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Fl(\021)19 b Fm(j)c Fl(!)s Fn(\))26 b Fl(>)f Fm(\0001)p Fb(,)d(one)i(has)928 2122 y Fn(En)m(t\()p Fl(\021)g Fm(\016)d Fl(\034)1293 2084 y Fi(t)1283 2144 y(V)1359 2122 y Fm(j)15 b Fl(!)s Fn(\))26 b(=)f(En)m(t\()p Fl(\021)19 b Fm(j)c Fl(!)s Fn(\))21 b Fm(\000)2106 1998 y Fh(Z)2197 2025 y Fi(t)2157 2204 y Fg(0)2242 2122 y Fn(Ep)2354 2144 y Fi(V)2415 2122 y Fn(\()p Fl(\021)j Fm(\016)d Fl(\034)2635 2084 y Fi(s)2625 2144 y(V)2686 2122 y Fn(\))15 b Fl(ds:)244 2506 y Ft(Remark.)28 b Fs(The)23 b(same)h(result)g(\(with)g(the)g(same)f(proof\))i(holds)g (for)f Fl(W)2467 2473 y Fk(\003)2505 2506 y Fs(-dynamical)i(systems.) 380 2618 y(In)j(the)h(rest)g(of)f(this)g(section)i(we)e(will)g(discuss) i(some)e(elementary)i(properties)h(of)d Fn(Ep)3190 2640 y Fi(V)3251 2618 y Fn(\()p Fm(\001)p Fn(\))p Fs(.)45 b(Let)244 2731 y Fl(!)304 2693 y Fg(+)301 2760 y Fi(V)388 2731 y Fm(2)25 b Fn(\006)540 2693 y Fg(+)540 2760 y Fi(V)600 2731 y Fn(\()p Fl(!)s Fn(\))f Fs(and)g Fl(T)961 2745 y Fi(n)1033 2731 y Fm(!)h(1)e Fs(be)g(such)i(that)1359 3002 y Fn(lim)1400 3057 y Fi(n)1537 2941 y Fn(1)p 1510 2982 101 4 v 1510 3065 a Fl(T)1563 3079 y Fi(n)1635 2879 y Fh(Z)1726 2905 y Fi(T)1767 2913 y Fe(n)1686 3085 y Fg(0)1814 3002 y Fl(!)e Fm(\016)e Fl(\034)2010 2965 y Fi(t)2000 3025 y(V)2075 3002 y Fl(dt)26 b Fn(=)f Fl(!)2337 2964 y Fg(+)2334 3031 y Fi(V)2396 3002 y Fl(:)940 b Fs(\(1.1\))244 3246 y(Then,)23 b(with)h(the)f(particular)j(choice)f Fl(\021)k Fn(=)c Fl(!)s Fs(,)d(Theorem)i(1.1)g(gi)n(v)o(es)476 3508 y Fn(lim)518 3563 y Fi(n)655 3447 y Fn(1)p 627 3487 V 627 3571 a Fl(T)680 3585 y Fi(n)737 3508 y Fn(En)m(t)q(\()p Fl(!)f Fm(\016)e Fl(\034)1114 3469 y Fi(T)1155 3477 y Fe(n)1104 3537 y Fi(V)1216 3508 y Fm(j)15 b Fl(!)s Fn(\))26 b(=)f Fm(\000)15 b Fn(lim)1601 3563 y Fi(n)1738 3447 y Fn(1)p 1711 3487 V 1711 3571 a Fl(T)1764 3585 y Fi(n)1836 3385 y Fh(Z)1927 3411 y Fi(T)1968 3419 y Fe(n)1886 3591 y Fg(0)2030 3508 y Fl(!)s Fn(\()p Fl(\034)2175 3471 y Fi(s)2165 3531 y(V)2225 3508 y Fn(\()p Fl(\033)2312 3522 y Fi(V)2374 3508 y Fn(\)\))g Fl(ds)26 b Fn(=)f Fm(\000)p Fn(Ep)2854 3530 y Fi(V)2915 3508 y Fn(\()p Fl(!)3010 