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Fm(0)2457 651 y Fk(\021)2499 627 y Fl(e)2542 586 y Fe(i)p Fk(tL)2635 597 y Fi(V)2693 627 y Fn(:)0 847 y Fp(Let)d Fl(\001)239 862 y Fk(!)r Fm(j)p Fk(\021)372 847 y Fp(be)g(the)f(relati)n(v)o(e)g (modular)g(operator)-5 b(.)30 b(W)-8 b(e)25 b(recall)h(that)e Fl(Ker)17 b(\001)2527 862 y Fk(!)r Fm(j)p Fk(\021)2662 847 y Fl(=)28 b(Ker)17 b Fn(s)2986 811 y Fm(0)2986 871 y Fk(\021)3028 847 y Fp(,)1430 1093 y Fn(J)9 b Fl(\001)1574 1042 y Fe(1)p Fk(=)p Fe(2)1574 1124 y Fk(!)r Fm(j)p Fk(\021)1684 1093 y Fn(A)p Fl(\012)1827 1108 y Fk(\021)1898 1093 y Fl(=)27 b Fn(s)2047 1052 y Fm(0)2047 1118 y Fk(\021)2089 1093 y Fn(A)2162 1052 y Fm(\003)2202 1093 y Fl(\012)2272 1108 y Fk(!)2322 1093 y Fn(;)0 1324 y Fp(for)e(all)g Fn(A)i Fo(2)h Fa(M)565 1339 y Fk(!)640 1324 y Fp(and)d(that)f Fl(\001)1064 1339 y Fk(!)r Fm(j)p Fk(\021)1197 1324 y Fp(is)h(essentially)e(self-adjoint)h(on)h Fa(M)2443 1339 y Fk(!)2493 1324 y Fl(\012)2563 1339 y Fk(\021)2627 1324 y Fl(+)d(\()p Fb(1)g Fo(\000)h Fn(s)2987 1288 y Fm(0)2987 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2231 y Fp(where)e Fn(U)344 2195 y Fm(\003)412 2231 y Fo(\021)j Fl(\000)578 2195 y Fk(t)578 2256 y(V)639 2231 y Fp(.)0 2401 y(As)e(in)f(the)h (proof)g(of)g(Theorem)g(1.1,)g(we)g(set)g Fn(P)43 b Fo(\021)31 b Fn(\031)1849 2416 y Fk(!)1900 2401 y Fl(\()p Fo(\000)p Fl(i)p Fn(U)2119 2365 y Fm(\003)2158 2401 y Fn(\016)2201 2416 y Fk(!)2252 2401 y Fl(\()p Fn(U)10 b Fl(\)\))26 b Fp(and)g(denote)g(by)g Fn(\013)h Fp(the)f(perturbation)0 2522 y(of)h(the)f(modular)g(dynamics)f Fn(\033)1080 2537 y Fk(!)1158 2522 y Fp(by)h Fn(P)14 b Fp(.)36 b(It)26 b(follo)n(ws)f(that)h Fn(!)2069 2537 y Fk(U)2124 2518 y Fc(\003)2191 2522 y Fp(is)g(the)g(unique)g Fl(\()p Fn(\013)q(;)17 b Fo(\000)p Fl(1\))p Fp(-KMS)26 b(state.)36 b(Since)0 2642 y(also)c Fo(k)p Fl(e)286 2606 y Fe(\()p Fk(L)361 2614 y Fi(!)406 2606 y Fe(+)p Fk(P)10 b Fe(\))p Fk(=)p Fe(2)618 2642 y Fl(\012)688 2657 y Fk(!)739 2642 y Fo(k)41 b Fl(=)h(1)p Fp(,)34 b(the)e(basic)g(perturbation)f(formula)h (of)h(Araki-Donald)e(\(see)i(Lemma)f(5.7)g(in)0 2762 y([Don]\))25 b(yields)1194 2982 y Fn(s)1240 2941 y Fm(0)1240 3007 y Fk(\021)1299 2982 y Fl(log)16 b(\001)1522 2998 y Fk(!)1566 3013 y Fi(U)1613 2999 y Fc(\003)1654 2998 y Fm(j)p Fk(\021)1743 2982 y Fl(=)28 b Fn(s)1893 2941 y Fm(0)1893 3007 y Fk(\021)1951 2982 y Fl(log)17 b(\001)2175 2998 y Fk(!)r Fm(j)p Fk(\021)2305 2982 y Fo(\000)23 b Fn(s)2451 2941 y Fm(0)2451 3007 y Fk(\021)2493 2982 y Fn(P)s(:)0 3202 y Fp(Hence,)1015 3422 y Fl(e)1058 3381 y Fe(i)p Fk(tL)1151 3392 y Fi(V)1209 3422 y Fn(s)1255 3381 y Fm(0)1255 3447 y Fk(\021)1314 3422 y Fl(log)16 b(\001)1537 3438 y Fk(!)r Fm(j)p Fk(\021)1646 3422 y Fl(e)1689 3381 