This is a multi-part message in MIME format. ---------------0304070745672 Content-Type: text/plain; name="03-161.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-161.keywords" quantum statistical mechanics, reservoirs, time-dependent Hamiltonians, entropy, entropy production, CAR algebra, thermodynamic process, equilibrium state, return to equilibrium, time-periodic interaction, adiabatic theorem, adiabatic evolution, scattering morphism, free fermions ---------------0304070745672 Content-Type: application/postscript; name="fmsuarc.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fmsuarc.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: fmsuarc.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips fmsuarc -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.04.07:1332 %%BeginProcSet: 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m(y)1606 2044 y Fl( )1669 2059 y Fk(t)1727 2044 y Fs(=)27 b Fl(U)1906 2002 y Fr(\003)1960 2044 y Fs(\()p Fl(t;)17 b(s)p Fs(\))p Fl( )2224 2059 y Fk(s)2261 2044 y Fl(;)1077 b Fs(\(1.1\))328 2252 y(where)38 b Fl(U)690 2216 y Fr(\003)744 2252 y Fs(\()p Fl(t;)17 b(s)p Fs(\))36 b(denotes)i(the)g Fm(unitary)h(pr)-5 b(op)g(agator)36 b Fs(on)h Fn(H)2567 2216 y Fr(\003)2620 2252 y Fs(.)57 b(This)37 b(op)s(erator)f(is)g(the) 328 2373 y(solution)31 b(of)h(the)h(equation)1396 2558 y Fl(@)p 1379 2602 93 4 v 1379 2694 a(@)5 b(t)1481 2625 y(U)1557 2584 y Fr(\003)1610 2625 y Fs(\()p Fl(t;)17 b(s)p Fs(\))28 b(=)f Fn(\000)p Fl(iH)2141 2584 y Fr(\003)2133 2650 y Fk(t)2195 2625 y Fl(U)2271 2584 y Fr(\003)2324 2625 y Fs(\()p Fl(t;)17 b(s)p Fs(\))840 b(\(1.2\))328 2860 y(with)1684 2980 y Fl(U)1760 2939 y Fr(\003)1814 2980 y Fs(\()p Fl(s;)17 b(s)p Fs(\))27 b(=)g(1)-22 b(l)1154 b(\(1.3\))328 3149 y(and)33 b(has)g(the)g(prop)s(ert)m(y)g(that)1357 3358 y Fl(U)1433 3317 y Fr(\003)1487 3358 y Fs(\()p Fl(t;)17 b(s)p Fs(\))27 b(=)h Fl(U)1895 3317 y Fr(\003)1948 3358 y Fs(\()p Fl(t;)17 b(u)p Fs(\))p Fl(U)2235 3317 y Fr(\003)2289 3358 y Fs(\()p Fl(u;)g(s)p Fs(\))p Fl(;)827 b Fs(\(1.4\))328 3567 y(for)35 b(arbitrary)f(pairs)h(\()p Fl(t;)17 b(s)p Fs(\))p Fl(;)g Fs(\()p Fl(t;)g(u)p Fs(\))34 b(and)i(\()p Fl(u;)17 b(s)p Fs(\))34 b(of)h(times.)51 b(\(W)-8 b(e)36 b(are)f(using)g(units)g(suc)m(h)328 3687 y(that)d(Planc)m(k's)i (constan)m(t)f Fp(~)28 b Fs(=)f(1.\))474 3808 y(The)42 b(kinematics)d(of)h(the)h(system)g(is)f(enco)s(ded)h(in)f(an)g(algebra) f Fn(F)2988 3772 y Fr(\003)3081 3808 y Fs(of)h(b)s(ounded)328 3928 y(op)s(erators)32 b(on)h Fn(H)980 3892 y Fr(\003)1066 3928 y Fs(with)f(the)h(prop)s(erties)f(that)1658 4137 y Fn(F)1740 4096 y Fr(\003)1821 4137 y Fn(\022)c(B)s Fs(\()p Fn(H)2117 4096 y Fr(\003)2170 4137 y Fs(\))p Fl(;)1130 b Fs(\(1.5\))328 4346 y(and)1438 4466 y Fl(U)1514 4425 y Fr(\003)1568 4466 y Fs(\()p Fl(s;)17 b(t)p Fs(\))p Fl(AU)1918 4425 y Fr(\003)1971 4466 y Fs(\()p Fl(t;)g(s)p Fs(\))28 b Fn(2)g(F)2376 4425 y Fr(\003)2429 4466 y Fl(;)909 b Fs(\(1.6\))328 4636 y(for)32 b(all)e Fl(A)e Fn(2)g(F)889 4599 y Fr(\003)974 4636 y Fs(and)33 b(all)e(times)g Fl(s;)17 b(t)p Fs(.)474 4756 y(The)34 b(time)d(ev)m(olution)h(of)g(a)g(time-dep) s(enden)m(t)h(family)d(of)i(op)s(erators)1650 4965 y Fn(f)p Fl(A)1773 4980 y Fk(t)1802 4965 y Fn(g)1852 4980 y Fk(t)p Fj(2)p Fi(R)2004 4965 y Fn(\032)d(F)2192 4924 y Fr(\003)1922 5214 y Fs(2)p eop %%Page: 3 3 3 2 bop 328 631 a Fs(in)32 b(the)h Fm(Heisenb)-5 b(er)g(g)34 b(pictur)-5 b(e)33 b Fs(is)f(giv)m(en)g(b)m(y)1365 842 y Fl(A)p Fs(\()p Fl(t)p Fs(\))c(:=)g Fl(U)1784 801 y Fr(\003)1837 842 y Fs(\()p Fl(t)1910 857 y Fr(0)1950 842 y Fl(;)17 b(t)p Fs(\))p Fl(A)2140 857 y Fk(t)2170 842 y Fl(U)2246 801 y Fr(\003)2299 842 y Fs(\()p Fl(t;)g(t)2451 857 y Fr(0)2491 842 y Fs(\))836 b(\(1.7\))328 1054 y(where)38 b Fl(t)649 1069 y Fr(0)725 1054 y Fs(denotes)h(the)e(\\initial)c (time";)k(\(e.g.,)h(the)g(time)d(when)j(an)f(exp)s(erimen)m(t)g(in-)328 1174 y(v)m(olving)32 b(the)h(system)g(is)f(started\).)44 b(W)-8 b(e)33 b(also)f(denote)h(the)g(r.h.s.)44 b(of)33 b(\(1.7\))f(b)m(y)1758 1386 y Fl(\013)1821 1344 y Fr(\003)1820 1410 y Fk(t)1845 1419 y Fh(0)1880 1410 y Fk(;t)1930 1386 y Fs(\()p Fl(A)2041 1401 y Fk(t)2070 1386 y Fs(\))p Fl(:)1230 b Fs(\(1.8\))328 1597 y(Then)1210 1717 y Fl(A)p Fs(\()p Fl(t)p Fs(\))28 b(=)f Fl(\013)1588 1676 y Fr(\003)1587 1742 y Fk(t)1612 1751 y Fh(0)1648 1742 y Fk(;t)1697 1717 y Fs(\()p Fl(A)1808 1732 y Fk(t)1837 1717 y Fs(\))h(=)g Fl(\013)2070 1676 y Fr(\003)2069 1742 y Fk(t)2094 1751 y Fh(0)2129 1742 y Fk(;t)2174 1751 y Fh(1)2235 1717 y Fn(\016)22 b Fl(\013)2370 1676 y Fr(\003)2369 1742 y Fk(t)2394 1751 y Fh(1)2429 1742 y Fk(;t)2478 1717 y Fs(\()p Fl(A)2589 1732 y Fk(t)2619 1717 y Fs(\))p Fl(;)681 b Fs(\(1.9\))328 1888 y(for)32 b(an)g(arbitrary)g(time)f Fl(t)1284 1903 y Fr(1)1324 1888 y Fs(.)43 b(One)33 b(easily)f(v)m (eri\014es)i(that)1471 2084 y Fl(d)p 1454 2128 86 4 v 1454 2220 a(dt)1550 2151 y(A)p Fs(\()p Fl(t)p Fs(\))28 b(=)f Fl(\013)1928 2110 y Fr(\003)1927 2176 y Fk(t)1952 2185 y Fh(0)1987 2176 y Fk(;t)2053 2011 y Fg(\022)2136 2084 y Fl(D)s(A)2293 2099 y Fk(t)p 2136 2128 187 4 v 2170 2220 a Fl(D)s(t)2333 2011 y Fg(\023)2423 2151 y Fl(;)867 b Fs(\(1.10\))328 2420 y(where)34 b(the)f Fm(Heisenb)-5 b(er)g(g)34 b(derivative)p Fs(,)e Fl(D)s(A)1894 2435 y Fk(t)1923 2420 y Fl(=D)s(t)p Fs(,)g(is)h(de\014ned)h(b)m(y)1475 2612 y Fl(D)s(A)1632 2627 y Fk(t)p 1475 2657 V 1509 2748 a Fl(D)s(t)1699 2680 y Fs(=)28 b Fl(i)p Fs([)p Fl(H)1952 2638 y Fr(\003)1944 2704 y Fk(t)2005 2680 y Fl(;)17 b(A)2122 2695 y Fk(t)2151 2680 y Fs(])23 b(+)2335 2654 y(_)2299 2680 y Fl(A)2372 2695 y Fk(t)2401 2680 y Fl(:)889 b Fs(\(1.11\))474 2929 y(W)-8 b(e)33 b(assume)g(that)g(the)g(Hamiltonians)c Fl(H)2047 2893 y Fr(\003)2039 2953 y Fk(t)2132 2929 y Fs(are)k(of)f(the)h(form)1605 3140 y Fl(H)1694 3099 y Fr(\003)1686 3165 y Fk(t)1774 3140 y Fs(=)28 b Fl(H)1967 3099 y Fr(\003)1959 3165 y(0)2042 3140 y Fs(+)22 b Fl(W)2232 3155 y Fk(t)2261 3140 y Fl(;)1029 b Fs(\(1.12\))328 3352 y(where)38 b(the)f(term)f Fl(W)1115 3367 y Fk(t)1181 3352 y Fs(describ)s(es)i(a)f(time-dep)s(enden)m(t)f(p)s(erturbation)g (of)h(the)g(system.)328 3472 y(When)d(this)f(p)s(erturbation)f(is)h (turned)g(o\013)g(the)h(propagator)e(is)g(giv)m(en)i(b)m(y)g(the)f (unitary)328 3592 y(group)f Fn(f)p Fl(e)699 3556 y Fk(itH)811 3533 y Fh(\003)806 3577 y(0)862 3592 y Fn(g)912 3607 y Fk(t)p Fj(2)p Fi(R)1069 3592 y Fs(on)g Fn(H)1289 3556 y Fr(\003)1375 3592 y Fs(implemen)m(ting)e(the)j(Heisen)m(b)s(erg)g (time)e(ev)m(olution)1393 3815 y Fl(\013)1456 3768 y Fr(0)p Fk(;)p Fr(\003)1455 3838 y Fk(t)1564 3815 y Fs(\()p Fl(A)1675 3830 y Fk(t)1705 3815 y Fs(\))c(:=)h Fl(e)1946 3774 y Fk(itH)2058 3751 y Fh(\003)2053 3795 y(0)2108 3815 y Fl(A)2181 3830 y Fk(t)2211 3815 y Fl(e)2256 3774 y Fj(\000)p Fk(itH)2423 3751 y Fh(\003)2418 3795 y(0)2473 3815 y Fl(:)817 b Fs(\(1.13\))474 4027 y(The)30 b(unp)s(erturb)s(ed)g (system)g(ma)m(y)e(exhibit)h(a)f(group)h(of)f(dynamical)f(in)m(ternal)h (sym-)328 4147 y(metries)i(unitarily)e(represen)m(ted)33 b(on)e Fn(H)1799 4111 y Fr(\003)1852 4147 y Fs(.)43 b(F)-8 b(or)29 b(reasons)j(of)e(simplicit)m(y)e(of)i(our)g(exp)s(osi-)328 4267 y(tion,)h(w)m(e)h(assume)g(that)f(the)h(symmetry)f(group)g(is)g(a) g(connected)i(compact)e(Lie)g(group)328 4388 y Fn(G)6 b Fs(.)42 b(Let)28 b Fn(Z)36 b Fs(denote)29 b(an)f Fl(n)p Fs(-dimensional)d(con)m(tin)m(uous)j(connected)i(subgroup)f(con)m (tained)328 4508 y(in)i(or)g(equal)h(to)f(the)h(cen)m(tre)h(of)e(the)h (group)g Fn(G)6 b Fs(,)32 b(and)g(let)f Fl(Q)2473 4472 y Fr(\003)2473 4533 y(1)2526 4508 y Fl(;)17 b(:)g(:)g(:)f(;)h(Q)2822 4472 y Fr(\003)2822 4533 y Fk(n)2906 4508 y Fs(denote)33 b(the)f(gen-)328 4629 y(erators)i(of)g(the)g(unitra)m(y)g(represen)m (tation)h(of)e Fn(Z)42 b Fs(on)34 b Fn(H)2367 4592 y Fr(\003)2421 4629 y Fs(.)47 b(The)35 b(op)s(erators)f Fl(Q)3207 4592 y Fr(\003)3207 4653 y(1)3261 4629 y Fl(;)17 b(:)g(:)g(:)e(Q)3512 4592 y Fr(\003)3512 4653 y Fk(n)328 4749 y Fs(are)33 b(selfadjoin)m(t)e(op)s(erators)h(on)h Fn(H)1612 4713 y Fr(\003)1697 4749 y Fs(with)1178 4960 y([)p Fl(Q)1282 4919 y Fr(\003)1282 4985 y Fk(i)1335 4960 y Fl(;)17 b(Q)1456 4919 y Fr(\003)1456 4985 y Fk(j)1509 4960 y Fs(])28 b(=)f(0)p Fl(;)82 b Fs(for)32 b(all)e Fl(i;)17 b(j)34 b Fs(=)28 b(1)p Fl(;)17 b(:)g(:)g(:)e(;)i(n;)601 b Fs(\(1.14\))1922 5214 y(3)p eop %%Page: 4 4 4 3 bop 328 631 a Fs(\(in)27 b(the)h(sense)i(that)d(their)h(sp)s (ectral)f(pro)5 b(jections)28 b(comm)m(ute\),)h(and,)g(since)f Fn(G)34 b Fs(has)28 b(b)s(een)328 751 y(assumed)33 b(to)f(b)s(e)h(a)g (group)f(of)g Fm(dynamic)-5 b(al)31 b Fs(symmetries,)1414 971 y Fl(e)1459 930 y Fk(i)p Ff(\034)10 b Fj(\001)p Ff(Q)1610 905 y Fh(\003)1660 971 y Fl(e)1705 930 y Fk(itH)1817 907 y Fh(\003)1812 951 y(0)1895 971 y Fs(=)28 b Fl(e)2044 930 y Fk(itH)2156 907 y Fh(\003)2151 951 y(0)2206 971 y Fl(e)2251 930 y Fk(i)p Ff(\034)10 b Fj(\001)p Ff(Q)2403 905 y Fh(\003)2453 971 y Fl(;)837 b Fs(\(1.15\))328 1191 y(for)37 b(arbitrary)f Fe(\034)49 b Fs(=)35 b(\()p Fl(\034)1193 1206 y Fr(1)1233 1191 y Fl(;)17 b(:)g(:)g(:)f(;)h(\034)1494 1206 y Fk(n)1541 1191 y Fs(\))37 b(and)g(arbitrary)g Fl(t)p Fs(;)j(\(here)e Fe(\034)g Fn(\001)25 b Fe(Q)2816 1149 y Fr(\003)2904 1191 y Fs(:=)3043 1117 y Fg(P)3148 1143 y Fk(n)3148 1220 y(j)t Fr(=1)3292 1191 y Fl(\034)3334 1206 y Fk(j)3370 1191 y Fl(Q)3447 1155 y Fr(\003)3447 1216 y Fk(j)3501 1191 y Fs(\).)328 1312 y(W)-8 b(e)33 b(de\014ne)h Fm(gauge)g(tr)-5 b(ansformations)34 b(of)h(the)g(\014rst)g (kind)48 b Fs(b)m(y)1418 1532 y Fl(\036)1476 1491 y Fr(\003)1476 1556 y Ff(\034)1529 1532 y Fs(\()p Fl(A)p Fs(\))27 b(:=)h Fl(e)1881 1491 y Fk(i)p Ff(\034)10 b Fj(\001)p Ff(Q)2032 1466 y Fh(\003)2082 1532 y Fl(Ae)2200 1491 y Fj(\000)p Fk(i\034)e Fj(\001)p Ff(Q)2399 1466 y Fh(\003)2449 1532 y Fl(;)841 b Fs(\(1.16\))328 1752 y(for)30 b(arbitrary)f Fl(A)f Fn(2)g(F)1164 1716 y Fr(\003)1217 1752 y Fs(.)42 b(It)31 b(is)f(assumed)h(that)f Fl(\036)2142 1716 y Fr(\003)2142 1776 y Ff(\034)2225 1752 y Fs(are)h Fn(\003)p