3470 y Fg(+)3007 3537 y Fi(V)3069 3508 y Fn(\))p Fl(:)232 b Fs(\(1.2\))244 3752 y(Since)24 b(the)f(relati)n(v)o(e)i(entrop)o(y)h (is)d(non-positi)n(v)o(e,)j(we)d(immediately)i(get)244 3965 y Ft(Theor)n(em)46 b(1.2)k Fb(Assume)28 b(that)h Fs(\(A1\))g Fb(and)g Fs(\(A2\))g Fb(hold.)45 b(Then,)29 b(for)g(any)g Fs(NESS)d Fl(!)2906 3926 y Fg(+)2903 3993 y Fi(V)3000 3965 y Fm(2)34 b Fn(\006)3161 3926 y Fg(+)3161 3993 y Fi(V)3222 3965 y Fn(\()p Fl(!)s Fn(\))p Fb(,)29 b(one)244 4077 y(has)1612 4190 y Fn(Ep)1725 4212 y Fi(V)1786 4190 y Fn(\()p Fl(!)1881 4152 y Fg(+)1878 4219 y Fi(V)1940 4190 y Fn(\))d Fm(\025)e Fn(0)p Fl(:)380 4537 y Fs(W)l(ith)g(re)o(gard) g(to)g(\(1.2\),)f(on)h(physical)h(grounds)h(one)e(e)o(xpects)h(that)f (the)g(ratio)1564 4741 y Fn(En)m(t\()p Fl(\021)g Fm(\016)d Fl(\034)1929 4703 y Fi(t)1919 4763 y(V)1995 4741 y Fm(j)15 b Fl(!)s Fn(\))2130 4667 y Fh(\016)2183 4741 y Fl(t)244 4945 y Fs(becomes)25 b(independent)j(of)c(the)h(choice)g(of)g(the)f (reference)j(state)e Fl(!)h Fs(as)e Fl(t)i Fm(!)h(1)p Fs(.)j(More)24 b(precisely)-6 b(,)27 b(the)244 5058 y(follo)n(wing)e (result)g(holds:)p eop %%Page: 4 4 4 3 bop 3730 100 a Fp(4)244 407 y Ft(Pr)n(oposition)25 b(1.3)49 b Fb(Assume)25 b Fs(\(A1\))f Fb(and)h(that)h Fl(\021)31 b Fm(2)c(N)1939 421 y Fi(!)2013 407 y Fb(is)d(faithful.)34 b(Then)25 b(ther)m(e)h(is)e(a)h(norm-dense)i(set)244 520 y Fm(N)332 487 y Fk(0)319 542 y Fi(!)394 520 y Fm(\032)e(N)565 534 y Fi(!)638 520 y Fb(suc)o(h)f(that)g(for)g Fl(!)1179 487 y Fk(0)1227 520 y Fm(2)h(N)1401 487 y Fk(0)1388 542 y Fi(!)1438 520 y Fb(,)1089 709 y Fn(En)m(t\()p Fl(\021)f Fm(\016)d Fl(\034)1454 672 y Fi(t)1444 732 y(V)1520 709 y Fm(j)15 b Fl(!)1620 672 y Fk(0)1643 709 y Fn(\))26 b(=)f(En)m(t\()p Fl(\021)f Fm(\016)d Fl(\034)2165 672 y Fi(t)2155 732 y(V)2230 709 y Fm(j)15 b Fl(!)s Fn(\))21 b(+)f Fl(O)s Fn(\(1\))p Fl(;)244 899 y Fb(as)j Fl(t)i Fm(!)h(1)p Fb(.)380 1074 y Fs(One)i(also)g(e)o(xpects)h(that)g(in)f (thermal)g(equilibrium)j(the)d(entrop)o(y)h(production)i(is)d(zero,)h (that)g(is,)f(if)244 1187 y Fl(\021)h Fm(2)24 b(N)478 1201 y Fi(!)551 1187 y Fs(is)f(a)h Fn(\()p Fl(\034)773 1201 y Fi(V)834 1187 y Fl(;)15 b(\014)5 b Fn(\))p Fs(-KMS)23 b(state)h(then)g Fn(Ep)1697 1209 y Fi(V)1758 1187 y Fn(\()p Fl(\021)s Fn(\))i(=)f(0)p Fs(.)j(In)c(f)o(act,)f(a)h(much)f(stronger)j (result)f(holds.)244 1392 y Ft(Pr)n(oposition)g(1.4)49 b Fb(Assume)20 b Fs(\(A1\))p Fb(,)h Fs(\(A2\))f Fb(and)h(that)h Fl(\021)28 b Fm(2)d(N)2132 1406 y Fi(!)2202 1392 y Fb(is)20 b(a)g(faithful,)j Fl(\034)2693 1406 y Fi(V)2754 1392 y Fb(-in)l(variant)h(state)o(.)k(Then)1648 1582 y Fn(Ep)1760 1604 y Fi(V)1821 1582 y Fn(\()p Fl(\021)s Fn(\))e(=)f(0)p Fl(:)244 1947 y Ft(Remark.)j Fs(Again,)23 b(this)h(result)h(also)f (holds)h(for)e Fl(W)1875 1914 y Fk(\003)1914 1947 y Fs(-dynamical)j (systems.)380 2060 y(Let)j Fm(O)j Fs(be)d(the)h(CAR)d(algebra)32 b(o)o(v)o(er)d Fl(l)1616 2027 y Fg(2)1655 2060 y Fn(\()p Fj(Z)1754 2027 y Fg(3)1794 2060 y Fn(\))g Fs(describing)j(a)d(free)h (Fermi)f(gas)g(on)h(the)g(lattice)h Fj(Z)3474 2027 y Fg(3)3513 2060 y Fs(.)244 2173 y(Using)c(some)h(technical)h(results)g (pro)o(v)o(en)f(in)g([BM)o(])f(it)g(is)g(easy)h(to)f(construct)j(a)d (lar)n(ge)h(class)g(of)f(quasi-)244 2286 y(free)f(states)i Fl(!)g Fs(and)e(local)h(perturbations)j Fl(V)46 b Fs(such)27 b(that)f(\(A1\)-\(A2\))h(hold,)g(and)g(that)f Fn(\006)3031 2247 y Fg(+)3031 2314 y Fi(V)3092 2286 y Fn(\()p Fl(!)s Fn(\))g Fs(consists)244 2398 y(of)d(a)g(single)i(state)g Fl(!)895 2360 y Fg(+)892 2427 y Fi(V)953 2398 y Fs(.)j(In)c(these)g(e)o (xamples,)h Fn(Ep)1812 2420 y Fi(V)1873 2398 y Fn(\()p Fl(!)1968 2360 y Fg(+)1965 2427 y Fi(V)2027 2398 y Fn(\))e Fs(can)h(be)g(computed)h(perturbati)n(v)o(ely)i(\(similar)244 2511 y(calculations)g(are)d(done)g(in)f([HTP)o(]\),)g(and)h(one)g (easily)h(constructs)i(e)o(xamples)d(where)g Fn(Ep)3092 2533 y Fi(V)3153 2511 y Fn(\()p Fl(!)3248 2473 y Fg(+)3245 2540 y Fi(V)3307 2511 y Fn(\))h Fl(>)g Fn(0)p Fs(.)380 2624 y(In)f(the)g(ne)o(xt)f(e)o(xample)i(we)e(relate)h(entrop)o(y)i (production)g(to)e(heat)g(\003o)n(ws.)380 2737 y(Consider)32 b(tw)o(o)e(independent)k(systems)d Fn(\()p Fm(O)1801 2751 y Fi(i)1830 2737 y Fl(;)15 b(\034)1910 2751 y Fi(i)1938 2737 y Fl(;)g(!)2035 2751 y Fi(i)2063 2737 y Fn(\))p Fs(,)32 b Fl(i)37 b Fn(=)h(1)p Fl(;)15 b Fn(2)p Fs(,)32 