y Fm(\000)p Fe(i)p Fk(tL)1837 3392 y Fi(V)1922 3422 y Fl(=)28 b Fn(s)2072 3381 y Fm(0)2072 3447 y Fk(\021)2130 3422 y Fl(log)17 b(\001)2354 3438 y Fk(!)r Fm(j)p Fk(\021)2484 3422 y Fo(\000)23 b Fn(s)2630 3381 y Fm(0)2630 3447 y Fk(\021)2672 3422 y Fn(P)s(;)0 3642 y Fp(and)i(we)g(conclude)f(that)h (for)g(an)o(y)f(real)h(number)g Fn(\025)i Fo(6)p Fl(=)h(0)p Fp(,)595 3873 y 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Fl(\()p Fn(U)10 b Fl(\)\))28 b(=)f Fo(\000)1989 980 y Ff(Z)2089 1006 y Fk(t)2045 1205 y Fe(0)2119 1115 y Fn(\031)2174 1130 y Fk(!)2225 1115 y Fl(\()p Fn(\034)2316 1074 y Fk(s)2305 1140 y(V)2366 1115 y Fl(\()p Fn(\016)2447 1130 y Fk(!)2497 1115 y Fl(\()p Fn(V)22 b Fl(\)\)\))17 b(d)p Fn(s;)0 1377 y Fp(and)25 b(since)f Fn(\021)29 b Fp(is)24 b Fn(\034)608 1392 y Fk(V)670 1377 y Fp(-)h(in)l(v)n(ariant)e (we)i(conclude)g(that)744 1651 y Fl(0)i(=)h(\(\012)1032 1666 y Fk(\021)1074 1651 y Fn(;)17 b(P)d Fl(\012)1265 1666 y Fk(\021)1307 1651 y Fl(\))27 b(=)h Fo(\000)1570 1516 y Ff(Z)1670 1542 y Fk(t)1625 1741 y Fe(0)1699 1651 y Fn(\021)e Fo(\016)c Fn(\034)1898 1610 y Fk(s)1887 1676 y(V)1948 1651 y Fl(\()p Fn(\016)2029 1666 y Fk(!)2080 1651 y Fl(\()p Fn(V)f Fl(\)\))c(d)p Fn(s)28 b Fl(=)f Fo(\000)p Fn(t\021)t Fl(\()p Fn(\016)2765 1666 y Fk(!)2816 1651 y Fl(\()p Fn(V)22 b Fl(\)\))p Fn(;)0 1913 y Fp(for)j(all)g Fn(t)p Fp(.)31 b(This)24 b(yields)g(the)g(statement.)0 2301 y Fq(Refer)m(ences)0 2566 y 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b(Ojima,)27 b(I.:)34 b(Entrop)o(y)26 b(production)g(and)h(non-equilibrium)d(stationarity)h(in)i(quantum)e (dynamical)336 5268 y(systems:)k(physical)24 b(meaning)g(of)h(V)-11 b(an)25 b(Ho)o(v)o(e)e(limit.)g(J.)i(Stat.)g(Phys.)f Fh(56)p Fp(,)g(203)h(\(1989\).)p eop %%Page: 8 8 8 7 bop 3730 100 a Fp(8)0 407 y([O2])148 b(Ojima,)37 b(I.:)51 b(Entrop)o(y)34 b(production)f(and)i(non-equilibrium)e (stationarity)g(in)i(quantum)e(dynami-)336 527 y(cal)e(systems,)f(in)g Fj(Pr)l(oceedings)e(of)i(international)e(workshop)i(on)f(quantum)g (aspects)h(of)g(optical)336 648 y(communications.)23 b Fp(Lecture)i(Notes)g(in)f(Physics)g Fh(378)p Fp(,)g(164,)g(Springer)n (-V)-11 b(erlag,)26 b(Berlin)f(\(1991\).)0 851 y([OP])143 b(Ohya,)24 b(M.,)h(Petz,)g(D.:)30 b Fj(Quantum)24 b(Entr)l(opy)g(and)g (its)g(Use)o(.)g Fp(Springer)n(-V)-11 b(erlag,)25 b(Berlin)g(\(1993\).) 0 1054 y([Ru])154 b(Ruelle,)30 b(D.:)38 b(Entrop)o(y)27 b(production)h(in)g(quantum)f(spin)h(systems.)f(Commun.)g(Math.)h (Phys.)g Fh(224)p Fp(,)336 1175 y(3)d(\(2001\).)0 1378 y([Sp])165 b(Spohn,)25 b(H.:)31 b(Entrop)o(y)24 b(production)f(for)j (quantum)d(dynamical)h(semigroups.)g(J.)g(Math.)h(Phys.)f Fh(19)p Fp(,)336 1499 y(227)h(\(1978\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF