Fn(\021)t Fp(\()p Fn(i)p Fp(\)\))236 3009 y(The)43 b(hopping)f(rate)g(in)f(the)i(v)m(ertical)e(direction)g (is)h(constan)m(t)h(\(put)f Fn(E)50 b Fp(=)44 b(0)e(in)f(the)i(ab)s(o)m (v)m(e)g(if)236 3130 y Fn(j)49 b Fp(=)43 b(\()p Fn(i)515 3145 y Fi(1)554 3130 y Fn(;)17 b(i)631 3145 y Fi(2)699 3130 y Fo(\006)29 b Fp(1\)\).)70 b(T)-8 b(aking)41 b Fn(E)47 b Fp(large,)c(w)m(e)f(exp)s(ect)h(to)e(see)i(man)m(y)e(more)g (jumps)h(of)f(particles)236 3250 y(to)f(the)h(righ)m(t)f(than)g(to)g (the)h(left.)66 b(In)41 b(the)g(absence)h(of)e(reaction)g(rates,)i (that)f(is)f(for)f Fn(\015)3534 3265 y Fh(\006)3634 3250 y Fp(=)i(0,)236 3370 y(w)m(e)31 b(reco)m(v)m(er)h(the)e(so)h(called)d (asymmetric)h(exclusion)h(pro)s(cess)i(and)e(particle)f(n)m(um)m(b)s (er)h(is)f(strictly)236 3491 y(conserv)m(ed.)46 b(More)33 b(generally)-8 b(,)32 b(the)h(Master)g(Equation)f(is)376 3667 y Fn(d\032)477 3682 y Fg(t)506 3667 y Fp(\()p Fn(\021)t Fp(\))p 376 3711 259 4 v 462 3802 a Fn(dt)672 3734 y Fp(=)775 3640 y Ff(X)835 3849 y Fg(i)919 3734 y Fp([)p Fn(\032)996 3749 y Fg(t)1026 3734 y Fp(\()p Fn(\021)1116 3693 y Fg(i)1144 3734 y Fp(\))p Fn(c)p Fp(\()p Fn(i;)17 b(\021)1391 3693 y Fg(i)1419 3734 y Fp(\))22 b Fo(\000)g Fn(\032)1628 3749 y Fg(t)1658 3734 y Fp(\()p Fn(\021)t Fp(\))p Fn(c)p Fp(\()p Fn(i;)17 b(\021)t Fp(\)])22 b(+)2180 3640 y Ff(X)2196 3855 y Fh(h)p Fg(ij)t Fh(i)2324 3734 y Fp([)p Fn(\032)2401 3749 y Fg(t)2431 3734 y Fp(\()p Fn(\021)2521 3693 y Fg(ij)2581 3734 y Fp(\))p Fn(c)p Fp(\()p Fn(i;)17 b(j;)g(\021)2912 3693 y Fg(ij)2972 3734 y Fp(\))22 b Fo(\000)h Fn(\032)3182 3749 y Fg(t)3212 3734 y Fp(\()p Fn(\021)t Fp(\))p Fn(c)p Fp(\()p Fn(i;)17 b(j;)g(\021)t Fp(\)])236 4033 y(F)-8 b(or)25 b(this)g(mo)s(del,)h(the)g (stationary)e(measure)i Fn(\032)g Fp(is)f(the)h(pro)s(duct)g(measure)g (with)f(uniform)f(densit)m(y)236 4153 y(equal)31 b(to)f Fn(\015)660 4168 y Fi(+)719 4153 y Fn(=)p Fp(\()p Fn(\015)857 4168 y Fh(\000)934 4153 y Fp(+)18 b Fn(\015)1079 4168 y Fi(+)1138 4153 y Fp(\))30 b(corresp)s(onding)h(to)g(a)f(c)m(hemical)g (p)s(oten)m(tial)f(ln)16 b Fn(\015)2984 4168 y Fi(+)3042 4153 y Fn(=\015)3142 4168 y Fh(\000)3231 4153 y Fp(of)31 b(the)g(particle)236 4274 y(reserv)m(oir.)236 4394 y(F)-8 b(or)43 b(a)h(\014xed)h(nearest)f(neigh)m(b)s(or)g(pair)e Fo(h)p Fn(ij)6 b Fo(i)p Fp(,)47 b(with)c Fn(j)53 b Fp(=)47 b(\()p Fn(i)2484 4409 y Fi(1)2553 4394 y Fp(+)30 b(1)p Fn(;)17 b(i)2785 4409 y Fi(2)2824 4394 y Fp(\))43 b(to)h(the)g(righ)m (t)f(of)g Fn(i)p Fp(,)k(the)236 4515 y(time-in)m(tegrated)31 b(microscopic)g(curren)m(t)j(o)m(v)m(er)f(an)g(in)m(terv)-5 b(al)31 b([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])33 b(is)1630 4709 y Fn(J)1693 4667 y Fg(i)1684 4733 y(T)1767 4709 y Fo(\021)28 b Fn(N)1960 4661 y Fg(i)p Fh(!)p Fg(j)1950 4736 y(T)2114 4709 y Fo(\000)22 b Fn(N)2301 4661 y Fg(j)t Fh(!)p Fg(i)2291 4736 y(T)236 4903 y Fp(with)i Fn(N)538 4855 y Fg(i)p Fh(!)p Fg(j)528 4930 y(T)694 4903 y Fp(the)g(total)f(n)m (um)m(b)s(er)h(of)g(particles)f(that)h(ha)m(v)m(e)h(passed)h(from)d (site)g Fn(i)i Fp(to)e(site)h Fn(j)6 b Fp(.)41 b(W)-8 b(e)24 b(ha)m(v)m(e)236 5023 y(the)j(con)m(v)m(en)m(tion)g(to)f(tak)m (e)h(this)f(curren)m(t)h(p)s(ositiv)m(e)f(when)h(the)g(net)f(n)m(um)m (b)s(er)h(of)f(particles)f(jumping)236 5143 y(to)38 b(the)g(righ)m(t)g (\(i.e.,)h(in)e(the)h(direction)f(of)h(the)g(external)g(\014eld\))g(is) g(p)s(ositiv)m(e.)59 b(Multiplying)35 b(the)236 5264 y(sum)e(of)f(all)e(the)j(curren)m(t)h(con)m(tributions)e(in)g Fn(V)54 b Fp(with)32 b(the)h(\014eld)f Fn(E)39 b Fp(w)m(e)34 b(get)994 5470 y Fn(W)1086 5485 y Fg(V)t(;T)1205 5470 y Fp(\()p Fn(\021)1291 5485 y Fg(s)1328 5470 y Fn(;)17 b(s)27 b Fo(2)i Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]\))28 b Fo(\021)g Fn(E)2116 5375 y Ff(X)2124 5586 y Fg(i)p Fh(2)p Fg(V)2277 5470 y Fn(J)2340 5428 y Fg(i)2331 5494 y(T)2386 5470 y Fp(\()p Fn(\021)2472 5485 y Fg(s)2509 5470 y Fn(;)17 b(s)27 b Fo(2)h Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]\))2007 5800 y(7)p eop %%Page: 8 8 8 7 bop 236 548 a Fp(whic)m(h)32 b(is)g(a)f(random)g(v)-5 b(ariable)30 b(represen)m(ting)i(the)g(w)m(ork)h(done)f(on)g(our)f (system)i(o)m(v)m(er)f(the)h(time-)236 668 y(in)m(terv)-5 b(al)32 b([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(].)44 b(Its)34 b(exp)s(ectation)f(in)f(the)h(stationary)f(state)h(equals)h (\(up)f(to)f(a)h(temp)s(erature)236 789 y(factor\))f(the)h(exp)s(ected) i(heat)d(dissipated)h(in)f(the)h(en)m(vironmen)m(t)g(and)f(is)g(giv)m (en)h(b)m(y)1181 1007 y Fo(h)p Fn(W)1312 1022 y Fg(V)t(;T)1431 1007 y Fo(i)28 b Fp(=)f(2)p Fn(T)14 b Fo(j)p Fn(V)21 b Fo(j)c Fn(E)22 b Fp(sinh\()p Fn(E)6 b(=)p Fp(2\))2535 940 y Fn(\015)2586 955 y Fi(+)2645 940 y Fn(\015)2696 955 y Fh(\000)p 2418 984 455 4 v 2418 1075 a Fp(\()p Fn(\015)2507 1090 y Fi(+)2587 1075 y Fp(+)22 b Fn(\015)2736 1090 y Fh(\000)2795 1075 y Fp(\))2833 1047 y Fi(2)236 1253 y Fp(If)33 b(w)m(e)g(no)m(w)h(\014x)f(another)f(square)i(\003)28 b Fo(\032)g Fn(V)54 b Fp(inside)32 b(our)g(large)g(system,)h(then)1488 1460 y Fn(W)1580 1475 y Fi(\003)p Fg(;T)1732 1460 y Fo(\021)28 b Fn(E)2118 1365 y Ff(X)1932 1581 y Fg(i)p Fh(2)p Fi(\003:\()p Fg(i)2123 1590 y Fe(1)2157 1581 y Fi(+1)p Fg(;i)2291 1590 y Fe(2)2326 1581 y Fi(\))p Fh(2)p Fi(\003)2466 1460 y Fn(J)2529 1419 y Fg(i)2520 1485 y(T)236 1766 y Fp(is)d(the)h(random)f (v)-5 b(ariable)24 b(\\w)m(ork)i(done)g(on)f(the)h(system)h(in)d(\003)i (o)m(v)m(er)g(the)g(time-in)m(terv)-5 b(al)22 b([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]".)236 1886 y(That)25 b(constitutes)g(the)g (main)e(con)m(tribution)h(to)g(the)h(lo)s(cal)e(random)g(v)-5 b(ariable)23 b(\\en)m(trop)m(y)j(pro)s(duc-)236 2007 y(tion)34 b(in)g(the)i(space-time)e(windo)m(w)h(\003)23 b Fo(\002)i Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]".)50 b(Y)-8 b(et,)36 b(this)f(is)f(only)h(its)f(bulk)h(con)m(tribution.)236 2127 y(W)-8 b(e)43 b(ha)m(v)m(e)h(indeed)f(only)f(included)h(in)f Fn(W)1809 2142 y Fi(\003)p Fg(;T)1975 2127 y Fp(the)h(microscopic)e (curren)m(ts)k(b)s(et)m(w)m(een)f(the)f(sites)236 2247 y(strictly)30 b(inside)g(\003)h(while)f(particles)g(will)e(of)j(course) g(also)f(hop)h(in)f(and)h(out)g(of)f(\003)h(via)f(its)g(b)s(ound-)236 2368 y(ary)-8 b(.)43 b(In)30 b(other)g(w)m(ords,)i(the)e(region)f Fn(V)39 b Fo(n)16 b Fp(\003)30 b(acts)g(as)g(a)g(particle)f(reserv)m (oir)h(from)f(whic)m(h)i(particles)236 2488 y(can)j(en)m(ter)g(or)f (lea)m(v)m(e)g(\003.)45 b(That)34 b(also)e(con)m(tributes)i(to)f(the)g (en)m(trop)m(y)i(pro)s(duction)d(as,)i(quite)f(gen-)236 2608 y(erally)-8 b(,)34 b(the)h(c)m(hange)g(in)f(en)m(trop)m(y)i(in)e (the)h(particle)e(reserv)m(oir)i(equals)g(the)g(n)m(um)m(b)s(er)g(of)f (particles)236 2729 y(transferred)c(to)g(it,)f(m)m(ultiplied)d(b)m(y)k (its)f(c)m(hemical)f(p)s(oten)m(tial.)41 b(No)m(w)30 b(usually)-8 b(,)29 b(this)g(c)m(hemical)g(p)s(o-)236 2849 y(ten)m(tial)j(is)g(\014xed)i(and)g(constan)m(t,)g(i.e.,)e(not)h (dep)s(ending)g(on)g(whatev)m(er)i(happ)s(ens)f(in)e(the)i(system)236 2970 y(itself.)60 b(Here)39 b(this)f(is)g(not)g(the)h(case.)61 b(It)39 b(su\016ces)h(to)e(imagine)e(that)i(almost)f(all)f(particles)i (are)236 3090 y(in)f(fact)g(inside)g(\003)g(with)g(therefore)h(a)f(lo)m (w)g(densit)m(y)i(of)e(particles)g(in)f Fn(V)47 b Fo(n)26 b Fp(\003.)57 b(As)38 b(a)g(result,)g(the)236 3210 y Fk(e\013e)-5 b(ctive)27 b Fp(c)m(hemical)f(p)s(oten)m(tial)g(for)h (creating)g(or)g(destro)m(ying)h(particles)f(at)g(the)h(b)s(oundaries)f (of)g(\003)236 3331 y(will)j(dep)s(end)i(on)g(time)e(and)i(on)g(whatev) m(er)h(happ)s(ened)g(inside)e(\003)g(b)s(efore)h(that)f(time.)42 b(Moreo)m(v)m(er)236 3451 y(this)27 b(will)f(con)m(tribute)h(to)g(the)h (nonequilibrium)d(condition)h(only)h(for)g Fn(E)34 b Fo(6)p Fp(=)27 b(0)g(b)s(ecause)i(only)e(then)236 3572 y(will)g(there)k(b)s(e)e(a)g(di\013eren)m(t)h(rate)f(of)g(lea)m (ving/en)m(tering)g(\003)g(at)g(the)h(righ)m(t)e(v)m(ersus)k(the)e (left)e(v)m(ertical)236 3692 y(b)s(oundaries)34 b(of)f(\003.)47 b(This)34 b(is)f(not)h(the)g(case)h(for)e(the)h(upp)s(er)h(v)m(ersus)h (the)e(lo)m(w)m(er)g(b)s(oundaries)f(but)236 3812 y(also)27 b(there,)i(ev)m(en)g(when)g(there)f(w)m(ould)g(not)f(b)s(e)h(a)f (\014eld)h(strictly)f(inside)g(\003,)h(the)g(dynamics)f(inside)236 3933 y(will)j(b)s(e)h(in\015uenced)i(b)m(y)g(the)f(\014eld)f(outside.) 43 b(This)32 b(is)f(summarized)g(in)g(the)h(form)e(of)h(the)h(second) 236 4053 y(con)m(tribution)g(to)g(the)h(time-in)m(tegrated)e(en)m(trop) m(y)i(pro)s(duction)f(and)h(it)f(is)g(a)g(b)s(oundary)h(term:)1442 4248 y Fo(J)1510 4263 y Fg(@)t Fi(\003)p Fg(;T)1702 4248 y Fo(\021)28 b Fn(R)1881 4263 y Fg(`)1936 4248 y Fp(+)22 b Fn(R)2108 4263 y Fg(r)2169 4248 y Fp(+)g Fn(R)2341 4263 y Fg(u)2408 4248 y Fp(+)g Fn(R)2580 4263 y Fg(d)236 4444 y Fp(where)40 b(the)e(v)-5 b(arious)38 b(terms)g(corresp)s(ond)h (to)e(the)i(reactions)f(taking)f(place)h(at)g(the)h(left,)f(righ)m(t,) 236 4564 y(upp)s(er)k(and)g(lo)m(w)m(er)f(b)s(oundaries)h(of)f(the)g (square)i(\003.)70 b(W)-8 b(e)42 b(will)d(not)i(write)g(all)f(of)h (them)g(do)m(wn)236 4684 y(explicitly)31 b(but)i(here)g(is)f(for)g (example)271 4929 y Fn(R)345 4944 y Fg(r)411 4929 y Fo(\021)625 4835 y Ff(X)516 5050 y Fg(i)p Fh(2)p Fi(\003:)p Fg(j)t Fh(2)p Fg(V)15 b Fh(n)p Fi(\003)967 4835 y Ff(X)893 5046 y Fh(\000)p Fg(T)10 b Fh(\024)p Fg(t)p Fh(\024)p Fg(T)1185 4929 y Fp([)p Fn(\021)1260 4944 y Fg(t)1290 4929 y Fp(\()p Fn(i)p Fp(\))17 b(ln)1562 4862 y Fn(\015)1613 4877 y Fh(\000)1694 4862 y Fp(+)22 b Fn(q)1835 4877 y Fi(\003)p Fg(;t)1933 4862 y Fp(\()p Fn(j;)17 b(\021)t(;)g(E)6 b Fp(\))p 1523 4906 783 4 v 1523 4998 a Fn(\015)1574 5013 y Fh(\000)1655 4998 y Fp(+)22 b Fn(q)1796 5013 y Fi(\003)p Fg(;t)1895 4998 y Fp(\()p Fn(j;)17 b(\021)t(;)g Fo(\000)p Fn(E)6 b Fp(\))2338 4929 y(+)22 b(\(1)g Fo(\000)g Fn(\021)2692 4944 y Fg(t)2722 4929 y Fp(\()p Fn(i)p Fp(\)\))17 b(ln)3032 4862 y Fn(\015)3083 4877 y Fi(+)3164 4862 y Fp(+)22 b Fn(p)3311 4877 y Fi(\003)p Fg(;t)3409 4862 y Fp(\()p Fn(j;)17 b(\021)t(;)g(E)6 b Fp(\))p 2994 4906 788 4 v 2994 4998 a Fn(\015)3045 5013 y Fi(+)3125 4998 y Fp(+)22 b Fn(p)3272 5013 y Fi(\003)p Fg(;t)3371 4998 y Fp(\()p Fn(j;)17 b(\021)t(;)g Fo(\000)p Fn(E)6 b Fp(\))236 5235 y(where)29 b Fn(j)k Fp(=)28 b(\()p Fn(i)761 5250 y Fi(1)811 5235 y Fp(+)11 b(1)p Fn(;)17 b(i)1024 5250 y Fi(2)1063 5235 y Fp(\),)28 b(the)g(sum)f(o)m(v)m(er)h(times)e Fn(t)h Fp(is)g(o)m(v)m(er)h(the)f(times)g(when)h(a)f(particle)e(is)i(created) 236 5355 y(or)32 b(destro)m(y)m(ed)j(at)d Fn(i)p Fp(,)h(and)g(the)g (rates)g Fn(p)g Fp(and)f Fn(q)37 b Fp(are)32 b(giv)m(en)h(b)m(y)811 5551 y Fn(q)854 5566 y Fi(\003)p Fg(;t)952 5551 y Fp(\()p Fn(j;)17 b(\021)t(;)g(E)6 b Fp(\))27 b Fo(\021)i Fn(e)1464 5510 y Fg(E)t(=)p Fi(2)1594 5551 y Fp(Prob[)p Fn(\021)1876 5566 y Fg(t)1906 5551 y Fp(\()p Fn(j)6 b Fp(\))28 b(=)f(0)p Fo(j)p Fn(\021)2284 5566 y Fg(s)2321 5551 y Fp(\()p Fn(k)s Fp(\))p Fn(;)17 b(k)30 b Fo(2)e Fp(\003)p Fn(;)17 b(s)27 b Fo(2)h Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(t)p Fp(]])2007 5800 y(8)p eop %%Page: 9 9 9 8 bop 236 548 a Fp(and)781 768 y Fn(p)830 783 y Fi(\003)p Fg(;t)928 768 y Fp(\()p Fn(j;)17 b(\021)t(;)g(E)6 b Fp(\))27 b Fo(\021)h Fn(e)1439 727 y Fh(\000)p Fg(E)t(=)p Fi(2)1624 