Fs(automorphisms)d(of)i Fn(F)3308 1716 y Fr(\003)3361 1752 y Fs(.)42 b(W)-8 b(e)328 1872 y(de\014ne)38 b(the)g Fl(C)864 1836 y Fj(\003)903 1872 y Fs(-algebra)d Fn(A)1361 1836 y Fr(\003)1451 1872 y Fs(of)i(\\observ)-5 b(ables")37 b(to)f(b)s(e)i(the)f(\014xed-p)s(oin) m(t)g(subalgebra)328 1993 y(of)32 b(the)h(algebra)e Fn(F)1030 1956 y Fr(\003)1116 1993 y Fs(with)h(resp)s(ect)i(to)e(the)h (automorphism)d(group)j Fn(f)p Fl(\036)2989 1956 y Fr(\003)2989 2017 y Ff(\034)3041 1993 y Fn(g)3091 2008 y Ff(\034)10 b Fj(2)p Fi(R)3233 1989 y Fd(n)3279 1993 y Fs(;)33 b(i.e.,)1196 2213 y Fn(A)1276 2171 y Fr(\003)1357 2213 y Fs(:=)27 b Fn(f)p Fl(A)h Fn(2)g(F)1814 2171 y Fr(\003)1899 2213 y Fn(j)k Fl(\036)2017 2171 y Fr(\003)2017 2237 y Ff(\034)2070 2213 y Fs(\()p Fl(A)p Fs(\))c(=)f Fl(A;)50 b Fn(8)p Fe(\034)13 b Fn(g)p Fl(:)620 b Fs(\(1.17\))474 2433 y Fm(Mixe)-5 b(d)38 b(states)e Fs(of)f(the)h(system)h(are)f(describ)s(ed)g(b)m(y)h Fm(density)h(matric)-5 b(es)p Fs(,)36 b Fl(\045)p Fs(,)h(on)f Fn(H)3486 2396 y Fr(\003)3539 2433 y Fs(,)328 2553 y(\(i.e.,)28 b(b)m(y)h(non-negativ)m(e)e(trace-class)h(op)s(erators)g(with)f(tr)p Fl(\045)h Fs(=)f(1\).)42 b(If)27 b(the)i(p)s(erturbation)328 2673 y Fl(W)420 2688 y Fk(t)482 2673 y Fs(of)j(the)h(system)h(v)-5 b(anishes,)33 b(i.e.,)1365 2893 y Fl(H)1454 2852 y Fr(\003)1446 2918 y Fk(t)1535 2893 y Fs(=)27 b Fl(H)1727 2852 y Fr(\003)1719 2918 y(0)1780 2893 y Fl(;)114 b Fs(for)32 b(all)e(times)i Fl(t)p Fs(,)789 b(\(1.18\))328 3113 y(then)24 b(the)f(notion)f(of)h Fm(thermal)i(e)-5 b(quilibrium)23 b Fs(of)f(the)i(system)g(is)e (meaningful.)38 b(A)m(t)24 b Fm(inverse)328 3234 y(temp)-5 b(er)g(atur)g(e)46 b Fl(\014)51 b Fs(and)45 b Fm(chemic)-5 b(al)45 b(p)-5 b(otentials)46 b Fl(\026)2099 3249 y Fr(1)2138 3234 y Fl(;)17 b(:)g(:)g(:)f(;)h(\026)2416 3249 y Fk(n)2463 3234 y Fs(,)48 b(the)e(equilibrium)c(state)j(is)328 3354 y(giv)m(en)33 b(b)m(y)g(the)g(densit)m(y)h(matrix)1127 3584 y Fl(\045)1177 3599 y Fk(\014)s(;)p Ff(\026)1322 3584 y Fs(:=)1452 3504 y Fg(\000)1498 3584 y Fs(\004)1563 3543 y Fr(\003)1563 3609 y Fk(\014)s(;)p Ff(\026)1680 3504 y Fg(\001)1725 3524 y Fj(\000)p Fr(1)1836 3584 y Fs(exp)18 b Fn(\000)p Fl(\014)6 b Fs([)p Fl(H)2256 3543 y Fr(\003)2248 3609 y(0)2331 3584 y Fn(\000)22 b Fe(\026)g Fn(\001)g Fe(Q)2659 3542 y Fr(\003)2713 3584 y Fs(])p Fl(:)550 b Fs(\(1.19\))328 3817 y(It)41 b(is)f(assumed,)j(here,)h(that) c(exp)18 b Fn(\000)p Fl(\014)6 b Fs([)p Fl(H)1864 3781 y Fr(\003)1856 3842 y(0)1945 3817 y Fn(\000)28 b Fe(\026)f Fn(\001)g Fe(Q)2290 3775 y Fr(\003)2343 3817 y Fs(])41 b(is)f(trace-class,)j(for)d(arbitrary)328 3938 y Fl(\014)f(>)34 b Fs(0,)j Fe(\026)d Fn(2)g Fp(R)916 3902 y Fk(n)969 3938 y Fs(;)k(the)f(normalization)32 b(factor)k(\004)2175 3902 y Fr(\003)2175 3964 y Fk(\014)s(;)p Ff(\026)2292 3938 y Fs(,)h(the)f(so-called)f Fm(gr)-5 b(and)38 b(p)-5 b(artition)328 4058 y(function)p Fs(,)28 b(is)f(c)m(hosen)i(suc)m(h)g (that)e(tr)p Fl(\045)1679 4073 y Fk(\014)s(;)p Ff(\026)1824 4058 y Fs(=)g(1,)i(and)e(one)h(commonly)e(assumes)i(that)f(the)328 4178 y(system)46 b(is)e Fm(thermo)-5 b(dynamic)g(al)5 b(ly)45 b(stable)p Fs(,)i(in)d(the)h(sense)i(that)d(the)h(thermo)s (dynamic)328 4299 y(p)s(oten)m(tial,)31 b Fl(G)844 4263 y Fr(\003)897 4299 y Fs(,)i(giv)m(en)f(b)m(y)1460 4519 y Fl(\014)6 b(G)1598 4478 y Fr(\003)1651 4519 y Fs(\()p Fl(\014)g(;)17 b Fe(\026)o Fs(\))27 b(:=)h Fn(\000)17 b Fs(ln)f(\004)2317 4478 y Fr(\003)2317 4544 y Fk(\014)s(;)p Ff(\026)3317 4519 y Fs(\(1.20\))328 4739 y(is)34 b Fm(extensive)p Fs(,)g(i.e.,)g(b)s(ounded)i(in)d(absolute)h(v)-5 b(alue)34 b(b)m(y)i(a)e(constan)m(t)h(times)e(the)i(v)m(olume)328 4859 y(of)d(\003,)g(for)g(arbitrary)g Fl(\014)h(>)28 b Fs(0)p Fl(;)17 b Fe(\026)26 b Fn(2)j Fp(R)1673 4823 y Fk(n)1725 4859 y Fs(,)k(and)g(\003)27 b Fn(\045)g Fp(R)2263 4823 y Fr(3)2309 4859 y Fs(.)1922 5214 y(4)p eop %%Page: 5 5 5 4 bop 474 631 a Fs(If,)31 b(at)g(time)e Fl(t)970 646 y Fr(0)1010 631 y Fs(,)i(the)g(system)h(is)e(in)g(a)h(mixed)f(state)h Fl(\045)p Fs(\()p Fl(t)2492 646 y Fr(0)2532 631 y Fs(\))g(then)g(its)g (state)g(at)f(time)g Fl(t)328 751 y Fs(is)i(giv)m(en)h(b)m(y)g(the)g (densit)m(y)h(matrix)1038 971 y Fl(\045)p Fs(\()p Fl(t)p Fs(\))28 b(=)f Fl(U)1406 930 y Fr(\003)1460 971 y Fs(\()p Fl(t;)17 b(t)1612 986 y Fr(0)1652 971 y Fs(\))p Fl(\045)p Fs(\()p Fl(t)1813 986 y Fr(0)1852 971 y Fs(\))p Fl(U)1966 930 y Fr(\003)2020 971 y Fs(\()p Fl(t)2093 986 y Fr(0)2133 971 y Fl(;)g(t)p Fs(\))27 b(=)h Fl(\013)2444 930 y Fr(\003)2443 996 y Fk(t;t)2513 1005 y Fh(0)2552 971 y Fs(\()p Fl(\045)p Fs(\()p Fl(t)2713 986 y Fr(0)2753 971 y Fs(\)\))p Fl(:)461 b Fs(\(1.21\))328 1191 y(Then,)34 b(using)e(equations)h(\(1.7\))f(and)h (\(1.21\),)e(w)m(e)j(\014nd)f(that)881 1411 y Fn(h)p Fl(A)993 1426 y Fk(t)1023 1411 y Fn(i)1038 1440 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))1186 1411 y Fs(:=)28 b(tr)16 b(\()p Fl(\045)p Fs(\()p Fl(t)p Fs(\))p Fl(A)1681 1426 y Fk(t)1711 1411 y Fs(\))28 b(=)f(tr)17 b(\()p Fl(\045)p Fs(\()p Fl(t)2134 1426 y Fr(0)2174 1411 y Fs(\))p Fl(A)p Fs(\()p Fl(t)p Fs(\)\))28 b(=:)f Fn(h)p Fl(A)p Fs(\()p Fl(t)p Fs(\))p Fn(i)2830 1440 y Fk(\045)p Fr(\()p Fk(t)2918 1449 y Fh(0)2954 1440 y Fr(\))2986 1411 y Fl(;)304 b Fs(\(1.22\))328 1631 y(as)33 b(exp)s(ected.)474 1752 y(The)h Fm(entr)-5 b(opy)32 b Fs(of)g(a)h(state)g(giv)m(en)f(b)m(y)i(a) e(densit)m(y)i(matrix)d Fl(\045)i Fs(is)f(de\014ned)i(b)m(y)1533 1972 y Fl(S)6 b Fs(\()p Fl(\045)p Fs(\))28 b(=)f Fn(\000)p Fs(tr)17 b(\()p Fl(\045)g Fs(ln)f Fl(\045)p Fs(\))h Fl(:)956 b Fs(\(1.23\))328 2192 y(\(W)-8 b(e)33 b(use)g(units)g(suc)m(h)h(that)e (Boltzmann's)g(constan)m(t)h Fl(k)2385 2207 y Fk(B)2473 2192 y Fs(=)28 b(1.\))474 2312 y(Since)h Fl(U)801 2276 y Fr(\003)855 2312 y Fs(\()p Fl(t;)17 b(t)1007 2327 y Fr(0)1046 2312 y Fs(\))28 b(is)g(unitary)-8 b(,)29 b(for)f(arbitrary)f Fl(t;)17 b(t)2235 2327 y Fr(0)2275 2312 y Fs(,)29 b(it)f(follo)m(ws)f (from)g(\(1.21\))h(and)g(the)328 2433 y(cyclicit)m(y)k(of)g(the)h (trace)g(that)1545 2653 y Fl(S)6 b Fs(\()p Fl(\045)p Fs(\()p Fl(t)p Fs(\)\))28 b(=)f Fl(S)6 b Fs(\()p Fl(\045)p Fs(\()p Fl(t)2206 2668 y Fr(0)2246 2653 y Fs(\)\))p Fl(;)968 b Fs(\(1.24\))328 2873 y(for)32 b(arbitrary)g Fl(t;)17 b(t)1006 2888 y Fr(0)1045 2873 y Fs(.)474 2993 y(Next,)30 b(w)m(e)f(in)m(tro)s(duce)f(the)h(notion)e(of)h(a)f(\(time-dep)s(enden) m(t\))i Fm(thermo)-5 b(dynamic)29 b(pr)-5 b(o-)328 3113 y(c)g(ess)p Fs(.)41 b(W)-8 b(e)26 b(imagine)d(that,)k(for)e(all)f (times)h Fl(t)j Fn(\024)g Fl(t)2039 3128 y Fr(0)2079 3113 y Fs(,)f(the)f(Hamiltonian)c Fl(H)2937 3077 y Fr(\003)2929 3138 y Fk(t)3017 3113 y Fs(=)28 b Fl(H)3210 3077 y Fr(\003)3202 3138 y Fk(t)3227 3147 y Fh(0)3294 3113 y Fs(=:)f Fl(H)3513 3077 y Fr(\003)3505 3138 y(0)328 3234 y Fs(is)j Fm(indep)-5 b(endent)29 b Fs(of)h(time)f Fl(t)p Fs(,)i(and)g(that)f(the)h(initial)c (state)k(at)f(time)f Fl(t)2834 3249 y Fr(0)2904 3234 y Fs(of)h(the)h(system)g(is)328 3354 y(giv)m(en)k(b)m(y)g(an)g Fm(e)-5 b(quilibrium)36 b(state)f Fl(\045)1650 3369 y Fk(\014)s(;)p Ff(\026)1799 3354 y Fs(=:)c Fl(\045)1983 3369 y Fk(\014)s(;)p Ff(\026)2100 3354 y Fs(\()p Fl(t)2173 3369 y Fr(0)2213 3354 y Fs(\),)k(as)g(de\014ned)h(in)e(equation)g (\(1.19\),)328 3474 y(for)27 b(some)g(in)m(v)m(erse)i(temp)s(erature)f Fl(\014)33 b Fs(and)27 b(c)m(hemical)g(p)s(oten)m(tials)f Fe(\026)o Fs(.)42 b(W)-8 b(e)28 b(are)g(in)m(terested)328 3595 y(in)34 b(studying)i(the)f(e\013ects)i(of)e Fm(lo)-5 b(c)g(al,)37 b(external)g(p)-5 b(erturb)g(ations)35 b Fs(acting)f(on)h(the)h(system.)328 3715 y(In)j(order)g(to)f(mak)m(e)g (more)g(precise)i(what)f(w)m(e)g(are)g(talking)e(ab)s(out,)i(w)m(e)h (assume)f(that)328 3836 y(the)f(systems)h(considered)f(in)f(this)g (note)h(ha)m(v)m(e)h(a)e Fm(lo)-5 b(c)g(al)39 b(structur)-5 b(e)p Fs(:)55 b(if)36 b(\003)3045 3851 y Fr(0)3122 3836 y Fs(is)h(an)g(arbi-)328 3956 y(trary)30 b(con)m(v)m(ex)j(subset)f(of)e (the)g(con)m(v)m(ex)j(region)c(\003)h(con)m(taining)f(the)i(system,)h (and)e(\003)p Fn(n)p Fs(\003)3527 3971 y Fr(0)328 4076 y Fs(denotes)47 b(its)f(complemen)m(t)g(then)h(the)f(Hilb)s(ert)f (space)j Fn(H)2530 4040 y Fr(\003)2629 4076 y Fs(of)e(the)h(system)g (can)g(b)s(e)328 4197 y(factorized)32 b(in)m(to)1523 4317 y Fn(H)1608 4276 y Fr(\003)1689 4317 y Fs(=)27 b Fn(H)1877 4276 y Fr(\003)1926 4285 y Fh(0)1987 4317 y Fn(\012)c(H)2172 4276 y Fr(\003)p Fj(n)p Fr(\003)2305 4285 y Fh(0)2344 4317 y Fl(;)328 4491 y Fs(where)39 b Fn(H)700 4455 y Fr(\003)749 4464 y Fh(0)826 4491 y Fs(can)f(b)s(e)h(in) m(terpreted)f(as)h(the)f(Hilb)s(ert)f(space)i(of)e(pure)i(state)g(v)m (ectors)g(of)328 4612 y(the)27 b(degrees)h(of)e(freedom)g(lo)s(calized) f(in)g(\003)1864 4627 y Fr(0)1904 4612 y Fs(.)41 b(Let)27 b Fn(F)2223 4576 y Fr(\003)2272 4585 y Fh(0)2338 4612 y Fn(\022)h(B)s Fs(\()p Fn(H)2634 4576 y Fr(\003)2683 4585 y Fh(0)2722 4612 y Fs(\))f(b)s(e)f(the)h(kinematical)328 4732 y(algebra)k(asso)s(ciated)i(to)f(the)h(region)f(\003)1782 4747 y Fr(0)1821 4732 y Fs(,)g(see)i(\(1.5\).)43 b(Then)34 b(the)f(subalgebra)1518 4952 y Fn(F)1600 4911 y Fr(\003)1649 4920 y Fh(0)1709 4952 y Fn(\012)23 b Fs(1)-22 b(l)p Fn(j)1892 4977 y Fj(H)1952 4955 y Fh(\003)p Fc(n)p Fh(\003)2066 4970 y(0)2136 4952 y Fn(\032)28 b(F)2323 4911 y Fr(\003)3317 4952 y Fs(\(1.25\))1922 5214 y(5)p eop %%Page: 6 6 6 5 bop 328 631 a Fs(is)43 b(naturally)f(iden)m(ti\014ed)i(with)f Fn(F)1620 595 y Fr(\003)1669 604 y Fh(0)1708 631 y Fs(.)77 b(Without)43 b(an)m(y)h(essen)m(tial)g(loss)f(of)h(generalit)m(y)-8 b(,)328 751 y(w)m(e)41 b(ma)m(y)f(assume)h(that)f(the)h(gauge)f (transformations)e Fl(\036)2490 715 y Fr(\003)2490 776 y Ff(\034)2583 751 y Fs(in)m(tro)s(duced)j(in)e(equation)328 872 y(\(1.16\))30 b(lea)m(v)m(e)h(the)g(subalgebra)f Fn(F)1581 836 y Fr(\003)1630 845 y Fh(0)1699 872 y Fs(of)g Fn(F)1890 836 y Fr(\003)1973 872 y Fs(in)m(v)-5 b(arian)m(t,)29 b(and)i(when)g(restricted)g(to)g Fn(F)3479 836 y Fr(\003)3528 845 y Fh(0)328 992 y Fs(coincide)26 b(with)f Fl(\036)969 956 y Fr(\003)1018 965 y Fh(0)969 1017 y Ff(\034)1066 992 y Fn(\012)9 b Fs(id)p Fn(j)1262 1017 y Fr(Aut\()p Fj(F)1465 995 y Fh(\003)p Fc(n)p Fh(\003)1579 1010 y(0)1619 1017 y Fr(\))1651 992 y Fs(.)41 b(Then)28 b(the)e(algebra)g Fn(A)2545 956 y Fr(\003)2594 965 y Fh(0)2658 992 y Fs(can)h(b)s(e)f (iden)m(ti\014ed)g(with)1437 1228 y Fn(f)p Fl(A)h Fn(2)h(F)1763 1187 y Fr(\003)1812 1196 y Fh(0)1883 1228 y Fn(j)k Fl(\036)2001 1187 y Fr(\003)2001 1253 y Ff(\034)2054 1228 y Fs(\()p Fl(A)p Fs(\))c(=)f Fl(A)p Fn(g)860 b Fs(\(1.26\))328 1448 y(and)33 b(will)d(b)s(e)j(view)m(ed)g(as)g(a)f(subalgebra)h(of)f Fn(A)2035 1412 y Fr(\003)2087 1448 y Fs(,)h(for)f(arbitrary)g(\003)2779 1463 y Fr(0)2846 1448 y Fn(\032)c Fs(\003.)474 1569 y(In)45 b(the)g(follo)m(wing,)h(w)m(e)f(shall)e(k)m(eep)k(\003)1941 1584 y Fr(0)2028 1569 y Fn(\032)i Fs(\003)44 b(\014xed)i(and)f(view)g (the)g(degrees)h(of)328 1689 y(freedom)32 b(lo)s(calized)f(in)g(\003) 1279 1704 y Fr(0)1351 1689 y Fs(as)i(a)f(\014nite)h(subsystem)h(of)e (the)h(en)m(tire)g(system,)h(while)d(the)328 1809 y(regions)h(\003)g (will)f(b)s(e)h(let)g(to)g(increase)i(to)e Fp(R)1891 1773 y Fr(3)1936 1809 y Fs(,)h(ev)m(en)m(tually)-8 b(.)474 1930 y(The)33 b(in)m(teraction)e(term)g Fl(W)1488 1945 y Fk(t)1518 1930 y Fs(,)h(giv)m(en)g(in)f(\(1.12\),)g(describ)s(es)i (the)f(dynamical)e(e\013ects)328 2050 y(of)f(an)h(external)g(p)s (erturbation)f(acting)g(on)h(the)g(system)h(and)f(is)f(assumed)i(to)e (ha)m(v)m(e)i(the)328 2171 y(follo)m(wing)f(prop)s(erties:)420 2374 y(\(i\))48 b Fl(W)664 2389 y Fk(t)721 2374 y Fs(=)28 b(0,)k(for)g Fl(t)c(<)g(t)1284 2389 y Fr(0)1323 2374 y Fs(;)33 b(and)393 2577 y(\(ii\))47 b Fl(W)664 2592 y Fk(t)729 2577 y Fs(is)34 b Fm(lo)-5 b(c)g(al)34 b Fs(in)h(the)g (sense)i(that)e Fl(W)1894 2592 y Fk(t)1955 2577 y Fn(2)e(F)2136 2541 y Fr(\003)2185 2550 y Fh(0)2223 2577 y Fs(,)i(for)g(all)e(times)h Fl(t)p Fs(,)i(where)g(\003)3288 2592 y Fr(0)3362 2577 y Fs(is)f(an)572 2698 y(arbitrary)-8 b(,)31 b(but)i(\014xed)h(b)s (ounded,)f(con)m(v)m(ex)i(subset)f(of)e Fp(R)2641 2662 y Fr(3)2719 2698 y Fs(\()p Fm(indep)-5 b(endent)31 b Fs(of)h Fl(t)p Fs(\).)328 2901 y(A)e(thermo)s(dynamic)e(pro)s(cess)j (is)e Fm(char)-5 b(ge-c)g(onserving)27 b Fs(i\013)i Fl(W)2544 2916 y Fk(t)2603 2901 y Fs(is)g Fm(gauge-invariant)p Fs(,)g(i.e.,)328 3022 y Fl(W)420 3037 y Fk(t)477 3022 y Fn(2)f(A)651 2985 y Fr(\003)700 2994 y Fh(0)739 3022 y Fs(,)k(for)g(all)f(times)g Fl(t)p Fs(.)474 3142 y(Later,)k(w)m(e)g (shall)e(also)g(assume)h(that)g Fl(W)1980 3157 y Fk(t)2044 3142 y Fs(is)g Fm(smal)5 b(l)33 b Fs(in)g(the)i(sense)h(that)e(a)g (suitable)328 3262 y(norm)e(of)g Fl(W)786 3277 y Fk(t)848 3262 y Fs(is)g(assumed)h(to)g(b)s(e)f(small,)f Fm(uniformly)h Fs(in)g Fl(t)p Fs(.)474 3383 y(The)k(p)s(erturbation)f Fl(W)1344 3398 y Fk(t)1409 3383 y Fs(ma)m(y)g(describ)s(e,)i(for)d (example,)i(the)f(e\013ects)i(of)e(shining)f(a)328 3503 y(fo)s(cussed)28 b(b)s(eam)e(of)h(ligh)m(t)e(in)m(to)h(the)i(system,)h (or)d(of)h(lo)s(cal,)f(time-dep)s(enden)m(t)h(v)-5 b(ariations)328 3623 y(of)40 b(an)h(external)g(magnetic)f(\014eld)g(applied)g(to)g(the) i(system,)i(or)c(of)h(the)g(motion)e(of)h(a)328 3744 y(piston)30 b(con\014ning)g(particles)g(to)g(a)g(time-dep)s(enden)m(t)h (subset,)i(\003)2704 3759 y Fk(t)2733 3744 y Fs(,)e(of)f(\003,)h(with)f (\003)p Fn(n)p Fs(\003)3432 3759 y Fk(t)3488 3744 y Fn(\022)328 3864 y Fs(\003)396 3879 y Fr(0)435 3864 y Fs(.)44 b(Th)m(us,)34 b Fl(W)872 3879 y Fk(t)934 3864 y Fs(is)e(t)m(ypically)f(of)i(the)g (form)1627 4084 y Fl(W)1719 4099 y Fk(t)1777 4084 y Fs(=)27 b Fl(W)14 b Fs(\()p Fe(\025)o Fs(\()p Fl(t)p Fs(\)\))p Fl(;)1051 b Fs(\(1.27\))328 4304 y(where)34 b Fe(\025)27 b Fs(=)h(\()p Fl(\025)903 4319 y Fr(1)942 4304 y Fl(;)17 b(:)g(:)g(:)f(;)h(\025)1218 4319 y Fk(k)1260 4304 y Fs(\))33 b(is)f(a)g(\014nite)g(set)i(of)e Fm(external)i(c)-5 b(ontr)g(ol)35 b(p)-5 b(ar)g(ameters)p Fs(,)32 b(and)g(the)328 4425 y(time-dep)s(endence)k(of)f Fl(W)1282 4440 y Fk(t)1348 4425 y Fs(is)g(en)m(tirely)g(due)h(to)g(a)f(p)s(ossible)g(time-dep)s (endence)h(of)f(the)328 4545 y(con)m(trol)d(parameters)g Fe(\025)p Fs(.)474 4665 y(If)47 b(the)h(ratio)d(v)m(olume\(\003)1426 4680 y Fr(0)1465 4665 y Fs(\):v)m(olume\(\003\))h(is)g(v)m(ery)j(small) c(the)i(subsystem)i(in)d(the)328 4786 y(region)24 b(\003)p Fn(n)p Fs(\003)799 4801 y Fr(0)862 4786 y Fs(can)h(b)s(e)g(in)m (terpreted)h(as)f(a)f Fm(thermostat)h Fs(for)f(the)h(small)e(subsystem) k(in)d(\003)3500 4801 y Fr(0)3539 4786 y Fs(,)328 4906 y(k)m(eeping)36 b(the)f(v)-5 b(alues)35 b(of)g(the)g(temp)s(erature)g (and)h(the)f(c)m(hemical)f(p)s(oten)m(tials)g(constan)m(t)1922 5214 y(6)p eop %%Page: 7 7 7 6 bop 328 631 a Fs(throughout)25 b(a)f(thermo)s(dynamic)f(pro)s (cess.)42 b(Since)25 b(w)m(e)h(ha)m(v)m(e)g(assumed)g(that)e(the)i (initial)328 751 y(state,)33 b Fl(\045)p Fs(\()p Fl(t)717 766 y Fr(0)757 751 y Fs(\))28 b(=)f Fl(\045)976 766 y Fk(t)1001 775 y Fh(0)1041 751 y Fs(,)32 b(of)g(the)h(en)m(tire)g (system)h(at)e(time)f Fl(t)2353 766 y Fr(0)2425 751 y Fs(is)h(an)h(equilibrium)c(state,)1561 967 y Fl(\045)p Fs(\()p Fl(t)1684 982 y Fr(0)1724 967 y Fs(\))e(=)h Fl(\045)1943 982 y Fk(t)1968 991 y Fh(0)2035 967 y Fs(=)g Fl(\045)2189 982 y Fk(\014)s(;)p Ff(\026)2306 967 y Fl(;)984 b Fs(\(1.28\))328 1183 y(as)41 b(de\014ned)h(in)e(equation)g(\(1.19\),)i(with)e Fl(H)1966 1147 y Fr(\003)1958 1208 y Fk(t)1983 1217 y Fh(0)2064 1183 y Fs(=)h Fl(H)2270 1147 y Fr(\003)2262 1208 y(0)2323 1183 y Fs(,)h(w)m(e)g(are)e(studying)h(thermo)s(dy-)328 1303 y(namic)33 b(pro)s(cesses)k(at)d(constan)m(t)i(temp)s(erature)e Fl(\014)2176 1267 y Fj(\000)p Fr(1)2305 1303 y Fs(and)g(constan)m(t)i (c)m(hemical)d(p)s(oten-)328 1424 y(tials)d Fe(\026)p Fs(;)i(\(at)g(least)g(after)g(passing)g(to)g(the)g(thermo)s(dynamic)f (limit)e(\003)e Fn(\045)g Fp(R)3132 1388 y Fr(3)3178 1424 y Fs(,)32 b(with)g(\003)3527 1439 y Fr(0)328 1544 y Fs(k)m(ept)i(\014xed\).)474 1664 y(The)g Fm(true)h(state)e Fs(of)f(the)h(system)h(at)e(time)f Fl(t)i Fs(is)f(giv)m(en)h(b)m(y)1563 1880 y Fl(\045)p Fs(\()p Fl(t)p Fs(\))28 b(=)g Fl(\013)1919 1839 y Fr(\003)1918 1905 y Fk(t;t)1988 1914 y Fh(0)2027 1880 y Fs(\()p Fl(\045)p Fs(\()p Fl(t)2188 1895 y Fr(0)2228 1880 y Fs(\)\))p Fl(;)986 b Fs(\(1.29\))328 2096 y(see)23 b(equation)e(\(1.21\).)39 b(If)22 b(the)g(time-dep)s(endence)h(of)e (the)h(p)s(erturbation)f Fl(W)3067 2111 y Fk(t)3124 2096 y Fs(=)28 b Fl(W)14 b Fs(\()p Fe(\025)o Fs(\()p Fl(t)p Fs(\)\))328 2216 y(is)37 b Fm(slow)g Fs(it)g(is)g(of)g(in)m(terest)h (to)f(compare)h(the)g(true)g(state)g Fl(\045)p Fs(\()p Fl(t)p Fs(\))g(of)f(the)h(system)h(with)e(a)328 2337 y(reference)d(state,)f Fl(\045)1053 2352 y Fk(t)1083 2337 y Fs(,)g(giv)m(en)f(b)m(y)1069 2553 y Fl(\045)1119 2568 y Fk(t)1177 2553 y Fs(:=)c Fl(e)1353 2511 y Fk(\014)s(G)1451 2488 y Fh(\003)1496 2511 y Fr(\()p Fk(\014)s(;)p Ff(\026)p Fr(;)p Ff(\025)p Fr(\()p Fk(t)p Fr(\)\))1832 2553 y Fs(exp)18 b Fn(\000)p 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Fs(\)\).)43 b(W)-8 b(e)32 b(shall)d(call)h(the)h(state)g Fl(\045)3162 3197 y Fk(t)3223 3182 y Fs(in)f(equa-)328 3302 y(tion)h(\(1.30\))h(the)h Fm(r)-5 b(efer)g(enc)g(e)34 b(state)f Fs(at)f(time)f Fl(t)p Fs(.)474 3422 y(An)46 b(imp)s(ortan)m(t)d(quan)m(tit)m(y)j(in)f(the)h(c)m(haracterization)e (of)h(thermo)s(dynamic)f(pro-)328 3543 y(cesses)31 b(is)d(the)i Fm(r)-5 b(elative)31 b(entr)-5 b(opy)28 b Fs(of)h(the)g(reference)h (state)g Fl(\045)2506 3558 y Fk(t)2564 3543 y Fs(with)f(resp)s(ect)h (to)e(the)h(true)328 3663 y(state)k Fl(\045)p Fs(\()p Fl(t)p Fs(\))g(of)f(the)h(system,)h(whic)m(h)f(is)f(giv)m(en)h(b)m(y) 419 3879 y Fl(S)6 b Fs(\()p Fl(\045)573 3894 y Fk(t)603 3879 y Fn(j)p Fl(\045)p Fs(\()p Fl(t)p Fs(\)\))28 b(:=)f Fn(\000)p Fs(tr)1158 3798 y Fg(\000)1204 3879 y Fl(\045)p Fs(\()p Fl(t)p Fs(\)[ln)16 b Fl(\045)1540 3894 y Fk(t)1592 3879 y Fn(\000)23 b Fs(ln)16 b Fl(\045)p Fs(\()p Fl(t)p Fs(\)])1978 3798 y Fg(\001)2052 3879 y Fs(=)27 b Fn(\000)p Fs(tr)33 b(\()p Fl(\045)p Fs(\()p Fl(t)p Fs(\))17 b(ln)f Fl(\045)2705 3894 y Fk(t)2735 3879 y Fs(\))22 b Fn(\000)h Fl(S)6 b Fs(\()p Fl(\045)p Fs(\()p Fl(t)p Fs(\)\))p Fl(;)92 b Fs(\(1.32\))328 4095 y(where)36 b(the)g(en)m(trop)m(y)h Fl(S)6 b Fs(\()p Fl(\045)p Fs(\()p Fl(t)p Fs(\)\))35 b(of)g Fl(\045)p Fs(\()p Fl(t)p Fs(\))h(has)f(b)s(een)h(de\014ned)h(in) e(\(1.23\).)51 b(If)35 b Fl(A)g Fs(is)g(a)g(non-)328 4215 y(negativ)m(e)c(trace-class)f(op)s(erator)g(and)h Fl(B)k Fs(is)30 b(a)g(strictly)g(p)s(ositiv)m(e)g(trace-class)g(op)s (erator)328 4336 y(then)1174 4456 y Fn(\000)p Fs(tr)j(\()p Fl(A)17 b Fs(ln)e Fl(B)28 b Fn(\000)22 b Fl(A)17 b Fs(ln)f Fl(A)p Fs(\))28 b Fn(\025)g Fs(tr)33 b(\()o Fl(A)23 b Fn(\000)f Fl(B)5 b Fs(\))17 b(;)328 4629 y(see)37 b(e.g.)54 b(Lemma)34 b(6.2.21)i(of)f([3].)54 b(Setting)35 b Fl(A)f Fs(=)f Fl(\045)p Fs(\()p Fl(t)p Fs(\))j(and)h Fl(B)h Fs(=)33 b Fl(\045)2841 4644 y Fk(t)2871 4629 y Fs(,)k(and)f(using)g (that)328 4749 y(tr)16 b Fl(\045)p Fs(\()p Fl(t)p Fs(\))28 b(=)g(tr)16 b Fl(\045)855 4764 y Fk(t)912 4749 y Fs(=)28 b(1,)k(w)m(e)i(conclude)f(that)1637 4965 y Fl(S)6 b Fs(\()p Fl(\045)1791 4980 y Fk(t)1821 4965 y Fn(j)p Fl(\045)p Fs(\()p Fl(t)p Fs(\)\))28 b Fn(\025)g Fs(0)p Fl(;)1060 b Fs(\(1.33\))1922 5214 y(7)p eop %%Page: 8 8 8 7 bop 328 631 a Fs(for)32 b(all)e(times)i Fl(t)p Fs(.)44 b(By)33 b(equation)f(\(1.24\),)1358 843 y Fl(S)6 b Fs(\()p Fl(\045)p Fs(\()p Fl(t)p Fs(\)\))27 b(=)h Fl(S)6 b Fs(\()p Fl(\045)p Fs(\()p Fl(t)2019 858 y Fr(0)2059 843 y Fs(\)\))27 b(=:)h Fl(S)6 b Fs(\()p Fl(t)2432 858 y Fr(0)2471 843 y Fs(\))p Fl(:)781 b Fs(\(1.34\))328 1056 y(It)33 b(then)g(follo)m(ws)e (from)g(\(1.32\))h(and)h(\(1.33\))f(that)1283 1268 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))27 b(:=)h Fn(\000)p Fs(tr)1788 1187 y Fg(\000)1833 1268 y Fl(\045)p Fs(\()p Fl(t)p Fs(\))17 b(ln)f Fl(\045)2159 1283 y Fk(t)2189 1187 y Fg(\001)2262 1268 y Fn(\025)29 b Fl(S)6 b Fs(\()p Fl(t)2507 1283 y Fr(0)2546 1268 y Fs(\))p Fl(;)706 b Fs(\(1.35\))328 1480 y(for)32 b(all)e(times)i Fl(t)c Fn(\025)g Fl(t)1076 1495 y