b(each)f(of)f(which)h(is)f(in)g(thermal)244 2850 y(equilibrium)36 b(at)d(temperature)j Fl(T)1319 2864 y Fi(i)1347 2850 y Fs(.)58 b(This)33 b(means)h(that)g Fl(!)2113 2864 y Fi(i)2173 2850 y Fs(is)f(a)g Fn(\()p Fl(\034)2414 2864 y Fi(i)2442 2850 y Fl(;)15 b(\014)2533 2864 y Fi(i)2562 2850 y Fn(\))p Fs(-KMS)33 b(state)h(on)f Fm(O)3252 2864 y Fi(i)3314 2850 y Fs(where)244 2963 y Fl(\014)295 2977 y Fi(i)349 2963 y Fn(=)25 b(1)p Fl(=T)588 2977 y Fi(i)617 2963 y Fs(.)j(Let)927 3152 y Fm(O)g Fn(=)d Fm(O)1195 3166 y Fg(1)1255 3152 y Fm(\012)20 b(O)1418 3166 y Fg(2)1458 3152 y Fl(;)197 b(\034)35 b Fn(=)25 b Fl(\034)1891 3166 y Fg(1)1951 3152 y Fm(\012)20 b Fl(\034)2082 3166 y Fg(2)2121 3152 y Fl(;)197 b(!)28 b Fn(=)d Fl(!)2581 3166 y Fg(1)2640 3152 y Fm(\012)20 b Fl(!)2788 3166 y Fg(2)2827 3152 y Fl(:)244 3342 y Fs(\()p Fm(\012)i Fs(is)h(the)g Fl(C)655 3309 y Fk(\003)694 3342 y Fs(-tensor)h(product,)h(see)e(Section)h (2.7.2)f(in)f([BR1]\).)28 b(Let)23 b Fl(\016)j Fs(be)c(the)i(generator) h(of)e Fl(\034)32 b Fs(and)23 b Fl(\016)3507 3356 y Fi(i)244 3455 y Fs(the)i(generator)j(of)c Fl(\034)886 3469 y Fi(i)914 3455 y Fs(.)33 b(Ob)o(viously)-6 b(,)26 b Fl(\016)32 b Fn(=)c Fl(\016)1599 3469 y Fg(1)1660 3455 y Fn(+)21 b Fl(\016)1792 3469 y Fg(2)1856 3455 y Fs(\(here)26 b(we)e(write)h Fl(\016)2448 3469 y Fg(1)2512 3455 y Fs(for)g Fl(\016)2682 3469 y Fg(1)2743 3455 y Fm(\012)c Fj(1)p Fs(,)j(etc\).)34 b(Let)24 b Fl(V)48 b Fm(2)28 b(O)244 3568 y Fs(be)23 b(such)i(that)f Fl(V)45 b Fm(2)25 b(D)s Fn(\()p Fl(\016)1033 3582 y Fi(i)1062 3568 y Fn(\))p Fs(.)j(Then)1160 3814 y Fl(!)23 b Fm(\016)d Fl(\034)1355 3776 y Fi(t)1345 3836 y(V)1406 3814 y Fn(\()p Fl(V)h Fn(\))f Fm(\000)g Fl(!)s Fn(\()p Fl(V)h Fn(\))k(=)1986 3690 y Fh(Z)2077 3716 y Fi(t)2037 3896 y Fg(0)2122 3814 y Fl(!)e Fm(\016)e Fl(\034)2318 3776 y Fi(s)2308 3836 y(V)2368 3814 y Fn(\(\010\))p Fl(ds;)244 4043 y Fs(where)j Fn(\010)e Fs(is)h(de\002ned)i(by)1526 4272 y Fl(\034)1576 4235 y Fi(t)1566 4295 y(V)1626 4272 y Fn(\(\010\))h(=)1910 4211 y Fl(d)p 1894 4251 81 4 v 1894 4335 a(dt)1984 4272 y(\034)2034 4235 y Fi(t)2024 4295 y(V)2085 4272 y Fn(\()p Fl(V)20 b Fn(\))p Fl(:)244 4488 y Fs(Ob)o(viously)-6 b(,)22 b Fn(\010)i(=)h(\010)910 4502 y Fg(1)954 4488 y Fn(+)5 b(\010)1096 4502 y Fg(2)1153 4488 y Fs(where)20 b Fn(\010)1460 4502 y Fi(i)1513 4488 y Fm(\021)25 b Fl(\016)1649 4502 y Fi(i)1678 4488 y Fn(\()p Fl(V)20 b Fn(\))f Fs(describes)j(the)e(ener)n(gy)h(\003ux)e(out)h(of)f (the)h Fl(i)p Fs(-th)f(system.)244 4600 y(Since)k(the)h(states)g Fl(!)883 4614 y Fi(i)933 4600 y Fs(are)g(KMS,)d(\(A1\))i(holds)h(with)f Fl(\016)1946 4614 y Fi(!)2022 4600 y Fn(=)i Fl(\016)2158 4614 y Fi(!)2202 4623 y Ff(1)2260 4600 y Fn(+)19 b Fl(\016)2390 4614 y Fi(!)2434 4623 y Ff(2)2495 4600 y Fs(and)24 b Fl(\016)2689 4614 y Fi(!)2733 4624 y Fe(i)2789 4600 y Fn(=)h Fm(\000)p Fl(\014)3007 4614 y Fi(i)3035 4600 y Fl(\016)3075 4614 y Fi(i)3104 4600 y Fs(.)j(Therefore,)244 4713 y(\(A1\))23 b(and)h(\(A2\))g(hold)g(and)1473 4826 y Fl(\014)1524 4840 y Fg(1)1564 4826 y Fn(\010)1630 4840 y Fg(1)1689 4826 y Fn(+)c Fl(\014)1831 4840 y Fg(2)1871 4826 y Fn(\010)1937 4840 y Fg(2)2001 4826 y Fn(=)25 b Fm(\000)p Fl(\033)2220 4840 y Fi(V)2281 4826 y Fl(:)244 4986 y Fs(It)e(follo)n(ws)h(that)g(in)g(a)f(NESS)e Fl(!)1236 4948 y Fg(+)1233 5014 y Fi(V)1320 4986 y Fm(2)k Fn(\006)1472 4948 y Fg(+)1472 5014 y Fi(V)1533 4986 y Fn(\()p Fl(!)s Fn(\))p Fs(,)e(the)g(ener)n(gy)j(\003ux)o(es)d(satisfy)1156 5178 y Fl(!)1216 5139 y Fg(+)1213 5206 y Fi(V)1275 5178 y Fn(\(\010)1376 5192 y Fg(1)1416 5178 y Fn(\))p 1156 5221 295 4 v 1257 5304 a Fl(T)1310 5318 y Fg(1)1481 5242 y Fn(+)1582 5178 y Fl(!)1642 5139 y Fg(+)1639 5206 y Fi(V)1701 5178 y Fn(\(\010)1802 5192 y Fg(2)1841 5178 y Fn(\))p 1582 5221 V 1683 5304 a Fl(T)1736 5318 y Fg(2)1912 5242 y Fn(=)i Fm(\000)p Fn(Ep)2191 5263 y Fi(V)2252 5242 y Fn(\()p Fl(!)2347 5203 y Fg(+)2344 5270 y Fi(V)2406 5242 y Fn(\))h Fm(\024)f Fn(0)p Fl(:)p eop %%Page: 5 5 5 4 bop 3730 100 a Fp(5)244 407 y Fs(Since)25 b Fl(!)530 369 y Fg(+)527 435 y Fi(V)589 407 y Fn(\(\010)690 421 y Fg(1)729 407 y Fn(\))c(+)g Fl(!)937 369 y Fg(+)934 435 y Fi(V)996 407 y Fn(\(\010)1097 421 y Fg(2)1136 407 y Fn(\))28 b(=)f(0)p Fs(,)d(if)h Fl(T)1522 421 y Fg(1)1588 407 y Fl(>)i(T)1739 421 y Fg(2)1779 407 y Fs(,)d(then)h Fn(\010)2072 421 y Fg(1)2138 407 y Fm(\025)i Fn(0)e