768 y Fp(Prob)q([)p Fn(\021)1907 783 y Fg(t)1936 768 y Fp(\()p Fn(j)6 b Fp(\))28 b(=)f(1)p Fo(j)p Fn(\021)2314 783 y Fg(s)2351 768 y Fp(\()p Fn(k)s Fp(\))p Fn(;)17 b(k)30 b Fo(2)e Fp(\003)p Fn(;)17 b(s)27 b Fo(2)h Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(t)p Fp(]])236 988 y(where)38 b(the)f(probabilities)d(refer)j(to)f(the)h(steady)h(state)f(in)f Fn(V)21 b Fp(.)56 b(In)37 b(other)g(w)m(ords,)i(the)e(external)236 1108 y(\014eld)g(do)s(es)f(not)h(only)f(w)m(ork)h(on)g(the)g(particles) e(in)h(\003)g(it)g(also)f(creates)j(a)e(gradien)m(t)g(in)g(c)m(hemical) 236 1229 y(p)s(oten)m(tial)25 b(\(large)g(at)g(the)i(left)e(b)s (oundary)i(and)f(smaller)e(at)i(the)g(righ)m(t\))g(in)f(\003.)41 b(The)27 b(total)e(random)236 1349 y(v)-5 b(ariable)31 b(\\en)m(trop)m(y)j(pro)s(duction)e(in)f(\003")h(no)m(w)i(reads)1606 1544 y(\026)1589 1569 y Fn(S)1649 1584 y Fi(\003)p Fg(;T)1801 1569 y Fo(\021)28 b Fn(W)1998 1584 y Fi(\003)p Fg(;T)2144 1569 y Fp(+)22 b Fo(J)2310 1584 y Fg(@)t Fi(\003)p Fg(;T)236 1789 y Fp(The)34 b(result)e(pro)m(v)m(ed)i(in)e(Section)h(4)f(is)g(the) h(\015uctuation)f(theorem)g(symmetry)h(for)3322 1764 y(\026)3305 1789 y Fn(S)3365 1804 y Fi(\003)p Fg(;T)3489 1789 y Fp(:)1268 2121 y(lim)1276 2183 y Fi(\003)p Fg(;T)1420 2121 y Fp(lim)1459 2183 y Fg(V)1654 2054 y Fp(1)p 1582 2098 194 4 v 1582 2190 a Fo(j)p Fp(\003)p Fo(j)p Fn(T)1802 2121 y Fp(ln)1949 2032 y(Prob[)2205 1963 y Fi(\026)2193 1980 y Fg(S)2236 1991 y Fe(\003)p Fd(;T)p 2193 2009 152 4 v 2199 2066 a Fh(j)p Fi(\003)p Fh(j)p Fg(T)2382 2032 y Fp(=)28 b Fn(a)p Fp(])p 1910 2098 694 4 v 1910 2217 a(Prob[)2166 2148 y Fi(\026)2154 2165 y Fg(S)2197 2176 y Fe(\003)p Fd(;T)p 2154 2194 152 4 v 2160 2251 a Fh(j)p Fi(\003)p Fh(j)p Fg(T)2344 2217 y Fp(=)f Fo(\000)p Fn(a)p Fp(])2641 2121 y(=)g Fn(a)782 b Fp(\(1.12\))236 2445 y(uniformly)31 b(in)g(the)i Fn(\015)1013 2460 y Fh(\006)1072 2445 y Fp(.)236 2565 y(One)43 b(ma)m(y)f(w)m(onder)i(whether)g(the)f(w) m(ork)g Fn(W)1924 2580 y Fi(\003)p Fg(;T)2091 2565 y Fp(satis\014es)g(a)f(similar)d(\015uctuation)k(symmetry)-8 b(.)236 2686 y(That)33 b(is)e(\(1.12\))h(with)g Fn(W)1173 2701 y Fi(\003)p Fg(;T)1328 2686 y Fp(replacing)1762 2661 y(\026)1745 2686 y Fn(S)1805 2701 y Fi(\003)p Fg(;T)1929 2686 y Fp(.)43 b(It)32 b(remains)g(uncertain)g(ho)m(w)m(ev)m(er)i (whether)g(that)236 2806 y(is)e(true)g(uniformly)d(in)i(the)i(v)-5 b(alues)31 b Fn(\015)1603 2821 y Fh(\006)1690 2806 y Fo(#)c Fp(0)32 b(but,)g(as)g(w)m(e)h(will)c(sho)m(w,)k(it)e(remains)g (true)h(whenev)m(er)236 2927 y Fn(\015)287 2942 y Fh(\006)374 2927 y Fo(6)p Fp(=)27 b(0.)383 3047 y(The)46 b(rest)h(of)f(our)g(pap)s (er)g(is)f(organized)h(as)g(follo)m(ws:)69 b(in)45 b(Section)h(2)g(w)m (e)h(giv)m(e)f(a)f(general)236 3167 y(strategy)d(to)f(obtain)g(LFT,)g (whic)m(h)i(w)m(e)f(apply)f(in)g(Section)g(3)g(for)g(spin\015ip)g(pro)s (cesses)j(and)d(in)236 3288 y(Section)33 b(4)f(for)g(the)h(asymmetric)e (exclusion)i(pro)s(cess.)236 3621 y Fv(2)161 b(Abstract)53 b(setting)236 3840 y Fp(W)-8 b(e)35 b(iden)m(tify)g(the)g(essen)m(tial) g(mathematical)c(structure,)37 b(needed)g(to)d(pass)i(from)d(a)i (global)e(to)h(a)236 3960 y(lo)s(cal)i(\015uctuation)i(theorem.)61 b(Our)39 b(later)e(sp)s(eci\014c)j(illustrations)35 b(will)h(then)j (just)g(b)s(e)g(applica-)236 4080 y(tions)32 b(of)g(the)h(same)g (theme.)236 4201 y(W)-8 b(e)36 b(consider)f(a)g(measurable)f(space)i (\(\012)p Fn(;)17 b Fo(F)10 b Fp(\))35 b(on)g(whic)m(h)g(t)m(w)m(o)h (sequences)i(of)d(probabilit)m(y)e(mea-)236 4321 y(sures)k Fn(P)547 4336 y Fg(n)630 4321 y Fp(and)e Fn(P)899 4285 y Fg(r)885 4346 y(n)937 4321 y Fp(.)52 b(Supp)s(ose)37 b(that)e(\002)1692 4336 y Fg(n)1775 4321 y Fp(is)g(an)h(in)m(v)m (olution)e(on)h(\012)h(suc)m(h)h(that)f Fn(P)3221 4336 y Fg(n)3303 4321 y Fp(and)g Fn(P)3559 4336 y Fg(n)3630 4321 y Fo(\016)24 b Fp(\002)3780 4336 y Fg(n)236 4441 y Fp(are)35 b(m)m(utually)e(absolutely)g(con)m(tin)m(uous)j(and)e(the)h (same)g(for)f(the)h(pairs)f Fn(P)3018 4405 y Fg(r)3004 4466 y(n)3089 4441 y Fp(and)h Fn(P)3358 4405 y Fg(r)3344 4466 y(n)3419 4441 y Fo(\016)23 b Fp(\002)3568 4456 y Fg(n)3615 4441 y Fp(.)49 b(W)-8 b(e)236 4562 y(write)1237 4806 y Fn(R)1311 4821 y Fg(n)1386 4806 y Fo(\021)28 b Fp(ln)1660 4739 y Fn(dP)1774 4754 y Fg(n)p 1599 4783 284 4 v 1599 4875 a Fn(dP)1713 4890 y Fg(n)1759 4875 y Fp(\002)1835 4890 y Fg(n)1892 4806 y Fn(;)17 b(F)1999 4821 y Fg(n)2074 4806 y Fo(\021)28 b Fn(R)2253 4821 y Fg(n)2322 4806 y Fp(+)22 b(ln)2528 4739 y Fn(dP)2656 4703 y Fg(r)2642 4764 y(n)2693 4739 y Fp(\002)2769 4754 y Fg(n)p 2528 4783 288 4 v 2590 4875 a Fn(dP)2718 4846 y Fg(r)2704 4899 y(n)236 5076 y Fp(then,)34 b(b)m(y)f(de\014nition,)f (for)g(all)e(functions)j Fn(f)11 b Fp(,)1295 5204 y Ff(Z)1411 5339 y Fn(dP)1525 5354 y Fg(n)1571 5339 y Fn(f)g Fp(\()p Fn(R)1742 5354 y Fg(n)1789 5339 y Fp(\))28 b(=)1958 5204 y Ff(Z)2075 5339 y Fn(dP)2189 5354 y Fg(n)2235 5339 y Fn(e)2280 5298 y Fh(\000)p Fg(R)2388 5306 y Fd(n)2435 5339 y Fn(f)11 b Fp(\()p Fo(\000)p Fn(R)2683 5354 y Fg(n)2730 5339 y Fp(\))809 b(\(2.13\))2007 5800 y(9)p eop %%Page: 10 10 10 9 bop 236 548 a Fp(and)1219 624 y Ff(Z)1335 760 y Fn(dP)1463 719 y Fg(r)1449 784 y(n)1500 760 y Fn(f)11 b Fp(\()p Fn(R)1671 775 y Fg(n)1718 760 y Fp(\))28 b(=)1887 624 y Ff(Z)2004 760 y Fn(dP)2132 719 y Fg(r)2118 784 y(n)2169 760 y Fn(e)2214 719 y Fh(\000)p Fg(R)2322 727 y Fd(n)2364 719 y Fi(+)p Fg(F)2464 727 y Fd(n)2511 760 y Fn(f)11 b Fp(\()p Fo(\000)p Fn(R)2759 775 y Fg(n)2806 760 y Fp(\))733 b(\(2.14\))236 997 y(The)29 b(iden)m(tit)m(y)f (\(2.13\))g(expresses)j(an)d(exact)h(symmetry)g(in)e(the)i (\015uctuations)f(of)g Fn(R)3300 1012 y Fg(n)3375 997 y Fp(but)h(should)236 1117 y(b)s(e)35 b(compared)e(with)h(the)h(global) d(symmetry)i(\(1.9,)g(1.10\).)47 b(The)35 b(next)g(equalit)m(y)f (\(2.14\))g(is)f(v)m(ery)236 1237 y(similar)e(but)k(there)g(is)f(the)h (correction)f(term)g Fn(F)2015 1252 y Fg(n)2062 1237 y Fp(.)49 b(T)-8 b(o)35 b(get)f(rid)g(of)g(it)f(\(at)h(least)g (asymptotically)236 1358 y(in)e Fn(n)p Fp(\))h(w)m(e)h(need)g(extra)f (assumptions.)43 b(This)33 b(will)e(then)i(yield)f(the)h(lo)s(cal)e (\015uctuation)h(theorem.)236 1478 y(Before)e(w)m(e)h(giv)m(e)f(a)g (general)f(w)m(a)m(y)i(of)f(expressing)h(these)g(assumptions,)f(the)h (reader)f(ma)m(y)g(appre-)236 1598 y(ciate)37 b(some)g(more)f (explication)g(concerning)h(our)g(c)m(hoice)h(of)e(`global')f(v)m (ersus)40 b('lo)s(cal')35 b(as)i(there)236 1719 y(is)32 b(of)g(course)i(no)f(natural)e(in)m(terpretation)g(of)i(this)f(within)f (the)i(prop)s(osed)g(abstraction.)236 1839 y(As)j(w)m(e)g(will)d(see)k 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1919 2038 V 1920 2036 V 1921 2034 V 1922 2032 V 1923 2030 V 1924 2027 V 1925 2025 V 1926 2023 V 1928 2021 V 1929 2019 V 1930 2016 V 1879 2027 4 14 v 1880 2023 4 9 v 1881 2021 4 7 v 1883 2019 4 5 v 1883 2016 4 3 v 1931 2080 4 14 v 1930 2080 4 9 v 1929 2080 4 7 v 1927 2080 4 5 v 1927 2080 4 3 v 1933 2044 a Fg(d)1973 2080 y Fp(;)k Fn(P)52 b Fp(will)37 b(b)s(e)i(an)f(in\014nite)g(v)m(olume)g(steady)i(state)236 2200 y(measure)30 b(\(i.e.,)f(the)h(path-space)g(measure)g(of)f(a)g (stationary)f(pro)s(cess)j(o)m(v)m(er)f(some)f(time)f(in)m(terv)-5 b(al)236 2321 y([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]\);)40 b Fn(n)e Fp(will)d(refer)j(to)f(a)g(\014nite)g(space-time)g(v)m(olume)g (\(corresp)s(onding)g(to)g(a)g(sequence)j(of)236 2441 y(cub)s(es)32 b(\003)571 2456 y Fg(n)647 2441 y Fp(cen)m(tered)g (around)e(the)h(origin)d(times)h(the)h(in)m(terv)-5 b(al)29 b([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]\))30 b(and)h(\002)3216 2456 y Fg(n)3293 2441 y Fp(will)c(b)s(e)k(time-)236 2562 y(rev)m(ersal)25 b(on)f(the)g(space-time)g(v)m(olume)f(\003)1742 2577 y Fg(n)1793 2562 y Fo(\002)5 b Fp([)p Fo(\000)p Fn(T)j(;)17 b(T)d Fp(].)42 b(The)25 b(pro)s(cess)g Fn(P)2841 2577 y Fg(n)2912 2562 y Fp(will)c(b)s(e)k(the)f(path-space)236 2682 y(measure)32 b(of)e(the)i(stationary)e(in)m(teracting)g(particle)f (pro)s(cess)k(on)e(this)f(\014nite)h(\003)3169 2697 y Fg(n)3235 2682 y Fo(\002)19 b Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(].)43 b Fn(P)3789 2646 y Fg(r)3775 2707 y(n)236 2802 y Fp(is)38 b(the)i(marginal)35 b(distribution)i(of)h(the)i (tra)5 b(jectories)38 b(restricted)i(to)e(the)h(space-time)f(windo)m(w) 236 2923 y(\003)304 2938 y Fg(n)374 2923 y Fo(\002)23 b Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])34 b(under)g Fn(P)14 b Fp(.)45 b(In)34 b(the)g(con)m(text)h(of)e(in)m(teracting)f (particle)g(systems,)j Fn(P)47 b Fp(and)34 b Fn(P)3595 2938 y Fg(n)3675 2923 y Fp(will)236 3043 y(b)s(e)f(path-space)h (measures)g(of)e(a)h(Mark)m(o)m(vian)g(pro)s(cess,)i(whereas)f Fn(P)2769 3007 y Fg(r)2755 3068 y(n)2840 3043 y Fp(will)c(b)s(e)k (non-Mark)m(o)m(vian.)236 3163 y(In)28 b(the)h(lo)s(cal)c (\015uctuation)j(theorem)g(it)f(is)g(attempted)h(to)f(reco)m(v)m(er)j (the)e(global)e(symmetry)i(of)g Fn(R)3780 3178 y Fg(n)236 3284 y Fp(under)i Fn(P)572 3299 y Fg(n)649 3284 y Fp(also)e(in)h(the)g (restrictions)g Fn(P)1695 3248 y Fg(r)1681 3308 y(n)1762 3284 y Fp(of)g Fn(P)42 b Fp(to)29 b(\014nite)h(v)m(olumes)f(\003)2777 3299 y Fg(n)2823 3284 y Fp(.)43 b(Clearly)28 b(then,)j(what)f(w)m(e)236 3404 y(need)h(is)d(that)h(the)h(di\013erence)g(b)s(et)m(w)m(een)i Fn(P)1798 3419 y Fg(n)1874 3404 y Fp(and)d Fn(P)2137 3368 y Fg(r)2123 3429 y(n)2204 3404 y Fp(is)g(a)g(b)s(oundary)g(term)g (but)h(this)f(is)f(more)h(or)236 3525 y(less)k(implied)c(b)m(y)k(ha)m (ving)f(our)g(in)m(teracting)f(particle)g(systems)i(enjo)m(y)g (Gibbsianness)f(on)g(space-)236 3645 y(time.)42 b(Finally)-8 b(,)29 b(the)j(meaning)e(of)i Fn(R)1587 3660 y Fg(n)1665 3645 y Fp(is)f(that)h(it)e(giv)m(es,)j(at)e(least)g(up)h(to)f 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5062 a Fn(dP)2170 5033 y Fg(r)2156 5087 y(n)2217 4853 y Ff(\023)2290 4875 y Fg(\025)2331 4884 y Fe(1)2386 4994 y Fn(e)2431 4952 y Fg(\025)2472 4961 y Fe(2)2507 4952 y Fg(B)2560 4960 y Fd(n)2635 4994 y Fp(=)28 b(0)789 b(\(2.15\))236 5234 y(for)32 b(all)f Fn(\025)578 5249 y Fi(1)617 5234 y Fn(;)17 b(\025)718 5249 y Fi(2)785 5234 y Fo(2)28 b Fp(I)-17 b(R.)44 b(Supp)s(ose)33 b(that)g(for)f(all)e Fn(z)j Fo(2)28 b Fp(I)-17 b(R)1435 5465 y Fn(p)p Fp(\()p Fn(z)t Fp(\))28 b(=)g(lim)1787 5525 y Fg(n)1928 5397 y Fp(1)p 1903 5442 99 4 v 1903 5533 a Fn(a)1954 5548 y Fg(n)2028 5465 y Fp(ln)2126 5329 y Ff(Z)2242 5465 y Fn(e)2287 5424 y Fh(\000)p Fg(z)2389 5407 y Fi(\026)2378 5424 y Fg(S)2421 5432 y Fd(n)2467 5465 y Fn(dP)2581 5480 y Fg(n)3577 5465 y Fp(\(2.16\))1983 5800 y(10)p eop %%Page: 11 11 11 10 bop 236 548 a Fp(exist)33 b(and)g(is)f(\014nite.)43 b(Then,)34 b(whenev)m(er)1434 808 y Fn(q)t Fp(\()p Fn(z)t Fp(\))28 b(=)g(lim)1784 868 y Fg(n)1924 741 y Fp(1)p 1900 786 99 4 v 1900 877 a Fn(a)1951 892 y Fg(n)2024 808 y Fp(ln)2122 673 y Ff(Z)2239 808 y Fn(e)2284 767 y Fh(\000)p Fg(z)2386 751 y Fi(\026)2375 767 y Fg(S)2418 775 y Fd(n)2464 808 y Fn(dP)2592 767 y Fg(r)2578 833 y(n)3577 808 y Fp(\(2.17\))236 1074 y(exists,)34 b(then)f Fn(p)p Fp(\()p Fn(z)t Fp(\))28 b(=)g Fn(q)t Fp(\()p Fn(z)t Fp(\))33 b(and)f Fn(q)t Fp(\()p Fn(z)t Fp(\))c(=)g Fn(q)t Fp(\(1)22 b Fo(\000)g Fn(z)t Fp(\).)236 1195 y Fc(Remarks:)236 1315 y Fp(1.)47 b(The)34 b(symmetry)g Fn(q)t Fp(\()p Fn(z)t Fp(\))c(=)g Fn(q)t Fp(\(1)22 b Fo(\000)i Fn(z)t Fp(\))34 b(is)f(dual)g(to)g(the)i(symmetry)f(as)f(expressed)k(in)c (\(1.7\).)46 b(Its)236 1436 y(Legendre)26 b(transform)f Fn(i)p Fp(\()p Fn(a)p Fp(\))j(=)f(sup)1526 1459 y Fg(z)1566 1436 y Fp(\()p Fo(\000)p Fn(q)t Fp(\()p Fn(z)t Fp(\))7 b Fo(\000)g Fn(z)t(a)p Fp(\))27 b(satis\014es)f Fn(i)p Fp(\()p Fn(a)p Fp(\))7 b Fo(\000)g Fn(i)p Fp(\()p Fo(\000)p Fn(a)p Fp(\))29 b(=)e Fo(\000)p Fn(a)p Fp(.)42 b(If)3385 1410 y(\026)3368 1436 y Fn(S)3428 1451 y Fg(n)3500 1436 y 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Fn(e)1363 3133 y Fh(\000)p Fg(z)1465 3116 y Fi(\026)1454 3133 y Fg(S)1497 3141 y Fd(n)1571 3174 y Fo(\024)1687 3107 y Fp(1)p 1686 3151 52 4 v 1686 3243 a Fn(a)1764 3174 y Fp(ln)1862 3039 y Ff(Z)1978 3174 y Fn(dP)2106 3133 y Fg(r)2092 3199 y(n)2143 3174 y Fp(\()2193 3107 y Fn(dP)2307 3122 y Fg(n)p 2191 3151 166 4 v 2191 3243 a Fn(dP)2319 3214 y Fg(r)2305 3267 y(n)2366 3174 y Fp(\))2404 3133 y Fg(a)2468 3174 y Fp(+)2576 3107 y(1)p 2576 3151 49 4 v 2580 3243 a Fn(b)2651 3174 y Fp(ln)2749 3039 y Ff(Z)2865 3174 y Fn(dP)2993 3133 y Fg(r)2979 3199 y(n)3030 3174 y Fn(e)3075 3133 y Fh(\000)p Fg(bz)3208 3116 y Fi(\026)3196 3133 y Fg(S)3239 3141 y Fd(n)943 3437 y Fo(\024)1060 3370 y Fp(1)p 1058 3414 52 4 v 1058 3505 a Fn(a)1136 3437 y Fp(ln)1234 3301 y Ff(Z)1350 3437 y Fn(dP)1478 3396 y Fg(r)1464 3462 y(n)1515 3437 y Fp(\()1565 3370 y Fn(dP)1679 3385 y Fg(n)p 1563 3414 166 4 v 1563 3505 a Fn(dP)1691 3477 y Fg(r)1677 3530 y(n)1738 3437 y Fp(\))1776 3396 y Fg(a)1840 3437 y Fp(+)1970 3370 y(1)p 1948 3414 93 4 v 1948 3505 a Fn(bv)2067 3437 y Fp(ln)2165 3301 y Ff(Z)2281 3437 y Fn(dP)2395 3452 y Fg(n)2441 3437 y Fn(e)2486 3396 y Fh(\000)p Fg(bv)r(z)2656 3379 y Fi(\026)2643 3396 y Fg(S)2686 3404 y Fd(n)2756 3437 y Fp(+)986 3632 y(1)p 953 3677 114 4 v 953 3768 a Fn(bw)1094 3700 y Fp(ln)1192 3564 y Ff(Z)1308 3700 y Fn(dP)1436 3659 y Fg(r)1422 3724 y(n)1473 3700 y Fp(\()1521 3632 y Fn(dP)1649 3596 y Fg(r)1635 3657 y(n)p 1521 3677 166 4 v 1523 3768 a Fn(dP)1637 3783 y Fg(n)1696 3700 y Fp(\))1734 3659 y Fg(w)r Fh(\000)p Fi(1)3577 3700 y Fp(\(2.19\))236 3966 y(Dividing)28 b(this)i(b)m(y)h Fn(a)1003 3981 y Fg(n)1081 3966 y Fp(and)f(taking)g(limits,)d(w)m(e)32 b(can)e(use)i(condition)d(\(2.15\))g(with)h Fn(\025)3330 3981 y Fi(2)3397 3966 y Fp(=)e(0)i(to)g(get)1551 4238 y Fn(p)p Fp(\()p Fn(z)t Fp(\))e Fo(\024)1868 4171 y Fn(q)t Fp(\()p Fn(bz)t Fp(\))p 1868 4215 214 4 v 1955 4307 a Fn(b)2120 4238 y Fo(\024)2235 4171 y Fn(p)p Fp(\()p Fn(bv)t(z)t Fp(\))p 2235 4215 267 4 v 2322 4307 a Fn(bv)236 4481 y Fp(Again)36 b(b)m(y)i(the)f(H\177)-49 b(older)37 b(inequalit)m(y)-8 b(,)37 b(b)s(oth)g(functions)f Fn(p)h Fp(and)g Fn(q)k Fp(are)c(con)m(v)m(ex,)j(and)e(hence)g(con-)236 4602 y(tin)m(uous.)54 b(Therefore)38 b(w)m(e)f(can)f(tak)m(e)h(the)f(limit)d (for)j Fn(b;)17 b(v)37 b Fo(!)c Fp(1)j(to)g(conclude)h(that)e Fn(p)p Fp(\()p Fn(z)t Fp(\))g(=)e Fn(q)t Fp(\()p Fn(z)t Fp(\).)236 4722 y(The)h(righ)m(t)d(hand)i(side)g(of)f(\(2.18\))g(can)h (b)s(e)g(treated)g(in)e(the)i(same)g(w)m(a)m(y:)632 4989 y(ln)730 4854 y Ff(Z)846 4989 y Fn(dP)960 5004 y Fg(n)1006 4989 y Fn(e)1051 4948 y Fh(\000)p Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))p Fg(S)1329 4956 y Fd(n)1373 4948 y Fi(+)p Fg(B)1481 4956 y Fd(n)1555 4989 y Fo(\024)1671 4922 y Fp(1)p 1670 4966 52 4 v 1670 5058 a Fn(a)1748 4989 y Fp(ln)1846 4854 y Ff(Z)1962 4989 y Fn(dP)2090 4948 y Fg(r)2076 5014 y(n)2127 4989 y Fp(\()2177 4922 y Fn(dP)2291 4937 y Fg(n)p 2175 4966 166 4 v 2175 5058 a Fn(dP)2303 5029 y Fg(r)2289 5082 y(n)2350 4989 y Fp(\))2388 4948 y Fg(a)2430 4989 y Fn(e)2475 4948 y Fg(aB)2565 4956 y Fd(n)2634 4989 y Fp(+)2742 4922 y(1)p 2742 4966 49 4 v 2746 5058 a Fn(b)2818 4989 y Fp(ln)2915 4854 y Ff(Z)3032 4989 y Fn(dP)3160 4948 y Fg(r)3146 5014 y(n)3197 4989 y Fn(e)3242 4948 y Fh(\000)p Fg(b)p Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))3519 4931 y(\026)3507 4948 y Fg(S)3550 4956 y Fd(n)632 5252 y Fo(\024)748 5185 y Fp(1)p 747 5229 52 4 v 747 5320 a Fn(a)825 5252 y Fp(ln)922 5116 y Ff(Z)1039 5252 y Fn(dP)1167 5211 y Fg(r)1153 5277 y(n)1204 5252 y Fp(\()1254 5185 y Fn(dP)1368 5200 y Fg(n)p 1252 5229 166 4 v 1252 5320 a Fn(dP)1380 5292 y Fg(r)1366 5345 y(n)1427 5252 y Fp(\))1465 5211 y Fg(a)1506 5252 y Fn(e)1551 5211 y Fg(aB)1641 5219 y Fd(n)1711 5252 y Fp(+)1841 5185 y(1)p 1819 5229 93 4 v 1819 5320 a Fn(bv)1938 5252 y Fp(ln)2036 5116 y Ff(Z)2152 5252 y Fn(dP)2266 5267 y Fg(n)2312 5252 y Fn(e)2357 5211 y Fh(\000)p Fg(bv)r Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))2672 5194 y(\026)2658 5211 y Fg(S)2701 5219 y Fd(n)2746 5211 y Fi(+)p Fg(B)2854 5219 y Fd(n)2922 5252 y Fp(+)674 5447 y(1)p 642 5492 114 4 v 642 5583 a Fn(bw)782 5515 y Fp(ln)880 5379 y Ff(Z)996 5515 y Fn(dP)1124 5474 y Fg(r)1110 5539 y(n)1161 5515 y Fp(\()1209 5447 y Fn(dP)1337 5411 y Fg(r)1323 5472 y(n)p 1209 5492 166 4 v 1211 5583 a Fn(dP)1325 5598 y Fg(n)1384 5515 y Fp(\))1422 5474 y Fg(w)r Fh(\000)p Fi(1)1569 5515 y Fn(e)1614 5474 y Fh(\000)p Fg(w)r(B)1775 5482 y Fd(n)1817 5474 y Fg(=v)3577 5515 y Fp(\(2.20\))1983 5800 y(11)p eop %%Page: 12 12 12 11 bop 236 548 a Fp(whic)m(h,)33 b(again)f(after)g(taking)g(limits)d Fn(n)f Fo(")g Fp(+)p Fo(1)p Fp(,)k(and)g(using)h Fn(B)2523 563 y Fg(n)2597 548 y Fp(=)2718 523 y(\026)2701 548 y Fn(S)2761 563 y Fg(n)2830 548 y Fo(\000)23 b Fn(R)3004 563 y Fg(n)3051 548 y Fp(,)32 b(giv)m(es)1031 798 y Fn(p)p Fp(\()p Fn(z)t Fp(\))c(=)f Fn(q)t Fp(\()p Fn(z)t Fp(\))i Fo(\024)1652 730 y Fn(q)t Fp(\()p Fn(b)p Fp(\(1)22 b Fo(\000)g Fn(z)t Fp(\)\))p 1652 775 461 4 v 1861 866 a Fn(b)2150 798 y Fo(\024)2265 730 y Fn(q)t Fp(\()p Fo(\000)p Fn(bv)t Fp(\(1)g Fo(\000)h Fn(z)t Fp(\))g(+)f(1\))p 2265 775 758 4 v 2597 866 a Fn(bv)236 1029 y Fp(and)33 b(w)m(e)h(can)e(tak)m (e)i(the)f(limits)c Fn(b;)17 b(v)32 b Fo(!)27 b Fp(1)32 b(to)h(get)f(the)h(desired)g Fn(q)t Fp(\()p Fn(z)t Fp(\))28 b(=)g Fn(q)t Fp(\(1)22 b Fo(\000)g Fn(z)t Fp(\).)p 3780 1029 48 48 v 236 1353 a Fc(Prop)s(osition)39 b(2.2:)49 b Fp(Let)35 b Fn(B)1319 1368 y Fg(n)1401 1353 y Fp(b)s(e)h(a)e (measurable)h(function)g(suc)m(h)h(that)f Fn(B)3028 1368 y Fg(n)3099 1353 y Fo(\016)24 b Fp(\002)3249 1368 y Fg(n)3328 1353 y Fp(=)32 b Fo(\000)p Fn(B)3587 1368 y Fg(n)3670 1353 y Fp(and)236 1474 y(de\014ne)535 1448 y(\026)518 1474 y Fn(S)578 1489 y Fg(n)653 1474 y Fp(=)27 b Fn(R)830 1489 y Fg(n)899 1474 y Fp(+)22 b Fn(B)1071 1489 y Fg(n)1118 1474 y Fp(.)43 b(Let)33 b(\()p Fn(a)1452 1489 y Fg(n)1499 1474 y Fp(\))f(b)s(e)h(a)f(sequence)j(of)d(p)s(ositiv)m(e)g(n)m(um)m(b) s(ers)h(tending)f(to)g(in\014nit)m(y)236 1594 y(with)g Fn(n)h Fp(so)g(that)f(for)g(all)f Fn(\025)c Fo(2)h Fp(I)-16 b(R)1417 1822 y(lim)1463 1882 y Fg(n)1604 1754 y Fp(1)p 1579 1799 99 4 v 1579 1890 a Fn(a)1630 1905 y Fg(n)1704 1822 y Fp(ln)1802 1686 y Ff(Z)1918 1822 y Fn(dP)2046 1781 y Fg(r)2032 1847 y(n)2083 1822 y Fn(e)2128 1781 y Fg(\025)p Fi(\()p Fg(B)2249 1789 y Fd(n)2292 1781 y Fi(+)p Fg(F)2392 1789 y Fd(n)2434 1781 y Fi(\))2494 1822 y Fp(=)27 b(0)931 b(\(2.21\))236 2065 y(Supp)s(ose)34 b(that)e(for)g(all)f Fn(z)h Fo(2)c Fp(I)-16 b(R)1434 2309 y Fn(q)t Fp(\()p Fn(z)t Fp(\))28 b(=)g(lim)1784 2369 y Fg(n)1924 2241 y Fp(1)p 1900 2286 V 1900 2377 a Fn(a)1951 2392 y Fg(n)2024 2309 y Fp(ln)2122 2173 y Ff(Z)2239 2309 y Fn(e)2284 2268 y Fh(\000)p Fg(z)2386 2251 y Fi(\026)2375 2268 y Fg(S)2418 2276 y Fd(n)2464 2309 y Fn(dP)2592 2268 y Fg(r)2578 2333 y(n)3577 2309 y Fp(\(2.22\))236 2558 y(exists)34 b(and)e(is)g(\014nite.)44 b(Then,)33 b Fn(q)t Fp(\()p Fn(z)t Fp(\))c(=)e Fn(q)t Fp(\(1)22 b Fo(\000)g Fn(z)t Fp(\).)236 2678 y Fc(Pro)s(of)37 b(of)h(Prop)s(osition)e(2.2:)43 b Fp(By)33 b(de\014nition)f(of)g Fn(F)2257 2693 y Fg(n)2304 2678 y Fp(,)h(w)m(e)g(ha)m(v)m(e)1163 2786 y Ff(Z)1279 2921 y Fn(dP)1407 2880 y Fg(r)1393 2946 y(n)1444 2921 y Fn(f)11 b Fp(\()1558 2896 y(\026)1541 2921 y Fn(S)1601 2936 y Fg(n)1648 2921 y Fp(\))28 b(=)1817 2786 y Ff(Z)1933 2921 y Fn(dP)2061 2880 y Fg(r)2047 2946 y(n)2099 2921 y Fn(e)2144 2880 y Fh(\000)2210 2864 y Fi(\026)2199 2880 y Fg(S)2242 2888 y Fd(n)2284 2880 y Fi(+)p Fg(F)2384 2888 y Fd(n)2426 2880 y Fi(+)p Fg(B)2534 2888 y Fd(n)2581 2921 y Fn(f)11 b Fp(\()p Fo(\000)2772 2896 y Fp(\026)2755 2921 y Fn(S)2815 2936 y Fg(n)2862 2921 y Fp(\))236 3165 y(W)-8 b(e)31 b(th)m(us)f(lea)m(v)m(e)h(the)f (left)f(hand)i(side)f(and)g(apply)f(a)h(similar)c(c)m(hain)k(of)g (inequalities)e(to)h(the)i(righ)m(t)236 3285 y(hand)i(side)g(as)g(w)m (as)g(used)h(in)e(the)h(pro)s(of)e(of)h(Prop)s(osition)f(2.1:)620 3529 y(ln)718 3393 y Ff(Z)834 3529 y Fn(dP)962 3488 y Fg(r)948 3553 y(n)999 3529 y Fn(e)1044 3488 y Fh(\000)p Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))p Fg(S)1322 3496 y Fd(n)1365 3488 y Fi(+)p Fg(F)1465 3496 y Fd(n)1507 3488 y Fi(+)p Fg(B)1615 3496 y Fd(n)1690 3529 y Fo(\024)1806 3461 y Fp(1)p 1805 3506 52 4 v 1805 3597 a Fn(a)1883 3529 y Fp(ln)1981 3393 y Ff(Z)2097 3529 y Fn(dP)2225 3488 y Fg(r)2211 3553 y(n)2262 3529 y Fn(e)2307 3488 y Fg(aF)2389 3496 y Fd(n)2432 3488 y Fi(+)p Fg(aB)2577 3496 y Fd(n)2646 3529 y Fp(+)2754 3461 y(1)p 2754 3506 49 4 v 2758 3597 a Fn(b)2829 3529 y Fp(ln)2927 3393 y Ff(Z)3044 3529 y Fn(dP)3172 3488 y Fg(r)3158 3553 y(n)3209 3529 y Fn(e)3254 3488 y Fh(\000)p Fg(b)p Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))3531 3471 y(\026)3519 3488 y Fg(S)3562 3496 y Fd(n)620 3783 y Fo(\024)736 3716 y Fp(1)p 735 3760 52 4 v 735 3852 a Fn(a)813 3783 y Fp(ln)911 3648 y Ff(Z)1027 3783 y Fn(dP)1155 3742 y Fg(r)1141 3808 y(n)1192 3783 y Fn(e)1237 3742 y Fg(a)p Fi(\()p Fg(F)1346 3750 y Fd(n)1389 3742 y Fi(+)p Fg(B)1497 3750 y Fd(n)1540 3742 y Fi(\))1594 3783 y Fp(+)1723 3716 y(1)p 1702 3760 93 4 v 1702 3852 a Fn(bv)1820 3783 y Fp(ln)1918 3648 y Ff(Z)2035 3783 y Fn(dP)2163 3742 y Fg(r)2149 3808 y(n)2200 3783 y Fn(e)2245 3742 y Fh(\000)p Fg(bv)r Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))2559 3725 y(\026)2546 3742 y Fg(S)2589 3750 y Fd(n)2633 3742 y Fi(+)p Fg(F)2733 3750 y Fd(n)2775 3742 y Fi(+)p Fg(B)2883 3750 y Fd(n)2952 3783 y Fp(+)662 3971 y(1)p 630 4015 114 4 v 630 4106 a Fn(bw)770 4038 y Fp(ln)868 3902 y Ff(Z)984 4038 y Fn(dP)1112 3997 y Fg(r)1098 4063 y(n)1150 4038 y Fn(e)1195 3997 y Fh(\000)p Fg(w)r Fi(\()p Fg(F)1375 4005 y Fd(n)1417 3997 y Fh(\000)p Fg(B)1525 4005 y Fd(n)1567 3997 y Fi(\))p Fg(=v)3577 4038 y Fp(\(2.23\))236 4281 y(W)-8 b(e)33 b(ma)m(y)g(th)m(us)g(again)e(divide)h(b)m(y)i Fn(a)1567 4296 y Fg(n)1646 4281 y Fp(and)f(tak)m(e)g(limits)d(\014rst)j Fn(n)28 b Fo(")f Fp(+)p Fo(1)32 b Fp(to)h(reac)m(h)1184 4531 y Fn(q)t Fp(\()p Fn(z)t Fp(\))28 b Fo(\024)1499 4464 y Fn(q)t Fp(\()p Fn(b)p Fp(\(1)22 b Fo(\000)h Fn(z)t Fp(\)\))p 1499 4508 461 4 v 1708 4600 a Fn(b)1997 4531 y Fo(\024)2112 4464 y Fn(q)t Fp(\()p Fo(\000)p Fn(bv)t Fp(\(1)f Fo(\000)h Fn(z)t Fp(\))g(+)f(1\))p 2112 4508 758 4 v 2445 4600 a Fn(bv)236 4757 y Fp(By)33 b(con)m(v)m(exit)m(y)i(w) m(e)e(can)g(tak)m(e)h(the)f(limits)c Fn(b;)17 b(v)32 b Fo(#)27 b Fp(1)33 b(to)f(obtain)f(the)i(desired)g(conclusion.)p 3780 4757 48 48 v 236 4998 a Fc(Remarks:)236 5118 y Fp(1.)61 b(Of)37 b(course,)k(if)c(it)h(happ)s(ens)h(that)f Fo(j)p Fn(F)1744 5133 y Fg(n)1817 5118 y