Fr(0)1116 1480 y Fs(.)43 b(Next,)34 b(w)m(e)f(note)g(that,)g(b)m(y)g (equation)f(\(1.30\),)981 1720 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))28 b(=)g Fl(\014)1367 1609 y Fg(h)1414 1639 y(\012)1461 1720 y Fl(H)1550 1679 y Fr(\003)1542 1744 y Fk(t)1625 1720 y Fn(\000)22 b Fe(\026)g Fn(\001)g Fe(Q)1953 1678 y Fr(\003)2006 1639 y Fg(\013)2030 1759 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))2173 1720 y Fn(\000)g Fl(G)p Fs(\()p Fl(\014)6 b(;)17 b Fe(\026)o Fs(;)g Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))2822 1609 y Fg(i)2885 1720 y Fl(:)405 b Fs(\(1.36\))328 1966 y(It)33 b(is)f(natural)f(to)h(de\014ne)i(the)f Fm(internal)i(ener)-5 b(gy)p Fs(,)32 b Fl(U)2218 1930 y Fr(\003)2272 1966 y Fs(\()p Fl(t)p Fs(\),)g(of)g(the)h(sytem)g(at)g(time)e Fl(t)i Fs(b)m(y)1567 2178 y Fl(U)1643 2137 y Fr(\003)1697 2178 y Fs(\()p Fl(t)p Fs(\))28 b(:=)1966 2098 y Fg(\012)2014 2178 y Fl(H)2103 2137 y Fr(\003)2095 2203 y Fk(t)2155 2098 y Fg(\013)2179 2217 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))2299 2178 y Fl(;)991 b Fs(\(1.37\))328 2403 y(and)33 b(the)g(v)-5 b(arious)31 b Fm(char)-5 b(ge)35 b(densities)c Fs(b)m(y)1588 2616 y Fl(q)1635 2575 y Fr(\003)1631 2640 y Fk(j)1688 2616 y Fs(\()p Fl(t)p Fs(\))d(:=)1957 2535 y Fg(\012)2004 2616 y Fl(Q)2081 2575 y Fr(\003)2081 2640 y Fk(j)2135 2535 y Fg(\013)2158 2657 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))2279 2616 y Fl(:)1011 b Fs(\(1.38\))328 2843 y(Dropping)31 b(sup)s(erscripts)j(\003,)e(it)g(follo)m(ws)f(that)1504 3056 y Fl(G)c Fs(=)h Fl(U)k Fn(\000)23 b Fe(\026)f Fn(\001)g Fe(q)j Fn(\000)e Fl(T)14 b(S;)927 b Fs(\(1.39\))328 3268 y(whic)m(h)37 b(is)f(the)g(usual)g(relation)f(b)s(et)m(w)m(een)j(the)f (Gibbs)f(p)s(oten)m(tial)e Fl(G)j Fs(and)f(the)h(in)m(ternal)328 3388 y(energy)-8 b(,)39 b(c)m(harge)f(densities)g(and)f(the)h(en)m (trop)m(y)-8 b(.)59 b(All)36 b(these)i(quan)m(tities)f(are)h Fm(extensive)328 3509 y Fs(and,)i(hence,)h(do)d(not)g(ha)m(v)m(e)h(a)f (limit,)f(as)h(\003)g(increases)h(to)f Fp(R)2597 3473 y Fr(3)2642 3509 y Fs(.)60 b(It)39 b(is)e(more)h(useful)g(to)328 3629 y(consider)28 b(their)g Fm(time)j(derivatives)p Fs(.)41 b(T)-8 b(aking)28 b(the)g(time)f(deriv)-5 b(ativ)m(e)28 b(of)g(equation)g(\(1.36\))328 3749 y(it)j(follo)m(ws)h(that)1143 3961 y(_)1115 3986 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))28 b(=)f Fl(\014)1511 3961 y Fs(_)1484 3986 y Fl(U)11 b Fs(\()p Fl(t)p Fs(\))22 b Fn(\000)g Fl(\014)6 b Fe(\026)22 b Fn(\001)2012 3985 y Fs(_)1997 3986 y Fe(q)s Fs(\()p Fl(t)p Fs(\))g Fn(\000)h Fl(\014)2357 3919 y(@)5 b(G)p 2357 3963 134 4 v 2362 4055 a(@)g Fe(\025)2524 3986 y Fn(\001)2593 3960 y Fs(_)2573 3986 y Fe(\025)p Fs(\()p Fl(t)p Fs(\))p Fl(:)539 b Fs(\(1.40\))328 4224 y(The)33 b(com)m(bination)1415 4369 y Fl(d)1428 4328 y Fj(\000)1454 4369 y Fl(A)27 b Fs(:=)h Fn(\000)p Fe(\026)22 b Fn(\001)g Fl(d)p Fe(q)k Fn(\000)2145 4301 y Fl(@)5 b(G)p 2145 4346 V 2150 4437 a(@)g Fe(\025)2311 4369 y Fn(\001)22 b Fl(d)p Fe(\025)838 b Fs(\(1.41\))328 4565 y(is)32 b(commonly)e(in)m(terpreted) k(as)e(the)h Fm(work)f Fs(done)h(b)m(y)g(the)g(system)g(during)f(a)g(c) m(hange)h(of)328 4686 y(state.)44 b(Hence)34 b(w)m(e)f(conclude)g(that) 1662 4904 y(_)1634 4930 y Fl(U)39 b Fs(=)27 b Fl(T)1941 4904 y Fs(_)1913 4930 y Fl(S)h Fn(\000)2111 4862 y Fl(d)2124 4826 y Fj(\000)2149 4862 y Fl(A)p 2111 4907 112 4 v 2124 4998 a(dt)2232 4930 y(;)1058 b Fs(\(1.42\))1922 5214 y(8)p eop %%Page: 9 9 9 8 bop 328 631 a Fs(whic)m(h)28 b(summarizes)f(the)h(\014rst)g(and)g (second)h(la)m(w)e(of)g(thermo)s(dynamics)g(\(for)g Fm(r)-5 b(eversible)328 751 y(pr)g(o)g(c)g(esses)p Fs(\).)474 872 y(It)27 b(is)f(imp)s(ortan)m(t)f(to)h(notice)g(that,)i(under)f(our) g(assumptions)f(on)h(the)g(p)s(erturbation)328 992 y(op)s(erator)32 b Fl(W)813 1007 y Fk(t)870 992 y Fs(=)c Fl(W)14 b Fs(\()p Fe(\025)o Fs(\()p Fl(t)p Fs(\)\),)33 b(the)g(quan)m(tities)1235 1146 y(_)1207 1171 y Fl(U)1283 1130 y Fr(\003)1337 1171 y Fs(\()p Fl(t)p Fs(\))p Fl(;)1507 1170 y Fs(_)1492 1171 y Fe(q)1550 1130 y Fr(\003)1603 1171 y Fs(\()p Fl(t)p Fs(\))65 b(and)g(\()p Fl(@)5 b(G)2172 1130 y Fr(\003)2226 1171 y Fl(=@)g Fe(\025)p Fs(\))23 b Fn(\001)2528 1145 y Fs(_)2508 1171 y Fe(\025)p Fs(\()p Fl(t)p Fs(\))328 1350 y(ha)m(v)m(e)39 b(\014nite)e(thermo)s(dynamic)g(limits.)55 b(By)38 b(equations)g(\(1.29\))f(and)h(\(1.37\),)g(and)g(b)s(e-)328 1470 y(cause)1163 1560 y Fl(D)s(H)1336 1524 y Fr(\003)1328 1585 y Fk(t)p 1163 1604 226 4 v 1216 1696 a Fl(D)s(t)1426 1627 y Fs(=)1565 1602 y(_)1529 1627 y Fl(H)1618 1586 y Fr(\003)1610 1652 y Fk(t)1698 1627 y Fs(=)1841 1602 y(_)1802 1627 y Fl(W)1894 1642 y Fk(t)1951 1627 y Fs(=)2065 1560 y Fl(@)5 b(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))p 2065 1604 417 4 v 2211 1696 a Fl(@)5 b Fe(\025)2513 1627 y Fn(\001)2583 1601 y Fs(_)2563 1627 y Fe(\025)p Fs(\()p Fl(t)p Fs(\))328 1810 y(it)31 b(follo)m(ws)h(that)1308 1937 y(_)1281 1962 y Fl(U)1357 1921 y Fr(\003)1411 1962 y Fs(\()p Fl(t)p Fs(\))27 b(=)1653 1822 y Fg(\034)1738 1895 y Fl(@)5 b(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))p 1738 1939 V 1884 2031 a Fl(@)5 b Fe(\025)2164 1822 y Fg(\035)2215 2061 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))2358 1962 y Fn(\001)2428 1936 y Fs(_)2408 1962 y Fe(\025)p Fs(\()p Fl(t)p Fs(\))p Fl(:)704 b Fs(\(1.43\))328 2192 y(Similarly)-8 b(,)639 2431 y(_)624 2432 y Fe(q)682 2391 y Fr(\003)735 2432 y Fs(\()p Fl(t)p Fs(\))83 b(=)1088 2292 y Fg(\034)1173 2365 y Fl(D)s Fe(Q)1343 2323 y Fr(\003)p 1173 2410 224 4 v 1225 2501 a Fl(D)s(t)1406 2292 y Fg(\035)1457 2531 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))929 2718 y Fs(=)g Fl(i)1121 2637 y Fg(\012)1168 2718 y Fs([)p Fl(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))p Fl(;)j Fe(Q)1685 2676 y Fr(\003)1738 2718 y Fs(])1765 2637 y Fg(\013)1789 2757 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))1937 2718 y Fs(=)28 b Fn(\000)2161 2650 y Fl(@)p 2128 2695 123 4 v 2128 2786 a(@)5 b Fe(\034)2260 2607 y Fg(D)2321 2718 y Fl(\036)2379 2677 y Fr(\003)2379 2742 y Ff(\034)2432 2607 y Fg(\020)2492 2718 y Fl(W)14 b Fs(\()p Fe(\025)o Fs(\()p Fl(t)p Fs(\)\))2851 2607 y Fg(\021E)2948 2787 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))3068 2603 y Fg(\014)3068 2663 y(\014)3068 2723 y(\014)3102 2787 y Ff(\034)9 b Fr(=0)3242 2718 y Fl(;)48 b Fs(\(1.44\))328 2951 y(where)42 b(the)g(gauge)f(transformations)e Fl(\036)1849 2915 y Fr(\003)1849 2976 y Ff(\034)1943 2951 y Fs(ha)m(v)m(e)k(b)s(een)f(de\014ned)g(in)f(equation)f(\(1.16\).) 328 3090 y(Under)28 b(our)g(h)m(yp)s(otheses)i(on)d Fl(W)14 b Fs(\()p Fe(\025)o Fs(\()p Fl(t)p Fs(\)\),)29 b(the)f(op)s(erators) 2423 3042 y Fk(@)t(W)10 b Fr(\()p Ff(\025)q Fr(\()p Fk(t)p Fr(\)\))p 2423 3067 301 4 v 2529 3124 a Fk(@)t Ff(\025)2761 3090 y Fs(and)2956 3042 y Fk(@)t(\036)3039 3019 y Fh(\003)3039 3059 y Fb(\034)3084 3042 y Fr(\()p Fk(W)g Fr(\()p Ff(\025)q Fr(\()p Fk(t)p Fr(\)\)\))p 2956 3067 443 4 v 3133 3124 a Fk(@)t Ff(\034)3436 3090 y Fs(are)328 3210 y(strictly)35 b Fm(lo)-5 b(c)g(al)p Fs(,)36 b(in)f(the)h(sense)i(that)e(they)h(are)e (elemen)m(ts)i(of)e(the)h(algebra)f Fn(F)3173 3174 y Fr(\003)3222 3183 y Fh(0)3293 3210 y Fn(\032)f(F)3486 3174 y Fr(\003)3539 3210 y Fs(,)328 3330 y(see)j(\(1.25\),)f(and)g Fm(indep)-5 b(endent)35 b Fs(of)g(\003.)53 b(In)37 b(the)f(next)h (section,)g(w)m(e)g(shall)d(see)k(that)d(the)328 3451 y(propagators)d(in)g(the)h Fm(inter)-5 b(action)34 b(pictur)-5 b(e)1271 3630 y Fl(U)1347 3589 y Fk(I)5 b(;)p Fr(\003)1456 3630 y Fs(\()p Fl(t;)17 b(s)p Fs(\))28 b(:=)f Fl(e)1860 3589 y Fk(itH)1972 3565 y Fh(\003)1967 3609 y(0)2023 3630 y Fl(U)2099 3589 y Fr(\003)2152 3630 y Fs(\()p Fl(t;)17 b(s)p Fs(\))p Fl(e)2398 3589 y Fj(\000)p Fk(isH)2573 3565 y Fh(\003)2568 3609 y(0)328 3808 y Fs(ha)m(v)m(e)33 b(thermo)s(dynamic)d(limits,)f(as)j(\003)c Fn(\045)f Fp(R)1956 3772 y Fr(3)2001 3808 y Fs(,)32 b(under)h(standard)f (assumptions)g(on)f(the)328 3929 y(unp)s(erturb)s(ed)39 b(ev)m(olution)f(generated)h(b)m(y)g Fl(H)2006 3893 y Fr(\003)1998 3953 y(0)2058 3929 y Fs(.)61 b(If)38 b(exp)s(ectations)h (of)e(lo)s(cal,)h(b)s(ounded)328 4049 y(op)s(erators)33 b(in)g(the)g(initial)d(state,)k Fl(\045)1643 4064 y Fk(t)1668 4073 y Fh(0)1736 4049 y Fs(=)29 b Fl(\045)1891 4064 y Fk(\014)s(;)p Ff(\026)2008 4049 y Fs(,)34 b(of)f(the)g(system)i(ha)m(v) m(e)f(thermo)s(dynamic)328 4170 y(limits,)39 b(as)h(\003)g Fn(\045)g Fp(R)1077 4133 y Fr(3)1122 4170 y Fs(,)i(whic)m(h)e(is)g(a)g (standard)g(assumption)f(\(or)h(result)g({)f(see)i([3])f(for)328 4290 y(examples)34 b({)g(\))g(then)g(it)f(follo)m(ws)g(that)h(the)g (thermo)s(dynamic)f(limit)d(of)k(the)g(quan)m(tities)328 4410 y(on)e(the)h(r.h.s.)45 b(of)32 b(equations)h(\(1.43\))e(and)i (\(1.44\))f(exist.)474 4531 y(Finally)-8 b(,)25 b(from)h(the)h (de\014nition)e(of)i Fl(G)1817 4495 y Fr(\003)1870 4531 y Fs(,)h(equation)e(\(1.31\),)h(and)g(using)f(the)h(cyclicit)m(y)328 4651 y(of)32 b(the)h(trace,)g(w)m(e)h(\014nd)f(that)753 4819 y Fl(@)5 b(G)886 4783 y Fr(\003)p 753 4863 187 4 v 785 4955 a Fl(@)g Fe(\025)950 4886 y Fs(\()p Fl(\014)h(;)17 b Fe(\026)o Fs(;)g Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))22 b Fn(\001)1514 4860 y Fs(_)1495 4886 y Fe(\025)o Fs(\()p Fl(t)p Fs(\))28 b(=)1804 4776 y Fg(D)1901 4861 y Fs(_)1865 4886 y Fl(H)1954 4845 y Fr(\003)1946 4911 y Fk(t)2006 4776 y Fg(E)2044 4955 y Fk(\045)2080 4963 y Fd(t)2139 4886 y Fs(=)2243 4746 y Fg(\034)2328 4819 y Fl(@)5 b(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))p 2328 4863 417 4 v 2474 4955 a Fl(@)5 b Fe(\025)2754 4746 y Fg(\035)2805 4985 y Fk(\045)2841 4993 y Fd(t)2895 4886 y Fn(\001)2965 4860 y Fs(_)2945 4886 y Fe(\025)p Fs(\()p Fl(t)p Fs(\))p Fl(;)1922 5214 y Fs(9)p eop %%Page: 10 10 10 9 bop 328 631 a Fs(hence)1255 716 y Fl(@)5 b(G)p 1255 761 134 4 v 1260 852 a(@)g Fe(\025)1399 784 y Fs(\()p Fl(\014)h(;)17 b(\026)p Fs(;)g Fe(\025)o Fs(\()p Fl(t)p Fs(\)\))27 b(=)1991 643 y Fg(\034)2076 716 y Fl(@)5 b(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))p 