Fs(and)g(the)g (heat)g(\003o)n(ws)e(from)i(the)g(hot)g(to)244 520 y(the)c(cold)h (reserv)n(oir)-5 b(.)30 b(This)21 b(calculation)j(is)c(easily)i (generalized)i(to)d(the)g(case)h(where)f Fl(N)10 b Fs(-le)n(v)o(el)21 b(quantum)244 633 y(system)j(is)g(coupled)h(to)f(se)n(v)o(eral)g (independent)j(thermal)e(rese)n(v)n(oirs.)380 746 y(W)-7 b(e)23 b(\002nish)h(this)g(section)h(with)e(the)h(follo)n(wing)h (remarks.)380 858 y(In)g([JP1)q(])f(we)g(pro)o(v)o(e)h(an)g(analog)i (of)e(Theorem)g(1.1)f(for)i(time-dependent)i(perturbations)h(and)c (dis-)244 971 y(cuss)f(the)g(relation)h(between)g(entrop)o(y)h (production)g(and)e(the)g(second)h(la)o(w)e(of)g(thermodynamics.)380 1084 y(In)e(the)h(forthcoming)i(paper)e([JP2],)f(we)g(will)f(study)j (NESS,)18 b(entrop)o(y)23 b(production)i(and)c(heat)h(\003o)n(ws)244 1197 y(for)27 b(a)f(model)h(of)g(an)g Fl(N)10 b Fs(-le)n(v)o(el)27 b(quantum)h(system)g(coupled)h(to)d(se)n(v)o(eral)i(independent)j(free) c(Fermi)f(gas)244 1310 y(reserv)n(oirs)g(\(similar)f(models)f(ha)n(v)o (e)g(been)h(studied)g(in)e([D,)g(Ru1]\).)380 1423 y(The)18 b(entrop)o(y)j(production)h(for)d(quantum)h(spin)f(systems)h(has)f (been)h(studied)g(in)f(the)g(recent)h(preprint)244 1536 y([Ru2)q(].)244 1739 y 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y(dense)d(in)e Fm(N)642 5098 y Fi(!)715 5084 y Fs(\(see,)h(e.g.,)e([R]\).)29 b(By)23 b(the)g(result)i(of)e(Araki)h(\(see)g(Proposition)i(6.2.32)e (in)g([BR2]\),)555 5268 y Fn(En)m(t\()p Fl(\021)g Fm(\016)d Fl(\034)920 5230 y Fi(t)910 5290 y(V)985 5268 y Fm(j)15 b Fl(!)1082 5282 y Fi(P)1141 5268 y Fn(\))26 b(=)f(En)m(t\()p Fl(\021)f Fm(\016)d Fl(\034)1663 5230 y Fi(t)1653 5290 y(V)1728 5268 y Fm(j)15 b Fl(!)s Fn(\))21 b(+)f Fl(\021)s Fn(\()p Fl(\034)2108 5230 y Fi(t)2098 5290 y(V)2160 5268 y Fn(\()p Fl(P)13 b Fn(\)\))21 b Fm(\000)f Fn(log)c Fm(k)p Fn(e)2665 5230 y Fg(\()p Fk(L)2741 5238 y Fe(!)2787 5230 y Fg(+)p Fi(P)10 b Fg(\))p Fi(=)p Fg(2)2998 5268 y Fn(\012)3064 5282 y Fi(!)3115 5268 y Fm(k)3160 5230 y Fg(2)3199 5268 y Fl(:)p eop %%Page: 9 9 9 8 bop 3730 100 a Fp(9)244 407 y Fs(The)25 b(statement)j(follo)n(ws)e (from)g(this)h(relation,)h(the)e(ob)o(vious)h(estimate)g Fm(j)p Fl(\021)s Fn(\()p Fl(\034)2697 374 y Fi(t)2687 434 y(V)2749 407 y Fn(\()p Fl(P)13 b Fn(\)\))p Fm(j)31 b(\024)e(k)p Fl(P)13 b Fm(k)26 b Fs(and)h(the)244 520 y(f)o(act)d(that)g Fn(0)i Fl(<)f Fm(k)p Fn(e)813 487 y Fg(\()p Fk(L)889 495 y Fe(!)934 487 y Fg(+)p Fi(P)10 b Fg(\))p Fi(=)p Fg(2)1146 520 y Fn(\012)1212 534 y Fi(!)1262 520 y Fm(k)26 b Fl(<)f Fm(1)p Fs(.)i Fa(\003)244 775 y Ft(Pr)n(oof)d(of)g(Pr)n(oposition)g(1.4.)29 b Fs(Since)23 b Fl(\021)h Fm(\016)d Fl(\034)1625 742 y Fi(t)1615 802 y(V)1701 775 y Fn(=)k Fl(\021)s Fs(,)d(Relation)j(\(2.17\))g(yields)f (that)h(for)e(all)h Fl(s)h(>)g Fn(0)p Fs(,)1006 909 y Fh(Z)1097 935 y Fi(t)1057 1115 y Fg(0)1142 1033 y Fn(\(\012)1243 1047 y Fi(\021)1285 1033 y Fl(;)15 b(\031)1377 1047 y Fi(!)1427 1033 y Fn(\()p Fl(\034)1512 995 y Fi(u)1502 1055 y(V)1563 1033 y Fn(\()p Fl(\033)1650 1047 y Fi(V)1712 1033 y Fn(\)\)[)p Fl(D)s(!)29 b Fn(:)c Fl(D)s(\021)s Fn(])2172 995 y Fi(s)2209 1033 y Fn(\012)2275 1047 y Fi(\021)2317 1033 y Fn(\))15 b Fl(du)26 b Fn(=)f Fl(o)p Fn(\(1\))p Fl(:)244 1271 y Fs(T)-7 b(aking)24 b Fl(s)h Fm(#)g Fn(0)f Fs(we)e(get)i(that)g(for)g(all)g Fl(t)p Fs(,)1367 1398 y Fh(Z)1458 1424 y Fi(t)1418 1604 y Fg(0)1503 1522 y Fl(\021)s Fn(\()p Fl(\033)1638 1536 y Fi(V)1700 1522 y Fn(\))p Fl(du)i Fn(=)f Fl(t\021)s Fn(\()p Fl(\033)2124 1536 y Fi(V)2185 1522 y Fn(\))h(=)f(0)p Fl(;)244 1765 y Fs(and)f(so)f Fn(Ep)614 1787 y Fi(V)675 1765 y Fn(\()p Fl(\021)s Fn(\))j(=)f(0)p Fs(.)j Fa(\003)244 2137 y Fo(Refer)m(ences) 244 2344 y Fs([Ar])155 b(Araki,)38 b(H.:)50 b(Relati)n(v)o(e)36 b(entrop)o(y)h(of)d(states)j(of)d(v)n(on)i(Neumann)g(algebras,)j(Pub)l (.)c(R.I.M.S.,)555 2457 y(K)n(yoto)24 b(Uni)n(v)-6 b(.)23 b Ft(11)p Fs(,)g(809)h(\(1976\).)244 2609 y([BM])109 b(Botvich,)27 b(D.D.,)d(Malyshe)n(v)-6 b(,)28 b(V)-12 b(.A.:)32 b(Unitary)27 b(equi)n(v)n(alence)i(of)d(temperature)i (dynamics)g(for)555 2722 y(ideal)c(and)g(locally)i(perturbed)g (Fermi-gas,)d(Commun.)g(Math.)h(Phys.)e Ft(91)p Fs(,)i(301)g(\(1983\).) 