Fp(+)26 b Fn(B)1993 5133 y Fg(n)2040 5118 y Fo(j)p Fn(=a)2168 5133 y Fg(n)2252 5118 y Fo(!)37 b Fp(0)h(uniformly)-8 b(,)37 b(then,)j(for)e(all)e(p)s (ositiv)m(e)236 5239 y(functions)d Fn(f)11 b Fp(,)1361 5497 y(lim)1407 5557 y Fg(n)1548 5430 y Fp(1)p 1523 5474 99 4 v 1523 5566 a Fn(a)1574 5581 y Fg(n)1648 5497 y Fp(ln)1889 5348 y Ff(R)1972 5428 y Fn(dP)2100 5392 y Fg(r)2086 5453 y(n)2137 5428 y Fn(f)g Fp(\()2250 5403 y(\026)2234 5428 y Fn(S)2294 5443 y Fg(n)2341 5428 y Fp(\))p 1755 5474 757 4 v 1755 5490 a Ff(R)1838 5570 y Fn(dP)1966 5541 y Fg(r)1952 5595 y(n)2004 5570 y Fn(e)2049 5541 y Fh(\000)2115 5524 y Fi(\026)2104 5541 y Fg(S)2147 5549 y Fd(n)2193 5570 y Fn(f)g Fp(\()p Fo(\000)2384 5545 y Fp(\026)2367 5570 y Fn(S)2427 5585 y Fg(n)2474 5570 y Fp(\))2550 5497 y(=)27 b(0)1983 5800 y(12)p eop %%Page: 13 13 13 12 bop 236 548 a Fp(without)32 b(further)h(ado.)236 668 y(2.)72 b(The)43 b(di\013erence)g(b)s(et)m(w)m(een)i(Prop)s (osition)40 b(2.1)i(and)g(Prop)s(osition)f(2.2)g(is)h(that)g(in)g(the)g (\014rst)236 789 y(w)m(e)d(supp)s(ose)g(that)e Fn(P)1036 804 y Fg(n)1121 789 y Fp(and)h Fn(P)1393 753 y Fg(r)1379 813 y(n)1468 789 y Fp(are)f(m)m(utually)f(absolutely)h(con)m(tin)m (uous)i(while)e(in)g(the)h(latter,)236 909 y(w)m(e)g(need)g(that)f Fn(P)909 873 y Fg(r)895 934 y(n)984 909 y Fp(and)g Fn(P)1255 873 y Fg(r)1241 934 y(n)1292 909 y Fp(\002)1368 924 y Fg(n)1452 909 y Fp(are)g(m)m(utually)f(absolutely)g(con)m(tin)m(uous.) 58 b(W)-8 b(e)37 b(will)e(follo)m(w)g(the)236 1029 y(second)f(strategy) f(in)f(Section)h(4)f(for)g(the)h(asymmetric)e(exclusion)i(pro)s(cess.) 236 1150 y(3.)42 b(The)31 b(condition)d(that)h(the)h(limits)c (de\014ning)k Fn(p)p Fp(\()p Fn(z)t Fp(\))g(and)f Fn(q)t Fp(\()p Fn(z)t Fp(\))h(exist)g(is)f(natural)f(in)h(the)g(con)m(text)236 1270 y(where)37 b(w)m(e)g(ha)m(v)m(e)g(a)f(large)f(deviation)f (principle)h(for)2225 1245 y(\026)2208 1270 y Fn(S)2268 1285 y Fg(n)2351 1270 y Fp(under)i Fn(P)2694 1285 y Fg(n)2776 1270 y Fp(and)f Fn(P)3046 1234 y Fg(r)3032 1295 y(n)3119 1270 y Fp(resp.)55 b(Ho)m(w)m(ev)m(er)38 b(if)236 1391 y(w)m(e)d(de\014ne)g Fn(p)713 1354 y Fi(+)772 1391 y Fn(;)17 b(p)865 1354 y Fh(\000)924 1391 y Fn(;)g(q)1015 1354 y Fi(+)1073 1391 y Fn(;)g(q)1164 1354 y Fh(\000)1256 1391 y Fp(b)m(y)35 b(the)f(corresp)s(onding)g(limsup,)e(resp.)j (liminf,)30 b(then)k(w)m(e)h(still)c(ha)m(v)m(e)236 1511 y(con)m(v)m(exit)m(y)g(of)c Fn(p)821 1475 y Fi(+)880 1511 y Fn(;)17 b(q)971 1475 y Fi(+)1058 1511 y Fp(\(the)29 b(limsups\),)f(but)g(not)h(necessarily)g(of)e Fn(p)2656 1475 y Fh(\000)2715 1511 y Fn(;)17 b(q)2806 1475 y Fh(\000)2865 1511 y Fp(.)42 b(W)-8 b(e)29 b(can)f(still)e(conclude)236 1631 y(ho)m(w)m(ev)m(er)35 b(the)e(equalit)m(y)f Fn(p)1198 1595 y Fi(+)1257 1631 y Fp(\()p Fn(z)t Fp(\))d(=)e Fn(q)1561 1595 y Fi(+)1620 1631 y Fp(\()p Fn(z)t Fp(\),)33 b(and)g Fn(q)2042 1595 y Fi(+)2101 1631 y Fp(\()p Fn(z)t Fp(\))28 b(=)g Fn(q)2405 1595 y Fi(+)2464 1631 y Fp(\(1)21 b Fo(\000)i Fn(z)t Fp(\).)236 1964 y Fv(3)161 b(LFT)55 b(for)f(spin\015ip)f(pro)t (cesses)236 2183 y Fp(W)-8 b(e)38 b(start)g(our)g(study)g(with)g(the,)h (for)e(ph)m(ysical)h(applications,)f(less)h(in)m(teresting)f(case)i(of) e(pure)236 2304 y(spin\015ip)h(pro)s(cesses.)64 b(F)-8 b(or)38 b(details)f(on)h(the)h(construction)g(of)f(spin\015ip)g(pro)s (cesses,)43 b(w)m(e)c(refer)g(to)236 2424 y([24].)236 2569 y(The)29 b(con\014guration)d(space)j(is)e Fn(K)35 b Fp(=)28 b Fo(f)p Fp(+1)p Fn(;)17 b Fo(\000)p Fp(1)p Fo(g)p 2003 2534 53 4 v 2003 2469 51 4 v 2003 2533 3 3 v 2004 2531 V 2005 2528 V 2006 2526 V 2007 2524 V 2008 2522 V 2009 2520 V 2010 2517 V 2011 2515 V 2013 2513 V 2014 2511 V 2015 2509 V 2016 2506 V 2017 2504 V 2018 2502 V 2019 2500 V 2020 2498 V 2021 2495 V 2022 2493 V 2023 2491 V 2025 2489 V 2026 2487 V 2027 2484 V 2028 2482 V 2029 2480 V 2030 2478 V 2031 2476 V 2032 2473 V 2033 2471 V 2034 2469 V 2020 2533 V 2021 2531 V 2022 2528 V 2023 2526 V 2025 2524 V 2026 2522 V 2027 2520 V 2028 2517 V 2029 2515 V 2030 2513 V 2031 2511 V 2032 2509 V 2033 2506 V 2034 2504 V 2036 2502 V 2037 2500 V 2038 2498 V 2039 2495 V 2040 2493 V 2041 2491 V 2042 2489 V 2043 2487 V 2044 2484 V 2045 2482 V 2047 2480 V 2048 2478 V 2049 2476 V 2050 2473 V 2051 2471 V 2052 2469 V 2001 2480 4 14 v 2002 2476 4 9 v 2004 2473 4 7 v 2005 2471 4 5 v 2005 2469 4 3 v 2054 2533 4 14 v 2052 2533 4 9 v 2051 2533 4 7 v 2050 2533 4 5 v 2049 2533 4 3 v 2055 2492 a Fd(d)2123 2569 y Fp(\(spins)28 b(on)g(the)g Fn(d)p Fo(\000)p Fp(dimensional)d(regular)h(lat-)236 2689 y(tice\))i(and)h(the)g(path)f(space)i(is)e(\012)g(=)g Fn(D)s Fp(\()p Fn(K)r(;)17 b Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]\))28 b(the)g(set)i(of)e(righ)m(t-con)m(tin)m(uous)g (tra)5 b(jectories)236 2810 y(ha)m(ving)26 b(left)g(limits,)f (parametrized)g(b)m(y)i(time)e Fn(t)j Fo(2)g Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])p Fn(;)j(T)42 b(>)27 b Fp(0)f(and)h(ha)m(ving) f(v)-5 b(alues)26 b Fn(!)3558 2825 y Fg(t)3615 2810 y Fo(2)j Fn(K)7 b Fp(.)236 2930 y(Our)31 b(pro)s(cesses)j(are)d(sp)s (eci\014ed)h(in)f(terms)g(of)g(spin\015ip)g(rates)h Fn(c)p Fp(\()p Fn(x;)17 b(\033)t Fp(\))p Fn(;)g(x)27 b Fo(2)p 2993 2932 53 4 v 2993 2866 51 4 v 2993 2930 3 3 v 2994 2928 V 2996 2926 V 2997 2923 V 2998 2921 V 2999 2919 V 3000 2917 V 3001 2915 V 3002 2912 V 3003 2910 V 3004 2908 V 3005 2906 V 3007 2904 V 3008 2901 V 3009 2899 V 3010 2897 V 3011 2895 V 3012 2893 V 3013 2891 V 3014 2888 V 3015 2886 V 3016 2884 V 3017 2882 V 3019 2880 V 3020 2877 V 3021 2875 V 3022 2873 V 3023 2871 V 3024 2869 V 3025 2866 V 3011 2930 V 3012 2928 V 3013 2926 V 3014 2923 V 3015 2921 V 3016 2919 V 3018 2917 V 3019 2915 V 3020 2912 V 3021 2910 V 3022 2908 V 3023 2906 V 3024 2904 V 3025 2901 V 3026 2899 V 3027 2897 V 3028 2895 V 3030 2893 V 3031 2891 V 3032 2888 V 3033 2886 V 3034 2884 V 3035 2882 V 3036 2880 V 3037 2877 V 3038 2875 V 3039 2873 V 3041 2871 V 3042 2869 V 3043 2866 V 2992 2877 4 14 v 2993 2873 4 9 v 2994 2871 4 7 v 2996 2869 4 5 v 2996 2866 4 3 v 3044 2930 4 14 v 3043 2930 4 9 v 3042 2930 4 7 v 3040 2930 4 5 v 3040 2930 4 3 v 3046 2894 a Fg(d)3087 2930 y Fn(;)17 b(\033)31 b Fo(2)d Fn(K)38 b Fp(for)31 b(whic)m(h)236 3050 y(our)41 b(\014rst)g(most)f (imp)s(ortan)m(t)f(assumption)h(is)g(that)g(they)i(are)f(p)s(ositiv)m (e)f(and)h(b)s(ounded:)60 b(there)236 3171 y(are)34 b(constan)m(ts)g Fn(b)873 3186 y Fi(1)942 3171 y Fn(>)29 b Fp(0)p Fn(;)17 b(b)1181 3186 y Fi(2)1249 3171 y Fn(<)29 b Fp(+)p Fo(1)j Fp(so)i(that)f Fn(b)1936 3186 y Fi(1)2005 3171 y Fn(<)c(c)p Fp(\()p Fn(x;)17 b(\033)t Fp(\))28 b Fn(<)h(b)2560 3186 y Fi(2)2633 3171 y Fp(for)k(all)e Fn(x;)17 b(\033)t Fp(.)46 b(F)-8 b(or)32 b(con)m(v)m(enience)236 3291 y(w)m(e)h(assume)f(that)g Fn(c)p Fp(\()p Fn(x;)17 b(\033)t Fp(\))31 b(only)h(dep)s(ends)h(on)e (the)i(neigh)m(b)s(oring)d(spins)i Fn(\033)t Fp(\()p Fn(y)t Fp(\))f(with)g Fo(j)p Fn(y)23 b Fo(\000)e Fn(x)p Fo(j)28 b(\024)g Fp(1.)236 3412 y(Thirdly)-8 b(,)40 b(w)m(e)g(assume)f (the)h(rates)f(to)g(b)s(e)g(translation)e(in)m(v)-5 b(arian)m(t:)54 b Fn(c)p Fp(\()p Fn(x;)17 b(\033)t Fp(\))39 b(=)f Fn(c)p Fp(\(0)p Fn(;)17 b(\034)3399 3427 y Fg(x)3443 3412 y Fn(\033)t Fp(\).)62 b(Here)236 3532 y(and)39 b(afterw)m(ards)h(w)m(e)g (put)g(\003)1320 3547 y Fg(n)1405 3532 y Fp(=)f([)p Fo(\000)p Fn(n;)17 b(n)p Fp(])1811 3496 y Fg(d)1879 3532 y Fo(\\)p 1987 3534 53 4 v 1987 3468 51 4 v 1987 3532 3 3 v 1989 3530 V 1990 3528 V 1991 3525 V 1992 3523 V 1993 3521 V 1994 3519 V 1995 3517 V 1996 3514 V 1997 3512 V 1998 3510 V 2000 3508 V 2001 3506 V 2002 3503 V 2003 3501 V 2004 3499 V 2005 3497 V 2006 3495 V 2007 3492 V 2008 3490 V 2009 3488 V 2010 3486 V 2012 3484 V 2013 3481 V 2014 3479 V 2015 3477 V 2016 3475 V 2017 3473 V 2018 3470 V 2019 3468 V 2005 3532 V 2006 3530 V 2007 3528 V 2008 3525 V 2009 3523 V 2011 3521 V 2012 3519 V 2013 3517 V 2014 3514 V 2015 3512 V 2016 3510 V 2017 3508 V 2018 3506 V 2019 3503 V 2020 3501 V 2021 3499 V 2023 3497 V 2024 3495 V 2025 3492 V 2026 3490 V 2027 3488 V 2028 3486 V 2029 3484 V 2030 3481 V 2031 3479 V 2032 3477 V 2034 3475 V 2035 3473 V 2036 3470 V 2037 3468 V 1986 3479 4 14 v 1987 3475 4 9 v 1988 3473 4 7 v 1990 3470 4 5 v 1990 3468 4 3 v 2038 3532 4 14 v 2037 3532 4 9 v 2036 3532 4 7 v 2035 3532 4 5 v 2034 3532 4 3 v 2040 3496 a Fg(d)2120 3532 y Fp(\002)2196 3547 y Fg(n)2281 3532 y Fp(denotes)41 b(time-rev)m(ersal)d(on)h(\003)3438 3547 y Fg(n)3523 3532 y Fp(de\014ned)236 3652 y(b)m(y)f(\(\002)490 3667 y Fg(n)537 3652 y Fn(!)t Fp(\))640 3667 y Fg(t)669 3652 y Fp(\()p Fn(x)p Fp(\))d Fo(\021)h Fn(!)1009 3667 y Fh(\000)p Fg(t)1093 3652 y Fp(\()p Fn(x)p Fp(\))h(if)f Fn(x)g Fo(2)f Fp(\003)1615 3667 y Fg(n)1662 3652 y Fp(,)j(and)f(\(\002) 2035 3667 y Fg(n)2082 3652 y Fn(!)t Fp(\))2185 3667 y Fg(t)2214 3652 y Fp(\()p Fn(x)p Fp(\))e Fo(\021)h Fn(!)2554 3667 y Fg(t)2583 3652 y Fp(\()p Fn(x)p Fp(\))h(if)f Fn(x)47 b(=)-60 b Fo(2)35 b Fp(\003)3105 3667 y Fg(n)3152 3652 y Fp(.)56 b(On)37 b(the)h(jump-)236 3773 y(times)c(w)m(e)j(adapt)e (\002)1001 3788 y Fg(n)1047 3773 y Fn(!)k Fp(so)c(that)g(it)f(b)s (ecomes)i(righ)m(t-con)m(tin)m(uous,)f(and)g(th)m(us)i(obtain)d(\002) 3520 3788 y Fg(n)3601 3773 y Fp(as)i(an)236 3893 y(in)m(v)m(olution)31 b(on)i(\012)383 4013 y(W)-8 b(e)22 b(de\014ne)h(\003)879 3977 y Fg(?)879 4038 y(n)953 4013 y Fo(\021)28 b(f)p Fn(x)g Fo(2)g Fp(\003)1353 4028 y Fg(n)1400 4013 y Fn(;)17 b(c)p Fp(\()p Fn(x;)g(\033)t Fp(\))27 b(=)h Fn(c)p Fp(\()p Fn(x;)17 b(\033)2089 3977 y Fh(0)2112 4013 y Fp(\))33 b(for)f(all)e Fn(\033)n(;)17 b(\033)2623 3977 y Fh(0)2674 4013 y Fo(2)28 b Fn(K)40 b Fp(with)32 b Fn(\033)t Fp(\()p Fn(y)t Fp(\))27 b(=)h Fn(\033)3490 3977 y Fh(0)3513 4013 y Fp(\()p Fn(y)t Fp(\))p Fn(;)17 b(y)29 b Fo(2)236 4134 y Fp(\003)304 4149 y Fg(n)351 4134 y Fo(g)d Fp(for)g(the)h(subset)h(of) e(sites)h(where)g(the)g(spin\015ip)f(rates)h(do)f(not)h(dep)s(end)g(on) g(the)g(con\014guration)236 4254 y(outside)33 b(\003)641 4269 y Fg(n)687 4254 y Fp(.)236 4375 y(W)-8 b(e)39 b(\014rst)f(describ) s(e)h(the)f(sequence)j(of)d(pro)s(cesses)i Fn(P)2189 4390 y Fg(n)2274 4375 y Fp(corresp)s(onding)e(to)g Fn(P)3094 4390 y Fg(n)3178 4375 y Fp(in)g(the)g(previous)236 4495 y(abstract)d(setting.)49 b(F)-8 b(or)34 b(this)g(w)m(e)i(\014x)f(a)f(b) s(oundary)h(condition)e Fn(\021)i Fo(2)d Fn(K)41 b Fp(and)35 b(w)m(e)h(de\014ne)f(spin\015ip)236 4615 y(rates)1027 4835 y Fn(c)1069 4850 y Fg(n)1116 4835 y Fp(\()p Fn(x;)17 b(\033)t Fp(\))28 b Fo(\021)g Fn(I)8 b Fp([)p Fn(x)28 b Fo(2)g Fp(\003)1806 4850 y Fg(n)1853 4835 y Fp(])p Fn(c)p Fp(\()p Fn(x;)17 b(\033)2114 4850 y Fi(\003)2163 4858 y Fd(n)2210 4835 y Fn(\021)2258 4850 y Fi(\003)2307 4831 y Fd(c)2307 4867 y(n)2354 4835 y Fp(\))p Fn(;)g(x)27 b Fo(2)p 2628 4837 53 4 v 2628 4771 51 4 v 2628 4835 3 3 v 2629 4833 V 2630 4831 V 2632 4828 V 2633 4826 V 2634 4824 V 2635 4822 V 2636 4820 V 2637 4818 V 2638 4816 V 2639 4813 V 2640 4811 V 2641 4809 V 2643 4807 V 2644 4805 V 2645 4802 V 2646 4800 V 2647 4798 V 2648 4796 V 2649 4794 V 2650 4791 V 2651 4789 V 2652 4787 V 2653 4785 V 2655 4783 V 2656 4780 V 2657 4778 V 2658 4776 V 2659 4774 V 2660 4772 V 2646 4835 V 2647 4833 V 2648 4831 V 2649 4828 V 2650 4826 V 2651 4824 V 2652 4822 V 2653 4820 V 2655 4818 V 2656 4816 V 2657 4813 V 2658 4811 V 2659 4809 V 2660 4807 V 2661 4805 V 2662 4802 V 2663 4800 V 2664 4798 V 2666 4796 V 2667 4794 V 2668 4791 V 2669 4789 V 2670 4787 V 2671 4785 V 2672 4783 V 2673 4780 V 2674 4778 V 2675 4776 V 2677 4774 V 2678 4772 V 2627 4783 4 14 v 2628 4778 4 9 v 2629 4776 4 7 v 2631 4774 4 5 v 2631 4772 4 3 v 2679 4835 4 14 v 2678 4835 4 9 v 2677 4835 4 7 v 2675 4835 4 5 v 2675 4835 4 3 v 2681 4794 a Fg(d)2721 4835 y Fn(;)17 b(\033)32 b Fo(2)c Fn(K)548 b Fp(\(3.24\))236 5055 y(where)26 b Fn(I)8 b Fp([)p Fo(\001)p Fp(])25 b(is)f(the)i(indicator)d(function)h (and)h Fn(\033)1934 5070 y Fi(\003)1983 5078 y Fd(n)2030 5055 y Fn(\021)2078 5070 y Fi(\003)2127 5051 y Fd(c)2127 5087 y(n)2202 5055 y Fo(2)j Fn(K)k Fp(coincides)25 b(with)f Fn(\033)29 b Fp(on)c(\003)3310 5070 y Fg(n)3381 5055 y Fp(and)g(equals)236 5176 y Fn(\021)33 b Fp(on)d(the)g(complemen)m(t)e (\003)1224 5140 y Fg(c)1224 5200 y(n)1298 5176 y Fo(\021)p 1419 5178 53 4 v 1419 5112 51 4 v 1419 5176 3 3 v 1421 5174 V 1422 5172 V 1423 5169 V 1424 5167 V 1425 5165 V 1426 5163 V 1427 5160 V 1428 5158 V 1429 5156 V 1430 5154 V 1432 5152 V 1433 5149 V 1434 5147 V 1435 5145 V 1436 5143 V 1437 5141 V 1438 5138 V 1439 5136 V 1440 5134 V 1441 5132 V 1443 5130 V 1444 5127 V 1445 5125 V 1446 5123 V 1447 5121 V 1448 5119 V 1449 5116 V 1450 5114 V 1451 5112 V 1437 5176 V 1438 5174 V 1439 5172 V 1440 5169 V 1441 5167 V 1443 5165 V 1444 5163 V 1445 5160 V 1446 5158 V 1447 5156 V 1448 5154 V 1449 5152 V 1450 5149 V 1451 5147 V 1452 5145 V 1453 5143 V 1455 5141 V 1456 5138 V 1457 5136 V 1458 5134 V 1459 5132 V 1460 5130 V 1461 5127 V 1462 5125 V 1463 5123 V 1464 5121 V 1466 5119 V 1467 5116 V 1468 5114 V 1469 5112 V 1418 5123 4 14 v 1419 5119 4 9 v 1420 5116 4 7 v 1422 5114 4 5 v 1422 5112 4 3 v 1470 5176 4 14 v 1469 5176 4 9 v 1468 5176 4 7 v 1467 5176 4 5 v 1466 5176 4 3 v 1472 5140 a Fg(d)1528 5176 y Fo(n)16 b Fp(\003)1662 5191 y Fg(n)1709 5176 y Fp(.)42 b Fn(P)1841 5191 y Fg(n)1917 5176 y Fp(is)29 b(the)h(stationary)f(pro)s(cess)h(on)g(\012)g(with)f (generator)1266 5413 y Fn(L)1332 5428 y Fg(n)1380 5413 y Fn(f)11 b Fp(\()p Fn(\033)t Fp(\))27 b Fo(\021)1706 5319 y Ff(X)1758 5528 y Fg(x)1867 5413 y Fn(c)1909 5428 y Fg(n)1956 5413 y Fp(\()p Fn(x;)17 b(\033)t Fp(\)[)p Fn(f)11 b Fp(\()p Fn(\033)2373 5372 y Fg(x)2416 5413 y Fp(\))22 b Fo(\000)h Fn(f)11 b Fp(\()p Fn(\033)t Fp(\)])1983 5800 y(13)p eop %%Page: 14 14 14 13 bop 236 548 a Fp(corresp)s(onding)35 b(to)f(a)h(spin\015ip)f(pro) s(cess)i(in)e(\003)1940 563 y Fg(n)2022 548 y Fp(with)g(rates)h Fn(c)p Fp(\()p Fn(x;)17 b(\033)t Fp(\))35 b(and)g(b)s(oundary)g (condition)236 668 y Fn(\021)t Fp(.)58 b(W)-8 b(e)37 b(call)f(the)i(\(unique\))f(stationary)g(measure)h Fn(\032)2203 683 y Fg(n)2286 668 y Fp(:)2349 588 y Ff(R)2432 668 y Fn(d\032)2533 683 y Fg(n)2580 668 y Fn(L)2646 683 y Fg(n)2693 668 y Fn(f)46 b Fp(=)36 b(0.)58 b(W)-8 b(e)37 b(alw)m(a)m(ys)h(assume) 236 789 y(that)31 b(for)g(all)f Fn(\033)h Fo(2)d(f\000)p Fp(1)p Fn(;)17 b Fp(+1)p Fo(g)1304 753 y Fi(\003)1353 761 y Fd(n)1399 789 y Fn(;)g(\032)1493 804 y Fg(n)1540 789 y Fp(\()p Fn(\033)t Fp(\))28 b Fo(\025)g Fn(b)1849 804 y Fi(1)1905 789 y Fp(exp)q([)p Fo(\000)p Fn(b)2199 804 y Fi(2)2240 789 y Fo(j)p Fp(\003)2336 804 y Fg(n)2382 789 y Fo(j)p Fp(].)43 b(W)-8 b(e)32 b(can)g(compute)f(the)h(densit)m(y) g(of)236 909 y Fn(P)299 924 y Fg(n)382 909 y Fp(with)j(resp)s(ect)i(to) e Fn(P)1129 924 y Fg(n)1176 909 y Fp(\002)1252 924 y Fg(n)1334 909 y Fp(via)g(a)h(Girsano)m(v)f(form)m(ula)e(for)j(p)s(oin)m (t)e(pro)s(cesses,)39 b(e.g.)53 b(in)34 b([8,)i(25].)236 1029 y(F)-8 b(or)33 b(giv)m(en)g Fn(!)f Fo(2)d Fp(\012)34 b(w)m(e)g(let)f Fn(N)1333 993 y Fg(x)1323 1054 y(s)1377 1029 y Fp(\()p Fn(!)t Fp(\))p Fn(;)17 b(s)28 b Fo(2)h Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])p Fn(;)j(x)29 b Fo(2)p 2282 1031 53 4 v 2282 965 51 4 v 2282 1029 3 3 v 2283 1027 V 2284 1025 V 2285 1022 V 2286 1020 V 2287 1018 V 2288 1016 V 2289 1014 V 2290 1012 V 2291 1010 V 2293 1008 V 2294 1005 V 2295 1003 V 2296 1001 V 2297 999 V 2298 997 V 2299 994 V 2300 992 V 2301 990 V 2302 988 V 2304 986 V 2305 983 V 2306 981 V 2307 979 V 2308 977 V 2309 975 V 2310 972 V 2311 970 V 2312 968 V 2313 966 V 2299 1029 V 2300 1027 V 2301 1025 V 2302 1022 V 2304 1020 V 2305 1018 V 2306 1016 V 2307 1014 V 2308 1012 V 2309 1010 V 2310 1008 V 2311 1005 V 2312 1003 V 2313 1001 V 2314 999 V 2316 997 V 2317 994 V 2318 992 V 2319 990 V 2320 988 V 2321 986 V 2322 983 V 2323 981 V 2324 979 V 2325 977 V 2327 975 V 2328 972 V 2329 970 V 2330 968 V 2331 966 V 2280 977 4 14 v 2281 972 4 9 v 2283 970 4 7 v 2284 968 4 5 v 2284 966 4 3 v 2333 1029 4 14 v 2331 1029 4 9 v 2330 1029 4 7 v 2329 1029 4 5 v 2328 1029 4 3 v 2334 993 a Fg(d)2408 1029 y Fp(denote)34 b(the)g(n)m(um)m(b)s(er)f(of)g(spin\015ips)g(at)236 1150 y Fn(x)g Fp(up)g(to)f(time)g Fn(s)p Fp(;)g(that)g(is,)h Fn(N)1337 1114 y Fg(x)1327 1174 y(s)1381 1150 y Fp(\()p Fn(!)t Fp(\))27 b Fo(\021)h(jf)p Fn(t)f Fo(2)i Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(s)p Fp(])p Fn(;)g(!)2280 1166 y Fg(t)2305 1147 y Fb(\000)2361 1150 y Fp(\()p Fn(x)p Fp(\))28 b(=)g Fo(\000)p Fn(!)2762 1165 y Fg(t)2792 1150 y Fp(\()p Fn(x)p Fp(\))p Fo(gj)p Fp(;)k(then,)722 1431 y Fn(R)796 1446 y Fg(n)871 1431 y Fo(\021)c Fp(ln)1145 1364 y Fn(dP)1259 1379 y Fg(n)p 1084 1408 284 4 v 1084 1499 a Fn(dP)1198 1514 y Fg(n)1244 1499 y Fp(\002)1320 1514 y Fg(n)1405 1431 y Fp(=)1526 1336 y Ff(X)1508 1548 y Fg(x)p Fh(2)p Fi(\003)1644 1556 y Fd(n)1703 1295 y Ff(Z)1803 1322 y Fg(T)1759 1521 y Fh(\000)p Fg(T)1885 1431 y Fn(dN)2024 1390 y Fg(x)2014 1456 y(s)2068 1431 y Fp(\()p Fn(!)t Fp(\))17 b(ln)2333 1364 y Fn(c)2375 1379 y Fg(n)2422 1364 y Fp(\()p Fn(x;)g(!)2620 1379 y Fg(s)p Fh(\000)2711 1364 y Fp(\))p 2333 1408 417 4 v 2360 1499 a Fn(c)2402 1514 y Fg(n)2449 1499 y Fp(\()p Fn(x;)g(!)2647 1514 y Fg(s)2684 1499 y Fp(\))2781 1431 y(+)22 b(ln)2987 1364 y Fn(\032)3037 1379 y Fg(n)3085 1364 y Fp(\()p Fn(!)3184 1379 y Fh(\000)p Fg(T)3293 1364 y Fp(\))p 2987 1408 344 4 v 3015 1499 a Fn(\032)3065 1514 y Fg(n)3112 1499 y Fp(\()p Fn(!)3211 1514 y Fg(T)3266 1499 y Fp(\))236 1737 y(As)40 b(a)g(consequence,)k(the)c(distribution)d (of)j Fn(R)1968 1752 y Fg(n)2054 1737 y Fp(as)g(induced)g(from)e Fn(P)2851 1752 y Fg(n)2938 1737 y Fp(satis\014es)i(immediately)236 1857 y(the)33 b(global)d(\015uctuation)j(symmetry)f(\(2.13\).)236 1977 y(W)-8 b(e)33 b(can)g(also)f(consider)h(the)g(observ)-5 b(able)1326 2209 y(\026)1310 2234 y Fn(S)1370 2249 y Fg(n)1444 2234 y Fo(\021)1567 2139 y Ff(X)1550 2350 y Fg(x)p Fh(2)p Fi(\003)1686 2332 y Fd(?)1686 2367 y(n)1744 2098 y Ff(Z)1844 2124 y Fg(T)1800 2324 y Fh(\000)p Fg(T)1926 2234 y Fn(dN)2065 2193 y Fg(x)2055 2258 y(s)2109 2234 y Fp(\()p Fn(!)t Fp(\))17 b(ln)2374 2166 y Fn(c)p Fp(\()p Fn(x;)g(!)2614 2181 y Fg(s)p Fh(\000)2705 2166 y Fp(\))p 2374 2211 370 4 v 2401 2302 a Fn(c)p Fp(\()p Fn(x;)g(!)2641 2317 y Fg(s)2678 2302 y Fp(\))236 2548 y(whic)m(h)34 b(is)f(measurable)f(inside)h(\003)1471 2563 y Fg(n)1517 2548 y Fp(.)45 b(This)34 b(is)e(not)h(an)g(arbitrary)g(c)m(hoice)g(but) g(there)h(is)f(to)s(o)f(little)236 2668 y(ph)m(ysics)37 b(here)f(to)e(call)g Fn(R)1167 2683 y Fg(n)1249 2668 y Fp(or)1387 2643 y(\026)1371 2668 y Fn(S)1431 2683 y Fg(n)1513 2668 y Fp(the)h(en)m(trop)m(y)i(pro)s(duction;)e(w)m(e)h (will)d(not)i(elab)s(orate)f(on)h(this.)236 2789 y(The)f(di\013erence)f Fn(B)947 2804 y Fg(n)1022 2789 y Fo(\021)1144 2763 y Fp(\026)1127 2789 y Fn(S)1187 2804 y Fg(n)1256 2789 y Fo(\000)23 b Fn(R)1430 2804 y Fg(n)1509 2789 y Fp(is)32 b(mainly)f(a)h(sum)h(o)m(v)m(er)g Fn(x)28 b Fo(2)h Fp(\003)2672 2804 y Fg(n)2740 2789 y Fo(n)22 b Fp(\003)2880 2752 y Fg(?)2880 2813 y(n)2927 2789 y Fp(.)236 2909 y(The)28 b(other)e(pro)s(cess)i Fn(P)1090 2873 y Fg(r)1076 2934 y(n)1154 2909 y Fp(w)m(e)g(need)f(to)f(lo)s(ok)g(at)g(is)g(v)m(ery)i (similar)23 b(but)k(it)e(is)h(in)g(general)g(not)h(Mark)m(o-)236 3029 y(vian.)43 b(T)-8 b(o)30 b(de\014ne)i(it,)e(w)m(e)h(tak)m(e)h(a)e (stationary)g(pro)s(cess)i Fn(P)44 b Fp(on)30 b(\012)h(and)g(w)m(e)g (tak)m(e)g(its)f(restriction)g(to)236 3150 y(\003)304 3165 y Fg(n)351 3150 y Fp(.)42 b(W)-8 b(e)29 b(write)g Fn(\032)g Fp(for)f(the)i(corresp)s(onding)e(stationary)h(measure)g(and) g(w)m(e)h(let)37 b(\026)-58 b Fn(\032)3191 3165 y Fg(n)3267 3150 y Fp(b)s(e)29 b(its)f(restric-)236 3270 y(tion)33 b(to)g(\003)626 3285 y Fg(n)673 3270 y Fp(.)46 b(W)-8 b(e)34 b(alw)m(a)m(ys)g(assume)g(that)g(for)f(all)e Fn(\033)i Fo(2)d(f\000)p Fp(1)p Fn(;)17 b Fp(+1)p Fo(g)2644 3234 y Fi(\003)2693 3242 y Fd(n)2739 3270 y Fn(;)26 b Fp(\026)-58 b Fn(\032)2833 3285 y Fg(n)2880 3270 y Fp(\()p Fn(\033)t Fp(\))29 b Fo(\025)h Fn(a)3202 3285 y Fi(1)3258 3270 y Fp(exp)q([)p Fo(\000)p Fn(a)3562 3285 y Fi(2)3603 3270 y Fo(j)p Fp(\003)3699 3285 y Fg(n)3745 3270 y Fo(j)p Fp(].)236 3390 y(Being)38 b(more)f(explicit,)h(w)m(e)h(let)f Fn(P)51 b Fp(b)s(e)38 b(an)g(in\014nite)f(v)m(olume)h(stationary)f(pro) s(cess)i(with)f(formal)236 3511 y(generator)1313 3736 y Fn(Lf)11 b Fp(\()p Fn(\033)t Fp(\))28 b Fo(\021)1706 3642 y Ff(X)1758 3851 y Fg(x)1867 3736 y Fn(c)p Fp(\()p Fn(x;)17 b(\033)t Fp(\)[)p Fn(f)11 b Fp(\()p Fn(\033)2326 3695 y Fg(x)2369 3736 y Fp(\))22 b Fo(\000)h Fn(f)11 b Fp(\()p Fn(\033)t Fp(\)])236 4032 y(and)33 b(put)g Fn(P)682 3995 y Fg(r)668 4056 y(n)752 4032 y Fp(the)g(unique)g (path-space)g(measure)g(suc)m(h)h(that)355 4229 y(1.)49 b(The)34 b(distribution)c(of)i Fo(f)p Fn(!)1434 4244 y Fg(t)1464 4229 y Fp(\()p Fn(x)p Fp(\))c(:)f Fn(x)h Fo(2)h Fp(\003)1923 4244 y Fg(n)1969 4229 y Fn(;)17 b(t)28 b Fo(2)g Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])p Fo(g)33 b Fp(under)g Fn(P)2917 4193 y Fg(r)2903 4254 y(n)2987 4229 y Fp(and)g Fn(P)46 b Fp(coincide.)355 4431 y(2.)j(Under)33 b Fn(P)852 4395 y Fg(r)838 4456 y(n)890 4431 y Fp(,)f Fn(!)1010 4446 y Fg(t)1040 4431 y Fp(\()p Fn(x)p Fp(\))c(=)f Fn(\021)t Fp(\()p Fn(x)p Fp(\))33 b(for)f(all)e Fn(x)e Fo(62)h Fp(\003)2048 4446 y Fg(n)2094 4431 y Fp(,)k Fn(t)28 b Fo(\025)g Fp(0.)236 4629 y Fc(Theorem)37 b(3.1)c Fp([LFT)g(for)f(spin\015ip)g(pro)s(cesses])i(F)-8 b(or)32 b(all)f Fn(z)h Fo(2)c Fp(I)-16 b(R,)1413 4916 y(lim)1424 4978 y Fg(n;T)1636 4848 y Fp(1)p 1575 4893 170 4 v 1575 4984 a Fn(n)1633 4955 y Fg(d)1674 4984 y Fn(T)1771 4916 y Fp(ln)1952 4766 y Ff(R)2035 4846 y Fn(dP)14 b(e)2208 4810 y Fh(\000)p Fg(z)2309 4793 y Fi(\026)2299 4810 y Fg(S)2342 4818 y Fd(n)p 1879 4893 581 4 v 1879 4908 a Ff(R)1962 4988 y Fn(dP)g(e)2135 4960 y Fh(\000)p Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))2382 4943 y(\026)2370 4960 y Fg(S)2413 4968 y Fd(n)2497 4916 y Fp(=)28 b(0)236 5190 y Fc(Pro)s(of)37 b(of)h(Theorem)f(3.1:)236 5310 y Fp(Ev)m(en)k(though)f Fn(P)903 5274 y Fg(r)889 5335 y(n)980 5310 y Fp(is)f(not)g(Mark)m(o)m(vian)h(\(in)f(general\),)i(it)d (remains)h(a)g(jump)g(pro)s(cess)i(and)e(the)236 5430 y(jump-in)m(tensities)30 b(can)h(b)s(e)g(computed)g(from)f(the)h (original)c(spin\015ip)k(rates.)43 b(In)31 b(order)g(to)g(ha)m(v)m(e)h (a)236 5551 y(Gibbsian)d(structure)j(these)h(in)m(tensities)d(m)m(ust)h (b)s(e)g(the)g(same)f(in)g(the)h(bulk)g(of)f(\003)3221 5566 y Fg(n)3299 5551 y Fp(as)h(they)g(w)m(ere)1983 5800 y(14)p eop %%Page: 15 15 15 14 bop 236 548 a Fp(for)38 b(the)h(in\014nite)e(v)m(olume)h(pro)s (cess)h Fn(P)14 b Fp(.)60 b(As)39 b(the)g(rates)f(are)h(lo)s(cal,)e (the)i(pro)s(cess)h Fn(P)3350 512 y Fg(r)3336 573 y(n)3425 548 y Fp(restricted)236 668 y(to)h(\003)432 683 y Fg(n)520 668 y Fp(indeed)h(has)g(the)g(same)f(in)m(tensities)g(as)h(the)g(pro)s (cess)g Fn(P)2636 683 y Fg(n)2725 668 y Fp(except)h(at)e(the)h(sites)f (of)g(the)236 789 y(b)s(oundary)33 b(\003)743 804 y Fg(n)812 789 y Fo(n)22 b Fp(\003)952 753 y Fg(?)952 813 y(n)999 789 y Fp(.)43 b(This)33 b(is)f(a)g(consequence)k(of)c(the)h(follo)m (wing)d(generally)h(stated)236 909 y Fc(Lemma)37 b(3.2:)236 1029 y Fp(Supp)s(ose)27 b Fn(N)690 1044 y Fg(t)745 1029 y Fp(is)e(a)h(p)s(oin)m(t)f(pro)s(cess)h(with)g(in)m(tensit)m(y)g Fn(c)2141 1044 y Fg(s)2177 1029 y Fp(,)i(i.e.,)e Fn(M)2504 1044 y Fg(t)2562 1029 y Fp(=)h Fn(N)2743 1044 y Fg(t)2781 1029 y Fo(\000)2866 949 y Ff(R)2933 976 y Fg(t)2914 1064 y Fi(0)2979 1029 y Fn(c)3021 1044 y Fg(s)3084 1029 y Fn(ds)e Fp(is)g(a)h(martingale)236 1150 y(for)32 b(the)h(\014ltration)e Fo(F)1032 1165 y Fg(t)1061 1150 y Fp(.)43 b(Supp)s(ose)34 b(that)e Fo(F)1807 1114 y Fh(0)1797 1174 y Fg(t)1858 1150 y Fo(\032)c(F)2035 1165 y Fg(t)2097 1150 y Fp(is)k(a)g (sub\014ltration)f(of)i Fo(F)3013 1165 y Fg(t)3042 1150 y Fp(,)f(and)h(de\014ne)1720 1362 y Fn(N)1808 1321 y Fh(0)1798 1387 y Fg(t)1860 1362 y Fp(=)27 b(I)-16 b(E[)p Fn(N)2153 1377 y Fg(t)2183 1362 y Fo(jF)2293 1321 y Fh(0)2283 1387 y Fg(t)2316 1362 y Fp(])1234 b(\(3.25\))236 1575 y(Then)34 b Fn(N)579 1539 y Fh(0)569 1599 y Fg(t)635 1575 y Fp(is)e(a)h(p)s(oin)m(t)e(pro)s(cess)j(with)e(in)m(tensit)m(y) 1750 1787 y Fn(c)1792 1746 y Fh(0)1792 1812 y Fg(s)1856 1787 y Fp(=)27 b(I)-16 b(E[)p Fn(c)2113 1802 y Fg(s)2150 1787 y Fo(jF)2260 1746 y Fh(0)2250 1812 y Fg(s)2286 1787 y Fp(])1264 b(\(3.26\))236 2000 y Fc(Pro)s(of)37 b(of)h(Lemma)f(3.2:) 236 2120 y Fp(It)28 b(is)e(easy)j(to)e(see)h(that)f Fn(M)1215 2084 y Fh(0)1205 2145 y Fg(t)1267 2120 y Fp(=)g Fn(N)1458 2084 y Fh(0)1448 2145 y Fg(t)1493 2120 y Fo(\000)11 b Fp(I)-15 b(E\()1705 2040 y Ff(R)1772 2066 y Fg(t)1752 2155 y Fi(0)1818 2120 y Fn(c)1860 2135 y Fg(s)1897 2120 y Fn(ds)p Fo(jF)2104 2084 y Fh(0)2094 2145 y Fg(t)2126 2120 y Fp(\))27 b(=)h(I)-17 b(E)q([)p Fn(M)2501 2135 y Fg(t)2531 2120 y Fo(jF)2641 2084 y Fh(0)2631 2145 y Fg(t)2663 2120 y Fp(])28 b(is)e(a)h Fo(F)2968 2084 y Fh(0)2958 2145 y Fg(t)3018 2120 y Fp(martingale.)39 b(Hence,)236 2240 y(it)32 b(su\016ces)i(to)f(sho)m(w)g(that)1258 2491 y Fn(B)1332 2506 y Fg(t)1390 2491 y Fp(=)27 b(I)-16 b(E[)1605 2355 y Ff(Z)1705 2382 y Fg(t)1661 2581 y Fi(0)1751 2491 y Fn(c)1793 2506 y Fg(s)1830 2491 y Fn(ds)p Fo(jF)2037 2450 y Fh(0)2027 2516 y Fg(t)2059 2491 y Fp(])22 b Fo(\000)2208 2355 y Ff(Z)2308 2382 y Fg(t)2263 2581 y Fi(0)2354 2491 y Fp(I)-16 b(E[)p Fn(c)2508 2506 y Fg(s)2545 2491 y Fo(jF)2655 2450 y Fh(0)2645 2516 y Fg(s)2681 2491 y Fp(])p Fn(ds)772 b Fp(\(3.27\))236 2754 y(is)32 b(a)h Fo(F)498 2718 y Fh(0)488 2779 y Fg(t)520 2754 y Fp(-martingale.)41 b(This)32 b(is)g(a)h(consequence)i(of)d(the)h(follo)m(wing)d(equalities:)1156 2967 y(I)-16 b(E[)p Fn(B)1342 2982 y Fg(t)1372 2967 y Fo(jF)1482 2926 y Fh(0)1472 2991 y Fg(s)1508 2967 y Fp(])1156 3175 y(=)28 b Fn(B)1334 3190 y Fg(s)1393 3175 y Fp(+)22 b(I)-16 b(E)1593 3034 y Ff(\024)1645 3039 y(Z)1745 3066 y Fg(t)1700 3265 y(s)1774 3175 y Fp(\()p Fn(c)1854 3190 y Fg(r)1914 3175 y Fo(\000)23 b Fp(I)-16 b(E[)p Fn(c)2168 3190 y Fg(r)2206 3175 y Fo(jF)2316 3134 y Fh(0)2306 3200 y Fg(r)2343 3175 y Fp(]\))p Fn(dr)s Fo(jF)2616 3134 y Fh(0)2606 3200 y Fg(s)2642 3034 y Ff(\025)1156 3453 y Fp(=)28 b Fn(B)1334 3468 y Fg(s)1393 3453 y Fp(+)22 b(I)-16 b(E[)1603 3317 y Ff(Z)1703 3344 y Fg(t)1658 3543 y(s)1749 3453 y Fn(c)1791 3468 y Fg(r)1829 3453 y Fn(dr)s Fo(jF)2037 3412 y Fh(0)2027 3478 y Fg(s)2062 3453 y Fp(])22 b Fo(\000)h Fp(I)-16 b(E)2313 3312 y Ff(\024)2365 3317 y(Z)2465 3344 y Fg(t)2421 3543 y(s)2511 3453 y Fp(I)g(E[)p Fn(c)2665 3468 y Fg(r)2703 3453 y Fo(jF)2813 3412 y Fh(0)2803 3478 y Fg(r)2840 3453 y Fp(])p Fo(jF)2977 3412 y Fh(0)2967 3478 y Fg(s)3003 3312 y Ff(\025)1156 3656 y Fp(=)28 b Fn(B)1334 3671 y Fg(s)3577 3656 y Fp(\(3.28\))p 3780 3868 48 48 v 236 3988 a(Therefore,)34 b(the)f(rates)g(of)f(the)h (restricted)g(pro)s(cess)h(on)e(\003)2368 4003 y Fg(n)2448 3988 y Fp(are)g(giv)m(en)h(b)m(y)833 4201 y(~)-51 b Fn(c)873 4216 y Fg(s)910 4201 y Fp(\()p Fn(x;)17 b(!)t Fp(\))27 b(=)g(I)-16 b(E[)p Fn(c)p Fp(\()p Fn(x;)17 b(\033)1626 4216 y Fg(s)1663 4201 y Fp(\))p Fo(j)p Fn(\033)1784 4216 y Fg(\034)1828 4201 y Fp(\()p Fn(y)t Fp(\))26 b(=)i Fn(!)2147 4216 y Fg(\034)2190 4201 y Fp(\()p Fn(y)t Fp(\))p Fn(;)17 b Fo(\000)p Fn(T)40 b Fo(\024)29 b Fn(\034)39 b Fo(\024)28 b Fn(s;)17 b(y)30 b Fo(2)e Fp(\003)3158 4216 y Fg(n)3205 4201 y Fp(])236 4413 y(where)34 b(the)f(exp)s(ectation)g(is)f(with)g (resp)s(ect)i(to)e Fn(P)14 b Fp(.)236 4534 y(Or,)34 b(for)g(all)d Fn(x)g Fo(2)f Fp(\003)948 4549 y Fg(n)995 4534 y Fp(,)k Fn(N)1144 4497 y Fg(x)1134 4558 y(t)1188 4534 y Fp(\()p Fn(!)t Fp(\))22 b Fo(\000)1452 4453 y Ff(R)1518 4480 y Fg(t)1499 4568 y Fi(0)1566 4534 y Fp(~)-51 b Fn(c)1606 4549 y Fg(s)1643 4534 y Fp(\()p Fn(x;)17 b(!)t Fp(\))33 b(is)h(a)f(martingale)e(under)k Fn(P)2942 4497 y Fg(r)2928 4558 y(n)2979 4534 y Fp(.)48 b(As)34 b(a)g(consequence,)238 4654 y(~)-51 b Fn(c)278 4669 y Fg(s)315 4654 y Fp(\()p Fn(x;)17 b(!)t Fp(\))27 b(=)h Fn(c)p Fp(\()p Fn(x;)17 b(\033)t Fp(\))32 b(when)i Fn(x)28 b Fo(2)g Fp(\003)1494 4618 y Fg(?)1494 4679 y(n)1573 4654 y Fp(and)33 b Fn(!)1824 4669 y Fg(s)1888 4654 y Fp(=)28 b Fn(\033)t Fp(.)236 4774 y(Just)d(as)f(for)f(the)i(pair)d Fn(P)1106 4789 y Fg(n)1153 4774 y Fn(;)17 b(P)1260 4789 y Fg(n)1307 4774 y Fp(\002)1383 4789 y Fg(n)1430 4774 y Fp(,)25 b(the)g(absolutely) e(con)m(tin)m(uit)m(y)h(of)f Fn(P)2728 4738 y Fg(r)2714 4799 y(n)2766 4774 y Fn(\022)2811 4789 y Fg(n)2882 4774 y Fp(with)h(resp)s(ect)h(to)e Fn(P)3608 4738 y Fg(r)3594 4799 y(n)3670 4774 y Fp(and)236 4895 y(vice)31 b(v)m(ersa)h(is)e (guaran)m(teed)h(b)m(y)h(the)f(p)s(ositivit)m(y)e(of)h(the)h (spin\015ip)f(rates)h(inside)f(\003)3202 4910 y Fg(n)3248 4895 y Fp(.)43 b(W)-8 b(e)31 b(are)g(th)m(us)236 5015 y(ready)38 b(to)e(apply)h(Prop)s(osition)e(2.1.)56 b(W)-8 b(e)37 b(m)m(ust)g(\014rst)g(v)m(erify)g(the)h(corresp)s(onding)e (assumption)236 5136 y(\(2.15\).)43 b(W)-8 b(e)33 b(\014nd)549 5403 y Fn(B)623 5418 y Fg(n)698 5403 y Fp(=)818 5378 y(\026)801 5403 y Fn(S)861 5418 y Fg(n)930 5403 y Fo(\000)23 b Fn(R)1104 5418 y Fg(n)1179 5403 y Fp(=)k Fo(\000)1457 5308 y Ff(X)1376 5524 y Fg(x)p Fh(2)p Fi(\003)1512 5532 y Fd(n)1555 5524 y Fh(n)p Fi(\003)1639 5505 y Fd(?)1639 5541 y(n)1698 5267 y Ff(Z)1814 5403 y Fn(dN)1953 5362 y Fg(y)1943 5428 y(s)1995 5403 y Fp(\()p Fn(!)t Fp(\))17 b(ln)2259 5336 y Fn(c)2301 5351 y Fg(n)2348 5336 y Fp(\()p Fn(x;)g(!)2546 5352 y Fg(s)2579 5333 y Fb(\000)2635 5336 y Fp(\))p 2259 5380 414 4 v 2286 5471 a Fn(c)2328 5486 y Fg(n)2374 5471 y Fp(\()p Fn(x;)g(!)2572 5486 y Fg(s)2609 5471 y Fp(\))2705 5403 y(+)22 b(ln)2938 5336 y Fn(\032)2988 5351 y Fg(n)3036 5336 y Fp(\()p Fn(!)3135 5351 y Fg(T)3189 5336 y Fp(\))p 2911 5380 344 4 v 2911 5471 a Fn(\032)2961 5486 y Fg(n)3008 5471 y Fp(\()p Fn(!)3107 5486 y Fh(\000)p Fg(T)3217 5471 y Fp(\))3577 5403 y(\(3.29\))1983 5800 y(15)p eop %%Page: 16 16 16 15 bop 236 548 a Fp(and)277 806 y(ln)385 739 y Fn(dP)513 703 y Fg(r)499 763 y(n)p 385 783 166 4 v 387 875 a Fn(dP)501 890 y Fg(n)588 806 y Fp(=)27 b(ln)832 739 y(\026)-58 b Fn(\032)p Fp(\()p Fn(!)972 754 y Fh(\000)p Fg(T)1082 739 y Fp(\))p 799 783 344 4 v 799 875 a Fn(\032)849 890 y Fg(n)896 875 y Fp(\()p Fn(!)995 890 y Fh(\000)p Fg(T)1105 875 y Fp(\))1175 806 y(+)1354 712 y Ff(X)1273 927 y Fg(x)p Fh(2)p Fi(\003)1409 935 y Fd(n)1451 927 y Fh(n)p Fi(\003)1535 909 y Fd(?)1535 944 y(n)1595 671 y Ff(Z)1711 806 y Fn(dN)1850 765 y Fg(x)1840 831 y(s)1894 806 y Fp(\()p Fn(!)t Fp(\))17 b(ln)2182 739 y(~)-51 b Fn(c)2222 754 y Fg(s)2259 739 y Fp(\()p Fn(x;)17 b(!)t Fp(\))p 2159 783 362 4 v 2159 875 a Fn(c)2201 890 y Fg(n)2248 875 y Fp(\()p Fn(x;)g(!)2446 890 y Fg(s)2482 875 y Fp(\))2552 806 y Fo(\000)2652 671 y Ff(Z)2751 697 y Fg(T)2707 896 y Fh(\000)p Fg(T)2834 806 y Fn(ds)p Fp([)r(~)-51 b Fn(c)3000 821 y Fg(s)3036 806 y Fp(\()p Fn(x;)17 b(!)t Fp(\))22 b Fo(\000)g Fn(c)3439 821 y Fg(n)3486 806 y Fp(\()p Fn(x;)17 b(!)3684 821 y Fg(s)3721 806 y Fp(\)])3577 1047 y(\(3.30\))236 1261 y(Clearly)-8 b(,)29 b(b)s(oth)g Fo(j)p Fn(B)920 1276 y Fg(n)967 1261 y Fo(j)f Fp(and)i Fo(j)17 b Fp(ln)e Fn(dP)1480 1225 y Fg(r)1466 1286 y(n)1517 1261 y Fn(=dP)1680 1276 y Fg(n)1726 1261 y Fo(j)29 b Fp(are)g(b)s(ounded)h(b)m(y)g Fn(c)2511 1276 y Fi(1)2550 1261 y Fn(N)10 b Fp(\([)p Fo(\000)p Fn(T)e(;)17 b(T)d Fp(])p Fn(;)j Fp(\003)3099 1276 y Fg(n)3161 1261 y Fo(n)e Fp(\003)3294 1225 y Fg(?)3294 1286 y(n)3341 1261 y Fp(\))g(+)g Fn(c)3527 1276 y Fi(2)3566 1261 y Fo(j)p Fp(\003)3662 1276 y Fg(n)3708 1261 y Fo(j)g Fp(+)236 1381 y Fn(c)278 1396 y Fi(3)318 1381 y Fn(T)f Fo(j)p Fp(\003)485 1396 y Fg(n)562 1381 y Fo(n)31 b Fp(\003)711 1345 y Fg(?)711 1406 y(n)757 1381 y Fo(j)46 b Fp(for)f(some)g(constan)m (ts)i Fn(c)1737 1396 y Fi(1)1777 1381 y Fn(;)17 b(c)1863 1396 y Fi(2)1902 1381 y Fn(;)g(c)1988 1396 y Fi(3)2077 1381 y Fn(<)50 b Fo(1)p Fp(,)f(where)e Fn(N)10 b Fp(\([)p Fo(\000)p Fn(T)e(;)17 b(T)d Fp(])p Fn(;)j Fp(\003)3223 1396 y Fg(n)3301 1381 y Fo(n)31 b Fp(\003)3450 1345 y Fg(?)3450 1406 y(n)3496 1381 y Fp(\))46 b(is)f(the)236 1502 y(n)m(um)m(b)s(er)29 b(of)g(spin\015ips)f(that)h(ha)m(v)m(e)h(o)s (ccurred)f(in)f(the)h(space-time)f(windo)m(w)i([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])g Fo(\002)g Fp(\(\003)3521 1517 y Fg(n)3583 1502 y Fo(n)g Fp(\003)3715 1466 y Fg(?)3715 1526 y(n)3762 1502 y Fp(\).)236 1622 y(It)33 b(remains)f(th)m(us)h(to)f (sho)m(w)i(for)e(all)f Fn(\025)1217 1861 y Fp(lim)1210 1924 y Fg(n)p Fh("1)1410 1794 y Fp(1)p 1385 1838 99 4 v 1385 1930 a Fn(n)1443 1901 y Fg(d)1511 1861 y Fp(ln)1608 1726 y Ff(Z)1725 1861 y Fn(dP)1853 1820 y Fg(r)1839 1886 y(n)1890 1861 y Fn(e)1935 1820 y Fg(\025N)7 b Fi(\([)p Fh(\000)p Fg(T)f(;T)k Fi(])p Fg(;)p Fi(\003)2348 1828 y Fd(n)2389 1820 y Fh(n)p Fi(\003)2473 1797 y Fb(\003)2473 1837 y Fd(n)2516 1820 y Fi(\))1210 2117 y Fp(=)34 b(lim)1314 2180 y Fg(n)p Fh("1)1514 2049 y Fp(1)p 1489 2094 V 1489 2185 a Fn(n)1547 2156 y Fg(d)1614 2117 y Fp(ln)1712 1981 y Ff(Z)1828 2117 y Fn(dP)1956 2076 y Fi(0)2005 2049 y Fn(dP)2133 2013 y Fg(r)2119 2074 y(n)p 2005 2094 167 4 v 2005 2185 a Fn(dP)2133 2156 y Fi(0)2119 2210 y Fg(n)2181 2117 y Fn(e)2226 2076 y Fg(\025N)7 b Fi(\([)p Fh(\000)p Fg(T)f(;T)k Fi(])p Fg(;)p Fi(\003)2639 2084 y Fd(n)2681 2076 y Fh(n)p Fi(\003)2765 2052 y Fb(\003)2765 2092 y Fd(n)2807 2076 y Fi(\))2867 2117 y Fp(=)27 b(0)236 2392 y(where)j(w)m(e)g(inserted)f(the)h(reference)g(pro)s(cess)g Fn(P)2002 2356 y Fi(0)2070 2392 y Fp(\(and)f(its)f(restriction)g Fn(P)2966 2356 y Fi(0)2952 2417 y Fg(n)3033 2392 y Fp(to)h(\003)3217 2407 y Fg(n)3264 2392 y Fp(\))f(corresp)s(ond-)236 2513 y(ing)k(to)g(the)h(pro)s(duct)g(pro)s(cess)h(of)e(indep)s(enden)m(t)i (spin\015ips)e(\(rate)g(1\).)44 b(In)32 b(particular,)578 2783 y(lim)572 2846 y Fg(n)p Fh("1)772 2716 y Fp(1)p 747 2760 99 4 v 747 2852 a Fn(n)805 2823 y Fg(d)872 2783 y Fp(ln)970 2648 y Ff(Z)1086 2783 y Fn(dP)1214 2742 y Fi(0)1253 2783 y Fn(e)1298 2742 y Fg(\025N)7 b Fi(\([)p Fh(\000)p Fg(T)f(;T)k Fi(])p Fg(;)p Fi(\003)1711 2750 y Fd(n)1752 2742 y Fh(n)p Fi(\003)1836 2719 y Fb(\003)1836 2759 y Fd(n)1879 2742 y Fi(\))1938 2783 y Fp(=)34 b(lim)2042 2846 y Fg(n)p Fh("1)2217 2716 y Fo(j)p Fp(\003)2313 2731 y Fg(n)2381 2716 y Fo(n)22 b Fp(\003)2521 2680 y Fh(\003)2521 2741 y Fg(n)2568 2716 y Fo(j)p 2217 2760 379 4 v 2357 2852 a Fn(n)2415 2823 y Fg(d)2606 2783 y Fp(2)p Fn(T)14 b Fp(\()p Fn(e)2809 2742 y Fg(\025)2876 2783 y Fo(\000)22 b Fp(1\))28 b(=)f(0)236 3042 y(and)39 b(w)m(e)h(can)f(apply)f(the)h (same)g(argumen)m(t)f(as)h(in)f(Prop)s(osition)f(2.1.)61 b(Finally)-8 b(,)38 b(the)h(condition)236 3162 y(\(2.16\))32 b(of)g(Prop)s(osition)f(2.1)h(is)g(a)h(consequence)i(of)d(the)h(large)f (deviation)f(results)i(of)f([23].)p 3780 3162 48 48 v 236 3615 a Fv(4)161 b(LFT)55 b(for)f(the)e(asymmetric)h(exclusion)h (pro)t(cess)236 3859 y Fp(The)23 b(con\014guration)e(space)i(is)e(no)m (w)h Fn(K)35 b Fp(=)28 b Fo(f)p Fp(0)p Fn(;)17 b Fp(1)p Fo(g)p 2018 3824 53 4 v 2018 3758 51 4 v 2018 3823 3 3 v 2019 3820 V 2020 3818 V 2022 3816 V 2023 3814 V 2024 3812 V 2025 3810 V 2026 3807 V 2027 3805 V 2028 3803 V 2029 3801 V 2030 3799 V 2031 3796 V 2033 3794 V 2034 3792 V 2035 3790 V 2036 3788 V 2037 3785 V 2038 3783 V 2039 3781 V 2040 3779 V 2041 3777 V 2042 3774 V 2043 3772 V 2045 3770 V 2046 3768 V 2047 3766 V 2048 3763 V 2049 3761 V 2050 3759 V 2036 3823 V 2037 3820 V 2038 3818 V 2039 3816 V 2040 3814 V 2041 3812 V 2042 3810 V 2043 3807 V 2045 3805 V 2046 3803 V 2047 3801 V 2048 3799 V 2049 3796 V 2050 3794 V 2051 3792 V 2052 3790 V 2053 3788 V 2054 3785 V 2056 3783 V 2057 3781 V 2058 3779 V 2059 3777 V 2060 3774 V 2061 3772 V 2062 3770 V 2063 3768 V 2064 3766 V 2065 3763 V 2067 3761 V 2068 3759 V 2017 3770 4 14 v 2018 3766 4 9 v 2019 3763 4 7 v 2021 3761 4 5 v 2021 3759 4 3 v 2069 3823 4 14 v 2068 3823 4 9 v 2067 3823 4 7 v 2065 3823 4 5 v 2065 3823 4 3 v 2071 3782 a Fe(2)2131 3859 y Fp(\(o)s(ccupation)k(v)-5 b(ariables)21 b(on)g(the)i(2-dimensional)236 3979 y(regular)34 b(lattice\))f(and)i(the)h(pathspace)g(\012)c(=)f Fn(D)s Fp(\()p Fn(K)r(;)17 b Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(]\))35 b(is)f(essen)m(tially)h(unc)m(hanged)h(from)236 4100 y(that)g(in)f(the)i(previous)f(section.)55 b(F)-8 b(or)35 b Fn(\021)i Fo(2)d Fn(K)7 b Fp(,)37 b Fn(\021)t Fp(\()p Fn(x)p Fp(\))d(=)f(1)p Fn(;)17 b Fp(0)36 b(indicates)f(the)i (presence,)i(resp)s(ec-)236 4220 y(tiv)m(ely)25 b(absence)h(of)e(a)h (particle)e(at)i(the)g(site)g Fn(x)j Fo(2)p 2005 4222 53 4 v 2005 4156 51 4 v 2005 4220 3 3 v 2006 4218 V 2007 4216 V 2008 4213 V 2009 4211 V 2010 4209 V 2011 4207 V 2012 4205 V 2013 4202 V 2015 4200 V 2016 4198 V 2017 4196 V 2018 4194 V 2019 4191 V 2020 4189 V 2021 4187 V 2022 4185 V 2023 4183 V 2024 4180 V 2026 4178 V 2027 4176 V 2028 4174 V 2029 4172 V 2030 4170 V 2031 4167 V 2032 4165 V 2033 4163 V 2034 4161 V 2035 4159 V 2037 4156 V 2022 4220 V 2023 4218 V 2024 4216 V 2026 4213 V 2027 4211 V 2028 4209 V 2029 4207 V 2030 4205 V 2031 4202 V 2032 4200 V 2033 4198 V 2034 4196 V 2035 4194 V 2037 4191 V 2038 4189 V 2039 4187 V 2040 4185 V 2041 4183 V 2042 4180 V 2043 4178 V 2044 4176 V 2045 4174 V 2046 4172 V 2047 4170 V 2049 4167 V 2050 4165 V 2051 4163 V 2052 4161 V 2053 4159 V 2054 4156 V 2003 4167 4 14 v 2004 4163 4 9 v 2006 4161 4 7 v 2007 4159 4 5 v 2007 4156 4 3 v 2056 4220 4 14 v 2054 4220 4 9 v 2053 4220 4 7 v 2052 4220 4 5 v 2051 4220 4 3 v 2057 4184 a Fg(d)2098 4220 y Fp(.)41 b(This)25 b(hopping)f(dynamics)g(will)f(b)s (e)i(mo)s(deled)236 4340 y(b)m(y)35 b(an)f(asymmetric)f(exclusion)h (pro)s(cess.)48 b(This)35 b(is)e(a)h(bulk)f(driv)m(en)i(di\013usiv)m(e) f(lattice)e(gas.)48 b(The)236 4461 y(hopping)40 b(rates)g(for)f(v)m (ertical)g(\()p Fn(v)t Fp(\))h(and)g(horizon)m(tal)e(b)s(onds)j(\()p Fn(h)p Fp(\))e(dep)s(end)j(on)d(the)i(direction)d(in)236 4581 y(the)33 b(follo)m(wing)d(w)m(a)m(y:)1031 4836 y Fn(c)1073 4795 y Fg(v)1114 4836 y Fp(\()p Fn(x;)17 b(\021)t Fp(\))27 b Fo(\021)1483 4768 y Fp(1)p 1483 4813 49 4 v 1483 4904 a(2)1542 4836 y([)p Fn(\021)t Fp(\()p Fn(x)p Fp(\)\(1)22 b Fo(\000)h Fn(\021)t Fp(\()p Fn(x)f Fp(+)g Fn(e)2271 4851 y Fi(2)2310 4836 y Fp(\)\))g(+)g Fn(\021)t Fp(\()p Fn(x)h Fp(+)f Fn(e)2817 4851 y Fi(2)2856 4836 y Fp(\)\(1)g Fo(\000)h Fn(\021)t Fp(\()p Fn(x)p Fp(\)\)])546 5069 y Fn(c)588 5028 y Fg(h;E)708 5069 y Fp(\()p Fn(x;)17 b(\021)t Fp(\))27 b Fo(\021)1078 5001 y Fp(1)p 1078 5046 V 1078 5137 a(2)1136 5069 y([)p Fn(e)1208 5028 y Fg(E)t(=)p Fi(2)1339 5069 y Fn(\021)t Fp(\()p Fn(x)p Fp(\)\(1)22 b Fo(\000)g Fn(\021)t Fp(\()p Fn(x)g Fp(+)g Fn(e)2040 5084 y Fi(1)2080 5069 y Fp(\)\))g(+)g Fn(e)2321 5028 y Fh(\000)p Fg(E)t(=)p Fi(2)2506 5069 y Fn(\021)t Fp(\()p Fn(x)h Fp(+)f Fn(e)2817 5084 y Fi(1)2856 5069 y Fp(\)\(1)g Fo(\000)h Fn(\021)t Fp(\()p Fn(x)p Fp(\)\)])236 5310 y(where)34 b Fn(e)563 5325 y Fi(1)603 5310 y Fn(;)17 b(e)692 5325 y Fi(2)764 5310 y Fp(are)32 b(the)h(unit)f(v)m(ectors)i (in)e(the)h(p)s(ositiv)m(e)f(horizon)m(tal)f(and)i(v)m(ertical)e (direction.)236 5430 y(In)45 b(addition,)h(for)f(the)g(momen)m(t,)i(w)m (e)f(allo)m(w)d(for)h(the)h(p)s(ossibilit)m(y)e(of)h(particle)f (creation)i(and)236 5551 y(destruction.)d(W)-8 b(e)25 b(put)h(the)g(birth/death)f(rate)g Fn(c)p Fp(\()p Fn(x;)17 b(\021)t Fp(\))27 b Fo(\021)i Fn(\017)c Fp(indep)s(enden)m(t)i(of)e (the)h(con\014guration)1983 5800 y(16)p eop %%Page: 17 17 17 16 bop 236 548 a Fn(\021)36 b Fp(and)d(the)g(site)g Fn(x)p Fp(.)236 668 y(The)38 b(formal)c(Mark)m(o)m(v)j(generator)g Fn(L)f Fp(to)h(the)f(in\014nite)g(v)m(olume)g(pro)s(cess)h(is)f(then)h (found)g(as)f(the)236 789 y(sum)806 977 y Fn(Lf)11 b Fp(\()p Fn(\021)t Fp(\))27 b Fo(\021)h Fn(\017)1247 883 y Ff(X)1299 1092 y Fg(x)1391 977 y Fp([)p Fn(f)11 b Fp(\()p Fn(\021)1567 936 y Fg(x)1610 977 y Fp(\))22 b Fo(\000)h Fn(f)11 b Fp(\()p Fn(\021)t Fp(\)])22 b(+)2104 883 y Ff(X)2110 1099 y Fh(h)p Fg(xy)r Fh(i)2264 977 y Fn(c)p Fp(\()p Fn(x;)17 b(y)t(;)g(\021)t Fp(\)[)p Fn(f)11 b Fp(\()p Fn(\021)2805 936 y Fg(xy)2884 977 y Fp(\))22 b Fo(\000)h Fn(f)11 b 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b(is)g(a)g(non-rev)m(ersible)g(stationary)g(measure.)70 b(The)43 b(corresp)s(onding)e(pathspace)h(measure)236 1993 y(o)m(v)m(er)f(the)f(time-in)m(terv)-5 b(al)36 b([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])40 b(is)e Fn(P)53 b Fp(=)39 b Fn(P)1971 1957 y Fg(E)2070 1993 y Fp(and)g(w)m(e)i(put)e Fn(P)2679 1957 y Fg(E)2665 2018 y(n)2778 1993 y Fp(the)h(pro)s(cess)g (restricted)g(to)236 2113 y(the)c(\014nite)g(square)h(\003)1038 2128 y Fg(n)1120 2113 y Fp(This)f Fn(P)1423 2077 y Fg(E)1409 2138 y(n)1518 2113 y Fp(will)d(no)m(w)k(pla)m(y)e(the)h(role)f(of)g Fn(P)2681 2077 y Fg(r)2667 2138 y(n)2755 2113 y Fp(of)g(Section)g(2.)53 b(F)-8 b(or)35 b(a)h(giv)m(en)236 2234 y(tra)5 b(jectory)36 b Fn(!)f Fo(2)e Fp(\012)i(w)m(e)i(let)d Fn(N)1372 2198 y Fg(xy)1362 2258 y(s)1454 2234 y Fp(\()p Fn(!)t Fp(\))p Fn(;)17 b(s)31 b Fo(2)h Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(])36 b(b)s(e)f(the)h(n)m(um)m(b)s(er)f(of)g(hopping)f(times)g (where)236 2354 y(the)k(o)s(ccupation)f(at)g(the)h(nearest)g(neigh)m(b) 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b(s)27 b Fo(2)h Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(t)p Fp(]])1983 5800 y(17)p eop %%Page: 18 18 18 17 bop 236 548 a Fp(and)1074 764 y Fn(\024)1130 723 y Fg(E)1130 789 y(y)1190 764 y Fp(\()p Fn(\021)t(;)17 b(t)p Fp(\))27 b Fo(\021)i Fp(I)-17 b(E)1615 722 y Fg(E)1674 764 y Fp([)p Fn(\021)1749 779 y Fg(t)1779 764 y Fp(\()p Fn(y)t Fp(\))p Fo(j)p Fn(\021)1983 779 y Fg(s)2019 764 y Fp(\()p Fn(z)t Fp(\))p Fn(;)17 b(z)32 b Fo(2)c Fp(\003)2427 779 y Fg(n)2474 764 y Fn(;)17 b(s)27 b Fo(2)i Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(t)p Fp(]])236 981 y(for)40 b Fn(E)465 996 y Fg(j)543 981 y Fp(=)h Fo(\006)p Fn(E)47 b Fp(if)39 b Fn(y)44 b Fp(=)d Fn(x)28 b Fo(\006)g Fn(e)1395 996 y Fi(1)1435 981 y Fn(;)17 b(E)1551 996 y Fg(j)1628 981 y Fp(=)41 b(0)f(if)g Fn(y)k Fp(=)d Fn(x)28 b Fo(\006)g Fn(e)2374 996 y Fi(2)2454 981 y Fp(and)40 b(the)h(exp)s(ectations)g (are)g(in)e(the)236 1101 y(pro)s(cess)34 b Fn(P)41 b Fp(=)28 b Fn(P)861 1065 y Fg(E)920 1101 y Fp(.)43 b(The)34 b(v)-5 b(ariable)30 b(en)m(trop)m(y)k(pro)s(duction)e(is)g(put)1742 1292 y(\026)1726 1317 y Fn(S)1786 1332 y Fg(n)1860 1317 y Fp(=)1992 1292 y(\026)1964 1317 y Fn(W)2056 1332 y Fg(n)2125 1317 y Fp(+)22 b Fo(J)2291 1332 y Fg(n)236 1534 y Fp(The)34 b(symmetry)f(in)e(the)i(\015uctuations)g(of)1834 1508 y(\026)1817 1534 y Fn(S)1877 1549 y Fg(n)1957 1534 y Fp(is)f(giv)m(en)g(b)m(y)236 1654 y Fc(Theorem)j(4.1)30 b Fp([LFT)g(for)g(the)g(a)m(ymmetric)f(exclusion)i(pro)s(cess])g(F)-8 b(or)29 b(all)g Fn(\017)h Fp(\(including)e Fn(\017)g Fp(=)g(0\),)236 1774 y(for)k(all)f Fn(z)h Fo(2)c Fn(R)q Fp(,)1414 2058 y(lim)1425 2120 y Fg(n;T)1636 1991 y Fp(1)p 1576 2035 169 4 v 1576 2127 a Fn(n)1634 2098 y Fi(2)1674 2127 y Fn(T)1771 2058 y Fp(ln)1951 1909 y Ff(R)2034 1989 y Fn(dP)14 b(e)2207 1953 y Fh(\000)p Fg(z)2309 1936 y Fi(\026)2298 1953 y Fg(S)2341 1961 y Fd(n)p 1879 2035 581 4 v 1879 2051 a Ff(R)1962 2131 y Fn(dP)g(e)2135 2102 y Fh(\000)p Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))2381 2086 y(\026)2370 2102 y Fg(S)2413 2110 y Fd(n)2497 2058 y Fp(=)27 b(0)236 2335 y Fc(Pro)s(of)37 b(of)h(Theorem)f(4.1)c Fp(W)-8 b(e)33 b(start)f(b)m(y)i(noting)d(that)i(for)f(the)h(time-rev)m (ersal)e(\002)3349 2350 y Fg(n)3396 2335 y Fp(,)1741 2551 y Fn(P)1818 2510 y Fg(E)1804 2576 y(n)1877 2551 y Fp(\002)1953 2566 y Fg(n)2028 2551 y Fp(=)c Fn(P)2208 2510 y Fh(\000)p Fg(E)2194 2576 y(n)236 2768 y Fp(Ob)m(viously)33 b(then,)g(for)f(a)g(function)g Fn(f)44 b Fp(measurable)32 b(in)f(\003)2339 2783 y Fg(n)2408 2768 y Fo(\002)23 b Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(],)985 2910 y Ff(Z)1101 3046 y Fn(dP)g(f)d Fp(\()p Fn(!)t Fp(\))26 b(=)1559 2910 y Ff(Z)1675 3046 y Fn(dP)1803 3005 y Fg(E)1789 3070 y(n)1862 3046 y Fn(f)11 b Fp(\()p Fn(!)t Fp(\))26 b(=)2192 2910 y Ff(Z)2308 3046 y Fn(dP)2436 3005 y Fg(E)2422 3070 y(n)2505 2978 y Fn(dP)2633 2942 y Fh(\000)p Fg(E)2619 3003 y(n)p 2505 3023 242 4 v 2532 3114 a Fn(dP)2660 3085 y Fg(E)2646 3139 y(n)2756 3046 y Fn(f)11 b Fp(\(\002)2929 3061 y Fg(n)2976 3046 y Fn(!)t Fp(\))236 3324 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4877 y(y)1235 4853 y Fp(\()p Fn(!)t(;)17 b(s)p Fp(\)\))k Fo(\000)h Fn(\024)1680 4812 y Fh(\000)p Fg(E)1680 4877 y(y)1795 4853 y Fp(\()p Fn(!)t(;)17 b(s)p Fp(\)])1524 b(\(4.34\))236 5069 y(No)m(w,)34 b Fo(j)p Fn(F)577 5084 y Fg(n)624 5069 y Fo(j)27 b(\024)i Fn(c)p Fo(j)p Fn(@)5 b Fp(\003)p Fo(j)p Fn(T)46 b Fp(b)s(ecause)34 b(the)f(\(\014rst\))h (bulk)e(term)h(in)f(\(4.34\))g(telescop)s(es)h(to)g(a)f(b)s(oundary)236 5190 y(term.)43 b(W)-8 b(e)33 b(can)g(th)m(us)h(apply)e(Remark)g(1)g (after)h(Prop)s(osition)d(2.2)j(to)f(\014nish)h(the)g(pro)s(of.)p 3780 5190 48 48 v 383 5430 a(Next,)44 b(w)m(e)e(in)m(v)m(estigate)g (whether)h(the)e(v)-5 b(ariable)40 b(w)m(ork)i Fn(W)2579 5445 y Fg(n)2668 5430 y Fp(of)e(\(4.31\))h(itself)f(satis\014es)i(the) 236 5551 y(same)33 b(lo)s(cal)d(symmetry)j(as)g(the)g(en)m(trop)m(y)g (pro)s(duction.)1983 5800 y(18)p eop %%Page: 19 19 19 18 bop 236 548 a Fc(Theorem)37 b(4.2)c Fp([LFT)g(for)f(the)h(w)m (ork)g(done])g(F)-8 b(or)32 b(all)f Fn(\017)d(>)f Fp(0,)33 b(for)f(all)e Fn(z)j Fo(2)28 b Fp(I)-17 b(R,)1402 789 y(lim)1413 851 y Fg(n;T)1624 722 y Fp(1)p 1564 766 169 4 v 1564 858 a Fn(n)1622 829 y Fi(2)1662 858 y Fn(T)1759 789 y Fp(ln)1939 640 y Ff(R)2022 720 y Fn(dP)14 b(e)2195 684 y Fh(\000)p Fg(z)2306 667 y Fi(\026)2286 684 y Fg(W)2353 692 y Fd(n)p 1867 766 605 4 v 1867 782 a Ff(R)1950 862 y Fn(dP)g(e)2123 833 y Fh(\000)p Fi(\(1)p Fh(\000)p Fg(z)s Fi(\))2378 816 y(\026)2358 833 y Fg(W)2425 841 y Fd(n)2509 789 y Fp(=)27 b(0)236 1023 y Fc(Pro)s(of)44 b(of)g(Theorem)f(4.2)c Fp(Clearly)-8 b(,)38 b(since)h Fn(\017)f(>)f Fp(0,)i Fo(jJ)2359 1038 y Fg(n)2405 1023 y Fo(j)e(\024)h Fn(cN)10 b Fp(\([)p Fo(\000)p Fn(T)e(;)17 b(T)d Fp(])p Fn(;)j Fp(\003)3176 1038 y Fg(n)3249 1023 y Fo(n)26 b Fp(\003)3393 987 y Fh(\003)3393 1048 y Fg(n)3440 1023 y Fp(\),)39 b(that)f(is)236 1144 y(b)s(ounded,)25 b(up)e(to)f(a)g(constan)m(t,)k(b) m(y)d(the)g(n)m(um)m(b)s(er)f(of)g(\015ips)h(in)e(the)i(tra)5 b(jectory)23 b(on)f(sites)h Fn(x)28 b Fo(2)g Fp(\003)3586 1159 y Fg(n)3634 1144 y Fo(n)q Fp(\003)3753 1108 y Fh(\003)3753 1168 y Fg(n)3800 1144 y Fp(,)236 1264 y(for)37 b(times)g Fn(t)g Fo(2)f Fp([)p Fo(\000)p Fn(T)8 b(;)17 b(T)d Fp(].)60 b(W)-8 b(e)38 b(can)g(therefore)g(v)m(erify)g(condition)e(\(2.21\))h (in)g(the)h(same)g(w)m(a)m(y)h(as)236 1385 y(w)m(e)e(did)f(for)f (Theorem)i(3.1.)54 b(Finally)-8 b(,)34 b(the)i(large)f(deviation)g (results)i(of)e([23])h(remain)f(v)-5 b(alid)34 b(for)236 1505 y Fn(\017)28 b(>)g Fp(0,)k(so)h(that)f(w)m(e)i(can)f(\014nish)f (the)h(pro)s(of)f(along)f(the)i(lines)f(of)g(Prop)s(osition)f(2.2.)p 3780 1505 48 48 v 236 1950 a Fv(5)161 b(Remarks)355 2169 y Fp(1.)49 b(It)e(is)f(clear)g(from)g(the)h(preceding)g(analysis)f (that)h(the)g(reasons)h(for)e(ha)m(ving)h(a)f(global)480 2290 y(or)35 b(lo)s(cal)e(\015uctuation)i(theorem)g(do)g(not)g(in)f(an) m(y)i(w)m(a)m(y)h(dep)s(end)f(on)f(the)h(\002)3257 2305 y Fg(n)3339 2290 y Fp(b)s(eing)f(time-)480 2410 y(rev)m(ersal.)42 b(Th)m(us,)29 b(the)e(same)f(results)g(will)e(b)s(e)i(repro)s(duced)h (in)f(exactly)g(the)h(same)e(form)g(for)480 2531 y(an)m(y)34 b(other)f(in)m(v)m(olution.)44 b(Of)33 b(course,)i(the)f(symmetry)f (breaking)g(part)g(in)g(the)h(pathspace)480 2651 y(action)c(functional) f(will)f(b)s(e)j(the)f(v)-5 b(ariable)29 b(for)h(whic)m(h)h(the)g (\015uctuation)f(symmetry)h(holds)480 2771 y(\(replacing)f(en)m(trop)m (y)j(pro)s(duction)d(corresp)s(onding)h(to)g(time-rev)m(ersal)f (symmetry)i(break-)480 2892 y(ing\).)60 b(As)39 b(an)f(example,)h(if)e (for)h(a)g(spin\015ip)g(pro)s(cess,)j(the)e(rates)f(are)h(not)f(ev)m (en)i(under)f(a)480 3012 y(global)28 b(spin\015ip)i(\(b)m(y)h(the)g (presence)h(of)e(a)g(bias)g(or)g(magnetic)f(\014eld\),)h(then)h(a)f(lo) s(cal)e(\015uctu-)480 3133 y(ation)i(theorem)h(will)d(b)s(e)k (established)f(for)f(the)i(v)-5 b(ariable)29 b(magnetization.)40 b(F)-8 b(urthermore,)480 3253 y(w)m(e)32 b(ma)m(y)f(consider)h(the)g (comp)s(osition)d(of)h(t)m(w)m(o)i(or)f(more)g(in)m(v)m(olutions)f(|)h (in)f(this)h(w)m(a)m(y)-8 b(,)33 b(w)m(e)480 3373 y(could)h(e.g.)50 b(obtain)33 b(a)h(lo)s(cal)e(\015uctuation)i(theorem)h(for)f(the)h(o)s (dd)f(part)g(\(under)h(spin\015ip\))480 3494 y(of)40 b(the)g(v)-5 b(ariable)38 b(en)m(trop)m(y)k(pro)s(duction.)65 b(Finally)-8 b(,)39 b(w)m(e)j(can)e(ev)m(en)i(go)d(b)s(ey)m(ond)i(the)g (case)480 3614 y(of)d(in)m(v)m(olutions)f(and)i(consider)g(instead)f (the)h(generators)g(of)f(the)h(symmetry)g(group)f(for)480 3734 y(the)44 b(unp)s(erturb)s(ed)g(dynamics.)75 b(In)44 b(this)f(case,)k(the)d(precise)g(form)e(of)g(the)i(\015uctuation)480 3855 y(symmetry)33 b(is)f(not)g(preserv)m(ed)j(but)e(its)f(mo)s (di\014cation)e(presen)m(ts)35 b(no)d(real)g(problem.)355 4043 y(2.)49 b(W)-8 b(e)28 b(restricted)g(our)g(discussion)g(to)f(in)m (teracting)f(particle)h(systems)i(where)g(the)f(ev)m(olution)480 4164 y(is)43 b(Mark)m(o)m(vian.)77 b(Within)43 b(the)h(Gibbsian)e (space-time)h(picture,)j(this)d(means)h(that)f(the)480 4284 y(in)m(teraction)38 b(is)g(\\nearest)i(neigh)m(b)s(or")e(in)g(the) h(time)f(direction)f(\(the)j(jump)e(in)m(tensit)m(y)h(at)480 4405 y(time)22 b Fn(t)h Fp(dep)s(ends)i(only)d(on)h(the)h (con\014guration)e(at)h(time)e Fn(t)2546 4368 y Fh(\000)2606 4405 y Fp(\).)40 b(Ho)m(w)m(ev)m(er,)27 b(this)c(restriction)f(is)480 4525 y(not)27 b(at)g(all)e(necessary)-8 b(.)44 b(If)27 b(the)g(jump)g(in)m(tensities)f(are)i(lo)s(cal)c(in)j(space)h(and)f(b)s (ounded)h(from)480 4645 y(ab)s(o)m(v)m(e)36 b(and)f(from)f(b)s(elo)m (w,)i(then)g(w)m(e)g(can)f(still)e(apply)i(the)g(Girsano)m(v)g(form)m (ula)e(for)i(p)s(oin)m(t)480 4766 y(pro)s(cesses)43 b(to)e(obtain)f (the)i(lo)s(cal)c(\015uctuation)j(theorem)g(from)f(the)h(global)e (\015uctuation)480 4886 y(theorem.)236 5211 y Fv(References)285 5430 y Fp([1])49 b(Minlos)25 b(R.A.,)i Fk(Limiting)h(Gibbs)h (distribution)p Fp(,)e(F)-8 b(unct.)26 b(Anal.)g(Appl.)f Fc(1)p Fp(,140-150)f(,)k(206-217)437 5551 y(\(1967\).)1983 5800 y(19)p eop %%Page: 20 20 20 19 bop 285 548 a Fp([2])49 b(Minlos)31 b(R.A.,)i(Ro)s(elly)e(S.,)i (Zessin)g(H.,)g Fk(Gibbs)i(states)g(on)g(sp)-5 b(ac)g(etime)p Fp(,)31 b(J.)i(P)m(oten)m(tial)f(Anal-)437 668 y(ysis,)h Fc(13)p Fp(,)g(Issue)h(4)e(\(2001\).)285 872 y([3])49 b(Maes)39 b(C.,)h Fk(Fluctuation)g(the)-5 b(or)g(em)40 b(as)f(a)h(Gibbs)g(pr)-5 b(op)g(erty)9 b Fp(,)39 b(J.)g(Stat.)f(Ph)m (ys.)i Fc(95)p Fp(,)f(367{392)437 992 y(\(1999\).)285 1196 y([4])49 b(Maes)41 b(C.,)i(Redig)c(F.,)j(V)-8 b(ersc)m(h)m(uere)43 b(M.,)f Fk(Entr)-5 b(opy)43 b(pr)-5 b(o)g(duction)41 b(for)h(inter)-5 b(acting)41 b(p)-5 b(article)437 1316 y(systems)p Fp(,)32 b(Mark)m(o)m(v.)i(Pro)s(c.)f(Rel.)f(Fields.,)f(to)i (app)s(ear.)285 1519 y([5])49 b(Maes)33 b(C.,)g(Redig)f(F.,)h(V)-8 b(ersc)m(h)m(uere)35 b(M.,)e Fk(No)i(curr)-5 b(ent)35 b(without)g(he)-5 b(at)p Fp(,)33 b(Preprin)m(t)g(\(2000\).)285 1723 y([6])49 b(Maes)30 b(C.,)h(Redig)e(F.,)h(V)-8 b(an)30 b(Mo\013aert)f(A.,)i Fk(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 1843 y(examples)p Fp(,)f(J.)i(Math.)g(Ph)m(ys.)h Fc(41)p Fp(,)f(1528{1554)d(\(2000\).)285 2046 y([7])49 b(Maes)42 b(C.,)h(Redig)e(F.,)i Fk(Positivity)g(of)f (entr)-5 b(opy)43 b(pr)-5 b(o)g(duction)p Fp(,)42 b(J.)g(Stat.)e(Ph)m (ys.)j Fc(101)p Fp(,)h(3{16)437 2167 y(\(2000\))285 2370 y([8])49 b(Br)m(\023)-46 b(emaud)42 b(P)-8 b(.,)45 b Fk(Point)e(Pr)-5 b(o)g(c)g(esses)43 b(and)g(Queues,)j(the)d(martingale) g(appr)-5 b(o)g(ach)p Fp(,)44 b(Springer-)437 2491 y(V)-8 b(erlag,)31 b(New)j(Y)-8 b(ork,)33 b(Heidelb)s(erg,)e(Berlin)h (\(1981\).)285 2694 y([9])49 b(Kurc)m(han)24 b(J.,)i Fk(Fluctuation)h(the)-5 b(or)g(em)26 b(for)h(sto)-5 b(chastic)26 b(dynamics)p Fp(,)f(J.)f(Ph)m(ys.)h(A:)f(Math.)h(Gen.)437 2814 y Fc(31)p Fp(,)32 b(3719{3729)f(\(1998\).)236 3018 y([10])49 b(Leb)s(o)m(witz)29 b(J.L.,)i(Maes)g(C.,)f(Sp)s(eer)h(E.R.,)f Fk(Statistic)-5 b(al)32 b(me)-5 b(chanics)31 b(of)h(Pr)-5 b(ob)g(abilistic)31 b(Cel)5 b(lu-)437 3138 y(lar)34 b(A)n(utomata)p Fp(,)f(J.)g(Stat.)f(Ph)m(ys.)j Fc(59)p Fp(,)d(117{170)f(\(1990\).)236 3342 y([11])49 b(Leb)s(o)m(witz)32 b(J.L.,)h(Sp)s(ohn)g(H.,)f Fk(A)k(Gal)5 b(lavotti-Cohen)33 b(typ)-5 b(e)35 b(symmetry)g(in)g(the)f (lar)-5 b(ge)35 b(devia-)437 3462 y(tions)f(functional)h(of)f(sto)-5 b(chastic)35 b(dynamics)p Fp(,)c(J.)i(Stat.)f(Ph)m(ys.)j Fc(95)p Fp(,)d(333{365)f(\(1999\).)236 3665 y([12])49 b(Ev)-5 b(ans)33 b(D.J.,)f(Cohen)h(E.G.D.,)f(Morriss)g(G.P)-8 b(.,)32 b Fk(Pr)-5 b(ob)g(ability)34 b(of)g(se)-5 b(c)g(ond)34 b(law)g(violations)f(in)437 3786 y(ste)-5 b(ady)35 b(\015ows)p Fp(,)d(Ph)m(ys.)i(Rev.)g(Lett.)e Fc(71)p Fp(,)h(2401-2404)d(\(1993\).) 236 3989 y([13])49 b(Galla)m(v)m(otti)25 b(G.,)k(Cohen)g(E.G.D.,)g Fk(Dynamic)-5 b(al)30 b(ensembles)f(in)i(none)-5 b(quilibrium)30 b(Statistic)-5 b(al)437 4110 y(Me)g(chanics)p Fp(,)31 b(Ph)m(ys.)k(Rev.)e(Letters)h Fc(74)p Fp(,)e(2694-2697)e(\(1995\).)236 4313 y([14])49 b(Galla)m(v)m(otti)26 b(G.,)k(Cohen)h(E.G.D.,)f Fk(Dynamic)-5 b(al)30 b(ensembles)h(in)h(stationary)g(states)p Fp(,)e(J.)g(Stat.)437 4433 y(Ph)m(ys.)k Fc(80)p Fp(,)f(931-970)e (\(1995\).)236 4637 y([15])49 b(Galla)m(v)m(otti)27 b(G.,)k Fk(Chaotic)h(hyp)-5 b(othesis:)43 b(Onsager)31 b(r)-5 b(e)g(cipr)g(o)g(city)33 b(and)f(the)h(\015uctuation)h(dissi-)437 4757 y(p)-5 b(ation)34 b(the)-5 b(or)g(em)p Fp(,)32 b(J.)h(Stat.)f(Ph)m (ys.)j Fc(84)p Fp(,)e(899-926)d(\(1996\).)236 4961 y([16])49 b(Galla)m(v)m(otti)27 b(G.,)j Fk(Chaotic)h(dynamics,)h(\015uctuations,) h(none)-5 b(quilibrium)31 b(ensembles)p Fp(,)e(Chaos)437 5081 y Fc(8)p Fp(,)j(384-392)f(\(1998\).)236 5284 y([17])49 b(G.)32 b(Galla)m(v)m(otti:)40 b Fk(A)c(lo)-5 b(c)g(al)34 b(\015uctuation)i(the)-5 b(or)g(em)p Fp(,)32 b(Ph)m(ysica)h(A)g Fc(263)p Fp(,)g(39{50)e(\(1999\).)1983 5800 y(20)p eop %%Page: 21 21 21 20 bop 236 548 a Fp([18])49 b(G.)j(Galla)m(v)m(otti)e(and)j(F.)f(P)m (erroni:)84 b Fk(A)n(n)53 b(exp)-5 b(erimental)52 b(test)i(of)f(the)g (lo)-5 b(c)g(al)53 b(\015uctuation)437 668 y(the)-5 b(or)g(em)55 b(in)g(chains)g(of)g(we)-5 b(akly)55 b(inter)-5 b(acting)55 b(A)n(nosov)g(systems)p Fp(,)61 b Fa(mp)p 3147 668 31 4 v 37 w(arc)56 b Fp(#)f(99-320,)437 789 y Fa(chao-dyn)p Fp(/9909007.)236 992 y([19])49 b(G.)c(Gen)m(tile:)68 b Fk(L)-5 b(ar)g(ge)46 b(deviation)g(rule)h(for)f(A)n(nosov)g(\015ows)p Fp(,)i(F)-8 b(orum)45 b(Math.)g Fc(10)p Fp(,)k(89{118)437 1112 y(\(1998\).)236 1316 y([20])g(Ruelle)24 b(D.,)j Fk(Positivity)i(of)f(entr)-5 b(opy)29 b(pr)-5 b(o)g(duction)28 b(in)h(none)-5 b(quilibrium,)28 b(statistic)-5 b(al)29 b(me)-5 b(chan-)437 1436 y(ics)p Fp(,)32 b(J.)h(Stat.)f(Ph)m(ys.)j Fc(85)p Fp(,)d(1{25)g(\(1996\).)236 1640 y([21])49 b(Ruelle)24 b(D.,)j Fk(Entr)-5 b(opy)29 b(pr)-5 b(o)g(duction)29 b(in)f(none)-5 b(quilibrium)28 b(statistic)-5 b(al)29 b(me)-5 b(chanics)p Fp(,)25 b(Comm)m(un.)437 1760 y(Math.)33 b(Ph)m(ys.)h Fc(189)p Fp(,)f(365{371)e(\(1997\).)236 1963 y([22])49 b(Ruelle)31 b(D.,)h Fk(Smo)-5 b(oth)33 b(Dynamics)h(and)f(New)i(The)-5 b(or)g(etic)g(al)33 b(Ide)-5 b(as)34 b(in)g(None)-5 b(quilibrium)34 b(Sta-)437 2084 y(tistic)-5 b(al)35 b(Me)-5 b(chanics)p Fp(,)31 b(J.)i(Stat.)f(Ph)m (ys.)j Fc(95)p Fp(,)e(393{468)d(\(1999\).)236 2287 y([23])49 b(Dai)31 b(Pra)i(P)-8 b(.,)34 b Fk(Sp)-5 b(ac)g(e-time)34 b(lar)-5 b(ge)35 b(deviations)f(for)h(inter)-5 b(acting)35 b(p)-5 b(article)35 b(systems)p Fp(,)e(Comm.)437 2408 y(Pure)g(and)g(Appl.)f(Math.)h Fc(46)p Fp(,)g(387-422)d(\(1993\).)236 2611 y([24])49 b(Liggett)33 b(T.)h(M.,)h Fk(Inter)-5 b(acting)36 b(p)-5 b(article)35 b(systems)p Fp(,)g(Springer-V)-8 b(erlag,)32 b(New)j(Y)-8 b(ork,)35 b(Heidel-)437 2731 y(b)s(erg,)d(Berlin)g(\(1985\).)236 2935 y([25])49 b(Lipster)i(R.S.,)56 b(Shiry)m(a)m(y)m(ev)c(A.N.,)57 b Fk(Statistics)52 b(of)f(R)-5 b(andom)51 b(Pr)-5 b(o)g(c)g(esses)51 b(I,II)p Fp(,)f(Springer-)437 3055 y(V)-8 b(erlag,)31 b(New)j(Y)-8 b(ork,)33 b(Heidelb)s(erg,)e (Berlin)h(\(1978\).)1983 5800 y(21)p eop %%Trailer end userdict /end-hook 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