2076 761 417 4 v 2222 852 a Fl(@)5 b Fe(\025)2502 643 y Fg(\035)2553 883 y Fk(\045)2589 891 y Fd(t)2621 784 y Fl(;)669 b Fs(\(1.45\))328 1020 y(whic)m(h,)36 b(under)f(the)g(same)f(standard)h(assumptions,)g (has)g(a)g(w)m(ell)e(de\014ned)j(thermo)s(dy-)328 1141 y(namic)31 b(limit.)40 b(W)-8 b(e)33 b(ha)m(v)m(e)h(th)m(us)g(pro)m(v)m (en)g(the)f(equation)826 1402 y(_)798 1427 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))83 b(=)g Fl(\014)1294 1257 y Fg(")1352 1287 y(\034)1437 1360 y Fl(@)5 b(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))p 1437 1404 V 1583 1495 a Fl(@)5 b Fe(\025)1863 1287 y Fg(\035)1915 1526 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))2057 1427 y Fn(\000)2157 1287 y Fg(\034)2242 1360 y Fl(@)g(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))p 2242 1404 V 2388 1495 a Fl(@)5 b Fe(\025)2668 1287 y Fg(\035)2719 1526 y Fk(\045)2755 1534 y Fd(t)2787 1257 y Fg(#)2868 1427 y Fn(\001)2937 1401 y Fs(_)2918 1427 y Fe(\025)o Fs(\()p Fl(t)p Fs(\))1217 1721 y(+)p Fl(\014)h Fe(\026)21 b Fn(\001)1539 1654 y Fl(@)p 1506 1699 123 4 v 1506 1790 a(@)5 b Fe(\034)1638 1611 y Fg(D)1699 1721 y Fl(\036)1757 1680 y Fr(\003)1757 1746 y Ff(\034)1810 1611 y Fg(\020)1869 1721 y Fl(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t)p Fs(\)\))2229 1611 y Fg(\021E)2326 1790 y Fk(\045)p Fr(\()p Fk(t)p Fr(\))2446 1607 y Fg(\014)2446 1667 y(\014)2446 1726 y(\014)2480 1790 y Ff(\034)9 b Fr(=0)3317 1721 y Fs(\(1.46\))328 1974 y(for)39 b(the)i(rate)f(of)f(c)m (hange)i(in)e(time)f(of)i(the)g(en)m(trop)m(y)i Fl(S)6 b Fs(\()p Fl(t)p Fs(\),)41 b(and)f(w)m(e)h(ha)m(v)m(e)g(con)m(vinced) 328 2095 y(ourselv)m(es)24 b(that)f(all)e(three)i(terms)g(on)g(the)h (r.h.s.)41 b(of)22 b(equation)h(\(1.46\))f(ha)m(v)m(e)i(w)m(ell)e (de\014ned)328 2215 y(thermo)s(dynamic)31 b(limits.)474 2336 y(Returning)26 b(to)g(equations)h(\(1.32\),)g(\(1.33\))f(and)g (\(1.35\),)h(one)g(ma)m(y)f(ask)h(under)g(what)328 2456 y(conditions)32 b(the)h(inequalities)d(in)i(\(1.33\))g(and)g(\(1.35\))g (are)h(saturated.)44 b(The)33 b(answ)m(er)h(is)328 2576 y(giv)m(en)f(in)e(Lemma)h(6.2.21)f(of)i([3)o(]:)1280 2781 y Fl(S)6 b Fs(\()p Fl(\045)1434 2796 y Fk(t)1463 2781 y Fn(j)p Fl(\045)p Fs(\()p Fl(t)p Fs(\)\))28 b(=)g(0)129 b(i\013)g Fl(\045)2264 2796 y Fk(t)2322 2781 y Fs(=)28 b Fl(\045)p Fs(\()p Fl(t)p Fs(\))p Fl(;)703 b Fs(\(1.47\))328 2986 y(i.e.,)40 b(i\013)e(the)h(reference)i(state)e Fl(\045)1528 3001 y Fk(t)1597 2986 y Fs(coincides)g(with)g(the)g(true)g(state)h Fl(\045)p Fs(\()p Fl(t)p Fs(\).)63 b(In)39 b(view)g(of)328 3106 y(equation)32 b(\(1.46\))f(one)h(ma)m(y)g(exp)s(ect)h(that)f(a)g (similar)c(result)k(holds)g(in)f(the)h(thermo)s(dy-)328 3227 y(namic)f(limit.)40 b(Equation)32 b(\(1.46\))g(pro)m(v)m(es)j (that)d(the)h(thermo)s(dynamic)e(limit)e(of)1188 3494 y(\001)p Fl(S)6 b Fs(\()p Fl(t)p Fs(\))28 b(:=)f Fl(S)6 b Fs(\()p Fl(t)p Fs(\))22 b Fn(\000)h Fl(S)6 b Fs(\()p Fl(t)2042 3509 y Fr(0)2081 3494 y Fs(\))28 b(=)2251 3359 y Fg(Z)2350 3385 y Fk(t)2306 3584 y(t)2331 3593 y Fh(0)2424 3469 y Fs(_)2396 3494 y Fl(S)6 b Fs(\()p Fl(t)2535 3453 y Fj(0)2559 3494 y Fs(\))p Fl(dt)2683 3453 y Fj(0)3317 3494 y Fs(\(1.48\))328 3761 y(exists,)37 b(and)f(one)f(exp)s(ects)j (that)d(if)f(\001)p Fl(S)6 b Fs(\()p Fl(t)p Fs(\))33 b(=)g(0)i(then)h(the)g(true)g(state)g(of)f(the)h(system)328 3881 y(at)45 b(time)f Fl(t)i Fs(is)f(giv)m(en)g(b)m(y)i(the)f(thermo)s (dynamic)e(limit)d(of)k(the)h(reference)h(states)g Fl(\045)3509 3896 y Fk(t)3539 3881 y Fs(;)328 4002 y(\(see)f([4]\).)82 b(F)-8 b(urthermore,)49 b(if,)e(at)e(time)f Fl(t)p Fs(,)49 b(the)d(restrictions)f(of)g(the)h(true)g(and)f(the)328 4122 y(reference)40 b(state)f(to)e(the)i(subalgebra)f Fn(F)1862 4086 y Fr(\003)1911 4095 y Fh(0)1988 4122 y Fs(coincide)f(in)h(the)g(thermo)s(dynamic)f(limit)328 4242 y(then)c(\001)659 4217 y(_)631 4242 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))28 b(=)967 4217 y(_)940 4242 y Fl(S)5 b Fs(\()p Fl(t)p Fs(\))28 b(=)g(0,)k(as)h(follo)m(ws)e(from)g(equation)i (\(1.46\).)474 4363 y(Of)47 b(course,)k(if)46 b Fl(W)1166 4378 y Fk(t)1243 4363 y Fs(dep)s(ends)i(non-trivially)c(on)j(time)e Fl(t)p Fs(,)51 b(for)46 b Fl(t)52 b(>)g(t)3119 4378 y Fr(0)3159 4363 y Fs(,)f(there)c(is)328 4483 y(no)31 b(reason)g(wh)m(y)h (the)f(true)g(and)g(the)g(reference)h(state)f(should)g(ev)m(er)h (coincide)e(at)h(times)328 4604 y Fl(t)d(>)f(t)529 4619 y Fr(0)569 4604 y Fs(.)41 b(Then)27 b(a)f(relev)-5 b(an)m(t)26 b(and)g(in)m(teresting)f(problem)g(is)g(to)g(study)i(the)g(rate)e(of)h (c)m(hange)328 4724 y(of)d(the)h(en)m(trop)m(y)h(under)f(the)g (assumption)f(that)g(the)h(p)s(erturbation)f Fl(W)2884 4739 y Fk(t)2937 4724 y Fs(dep)s(ends)i(slo)m(wly)328 4844 y(on)32 b(time,)1567 4965 y Fl(W)1659 4980 y Fk(t)1717 4965 y Fs(=)27 b Fl(W)14 b Fs(\()p Fe(\025)p Fs(\()p Fl(t=T)g Fs(\)\))p Fl(;)1898 5214 y Fs(10)p eop %%Page: 11 11 11 10 bop 328 631 a Fs(for)30 b(some)g(large)f Fl(T)14 b Fs(.)42 b(In)31 b(this)e(situation)g(one)h(w)m(ould)h(lik)m(e)e(to)h (pro)m(v)m(e)h(an)f Fm(adiab)-5 b(atic)32 b(the)-5 b(o-)328 751 y(r)g(em)26 b Fs(yielding)e(sharp)j(estimates)f(on)g(the)g(rate)g (at)g(whic)m(h)2454 726 y(_)2427 751 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))26 b(tends)h(to)f(0,)h(as)f Fl(T)42 b Fn(!)27 b(1)p Fs(.)328 872 y(More)h(in)m(terestingly)-8 b(,)28 b(suc)m(h)h(a)e(theorem)g(w)m(ould)h(tell)e(us)i(at)f(whic)m(h)h(rate)g (the)g(di\013erences)328 992 y(of)h(exp)s(ectation)h(v)-5 b(alues)30 b(of)f(lo)s(cal)e(op)s(erators)j(in)f(the)h(true)g(state)g (and)g(in)f(the)h(reference)328 1112 y(state)j(tend)g(to)f(0,)h(as)g Fl(T)41 b Fn(!)27 b(1)p Fs(.)43 b(In)33 b(this)f(note)h(w)m(e)h(shall)d (not)h(address)i(this)f(problem.)474 1233 y(The)38 b(essen)m(tial)g (features)g(of)f(our)g(de\014nition)f(of)h(the)h(en)m(trop)m(y)g Fl(S)6 b Fs(\()p Fl(t)p Fs(\))37 b(can)h(b)s(e)f(sum-)328 1353 y(marized)31 b(as)i(follo)m(ws:)474 1474 y(\(1\))c(It)g(is)g (compatible)e(with)h(the)i(\014rst)f(and)h(second)g(la)m(w)f(of)f (thermo)s(dynamics;)i(see)328 1594 y(equations)j(\(1.40\)-\(1.42\).)474 1714 y(\(2\))f Fl(S)6 b Fs(\()p Fl(t)p Fs(\))28 b Fn(\025)g Fl(S)6 b Fs(\()p Fl(t)1080 1729 y Fr(0)1120 1714 y Fs(\),)32 b(for)g(all)f Fl(t)d Fn(\025)g Fl(t)1705 1729 y Fr(0)1744 1714 y Fs(,)33 b(i.e.,)f(en)m(trop)m(y)i(tends)g(to)e(increase.)474 1835 y(\(3\))f(The)h(thermo)s(dynamic)d(limit)e(of)1894 1810 y(_)1867 1835 y Fl(S)5 b Fs(\()p Fl(t)p Fs(\),)32 b(and)f(hence)h(of)e(\001)p Fl(S)6 b Fs(\()p Fl(t)p Fs(\))28 b(=)g Fl(S)6 b Fs(\()p Fl(t)p Fs(\))18 b Fn(\000)h Fl(S)6 b Fs(\()p Fl(t)3488 1850 y Fr(0)3528 1835 y Fs(\))328 1955 y(exists,)33 b(for)f(all)f(times)h Fl(t)p Fs(.)474 2076 y(\(4\))38 b(If)g(the)g(time)f(ev)m(olution)g(is)g Fm(adiab)-5 b(atic)p Fs(,)39 b(in)e(the)i(sense)g(that)f(the)h(norm)e (of)g(the)328 2196 y(di\013erence)e(of)e(the)i(restrictions)e(of)h(the) g(true)g(and)g(of)g(the)g(reference)i(state)e(to)g(the)g(al-)328 2316 y(gebra)39 b Fn(F)682 2280 y Fr(\003)731 2289 y Fh(0)808 2316 y Fs(is)g(b)s(ounded)h(b)m(y)g(some)f(small,)f(p)s (ositiv)m(e)h(n)m(um)m(b)s(er)g Fl(\017)h(>)e Fs(0,)j(for)e(all)e (times)328 2437 y Fl(t)j Fn(\024)g Fl(t)555 2452 y Fr(1)595 2437 y Fs(,)h(then)g Fn(j)948 2412 y Fs(_)921 2437 y Fl(S)5 b Fs(\()p Fl(t)p Fs(\))p Fn(j)40 b Fl(<)f(O)s Fs(\()p Fl(\017)p Fs(\))g(and)h(\001)p Fl(S)6 b Fs(\()p Fl(t)p Fs(\))40 b Fl(<)g(O)s Fs(\()p Fl(\017)p Fs(\),)h(for)e Fl(t)h Fn(\024)g Fl(t)2767 2452 y Fr(1)2807 2437 y Fs(,)h(i.e.,)g(the)f (en)m(trop)m(y)328 2557 y(remains)32 b(appro)m(ximately)f(constan)m(t.) 44 b(F)-8 b(or)32 b(more)g(detail,)f(see)j([5)o(].)474 2677 y(In)49 b(the)f(follo)m(wing,)h(w)m(e)g(consider)f(t)m(w)m(o)h(t)m (ypical)e(examples)h(of)g(time-dep)s(enden)m(t)328 2798 y(thermo)s(dynamic)31 b(pro)s(cesses.)572 2998 y Fa(Pro)s(cess)51 b(\(I\))p Fs(.)43 b(The)j(p)s(erturbation)e Fl(W)2083 3013 y Fk(t)2157 2998 y Fs(con)m(v)m(erges)j(to)d(a)g(limiting)c(op)s (erator)572 3119 y Fl(W)664 3134 y Fj(1)766 3119 y Fn(6)p Fs(=)28 b(0,)k(as)h Fl(t)28 b Fn(!)f(1)p Fs(,)32 b(with)1546 3245 y Fg(Z)1646 3272 y Fj(1)1737 3381 y Fn(k)p Fl(W)1879 3396 y Fk(t)1931 3381 y Fn(\000)23 b Fl(W)2123 3396 y Fj(1)2198 3381 y Fn(k)p Fl(dt)k(<)g Fn(1)p Fl(:)726 b Fs(\(1.49\))572 3641 y(F)-8 b(or)33 b(suc)m(h)i(p)s(erturbations)f(w)m (e)h(study)g(the)g(phenomenon)f(of)f Fm(r)-5 b(eturn)37 b(to)f(e)-5 b(quilib-)572 3761 y(rium)p Fs(:)41 b(Under)28 b(suitable)f(assumptions)h(on)g(the)g(unp)s(erturb)s(ed)h(dynamics)e (in)g(the)572 3881 y(thermo)s(dynamic)35 b(limit)e(\(\\disp)s(ersiv)m (eness"\))38 b(and)f(assuming)f(that)g(a)h(suitable)572 4002 y(norm)44 b(of)g Fl(W)1054 4017 y Fj(1)1174 4002 y Fs(is)g(small)e(enough,)49 b(w)m(e)c(sho)m(w)h(that,)i(in)c(the)h (thermo)s(dynamic)572 4122 y(limit,)40 b(the)i(true)h(state)f(of)f(the) h(system)h(con)m(v)m(erges)h(to)d(an)h(equilibrium)c(state)572 4242 y(w.r.t.)63 b(the)40 b(dynamics)e(determined)h(b)m(y)h Fl(H)2221 4206 y Fr(\003)2213 4267 y Fj(1)2326 4242 y Fs(=)f Fl(H)2530 4206 y Fr(\003)2522 4267 y(0)2609 4242 y Fs(+)26 b Fl(W)2803 4257 y Fj(1)2917 4242 y Fs(at)39 b(temp)s(erature)572 4363 y Fl(\014)633 4327 y Fj(\000)p Fr(1)727 4363 y Fs(,)31 b(as)h Fl(t)c Fn(!)f(1)p Fs(,)k(if)f(the)i (initial)c(state)j(is)g(an)g(equilibrium)d(state)k(of)f(the)h(unp)s (er-)572 4483 y(turb)s(ed)h(dynamics)f(at)g(temp)s(erature)h Fl(\014)2056 4447 y Fj(\000)p Fr(1)2182 4483 y Fs(or)g(a)f(lo)s(cal)e (p)s(erturbation)i(thereof.)572 4604 y(F)-8 b(or)29 b(earlier)f (results,)j(see)g([11,)e(7,)h(1].)42 b(Our)30 b(analysis)f(is)h(an)f (extension)i(of)e(results)572 4724 y(in)j([6)o(,)h(11,)f(2])h(and)f(is) g(based)i(on)e(metho)s(ds)h(dev)m(elop)s(ed)g(in)f([4].)572 4844 y(The)27 b(result)e(describ)s(ed)i(here)f(is)g(a)f(kind)h(of)f (adiabatic)f(theorem)i(and)g(sho)m(ws)h(that)572 4965 y(thermo)s(dynamic)g(pro)s(cesses)j(with)e(p)s(erturbations)g Fl(W)2600 4980 y Fk(t)2659 4965 y Fs(as)g(sp)s(eci\014ed)h(ab)s(o)m(v)m (e)h(are)1898 5214 y(11)p eop %%Page: 12 12 12 11 bop 572 631 a Fm(r)-5 b(eversible)p Fs(.)42 b(It)33 b(implies)d(that,)j(in)e(the)i(thermo)s(dynamic)e(limit,)932 816 y(lim)917 876 y Fk(t)p Fj(!1)1128 791 y Fs(_)1100 816 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))28 b(=)f(0)p Fl(;)114 b Fs(and)33 b(hence)131 b(lim)2141 876 y Fk(t)p Fj(!1)2357 816 y Fs(lim)2324 884 y Fr(\003)p Fj(\045)p Fi(R)2492 865 y Fh(3)2558 815 y Fs(_)2542 816 y Fe(q)2600 775 y Fr(\003)2653 816 y Fs(\()p Fl(t)p Fs(\))28 b(=)g(0)p Fl(;)345 b Fs(\(1.50\))572 1049 y(i.e.,)33 b(the)g(en)m(trop)m(y)i(pro) s(duction)d(rate)i(and)f(the)g(rate)h(of)e(c)m(hange)i(of)f(the)h(c)m (harges)572 1169 y Fe(q)630 1133 y Fr(\003)683 1169 y Fs(\()p Fl(t)p Fs(\))e(v)-5 b(anish)33 b(in)e(the)i(thermo)s(dynamic)e (limit,)f(as)i(time)g(tends)h(to)f(in\014nit)m(y)-8 b(.)572 1483 y Fa(Pro)s(cess)49 b(\(I)s(I\))p Fs(.)43 b(F)-8 b(or)42 b Fl(t)k Fn(\025)h Fl(t)1662 1498 y Fr(0)1701 1483 y Fs(,)g(the)c(p)s(erturbation)g Fl(W)2628 1498 y Fk(t)2701 1483 y Fs(dep)s(ends)i Fm(p)-5 b(erio)g(dic)g(al)5 b(ly)572 1603 y Fs(on)38 b(time)f Fl(t)p Fs(,)k(with)d(p)s(erio)s(d)f Fl(T)14 b Fs(,)40 b(see)g([9],)g(and)f([10)o(,)g(12])f(for)g(recen)m(t) i(exp)s(erimen)m(ts)572 1724 y(in)m(v)m(olving)26 b(time-p)s(erio)s (dic)e(p)s(erturbations.)41 b(Under)29 b(the)e(same)h(assumptions)f(on) 572 1844 y(the)41 b(unp)s(erturb)s(ed)h(dynamics)f(in)f(the)h(thermo)s (dynamic)e(limit)f(as)j(in)f(\(I\))h(and)572 1965 y(if)g(a)i(suitable)e (norm)h(of)g Fl(W)1614 1980 y Fk(t)1686 1965 y Fs(is)g(small)f(enough,) k(for)d(all)f Fl(t)k Fn(2)g Fs([)p Fl(t)2993 1980 y Fr(0)3033 1965 y Fl(;)17 b(t)3112 1980 y Fr(0)3180 1965 y Fs(+)29 b Fl(T)14 b Fs(],)45 b(w)m(e)572 2085 y(pro)m(v)m(e)40 b(that,)g(in)d(the)i(thermo)s(dynamic)e(limit,)g(the)i(true)g(state)g (of)f(the)h(system)572 2205 y(con)m(v)m(erges)25 b(to)e(a)g Fm(time-p)-5 b(erio)g(dic)25 b(state)f Fs(of)e(p)s(erio)s(d)g Fl(T)14 b Fs(,)25 b(as)f(time)e Fl(t)h Fs(tends)h(to)f(in\014nit)m(y)-8 b(.)572 2326 y(It)32 b(is)f(not)h(hard)f(to)h(generalize)f(this)g(to)h (p)s(erturbations)f Fl(W)2750 2341 y Fk(t)2811 2326 y Fs(with)h(the)g(prop)s(ert)m(y)572 2446 y(that)1540 2457 y Fg(Z)1639 2483 y Fj(1)1731 2592 y Fn(k)p Fl(W)1873 2607 y Fk(t)1924 2592 y Fn(\000)23 b Fl(W)2130 2551 y Fj(1)2116 2617 y Fk(t)2204 2592 y Fn(k)p Fl(dt)28 b(<)f Fn(1)p Fl(;)719 b Fs(\(1.51\))572 2795 y(where)34 b Fl(W)960 2758 y Fj(1)946 2819 y Fk(t)1067 2795 y Fs(is)e(p)s(erio)s(dic)f(in)g Fl(t)i Fs(with)f(some)h(p)s(erio)s(d)e Fl(T)14 b Fs(.)328 3122 y Fq(2)161 b(Pro)t(cesses)34 b(\(I\))i(and)f(\(I)t(I\))h(in)h(the) e(thermo)t(dynamic)570 3305 y(limit)328 3524 y Fs(In)41 b(this)f(section)h(w)m(e)g(study)h(the)f(thermo)s(dynamic)e(limit,)g (\003)i Fn(\045)g Fp(R)2887 3488 y Fr(3)2933 3524 y Fs(,)h(of)e(thermo) s(dy-)328 3644 y(namic)33 b(pro)s(cesses,)k(in)c(particular)g(of)g(pro) s(cesses)k(\(I\))d(and)g(\(I)s(I\))g(describ)s(ed)h(at)f(the)h(end)328 3765 y(of)f(Section)g(1.)49 b(It)34 b(is)g(con)m(v)m(enien)m(t)i(to)e (in)m(tro)s(duce)h(a)f Fl(C)2317 3728 y Fj(\003)2356 3765 y Fs(-algebra,)g Fn(F)10 b Fs(,)34 b(of)g(op)s(erators)g(for)328 3885 y(the)f(in\014nite)f(system.)44 b(W)-8 b(e)33 b(de\014ne)1659 4087 y Fn(F)1706 4001 y Fc(\016)1778 4087 y Fs(=)1927 3992 y Fg(_)1881 4209 y Fr(\003)p Fj(\045)p Fi(R)2049 4190 y Fh(3)2100 4087 y Fn(F)2182 4045 y Fr(\003)3317 4087 y Fs(\(2.52\))328 4386 y(to)25 b(b)s(e)g(the)h(algebra)e (generated)i(b)m(y)h(all)c(the)j(algebras)e Fn(F)2369 4350 y Fr(\003)2422 4386 y Fs(,)j(for)d(an)i(increasing)e(sequence)328 4507 y(of)43 b(b)s(ounded)h(con)m(v)m(ex)i(regions)d(\003)j Fn(\045)g Fp(R)1858 4471 y Fr(3)1903 4507 y Fs(.)76 b(The)45 b Fl(C)2295 4471 y Fj(\003)2334 4507 y Fs(-algebra)d Fn(F)53 b Fs(is)43 b(de\014ned)i(as)e(the)328 4639 y(closure)j(of)g Fn(F)839 4558 y Fc(\016)929 4639 y Fs(in)g(the)h(op)s(erator)f(norm.)84 b(F)-8 b(or)45 b(an)i(op)s(erator)e Fl(A)52 b Fn(2)f(F)3048 4603 y Fr(\003)3147 4639 y Fs(and)c(a)f(set)328 4759 y(\003)396 4723 y Fj(0)447 4759 y Fn(\023)28 b Fs(\003,)k(one)h(can)g (de\014ne)1389 4965 y Fl(\013)1452 4917 y Fr(0)p Fk(;)p Fr(\003)1556 4894 y Fc(0)1451 4987 y Fk(t)1582 4965 y Fs(\()p Fl(A)p Fs(\))28 b(:=)g Fl(e)1935 4924 y Fk(itH)2047 4900 y Fh(\003)2088 4879 y Fc(0)2042 4944 y Fh(0)2119 4965 y Fl(Ae)2237 4924 y Fj(\000)p Fk(itH)2404 4900 y Fh(\003)2445 4879 y Fc(0)2399 4944 y Fh(0)2477 4965 y Fl(:)813 b Fs(\(2.53\))1898 5214 y(12)p eop %%Page: 13 13 13 12 bop 328 631 a Fs(It)38 b(is)f(a)h(standard)h(assumption)e(\(that) h(can)g(b)s(e)g(v)m(eri\014ed)h(in)e(ph)m(ysically)h(relev)-5 b(an)m(t)37 b(ex-)328 751 y(amples)32 b({)g(see)i(Section)e(3\))g(that) h(the)g(norm-limit)1361 949 y Fl(n)22 b Fn(\000)67 b Fs(lim)1541 1016 y Fr(\003)1590 997 y Fc(0)1612 1016 y Fj(\045)p Fi(R)1731 997 y Fh(3)1782 949 y Fl(\013)1845 902 y Fr(0)p Fk(;)p Fr(\003)1949 878 y Fc(0)1844 971 y Fk(t)1975 949 y Fs(\()p Fl(A)p Fs(\))27 b(=:)h Fl(\013)2345 908 y Fr(0)2344 973 y Fk(t)2384 949 y Fs(\()p Fl(A)p Fs(\))784 b(\(2.54\))328 1195 y(exists,)39 b(for)e(all)e Fl(A)h Fn(2)g(F)1217 1159 y Fr(\003)1269 1195 y Fs(,)i(for)f(an)g (arbitrary)f(b)s(ounded)i(con)m(v)m(ex)i(set)e(\003)d Fn(\032)h Fp(R)3214 1159 y Fr(3)3259 1195 y Fs(.)58 b(Since)328 1315 y Fl(\013)391 1268 y Fr(0)p Fk(;)p Fr(\003)495 1245 y Fc(0)390 1338 y Fk(t)521 1315 y Fs(\()p Fl(A)p Fs(\))30 b(b)s(elongs)g(to)g Fn(F)1247 1279 y Fr(\003)1296 1256 y Fc(0)1322 1315 y Fs(,)h(it)f(follo)m(ws)f(that)h Fl(\013)2066 1279 y Fr(0)2065 1340 y Fk(t)2105 1315 y Fs(\()p Fl(A)p Fs(\))e Fn(2)g(F)10 b Fs(.)42 b(By)31 b(con)m(tin)m(uit)m(y)-8 b(,)31 b Fl(\013)3219 1279 y Fr(0)3218 1340 y Fk(t)3289 1315 y Fs(can)g(b)s(e)328 1436 y(extended)k(to)d(a)g Fn(\003)p Fs(automorphism)e(group)j(of)f(the)h Fl(C)2271 1400 y Fj(\003)2310 1436 y Fs(-algebra)e Fn(F)10 b Fs(.)474 1556 y(F)-8 b(or)30 b(a)g(\014nite)g(system)i(con\014ned)g(to)e(a)g (region)g(\003)d Fn(\033)h Fs(\003)2433 1571 y Fr(0)2472 1556 y Fs(,)j(w)m(e)h(de\014ne)g(the)e Fm(pr)-5 b(op)g(agator)328 1677 y(in)34 b(the)h(inter)-5 b(action)35 b(pictur)-5 b(e)32 b Fs(b)m(y)1257 1863 y Fl(U)1333 1822 y Fk(I)5 b(;)p Fr(\003)1443 1863 y Fs(\()p Fl(t;)17 b(s)p Fs(\))27 b(:=)h Fl(e)1847 1822 y Fk(itH)1959 1798 y Fh(\003)1954 1842 y(0)2009 1863 y Fl(U)2085 1822 y Fr(\003)2139 1863 y Fs(\()p Fl(t;)17 b(s)p Fs(\))p Fl(e)2385 1822 y Fj(\000)p Fk(isH)2560 1798 y Fh(\003)2555 1842 y(0)2609 1863 y Fl(:)681 b Fs(\(2.55\))328 2049 y(Assuming)33 b(that)g(the)g(p)s (erturbation)f Fl(W)1821 2064 y Fk(t)1884 2049 y Fs(is)h(norm-con)m (tin)m(uous)f(in)h Fl(t)p Fs(,)g(w)m(e)h(can)g(expand)328 2169 y Fl(U)404 2133 y Fk(I)5 b(;)p Fr(\003)513 2169 y Fs(\()p Fl(t;)17 b(s)p Fs(\))29 b(in)f(a)h(Dyson)g(series)h(that)e (con)m(v)m(erges)k(in)c(norm,)h(uniformly)d(in)j(\003,)g(and)g(with)328 2290 y(the)e(prop)s(ert)m(y)g(that)f(all)f(terms)h(in)g(the)g(series)i (ha)m(v)m(e)f(a)f(thermo)s(dynamic)f(limit.)38 b(Setting)1563 2476 y Fl(W)1669 2429 y Fk(I)5 b(;)p Fr(\003)1655 2499 y Fk(t)1805 2476 y Fs(:=)27 b Fl(\013)1998 2429 y Fr(0)p Fk(;)p Fr(\003)1997 2499 y Fk(t)2106 2476 y Fs(\()p Fl(W)2236 2491 y Fk(t)2266 2476 y Fs(\))p Fl(;)328 2662 y Fs(and)1632 2783 y Fl(W)1738 2742 y Fk(I)1724 2807 y(t)1805 2783 y Fs(:=)g Fl(\013)1998 2742 y Fr(0)1997 2807 y Fk(t)2038 2783 y Fs(\()p Fl(W)2168 2798 y Fk(t)2197 2783 y Fs(\))p Fl(;)1055 b Fs(\(2.56\))328 2943 y(w)m(e)34 b(\014nd)f(that)689 3167 y Fl(U)765 3126 y Fk(I)5 b(;)p Fr(\003)874 3167 y Fs(\()p Fl(t;)17 b(s)p Fs(\))28 b(=)f(1)-22 b(l)21 b(+)1380 3072 y Fg(X)1386 3283 y Fk(n)p Fj(\025)p Fr(1)1524 3167 y Fs(\()p Fn(\000)p Fl(i)p Fs(\))1710 3126 y Fk(n)1774 3031 y Fg(Z)1874 3058 y Fk(t)1830 3257 y(s)1920 3167 y Fl(dt)2006 3182 y Fr(1)2062 3167 y Fn(\001)c(\001)g(\001)2195 3031 y Fg(Z)2294 3058 y Fk(t)2319 3067 y Fd(n)p Fc(\000)p Fh(1)2250 3257 y Fk(s)2462 3167 y Fl(dt)2548 3182 y Fk(n)2627 3167 y Fl(W)2733 3120 y Fk(I)5 b(;)p Fr(\003)2719 3189 y Fk(t)2744 3198 y Fh(1)2858 3167 y Fn(\001)17 b(\001)g(\001)e Fl(W)3097 3120 y Fk(I)5 b(;)p Fr(\003)3083 3189 y Fk(t)3108 3197 y Fd(n)328 3447 y Fs(and)33 b(this)f(op)s(erator)g(con)m(v)m (erges)j(in)c(norm)h(to)639 3691 y Fl(U)715 3650 y Fk(I)756 3691 y Fs(\()p Fl(t;)17 b(s)p Fs(\))27 b(=)h(1)-22 b(l)20 b(+)1261 3596 y Fg(X)1267 3807 y Fk(n)p Fj(\025)p Fr(1)1405 3691 y Fs(\()p Fn(\000)p Fl(i)p Fs(\))1591 3650 y Fk(n)1656 3555 y Fg(Z)1755 3582 y Fk(t)1711 3781 y(s)1801 3691 y Fl(dt)1887 3706 y Fr(1)1943 3691 y Fn(\001)d(\001)g(\001)2076 3555 y Fg(Z)2176 3582 y Fk(t)2201 3591 y Fd(n)p Fc(\000)p Fh(1)2132 3781 y Fk(s)2343 3691 y Fl(dt)2429 3706 y Fk(n)2508 3691 y Fl(W)2614 3650 y Fk(I)2600 3716 y(t)2625 3725 y Fh(1)2681 3691 y Fn(\001)g(\001)g(\001)e Fl(W)2920 3650 y Fk(I)2906 3716 y(t)2931 3724 y Fd(n)2978 3691 y Fl(;)312 b Fs(\(2.57\))328 3984 y(as)33 b(\003)27 b Fn(\045)g Fp(R)736 3948 y Fr(3)782 3984 y Fs(.)43 b(The)34 b(propagator)d Fl(U)1627 3948 y Fk(I)1668 3984 y Fs(\()p Fl(t;)17 b(s)p Fs(\))32 b(solv)m(es)i(the)f(di\013eren)m(tial)d (equation)1402 4129 y Fk(@)p 1389 4145 67 4 v 1389 4203 a(@)t(t)1465 4168 y Fl(U)1541 4132 y Fk(I)1582 4168 y Fs(\()p Fl(t;)17 b(s)p Fs(\))27 b(=)h Fn(\000)p Fl(iW)2130 4132 y Fk(I)2116 4193 y(t)2170 4168 y Fl(U)2246 4132 y Fk(I)2287 4168 y Fs(\()p Fl(t;)17 b(s)p Fs(\))p Fl(;)1690 4289 y(U)1766 4252 y Fk(I)1807 4289 y Fs(\()p Fl(s;)g(s)p Fs(\))27 b(=)h(1)-22 b(l)3317 4229 y(\(2.58\))328 4480 y(and)33 b Fl(U)594 4444 y Fk(I)634 4480 y Fs(\()p Fl(t;)17 b(s)p Fs(\))28 b Fn(2)g(F)10 b Fs(,)32 b(for)g(all)e(\014nite)j(times)e Fl(t;)17 b(s)p Fs(.)474 4601 y(These)25 b(remarks)d(enable)h(us)g(to)f (de\014ne)i(the)e(time)g(ev)m(olution)f(of)h(an)h(op)s(erator)e Fl(A)28 b Fn(2)g(F)328 4721 y Fs(in)k(the)h(Heisen)m(b)s(erg)g(picture) g(from)e(time)g Fl(s)i Fs(to)f(time)f Fl(t)i Fs(b)m(y)603 4907 y Fl(\013)665 4922 y Fk(s;t)747 4907 y Fs(\()p Fl(A)p Fs(\))28 b(=)f Fl(n)c Fn(\000)56 b Fs(lim)1207 4975 y Fr(\003)p Fj(\045)p Fi(R)1375 4956 y Fh(3)1426 4907 y Fl(\013)1489 4866 y Fr(\003)1488 4932 y Fk(s;t)1570 4907 y Fs(\()p Fl(A)p Fs(\))28 b(=)f Fl(\013)1913 4866 y Fr(0)1912 4932 y Fj(\000)p Fk(s)2020 4827 y Fg(\000)2066 4907 y Fl(U)2142 4866 y Fk(I)2183 4907 y Fs(\()p Fl(s;)17 b(t)p Fs(\))p Fl(\013)2447 4866 y Fr(0)2446 4932 y Fk(t)2486 4907 y Fs(\()p Fl(A)p Fs(\))p Fl(U)2711 4866 y Fk(I)2751 4907 y Fs(\()p Fl(t;)g(s)p Fs(\))2952 4827 y Fg(\001)3014 4907 y Fs(;)276 b(\(2.59\))1898 5214 y(13)p eop %%Page: 14 14 14 13 bop 328 631 a Fs(see)23 b(equations)f(\(2.55\))f(and)h(\(1.8\).) 39 b(Equation)21 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b(under)h(the)g(unp)s(erturb)s(ed)h(time)e(ev)m(olution)f Fl(\013)2934 2623 y Fr(0)2933 2683 y Fk(t)3005 2659 y Fs(\(see)j(equation)328 2779 y(\(2.54\),)f(\(2.61\)\),)g(it)f(follo)m (ws)h(that)453 3062 y Fl(!)518 3021 y Fr(0)557 3062 y Fs(\()p Fn(E)640 3077 y Fk(A)697 3062 y Fs(\()p Fl(t;)17 b(s)p Fs(\)\))27 b(=)h Fl(!)1132 3021 y Fr(0)1171 3062 y Fs(\()p Fl(\013)1272 3021 y Fr(0)1271 3087 y Fk(t)1296 3096 y Fh(0)1331 3087 y Fj(\000)p Fk(s)1422 3062 y Fs(\()p Fl(\013)1523 3021 y Fj(1)1522 3087 y Fk(s)p Fj(\000)p Fk(t)1635 3096 y Fh(0)1674 3062 y Fs(\()p Fl(A)p Fs(\)\))22 b(+)g Fl(i)2031 2927 y Fg(Z)2131 2953 y Fk(t)2086 3152 y(t)2111 3161 y Fh(0)2177 3062 y Fl(du)32 b(!)2377 3077 y Fk(u)2438 2982 y Fg(\000)2484 3062 y Fs([)p Fl(W)2603 3077 y Fk(u)2670 3062 y Fn(\000)23 b Fl(W)2862 3077 y Fj(1)2937 3062 y Fl(;)17 b(\013)3044 3021 y Fj(1)3043 3087 y Fk(s)p Fj(\000)p Fk(u)3175 3062 y Fs(\()p Fl(A)p Fs(\)])3351 2982 y Fg(\001)3413 3062 y Fl(;)3317 3244 y Fs(\(2.67\))328 3365 y(see)29 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y Fk(s)p Fj(\000)p Fk(u)1542 4292 y Fs(\()p Fl(A)p Fs(\)])1718 4212 y Fg(\001)1792 4292 y Fs(=)47 b(lim)1895 4352 y Fk(s)p Fj(!1)2086 4292 y Fl(!)2147 4307 y Fk(u)2208 4212 y Fg(\000)2254 4292 y Fs([)p Fl(W)2373 4307 y Fk(u)2440 4292 y Fn(\000)23 b Fl(W)2632 4307 y Fj(1)2707 4292 y Fl(;)17 b(\013)2814 4251 y Fr(0)2813 4317 y Fk(s)p Fj(\000)p Fk(u)2945 4292 y Fs(\()p Fl(\033)3038 4307 y Fr(+)3097 4292 y Fs(\()p Fl(A)p Fs(\)\)])3311 4212 y Fg(\001)3374 4292 y Fl(;)328 4545 y Fs(again)30 b(b)m(y)j(Hyp)s(othesis)g(\(A\),)e(and)h(it)f(follo) m(ws)g(from)f(the)i(prop)s(ert)m(y)h(of)e(return)i(to)e(equi-)328 4665 y(librium)e(for)j(the)h(unp)s(erturb)s(ed)h(time)d(ev)m(olution,)h Fl(\013)2285 4629 y Fr(0)2284 4690 y Fk(s)2324 4665 y Fs(,)h(that)1196 4885 y Fl(!)1257 4900 y Fk(u)1318 4805 y Fg(\000)1364 4885 y Fs([)p Fl(W)1483 4900 y Fk(u)1550 4885 y Fn(\000)23 b Fl(W)1742 4900 y Fj(1)1817 4885 y Fl(;)17 b(\013)1924 4844 y Fr(0)1923 4910 y Fk(s)p Fj(\000)p Fk(u)2055 4885 y Fs(\()p Fl(\033)2148 4900 y Fr(+)2207 4885 y Fs(\()p Fl(A)p Fs(\)\)])2421 4805 y Fg(\001)2495 4885 y Fn(!)27 b Fs(0)p Fl(;)619 b Fs(\(2.69\))1898 5214 y(15)p eop %%Page: 16 16 16 15 bop 328 631 a Fs(as)37 b Fl(s)e Fn(!)g(1)p Fs(;)k(see)g(Section)e (3)f(for)h(an)g(example)f(where)j(return)e(to)g(equilibrium)d(holds)328 751 y(for)e Fl(\013)540 715 y Fr(0)539 776 y Fk(s)579 751 y Fs(,)h(and)f([7,)h(4].)474 872 y(In)g(conclusion,)f(the)h(limit) 610 1034 y(lim)594 1094 y Fk(t)p Fj(!1)778 1034 y Fl(!)843 993 y Fr(0)881 1034 y Fs(\()p Fl(\013)981 1049 y Fk(t)1006 1058 y Fh(0)1041 1049 y Fk(;t)1091 1034 y Fs(\()p Fl(A)p Fs(\)\))27 b(=)43 b(lim)1409 1094 y Fk(t)p Fj(!1)1592 1034 y Fl(!)1657 993 y Fr(0)1696 1034 y Fs(\()p Fn(E)1779 1049 y Fk(A)1836 1034 y Fs(\()p Fl(t;)17 b(t)p Fs(\)\))27 b(=)h Fl(!)2260 993 y Fr(0)2299 1034 y Fs(\()p Fl(\033)2392 1049 y Fr(+)2451 1034 y Fs(\()p Fl(A)p Fs(\)\))p Fl(;)82 b(A)27 b Fn(2)i(F)10 b Fl(;)266 b Fs(\(2.70\))328 1229 y(exists,)52 b(b)m(y)c(equations)f(\(2.66\),)j(\(2.68\))d(and)g (\(2.69\).)87 b(It)47 b(is)f(kno)m(wn)j(from)d([3])h(that)328 1350 y(the)c(state)g Fl(!)820 1314 y Fr(0)858 1350 y Fs(\()p Fl(\033)951 1365 y Fr(+)1011 1350 y Fs(\()p Fn(\001)p Fs(\)\))f(is)g(an)g(equilibrium)e(\(i.e.)72 b(KMS\))43 b(state)g(for)f(the)g(asymptotic)328 1470 y(dynamics)32 b Fl(\013)822 1434 y Fj(1)821 1495 y Fk(t)897 1470 y Fs(.)474 1590 y(W)-8 b(e)44 b(note)h(that)e(equation)h(\(2.70\))f(is)g (v)-5 b(alid)42 b(for)h(an)m(y)i(initial)40 b(state)k Fl(!)3127 1554 y Fr(0)3210 1590 y Fs(whic)m(h)g(is)328 1711 y(in)m(v)-5 b(arian)m(t)39 b(under)i Fl(\013)1092 1675 y Fr(0)1091 1735 y Fk(t)1171 1711 y Fs(and)f(has)h(the)g(prop)s (ert)m(y)g(of)e(return)i(to)f(equilibrium,)f(see)j(also)328 1831 y(Section)32 b(3.)44 b(This)32 b(completes)h(our)f(discussion)h (of)f(the)h(thermo)s(dynamic)e(pro)s(cess)j(\(I\).)474 1952 y(Next,)k(w)m(e)f(examine)e Fa(pro)s(cess)41 b(\(I)s(I\))p Fs(.)35 b(Since)h(w)m(e)h(are)e(in)m(terested)i(in)e(times)g Fl(t)f Fn(\025)f Fl(t)3526 1967 y Fr(0)328 2072 y Fs(w)m(e)26 b(ma)m(y)f(consider)g(the)h(in)m(teraction)d Fl(W)1775 2087 y Fk(t)1830 2072 y Fs(to)i(b)s(e)g(p)s(erio)s(dic)e(with)i(p)s (erio)s(d)f Fl(T)39 b Fs(for)24 b Fm(al)5 b(l)25 b Fs(times,)328 2192 y Fl(W)420 2207 y Fk(t)p Fr(+)p Fk(T)583 2192 y Fs(=)j Fl(W)779 2207 y Fk(t)808 2192 y Fs(,)33 b(for)f Fl(t)c Fn(2)g Fp(R)5 b Fs(.)49 b(It)33 b(follo)m(ws)e(that)1428 2355 y Fl(\013)1490 2370 y Fk(s;t)1572 2355 y Fs(\()p Fl(A)p Fs(\))d(=)f Fl(\013)1914 2370 y Fk(s)p Fr(+)p Fk(nT)6 b(;t)p Fr(+)p Fk(nT)2289 2355 y Fs(\()p Fl(A)p Fs(\))p Fl(;)852 b Fs(\(2.71\))328 2518 y(for)37 b(arbitrary)f(times)h Fl(s;)17 b(t)p Fs(,)38 b Fl(n)e Fn(2)h Fp(Z)d Fs(and)k Fl(A)e Fn(2)g(F)10 b Fs(.)58 b(Decomp)s(osing)36 b(the)h(time)f(v)-5 b(ariable)328 2638 y Fl(t)28 b Fn(2)g Fp(R)39 b Fs(uniquely)29 b(as)f Fl(t)g Fs(=)g Fl(n)p Fs(\()p Fl(t)p Fs(\))p Fl(T)g Fs(+)14 b Fl(\034)d Fs(\()p Fl(t)p Fs(\),)29 b(with)f Fl(n)p Fs(\()p Fl(t)p Fs(\))g Fn(2)g Fp(Z)e Fs(and)i Fl(\034)11 b Fs(\()p Fl(t)p Fs(\))29 b Fn(2)f Fs([0)p Fl(;)17 b(T)d Fs(\))27 b(w)m(e)j(obtain)328 2758 y(from)h(\(2.71\))h (the)h(equation)1247 2921 y Fl(\013)1309 2936 y Fk(t)1334 2945 y Fh(0)1370 2936 y Fk(;t)1419 2921 y Fs(\()p Fl(A)p Fs(\))27 b(=)h Fl(\013)1761 2936 y Fk(t)1786 2945 y Fh(0)1821 2936 y Fj(\000)p Fk(n)p Fr(\()p Fk(t)p Fr(\))p Fk(T)6 b(;)p Fr(0)2122 2840 y Fg(\000)2167 2921 y Fl(\013)2229 2936 y Fr(0)p Fk(;\034)i Fr(\()p Fk(t)p Fr(\))2408 2921 y Fs(\()p Fl(A)p Fs(\))2557 2840 y Fg(\001)2619 2921 y Fl(:)328 3084 y Fs(The)33 b(in)m(v)-5 b(ariance)32 b(of)g Fl(!)1165 3047 y Fr(0)1236 3084 y Fs(under)i Fl(\013)1576 3047 y Fr(0)1575 3108 y Fk(t)1648 3084 y Fs(and)e(Hyp)s(othesis)i (\(A\))e(then)h(imply)e(that)1011 3246 y(lim)996 3306 y Fk(t)p Fj(!1)1179 3162 y Fg(\014)1179 3221 y(\014)1212 3246 y Fl(!)1277 3205 y Fr(0)1316 3246 y Fs(\()p Fl(\013)1416 3261 y Fk(t)1441 3270 y Fh(0)1476 3261 y Fk(;t)1525 3246 y Fs(\()p Fl(A)p Fs(\)\))22 b Fn(\000)h Fl(!)1899 3205 y Fr(0)1954 3166 y Fg(\000)2000 3246 y Fl(\033)2055 3261 y Fr(+)2131 3166 y Fg(\000)2177 3246 y Fl(\013)2239 3262 y Fr(0)p Fk(;\034)8 b Fr(\()p Fk(t)p Fr(\))2417 3246 y Fs(\()p Fl(A)p Fs(\))2566 3166 y Fg(\001\001)2658 3162 y(\014)2658 3221 y(\014)2718 3246 y Fs(=)28 b(0)p Fl(;)419 b Fs(\(2.72\))328 3436 y(for)32 b(all)e Fl(A)e Fn(2)g(F)10 b Fs(.)474 3556 y(The)34 b(state)f(giv)m(en)f(b)m(y)1138 3719 y Fl(!)1203 3678 y Fk(P)1199 3743 y(t)1261 3719 y Fs(\()p Fl(A)p Fs(\))c(:=)f Fl(!)1633 3678 y Fr(0)1672 3638 y Fg(\000)1718 3719 y Fl(\033)1773 3734 y Fr(+)1832 3719 y Fs(\()p Fl(\013)1932 3734 y Fr(0)p Fk(;\034)8 b Fr(\()p Fk(t)p Fr(\))2111 3719 y Fs(\()p Fl(A)p Fs(\)\))2298 3638 y Fg(\001)2343 3719 y Fl(;)82 b(A)28 b Fn(2)g(F)10 b Fl(;)328 3881 y Fs(is)42 b(p)s(erio)s(dic)g(in)g Fl(t)h Fs(with)f(p)s(erio)s(d)g Fl(T)57 b Fs(\(b)s(ecause)44 b Fl(t)i Fn(7!)f Fl(\034)11 b Fs(\()p Fl(t)p Fs(\))43 b(is\).)74 b(This)43 b(sho)m(ws)i(that)d Fl(!)3536 3896 y Fk(t)328 4002 y Fs(approac)m(hes)37 b(a)e Fm(time-p)-5 b(erio)g(dic)33 b Fs(state)j(as)g Fl(t)c Fn(!)g(1)p Fs(.)52 b(Notice)34 b(that)i(\(2.72\))e(holds)h(for)g(an)328 4122 y(arbitrary)d Fl(\013)806 4086 y Fr(0)805 4147 y Fk(t)845 4122 y Fs(-in)m(v)-5 b(arian)m(t)30 b(initial)f(state)k Fl(!)1872 4086 y Fr(0)1911 4122 y Fs(.)474 4363 y Fm(R)-5 b(emark.)131 b Fs(The)47 b(approac)m(h)g(to)f(the)g(asymptotic)g(state) g(\(whic)m(h)h(is)e(stationary)328 4483 y(for)33 b(Pro)s(cess)i(\(I\))f (and)f(time-p)s(erio)s(dic)e(for)i(Pro)s(cess)i(\(I)s(I\)\),)e(for)g (large)g(times,)g(holds)g(for)328 4604 y(arbitrary)h(initial)d(states) 36 b(whic)m(h)f(are)g(normal)e(w.r.t.)51 b(the)35 b(state)h Fl(!)2843 4567 y Fr(0)2916 4604 y Fs(giv)m(en)f(in)f(\(2.61\).)328 4724 y(In)e(other)g(w)m(ords,)h(relations)d(\(2.70\))h(and)h(\(2.72\))f (hold)f(if)h(w)m(e)i(replace)e Fl(!)2996 4688 y Fr(0)3035 4724 y Fs(\()p Fl(\013)3135 4739 y Fk(t)3160 4748 y Fh(0)3195 4739 y Fk(;t)3244 4724 y Fs(\()p Fl(A)p Fs(\)\))h(b)m(y)328 4844 y Fl(!)t Fs(\()p Fl(\013)493 4859 y Fk(t)518 4868 y Fh(0)552 4859 y Fk(;t)602 4844 y Fs(\()p Fl(A)p Fs(\)\),)j(for)g(an)m (y)g(state)h Fl(!)i Fs(on)d Fn(F)44 b Fs(whic)m(h)36 b(is)f(normal)e(w.r.t.)51 b Fl(!)2843 4808 y Fr(0)2882 4844 y Fs(.)f(The)36 b(pro)s(of)e(can)328 4965 y(b)s(e)f(found)f(in)g ([4].)1898 5214 y(16)p eop %%Page: 17 17 17 16 bop 328 631 a Fq(3)161 b(Thermo)t(dynamic)74 b(pro)t(cesses)f (for)i(a)h(reserv)l(oir)570 814 y(of)44 b(non-relativistic)h(non-in)l (teracting)e(fermions)328 1033 y Fs(W)-8 b(e)29 b(consider)g(an)g (ideal)e(quan)m(tum)i(gas)f(of)g(fermionic)f(particles,)i(e.g.)42 b(mo)s(delling)25 b(non-)328 1153 y(in)m(teracting,)j(non-relativistic) e(electrons)j(in)f(a)g(metal)f(or)h(a)g(semi-conductor,)h(sub)5 b(ject)328 1273 y(to)40 b(a)g(time-dep)s(enden)m(t)h(p)s(erturbation.) 67 b(F)-8 b(or)39 b(the)i(purp)s(ose)h(of)e(exp)s(osition,)h(w)m(e)h (con-)328 1394 y(cen)m(trate)30 b(here)g(on)f(spinless)g(fermions;)h(a) f(more)f(general)h(treatmen)m(t)g(can)g(b)s(e)g(found)h(in)328 1514 y([5,)i(4].)474 1755 y(The)45 b(Hilb)s(ert)c(space)k(of)e(pure)h (states)g(of)f(the)h(system)g(con\014ned)h(to)e(a)g(b)s(ounded)328 1875 y(region)32 b(\003)27 b Fn(\032)h Fp(R)887 1839 y Fr(3)965 1875 y Fs(is)k(giv)m(en)h(b)m(y)g(the)g(fermionic)e(F)-8 b(o)s(c)m(k)32 b(space)i(o)m(v)m(er)g Fl(L)2814 1839 y Fr(2)2854 1875 y Fs(\(\003)p Fl(;)17 b(d)3055 1839 y Fr(3)3093 1875 y Fl(x)p Fs(\),)924 2111 y Fn(H)1009 2070 y Fr(\003)1090 2111 y Fs(:=)28 b Fl(F)1284 2126 y Fj(\000)1343 2111 y Fs(\()p Fl(L)1447 2070 y Fr(2)1486 2111 y Fs(\(\003)p Fl(;)17 b(d)1687 2070 y Fr(3)1726 2111 y Fl(x)p Fs(\)\))28 b(=)1988 2017 y Fg(M)1997 2227 y Fk(n)p Fj(\025)p Fr(0)2156 2111 y Fl(P)2219 2126 y Fj(\000)2294 2031 y Fg(\000)2340 2111 y Fl(L)2406 2070 y Fr(2)2446 2111 y Fs(\(\003)p Fl(;)17 b(d)2647 2070 y Fr(3)2685 2111 y Fl(x)p Fs(\))2778 2031 y Fg(\001)2824 2053 y Fj(\012)p Fk(n)2943 2111 y Fl(;)347 b Fs(\(3.73\))328 2424 y(where)25 b Fl(P)664 2439 y Fj(\000)746 2424 y Fs(denotes)g(the)f(pro)5 b(jection)24 b(op)s(erator)f(on)m(to)g(the)h (subspace)i(of)d(an)m(tisymmetric)328 2544 y(functions,)33 b(and)f(where)i(the)f(subspace)i(for)d Fl(n)c Fs(=)f(0)32 b(is)g Fp(C)20 b Fs(.)474 2665 y(The)34 b(non-in)m(teracting)d (Hamiltonian)e(is)j(giv)m(en)h(b)m(y)1645 2895 y Fl(H)1734 2854 y Fr(\003)1726 2920 y(0)1814 2895 y Fs(:=)1945 2801 y Fg(M)1954 3011 y Fk(n)p Fj(\025)p Fr(0)2112 2895 y Fl(h)2168 2854 y Fr(\003)2168 2920 y Fk(n)2221 2895 y Fl(;)1069 b Fs(\(3.74\))328 3220 y(where)34 b Fl(h)666 3184 y Fr(\003)666 3245 y Fk(n)752 3220 y Fs(acts)f(on)f Fl(P)1151 3235 y Fj(\000)1210 3220 y Fs(\()p Fl(L)1314 3184 y Fr(2)1354 3220 y Fs(\(\003)p Fl(;)17 b(d)1555 3184 y Fr(3)1593 3220 y Fl(x)p Fs(\)\))1724 3184 y Fj(\012)p Fk(n)1859 3220 y Fs(as)1079 3515 y Fl(h)1135 3473 y Fr(\003)1135 3539 y Fk(n)1216 3515 y Fs(=)1370 3390 y Fk(n)1320 3420 y Fg(X)1327 3632 y Fk(k)r Fr(=1)1480 3515 y Fs(1)-22 b(l)21 b Fn(\012)i(\001)17 b(\001)g(\001)j(\012)j Fs(1)-22 b(l)20 b Fn(\012)j Fs(\()p Fn(\000)p Fs(\001\))g Fn(\012)f Fs(1)-22 b(l)17 b Fn(\001)g(\001)g(\001)i(\012)k Fs(1)-22 b(l)p Fl(;)328 3828 y Fs(and)48 b Fn(\000)p Fs(\001)i(is)d(the)i (Laplacian)e(on)h Fl(L)1717 3791 y Fr(2)1757 3828 y Fs(\(\003)p Fl(;)17 b(d)1958 3791 y Fr(3)1996 3828 y Fl(x)p Fs(\))49 b(with)f(selfadjoin)m(t)f(\(e.g.)91 b(Diric)m(hlet-,)328 3948 y(Neumann-,)33 b(or)h(p)s(erio)s(dic\))e(b)s(oundary)i (conditions,)f(acting)f(on)i(the)g Fl(k)s Fs(-th)f(factor.)46 b(W)-8 b(e)328 4068 y(set)33 b Fl(h)536 4032 y Fr(\003)536 4093 y(0)617 4068 y Fs(=)28 b(0.)474 4189 y(W)-8 b(e)48 b(de\014ne)h(the)f(\014eld)f(algebra)g(as)g(the)h(CAR)g(algebra)f (\(CAR)g(for)g(\\canonical)328 4309 y(an)m(ti-comm)m(utation)29 b(relations"\))1398 4528 y Fn(F)1480 4486 y Fr(\003)1560 4528 y Fs(:=)e(CAR)1922 4447 y Fg(\000)1968 4528 y Fl(L)2034 4486 y Fr(2)2074 4528 y Fs(\(\003)p Fl(;)17 b(d)2275 4486 y Fr(3)2313 4528 y Fl(x)p Fs(\))2406 4447 y Fg(\001)2469 4528 y Fl(;)328 4746 y Fs(whic)m(h)33 b(is)f(the)h Fl(C)950 4710 y Fj(\003)990 4746 y Fs(-algebra)d(generated)k(b)m(y)f(creation-)f (and)g(annihilation)d(op)s(erators,)1403 4965 y Fn(f)p Fl(a)1504 4924 y Fr(#)1567 4965 y Fs(\()p Fl(f)11 b Fs(\))32 b Fn(j)g Fl(f)39 b Fn(2)28 b Fl(L)2041 4924 y Fr(2)2081 4965 y Fs(\(\003)p Fl(;)17 b(d)2282 4924 y Fr(3)2320 4965 y Fl(x)p Fs(\))p Fn(g)p Fl(:)1898 5214 y Fs(17)p eop %%Page: 18 18 18 17 bop 328 631 a Fs(The)36 b(sym)m(b)s(ol)f Fl(a)919 595 y Fr(#)1017 631 y Fs(denotes)h(either)f Fl(a)h Fs(or)e Fl(a)1910 595 y Fj(\003)1950 631 y Fs(;)j(recall)c(that)i(the)h (annihilation)31 b(op)s(erator)328 751 y Fl(a)p Fs(\()p Fl(f)11 b Fs(\))32 b(acts)h(on)g(a)f(w)m(a)m(v)m(e-function)i Fl( )d Fn(2)d Fl(F)1836 766 y Fj(\000)1896 751 y Fs(\()p Fl(L)2000 715 y Fr(2)2039 751 y Fs(\(\003)p Fl(;)17 b(d)2240 715 y Fr(3)2279 751 y Fl(x)p Fs(\)\))33 b(as)443 1009 y(\()p Fl(a)p Fs(\()p Fl(f)11 b Fs(\))p Fl( )t Fs(\))772 1024 y Fk(n)819 1009 y Fs(\()p Fl(x)912 1024 y Fr(1)952 1009 y Fl(;)17 b(:)g(:)g(:)f(;)h(x)1226 1024 y Fk(n)1273 1009 y Fs(\))27 b(=)1442 925 y Fn(p)p 1525 925 228 4 v 84 x Fl(n)22 b Fs(+)g(1)1769 873 y Fg(Z)p 1885 922 328 4 v 1885 1009 a Fl(f)11 b Fs(\()p Fl(x)2037 1024 y Fk(n)p Fr(+1)2174 1009 y Fs(\))p Fl( )2275 1024 y Fk(n)p Fr(+1)2413 1009 y Fs(\()p Fl(x)2506 1024 y Fr(1)2545 1009 y Fl(;)17 b(:)g(:)g(:)f(;)h(x)2819 1024 y Fk(n)2866 1009 y Fl(;)g(x)2965 1024 y Fk(n)p Fr(+1)3103 1009 y Fs(\))p Fl(d)3192 968 y Fr(3)3231 1009 y Fl(x)3286 1024 y Fk(n)p Fr(+1)3423 1009 y Fl(;)328 1261 y Fs(where)37 b Fl( )676 1276 y Fk(n)758 1261 y Fs(is)e(the)h(pro)5 b(jection)35 b(of)g Fl( )40 b Fs(on)m(to)35 b(the)h Fl(n)p Fs(-particle)e(subspace)j(of)e(F)-8 b(o)s(c)m(k)36 b(space.)328 1381 y(The)45 b(creation)f(op)s(erators)h Fl(a)1421 1345 y Fj(\003)1460 1381 y Fs(\()p Fl(f)11 b Fs(\))44 b(\(adjoin)m(t)g(of)g Fl(a)p Fs(\()p Fl(f)11 b Fs(\)\))44 b(and)h(annihilation)c(op)s (erators)328 1502 y(satisfy)33 b(the)g(canonical)e(an)m(ti-comm)m 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Fn(k)p Fl(W)1239 2126 y Fk(t)1269 2111 y Fn(k)1319 2070 y Fj(0)1319 2136 y(1)1421 2111 y Fs(:=)1556 2016 y Fg(X)1552 2228 y Fk(N)7 b Fj(\025)p Fr(1)1721 2111 y Fs(2)1770 2070 y Fr(5)p Fk(N)1873 2111 y Fl(N)59 b Fs(sup)2024 2193 y Fk(t)p Fj(2)p Fi(R)2174 2111 y Fn(k)p Fl(w)2297 2070 y Fk(N)2363 2111 y Fs(\()p Fl(t;)17 b Fn(\001)p Fl(;)g Fn(\001)p Fs(\))p Fn(k)2668 2070 y Fj(0)2668 2136 y Fr(6)p Fk(N)2769 2111 y Fl(;)328 2426 y Fs(with)817 2638 y Fn(k)p Fl(f)11 b Fn(k)976 2597 y Fj(0)976 2662 y Fk(M)1082 2638 y Fs(:=)1313 2570 y(1)p 1222 2615 230 4 v 1222 2706 a(2)1271 2677 y Fr(3)p Fk(M)s(=)p Fr(2)1479 2467 y Fg(*)1559 2638 y Fl(f)5 b(;)1683 2513 y Fk(M)1657 2543 y Fg(Y)1656 2755 y Fk(k)r Fr(=1)1801 2497 y Fg(\022)1875 2638 y Fn(\000)1991 2570 y Fl(d)2042 2534 y Fr(2)p 1962 2615 149 4 v 1962 2706 a Fl(dx)2068 2672 y Fr(2)2068 2734 y Fk(k)2143 2638 y Fs(+)22 b Fl(x)2296 2597 y Fr(2)2296 2662 y Fk(k)2361 2638 y Fs(+)g(1)2508 2497 y Fg(\023)2581 2518 y Fr(3)2637 2638 y Fl(f)2696 2467 y Fg(+)2776 2484 y Fr(1)p Fk(=)p Fr(2)2776 2769 y Fk(L)2824 2751 y Fh(2)2859 2769 y Fr(\()p Fi(R)2934 2751 y Fd(M)3002 2769 y Fr(\))3050 2638 y Fl(;)328 2930 y Fs(for)44 b(a)h(function)g Fl(f)59 b Fn(2)50 b Fl(L)1267 2894 y Fr(2)1307 2930 y Fs(\()p Fp(R)1410 2894 y Fk(M)1495 2930 y Fs(\))45 b(and)h(where)g Fn(h\001)p Fl(;)17 b Fn(\001i)2253 2949 y Fk(L)2301 2930 y Fh(2)2334 2949 y Fr(\()p Fi(R)2409 2930 y Fd(M)2477 2949 y Fr(\))2553 2930 y Fs(is)45 b(the)g(inner)g(pro)s(duct)h(in)328 3050 y Fl(L)394 3014 y Fr(2)434 3050 y Fs(\()p Fp(R)538 3014 y Fk(M)623 3050 y Fs(\).)474 3171 y(It)33 b(is)f(sho)m(wn)h(in)f ([4])g(that)g(the)h(limit)c(\(2.60\))j(exists,)h(for)f(time-indep)s (enden)m(t)f(in)m(ter-)328 3291 y(actions)g(in)f Fn(W)8 b Fs(;)32 b(the)g(pro)s(of)e(of)h(con)m(v)m(ergence)j(giv)m(en)d(there) h(generalizes)f(readily)f(to)h(the)328 3411 y(time-dep)s(enden)m(t)i (case,)g(hence)h(Hyp)s(othesis)g(\(A\))e(holds.)474 3652 y Fm(R)-5 b(emark.)90 b Fs(One)36 b(can)g(explicitly)e(calculate)h (relev)-5 b(an)m(t)35 b(ph)m(ysical)h(quan)m(tities,)g(suc)m(h)328 3773 y(as)g(rates)h(of)e(c)m(hange)i(in)e(time)g(of)h(in)m(ternal)f (energy)-8 b(,)2318 3747 y(_)2290 3773 y Fl(U)11 b Fs(\()p Fl(t)p Fs(\),)37 b(c)m(harge,)56 b(_)-45 b Fl(q)t Fs(\()p Fl(t)p Fs(\),)37 b(or)e(en)m(trop)m(y)-8 b(,)355 3868 y(_)328 3893 y Fl(S)6 b Fs(\()p Fl(t)p Fs(\))31 b(\(see)i(\(1.43\),)e (\(1.44\),)g(\(1.46\)\))f(in)h(the)h(thermo)s(dynamic)e(limit)e(in)i(a) i Fm(p)-5 b(erturb)g(ative)328 4013 y(way)31 b Fs(b)m(y)h(using)f(a)g (Dyson)h(series)g(expansion)g(\(whic)m(h)g(is)f(norm-con)m(v)m(ergen)m (t)h(uniformly)328 4134 y(in)d(time\);)g(see)i(equations)f (\(2.63\)-\(2.70\).)41 b(F)-8 b(urthermore,)29 b(exp)s(ectations)i(in)e (the)h(refer-)328 4254 y(ence)38 b(state)f Fl(\045)842 4269 y Fk(t)908 4254 y Fs(ha)m(v)m(e)h(a)e(con)m(v)m(ergen)m(t)j(p)s (erturbation)c(expansion,)j(as)f(w)m(ell.)55 b(Th)m(us,)39 b(our)328 4374 y(metho)s(ds)32 b(are)h(quan)m(titativ)m(e.)474 4615 y Fa(Ac)m(kno)m(wledgemen)m(ts.)66 b Fs(W)-8 b(e)24 b(thank)g(W)-8 b(alter)23 b(Kohn)g(for)g(attracting)f(our)h(in)m (terest)328 4736 y(to)46 b(problems)h(of)f(statistical)f(mec)m(hanics)i (in)m(v)m(olving)f(time-p)s(erio)s(dic)d(Hamiltonians)328 4856 y(and)33 b(for)f(dra)m(wing)g(our)g(atten)m(tion)g(to)h(reference) h([9].)1898 5214 y(20)p eop %%Page: 21 21 21 20 bop 328 631 a Fq(References)425 850 y Fs([1])49 b(V.)33 b(Bac)m(h,)g(J.)g(F)-8 b(r\177)-49 b(ohlic)m(h,)31 b(I.)i(M.)g(Sigal,)d Fm(J.)35 b(Math.)g(Phys.)e Fa(41)p Fs(,)g(3985)e(\(2000\).)425 1053 y([2])49 b(D.)32 b(D.)g(Botvic)m(h,)h (V.)g(A.)g(Malishev,)f Fm(Comm.)i(Math.)h(Phys.)e Fa(91)p Fs(,)f(301)g(\(1983\).)425 1257 y([3])49 b(O.)37 b(Bratteli,)g(D.)g (Robinson,)h Fm(Op)-5 b(er)g(ator)39 b(A)n(lgebr)-5 b(as)38 b(and)h(Quantum)g(Statistic)-5 b(al)577 1377 y(Me)g(chanics)42 b(I,)g(II)d Fs(\(T)-8 b(exts)43 b(and)d(Monographs)i(in)e(Ph)m(ysics,)k (Springer)d(V)-8 b(erlag,)577 1498 y(1987,)32 b(1997\).)425 1701 y([4])49 b(J.)33 b(F)-8 b(r\177)-49 b(ohlic)m(h,)31 b(M.)i(Merkli,)f(D.)g(Ueltsc)m(hi,)h(submitted)f(to)g(Ann.)h(H.)g(P)m (oincar)m(\023)-46 b(e.)425 1904 y([5])49 b(J.)33 b(F)-8 b(r\177)-49 b(ohlic)m(h,)31 b(M.)i(Merkli,)f(S.)h(Sc)m(h)m(w)m(arz,)i (D.)d(Ueltsc)m(hi,)g(in)g(preparation.)425 2108 y([6])49 b(K.)33 b(Hepp,)g Fm(Solid)h(State)h(Communic)-5 b(ations)31 b Fa(8)p Fs(,)i(2087)e(\(1970\).)425 2311 y([7])49 b(V.)33 b(Jak)-5 b(\025)-44 b(si)m(\023)e(c,)33 b(C.-A.)g(Pillet,)d Fm(Comm.)k(Math.)h(Phys.)e Fa(178)p Fs(,)f(627)g(\(1996\).)425 2515 y([8])49 b(T.)39 b(Kato,)g Fm(Perturb)-5 b(ation)41 b(The)-5 b(ory)40 b(for)g(Line)-5 b(ar)40 b(Op)-5 b(er)g(ators)38 b Fs(\(Springer)g(V)-8 b(erlag,)577 2635 y(1966\).)425 2838 y([9])49 b(W.)33 b(Kohn,)g Fm(J.)i(Stat.)g(Phys.)d Fa(103)p Fs(,)h(417)f(\(2001\).)377 3042 y([10])48 b(R.)33 b(Mani,)g(J.)h(H.)f(Smet,)g(K.)g(v)m(on)h(Klitzing,)d(V.)j(Nara)m(y)m (anam)m(urti,)e(W.B.)i(John-)577 3162 y(son,)f(and)g(V.)g(Umansky)-8 b(,)33 b Fm(Natur)-5 b(e)33 b Fa(420)p Fs(,)g(646)f(\(2002\).)377 3366 y([11])48 b(D.)32 b(Robinson,)g Fm(Comm.)i(Math.)h(Phys.)e Fa(31)p Fs(,)f(171)g(\(1973\).)377 3569 y([12])48 b(M.A.)79 b(Zudo)m(v,)90 b(R.R.)77 b(Du,)89 b(L.N.)78 b(Pfei\013er,)89 b(and)78 b(K.W.)h(W)-8 b(est,)90 b(cond-)577 3689 y(mat/0210034.)1898 5214 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0304070745672--