244 2874 y([BR1])84 b(Brattelli,)36 b(O,)31 b(Robinson,)37 b(D.)31 b(W)-8 b(.:)46 b Fb(Oper)o(ator)34 b(Alg)o(ebr)o(as)g(and)f (Quantum)h(Statistical)h(Me-)555 2987 y(c)o(hanics)25 b(1)p Fs(.)e(Springer)n(-V)-10 b(erlag,)26 b(Berlin,)e(second)h (edition)g(\(1987\))244 3139 y([BR2])84 b(Brattelli,)36 b(O,)31 b(Robinson,)37 b(D.)31 b(W)-8 b(.:)46 b Fb(Oper)o(ator)34 b(Alg)o(ebr)o(as)g(and)f(Quantum)h(Statistical)h(Me-)555 3252 y(c)o(hanics)25 b(2)p Fs(.)e(Springer)n(-V)-10 b(erlag,)26 b(Berlin,)e(second)h(edition)g(\(1996\).)244 3404 y([D])185 b(Da)n(vies,)24 b(E.B.:)j(Mark)o(o)o(vian)f(master)e(equations,)i (Commun.)c(Math.)i(Phys.)f Ft(39)p Fs(,)g(91)h(\(1974\).)244 3556 y([H])185 b(Haag,)23 b(R.:)28 b Fb(Local)c(Quantum)g(Physics)p Fs(.)g(Springer)n(-V)-10 b(erlag,)26 b(Berlin)e(\(1993\).)244 3708 y([HTP])78 b(Haag,)36 b(R.,)e(T)m(rych-Pohlme)o(yer)h(E.:)48 b(Stability)34 b(properties)j(of)c(equilibrium)j(states,)h(Com-)555 3821 y(mun.)23 b(Math.)g(Phys.)g Ft(56)p Fs(,)h(213)g(\(1977\).)244 3973 y([JP1])120 b(Jak)-5 b Fn(\024)-40 b(s)q Fs(i)742 3972 y(\264)737 3973 y(c,)30 b(V)-12 b(.,)29 b(Pillet,)h(C.-A.:)38 b(On)28 b(entrop)o(y)j(production)h(in)d(quantum)h(statistical)i (mechanics)555 4086 y(II.)23 b(T)m(ime-dependent)j(perturbations,)i (preprint.)244 4238 y([JP2])120 b(Jak)-5 b Fn(\024)-40 b(s)q Fs(i)742 4237 y(\264)737 4238 y(c,)23 b(V)-12 b(.,)22 b(Pillet,)i(C.-A.:)j(In)d(preparation.)244 4390 y([OP])134 b(Ohya,)21 b(M.,)f(Petz,)i(D.:)k Fb(Quantum)c(Entr)l(opy)h(and)f(its)f (Use)p Fs(,)g(Springer)n(-V)-10 b(erlag,)25 b(Berlin)d(\(1993\).)244 4542 y([R])190 b(Robinson,)38 b(D.W)-8 b(.:)47 b Fl(C)1296 4509 y Fk(\003)1335 4542 y Fs(-algebras)36 b(in)e(quantum)h (statistical)i(mechanics,)h(in)c Fl(C)3151 4509 y Fk(\003)3189 4542 y Fb(-alg)o(ebr)o(as)555 4655 y(and)g(their)h(Applications)h(to)d (Statistical)k(Mec)o(hanics)e(and)f(Quantum)g(F)l(ield)g(Theory)-5 b(,)36 b Fs(\(D.)555 4768 y(Kastler)24 b(editor\),)h(Amsterdam,)f (North-Holand)h(\(1976\).`)244 4920 y([Ru1])100 b(Ruelle,)22 b(D.:)k(Natural)c(nonequilibrium)j(states)e(in)e(quantum)h(statistical) i(mechanics,)f(J.)d(Stat.)555 5033 y(Phys.)j Ft(98)p Fs(,)g(57)h(\(2000\).)244 5185 y([Ru2])100 b(Ruelle,)24 b(D.:)k(Entrop)o(y)c(production)j(in)c(quantum)i(spin)g(systems,)f (preprint.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF