Content-Type: multipart/mixed; boundary="-------------0304150253571" This is a multi-part message in MIME format. ---------------0304150253571 Content-Type: text/plain; name="03-174.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-174.comments" Revised Version ---------------0304150253571 Content-Type: text/plain; name="03-174.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-174.keywords" non equilibrium quantum statistical mechanics, transport process, heat flow, particle flow, thermal junction, tunnelling junction, non equilibrium stationary state, positive temperature Fermi reservoir, algebraic scattering, entropy production, Ohm law, Onsager relation ---------------0304150253571 Content-Type: application/postscript; name="FMU3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="FMU3.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: FMU3.dvi %%Pages: 54 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Bold Times-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips FMU3 -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.04.10:1844 %%BeginProcSet: texc.pro %! 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b(approach)h(in)g(a)g(study)f(of)h(ener)n(gy)g (transport)f(in)g(the)h(XY)g(spin)f(chain)g(and)h(of)g(tunnelling)257 4786 y(between)j(free-fermion)h(reserv)n(oirs)e(appeared)i(in)e([DFG].) 404 4907 y(In)19 b(Section)f(2.5,)i(long-time)d(stability)g(properties) i(of)g(equilibrium)d(and)j(non-equilibrium)257 5027 y(stationary)j (states)h(against)f(perturbations)g(of)h(the)g(initial)e(state)i(of)g (the)g(coupled)g(system)f(are)257 5147 y(studied,)g(and)g(conditions)f (for)i(the)f(e)o(xistence)g(of)g(temperature)g(or)h(density)e (pro\002les)i(in)f(non-)257 5268 y(equilibrium)h(stationary)h(states)g (are)i(identi\002ed.)404 5388 y(The)h(general)h(theory)f(of)g(Section)g (2)h(is)f(illustrated)f(in)h(Sections)g(3,4)g(and)g(5)g(on)g(the)g(e)o (x-)p eop %%Page: 4 5 4 4 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2362 b FB(4)257 573 y(ample)25 b(of)g(tw)o(o)f(coupled)g(free-electron)i (reserv)n(oirs.)404 693 y(In)i(Section)h(3,)g(the)g(quantum)e(theory)h 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b(gi)n(v)o(en)d(by)i(a)g(\(norm-\))g(con)l(v)o(er)n(gent) f(Dyson)g(series.)29 b(As)18 b(a)i(consequence,)g(non-equilibrium)257 1897 y(stationary)j(states)g(can)h(be)g(constructed)g(with)f(the)g (help)h(of)f(a)h(con)l(v)o(er)n(gent)g(perturbation)e(e)o(x-)257 2017 y(pansion.)404 2138 y(The)g(results)g(of)g(Section)h(4)f(are)h (used)f(in)g(Section)h(5)f(to)g(deri)n(v)o(e)f(e)o(xplicit)g(e)o (xpressions)g(for)257 2258 y(the)26 b(ener)n(gy)h(and)f(char)n(ged)h (particle-currents)g(to)e(leading)h(order)h(in)f(the)g(man)o(y-body)e (inter)n(-)257 2379 y(action)g(terms.)30 b(These)25 b(e)o(xpresssions,) d(along)i(with)g(the)g(con)l(v)o(er)n(gence)g(of)h(the)f(Dyson)f (series)257 2499 y(pro)o(v)o(e)36 b(that,)k(for)d(small)f(coupling)g (constants,)j(the)d(entrop)o(y)h(production)f(rate)h(is)f(strictly)257 2619 y(positi)n(v)o(e,)25 b(Ohm')-5 b(s)25 b(la)o(w)h(holds)f(to)h (leading)g(order)h(in)f(the)g(v)n(oltage)g(drop)g(between)g(the)g (reser)n(-)257 2740 y(v)n(oirs,)37 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Fs(2)p Fx(\031)624 3554 y Ft(Z)680 3780 y Fq(R)728 3761 y Fp(6)783 3690 y Fx(d)p Fr(k)26 b Fx(d)p Fr(l)g Fx(\016)t Fs(\()p Fv(j)p Fr(k)q Fv(j)1234 3649 y Fo(2)1295 3690 y Fv(\000)d(j)p Fr(l)p Fv(j)1486 3649 y Fo(2)1525 3690 y Fs(\))17 b Fv(j)f Ft(b)-71 b Fx(w)r Fs(\(\()p Fv(\000)p Fr(k)r Fx(;)17 b(I)8 b(I)g Fs(\))p Fx(;)17 b Fs(\()p Fr(l)p Fx(;)g(I)8 b Fs(\)\))p Fv(j)2394 3642 y Fo(2)2450 3690 y Fs(\()p Fx(\032)2538 3705 y Fn(I)d(I)2614 3690 y Fs(\()p Fr(k)r Fs(\))22 b Fv(\000)h Fx(\032)2924 3705 y Fn(I)2964 3690 y Fs(\()p Fr(k)r Fs(\)\))16 b Fx(;)279 3949 y Fv(P)58 b(')50 b Fs(2)p Fx(\031)624 3813 y Ft(Z)680 4039 y Fq(R)728 4020 y Fp(6)783 3949 y Fx(d)p Fr(k)26 b Fx(d)p Fr(l)g Fv(j)p Fr(k)q Fv(j)1149 3908 y Fo(2)1188 3949 y Fx(\016)t Fs(\()p Fv(j)p Fr(k)r Fv(j)1391 3908 y Fo(2)1452 3949 y Fv(\000)d(j)p Fr(l)p Fv(j)1643 3908 y Fo(2)1682 3949 y Fs(\))17 b Fv(j)f Ft(b)-72 b Fx(w)s Fs(\(\()p Fv(\000)p Fr(k)r Fx(;)17 b(I)8 b(I)g Fs(\))p Fx(;)17 b Fs(\()p Fr(l)p Fx(;)g(I)8 b Fs(\)\))p 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257 266 a FC(FMU)25 b(April)f(10,)h(2003)2362 b FB(5)257 573 y(same)25 b(chemical)g(potential.)404 693 y(Another)19 b(consequence)h(of)g(\()p Fv(\003)p FB(\))g(is)f(that,)h(to)g(leading)f(order)h(in)g(the)f(interaction,)h (and)g(for)g(a)257 814 y(small)h(v)n(oltage)f(drop,)i Fs(\001)p Fx(\026)28 b Fs(=)f Fx(\026)1367 777 y Fn(I)5 b(I)1452 814 y Fv(\000)k Fx(\026)1597 777 y Fn(I)1658 814 y FB(\(at)22 b(a)g(\002x)o(ed)f(temperature,)g Fx(T)14 b FB(,)22 b(for)g(both)e(reserv)n(oirs\),)257 934 y Fm(Ohm')l(s)25 b(law)g FB(is)f(v)n(alid,)g(i.e.,)g(the)h(v)n(oltage)f(drop)h(is)f (proportional)f(to)i(the)f(current,)1501 1154 y Fs(\001)p Fx(\026)k Fv(')g Fx(R)q Fs(\()p Fx(\026)1946 1113 y Fn(I)1986 1154 y Fx(;)17 b(T)d Fs(\))p Fv(J)j Fx(:)257 1374 y FB(Our)28 b(calculations)g(sho)n(w)f(that)g(the)h(resistance)g Fx(R)q Fs(\()p Fx(\026)2101 1338 y Fn(I)2141 1374 y Fx(;)17 b(T)d Fs(\))27 b FB(gro)n(ws)h(linearly)f(in)h Fx(T)14 b FB(,)29 b(for)f(lar)n(ge)257 1494 y Fx(T)14 b FB(,)33 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Fn(r)1905 1494 y FB(,)h(called)e(the)g Fm(\223\002eld)g(alg)o(ebr)o(a\224)p FB(,)h(con)l(v)o(enient)257 1615 y(for)25 b(the)g(description)f(of)h(the)f(thermodynamic)g(limit)f (of)i(a)g(reserv)n(oir:)1446 1869 y Fv(F)1528 1828 y Fn(r)1621 1869 y Fs(:=)p 1779 1752 500 4 v 1819 1774 a Ft(_)1779 1986 y Fo(\003)p Fu(\0451)1986 1869 y Fx(B)5 b Fs(\()p Fv(H)2188 1840 y Fo(\003)2241 1869 y Fs(\))p Fx(;)949 b FB(\(2.15\))257 2199 y(where)521 2124 y Ft(W)604 2228 y Fo(\003)p Fu(\0451)814 2199 y Fx(B)5 b Fs(\()p Fv(H)1016 2163 y Fo(\003)1070 2199 y Fs(\))20 b FB(is)f(the)g(algebra)i (generated)f(by)f(all)h(the)f(operators)h(in)f(the)h(increasing)257 2319 y(sequence)25 b(of)g(algebras)1152 2539 y Fx(:)17 b(:)g(:)27 b Fv(\022)56 b Fx(B)1506 2458 y Ft(\000)1551 2539 y Fv(H)1636 2498 y Fo(\003)1690 2458 y Ft(\001)1791 2539 y Fv(\022)g Fx(B)2003 2458 y Ft(\000)2048 2539 y Fv(H)2133 2498 y Fo(\003)2182 2474 y Ff(0)2209 2458 y Ft(\001)2310 2539 y Fv(\022)g Fx(:)17 b(:)g(:)f(;)257 2777 y Fs(\003)36 b 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b(of)g(the)g(thermodynamic)h(limit)e(of)h(the)g(dynamics)g(and)g(the)g (gauge)257 4592 y(transf)n(ormations)p FB(.)404 4713 y Fm(F)-10 b(or)24 b(e)o(very)h(oper)o(ator)e Fx(a)28 b Fv(2)g(F)1434 4677 y Fn(r)1471 4713 y Fm(,)d(the)g(limits)e(in)h (oper)o(ator)g(norm)1359 4933 y Fx(n)e Fv(\000)h Fs(lim)1421 5004 y Fo(\003)p Fu(\0451)1719 4933 y Fx(\013)1782 4892 y Fo(\003)1781 4957 y Fn(t)1834 4933 y Fs(\()p Fx(a)p Fs(\))56 b(=:)f Fx(\013)2237 4948 y Fn(t)2267 4933 y Fs(\()p Fx(a)p Fs(\))861 b FB(\(2.17\))257 5203 y Fm(and)1351 5323 y Fx(n)23 b Fv(\000)f Fs(lim)1414 5394 y Fo(\003)p Fu(\0451)1711 5323 y Fx(')1775 5282 y Fo(\003)1775 5348 y Fj(s)1828 5323 y Fs(\()p Fx(a)p Fs(\))55 b(=:)h Fx(')2233 5338 y Fj(s)2274 5323 y Fs(\()p Fx(a)p Fs(\))854 b FB(\(2.18\))p eop %%Page: 9 10 9 9 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2362 b FB(9)257 573 y Fm(e)n(xist,)27 b(for)f(all)h Fx(t)k Fv(2)h Fl(R)5 b Fx(;)49 b Fr(s)32 b Fv(2)f Fl(R)1320 537 y Fn(M)1405 573 y Fm(,)d(and)e(de\002ne)1899 537 y 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Fv(F)3200 1259 y Fn(r)3257 1295 y Fm(point-)257 1416 y(wise)25 b(in)l(variant)f FB(under)g Fv(f)p Fx(')1211 1431 y Fj(s)1253 1416 y Fv(g)1303 1434 y Fj(s)p Fu(2)p Fq(R)1435 1415 y Fe(M)1506 1416 y FB(,)h(i.e.,)861 1620 y Fv(A)941 1579 y Fn(r)1034 1620 y Fs(:=)1192 1539 y Ft(\010)1250 1620 y Fx(a)j Fv(2)g(F)1505 1579 y Fn(r)1570 1535 y Ft(\014)1570 1595 y(\014)1631 1620 y Fx(')1695 1635 y Fj(s)1737 1620 y Fs(\()p Fx(a)p Fs(\))55 b(=)g Fx(a;)42 b Fs(for)24 b(all)48 b Fr(s)28 b Fv(2)g Fl(R)2705 1579 y Fn(M)2790 1539 y Ft(\011)2865 1620 y Fx(:)363 b FB(\(2.19\))257 1825 y(Since,)38 b(by)d(\(2.5\),)j(\(2.17\))d(and)g (\(2.18\),)i Fx(\013)1750 1840 y Fn(t)1815 1825 y FB(and)e Fx(')2058 1840 y Fj(s)2134 1825 y FB(commute,)i Fx(\013)2630 1840 y Fn(t)2660 1825 y Fs(\()p Fx(a)p Fs(\))46 b Fv(2)h(A)3026 1788 y Fn(r)3064 1825 y FB(,)38 b(for)d(e)n(v)o(ery)257 1945 y Fx(a)28 b Fv(2)g(A)510 1909 y Fn(r)548 1945 y FB(.)404 2183 y Fz(\(A2\))50 b(Existence)25 b(of)g(the)h(thermodynamic) g(limit)e(of)h(the)h(equilibrium)g(state)p FB(.)404 2304 y Fm(F)-10 b(or)24 b(e)o(very)h Fx(a)j Fv(2)g(F)1062 2268 y Fn(r)1099 2304 y Fm(,)1375 2424 y Fs(lim)1348 2487 y Fo(\003)p Fu(\0451)1582 2424 y Fx(!)1647 2383 y Fo(\003)1643 2449 y Fn(\014)s(;)p Fj(\026)1760 2424 y Fs(\()p Fx(a)p Fs(\))55 b(=:)g Fx(!)2161 2439 y Fn(\014)s(;)p Fj(\026)2278 2424 y Fs(\()p Fx(a)p Fs(\))850 b FB(\(2.20\))257 2634 y Fm(e)n(xists)25 b(and)f(is)g(time-tr)o(anslation)e(in)l (variant,)h(i.e)o(.,)1365 2838 y Fx(!)1426 2853 y Fn(\014)s(;)p Fj(\026)1575 2838 y Fs(\()p Fx(\013)1675 2853 y Fn(t)1705 2838 y Fs(\()p Fx(a)p Fs(\)\))55 b(=)g Fx(!)2117 2853 y Fn(\014)s(;)p Fj(\026)2234 2838 y Fs(\()p Fx(a)p Fs(\))p Fx(;)867 b FB(\(2.21\))257 3043 y Fm(for)25 b(all)f Fx(a)k Fv(2)g(F)784 3006 y Fn(r)821 3043 y Fx(;)42 b(t)28 b Fv(2)g Fl(R)22 b Fm(.)257 3300 y FB(W)-8 b(e)31 b(assume)e(that)h Fv(F)1003 3263 y Fn(r)1070 3300 y FB(contains)g(a)g(norm-dense)g (subalgebra)2521 3212 y Fu(\016)2469 3300 y Fv(F)2551 3263 y 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Fs(\()p Fx(a)p Fs(\))17 b Fx(b)p Fs(\))55 b(=)g Fx(!)1931 4653 y Fn(\014)s(;)p Fj(\026)2048 4638 y Fs(\()p Fx(b)28 b(\013)2217 4653 y Fn(t)p Fo(+)p Fn(i\014)s Fq(~)2407 4638 y Fs(\()p Fx(a)p Fs(\)\))p Fx(:)656 b FB(\(2.23\))257 4931 y Fw(2.2)119 b(Thermal)24 b(contacts)f(and)h(tunnelling)i(junctions)e(between)h (macr)n(o-)526 5080 y(scopic)30 b(r)n(eser)o(v)o(oirs)257 5268 y FB(W)-8 b(e)22 b(consider)f(tw)o(o)f(reserv)n(oirs,)i Fx(I)29 b FB(and)21 b Fx(I)-9 b(I)8 b FB(,)22 b(with)f(all)g(the)g (properties)f(described)h(in)g(Sect.)h(2.1.)257 5388 y(These)i(reserv)n(oirs)g(may)f(or)h(may)f(not)g(ha)n(v)o(e)g(the)h (same)f(physical)g(properties.)30 b(F)o(or)23 b(e)o(xample,)p eop %%Page: 10 11 10 10 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(10)257 573 y(the)o(y)20 b(may)g(be)g(ordinary)g(metals)g(located)g (in)g(tw)o(o)g(complementary)f(half-spaces)i(of)f Fl(R)3246 537 y Fo(3)3291 573 y FB(;)i(or)e Fx(I)257 693 y FB(may)j(be)f(a)h (metal)f(and)h Fx(I)-9 b(I)31 b FB(a)23 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Fm(perturbed)g(Hamiltonian)p FB(,)e Fx(H)8 b FB(,)24 b(of)g(the)g(coupled)g(system.)30 b(The)24 b(operator)g Fx(H)32 b FB(has)257 3142 y(the)25 b(form)1382 3263 y Fx(H)63 b Fs(=)56 b Fx(H)1747 3222 y Fo(0)1808 3263 y Fs(+)22 b Fx(W)14 b Fs(\(\003)2118 3222 y Fn(I)2157 3263 y Fx(;)j Fs(\003)2269 3222 y Fn(I)-7 b(I)2332 3263 y Fs(\))885 b FB(\(2.27\))257 3437 y(where)29 b Fx(W)14 b Fs(\(\003)741 3401 y Fn(I)780 3437 y Fx(;)j Fs(\003)892 3401 y Fn(I)-7 b(I)955 3437 y Fs(\))28 b FB(is)g(a)g(bounded,)g(self)o(adjoint)f(operator)h(on)f Fv(H)j FB(for)e(each)g(choice)h(of)f Fs(\003)3456 3401 y Fn(I)257 3557 y FB(and)d Fs(\003)494 3521 y Fn(I)-7 b(I)558 3557 y FB(.)31 b(W)-8 b(e)25 b(shall)f Fm(always)h FB(require)g(the)f(follo)n(wing)f(assumption.)404 3796 y Fz(\(A3\))50 b(Existence)25 b(of)g(the)h(thermodynamic)g(limit)e(of)h (the)h(contact)g(interaction)p FB(.)1290 4016 y Fx(n)c Fv(\000)h Fs(lim)1335 4098 y Fo(\003)1384 4075 y Fe(I)1419 4098 y Fu(\0451)1325 4178 y Fo(\003)1374 4154 y Fe(I)-7 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Fs(\))22 b Fv(\012)h Fx(B)5 b Fs(\()p Fv(H)1992 537 y Fn(I)-7 b(I)2056 573 y Fs(\))p FB(.)404 693 y(It)32 b(follo)n(ws)e(from)i(assumptions)d(\(A1\))j(and)g(\(A3\),)i(Eqs.)51 b(\(2.17\))32 b(and)g(\(2.28\),)h(that)e(the)257 814 y(thermodynamic)g(limit)g(of)h(the)g(time)g(e)n(v)n(olution)e(of)i(the) g(coupled)g(reserv)n(oirs)g(e)o(xists:)44 b(F)o(or)257 934 y(arbitrary)25 b Fx(a)j Fv(2)g(F)35 b FB(the)24 b(limits)1203 1126 y Fx(\013)1266 1085 y Fo(0)1265 1151 y Fn(t)1305 1126 y Fs(\()p Fx(a)p Fs(\))55 b(=)h Fx(n)22 b Fv(\000)h Fs(lim)1664 1208 y Fo(\003)1713 1185 y Fe(I)1748 1208 y Fu(\0451)1653 1288 y Fo(\003)1702 1264 y Fe(I)-7 b(I)1758 1288 y Fu(\0451)1978 1126 y Fx(\013)2041 1085 y Fo(\003)2090 1062 y Fe(I)2040 1151 y Fn(t)2152 1126 y Fv(\012)22 b Fx(\013)2314 1085 y Fo(\003)2363 1062 y Fe(I)-7 b(I)2313 1151 y Fn(t)2423 1126 y Fs(\()p Fx(a)p Fs(\))705 b FB(\(2.30\))257 1459 y(and)1312 1579 y Fx(\013)1374 1594 y Fn(t)1403 1579 y Fs(\()p Fx(a)p Fs(\))56 b(=)f Fx(n)22 b Fv(\000)h 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4183 y(that)1333 4304 y Fx(')1397 4263 y Fn(I)1397 4328 y Fj(s)1439 4304 y Fs(\()p Fx(W)14 b Fs(\))55 b(=)g Fx(')1871 4263 y Fn(I)-7 b(I)1871 4328 y Fj(s)1935 4304 y Fs(\()p Fx(W)14 b Fs(\))55 b(=)g Fx(W)m(;)836 b FB(\(2.34\))257 4466 y(in)25 b(the)f (thermodynamic)g(limit.)404 4586 y(Ener)n(gy)g(appears)h(to)g(be)g(the) f(only)g(thermodynamic)f(quantity)h(that)g(can)h(be)g(e)o(xchanged)257 4707 y(through)f(a)h(thermal)g(contact.)404 4945 y Fz(\(J2\))54 b(T)-9 b(unnelling)28 b(junctions)p FB(.)66 b(There)28 b(are)g Fx(m)k Fv(\024)g Fs(min)16 b(\()p Fx(M)2547 4909 y Fn(I)2587 4945 y Fx(;)h(M)2735 4909 y Fn(I)-7 b(I)2800 4945 y Fs(\))27 b FB(linear)g(combina-)257 5076 y(tions,)516 5050 y Ft(e)496 5076 y Fx(Q)573 5039 y Fo(\003)622 5016 y Fe(I)573 5100 y Fo(1)662 5076 y Fx(;)17 b(:)g(:)g(:)f(;)900 5050 y Ft(e)881 5076 y Fx(Q)958 5039 y Fo(\003)1007 5016 y Fe(I)958 5100 y Fn(m)1066 5076 y FB(and)1248 5050 y Ft(e)1229 5076 y Fx(Q)1306 5039 y Fo(\003)1355 5016 y Fe(I)-7 b(I)1306 5100 y Fo(1)1415 5076 y Fx(;)17 b(:)g(:)g(:)f(;)1652 5050 y Ft(e)1634 5076 y Fx(Q)1711 5039 y Fo(\003)1760 5016 y Fe(I)-7 b(I)1711 5100 y Fn(m)1819 5076 y FB(,)21 b(of)e(conserv)n(ation)f(la)o(ws)g(of)i(the)f(tw)o(o)f(reserv)n(oirs) 257 5196 y(with)24 b(the)h(property)f(that)h(the)f(operators)1232 5388 y Fx(Q)1309 5347 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1309 5413 y(j)1512 5388 y Fs(:=)1689 5363 y Ft(e)1670 5388 y Fx(Q)1747 5347 y Fo(\003)1796 5324 y Fe(I)1747 5413 y Fn(j)1858 5388 y Fv(\012)23 b Fs(1)-22 b(l)20 b(+)i(1)-22 b(l)21 b Fv(\012)2326 5363 y Ft(e)2307 5388 y Fx(Q)2384 5347 y Fo(\003)2433 5324 y Fe(I)-7 b(I)2384 5413 y Fn(j)2493 5388 y Fx(;)735 b FB(\(2.35\))p eop %%Page: 12 13 12 12 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(12)257 573 y Fx(j)38 b Fs(=)32 b(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(m)p FB(,)28 b(are)f Fm(conservation)f(laws)h FB(of)g(the)g Fm(perturbed)g(dynamics)f FB(generated)i(by)e(the)257 693 y(Hamiltonian)35 b Fx(H)45 b FB(of)37 b(eq.)g(\(2.27\).)66 b(W)l(ithout)36 b(loss)f(of)i(generality)-6 b(,)39 b(we)e(may)g(assume) f(that)276 788 y Ft(e)257 814 y Fx(Q)334 777 y Fo(\003)383 754 y Fe(I)334 838 y Fn(j)455 814 y Fs(=)31 b Fx(Q)639 777 y Fo(\003)688 754 y Fe(I)639 838 y Fn(j)755 814 y FB(and)945 788 y Ft(e)926 814 y Fx(Q)1003 777 y Fo(\003)1052 754 y Fe(I)-7 b(I)1003 838 y Fn(j)1144 814 y Fs(=)31 b Fx(Q)1328 777 y Fo(\003)1377 754 y Fe(I)-7 b(I)1328 838 y Fn(j)1438 814 y FB(,)27 b(for)g Fx(j)38 b Fs(=)32 b(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(m)p FB(.)37 b(Of)28 b(course,)f(there)g (may)g(be)g(further)257 953 y(conserv)n(ation)18 b(la)o(ws)h(of)g(the)h (reserv)n(oirs,)g Fx(Q)1741 917 y Fo(\003)1790 894 y Fe(I)1741 978 y Fn(i)1831 953 y Fv(\012)r Fs(1)-22 b(l)o FB(,)21 b Fs(1)-22 b(l)p Fv(\012)r Fx(Q)2221 917 y Fo(\003)2270 894 y Fe(I)-7 b(I)2221 978 y Fn(j)2331 953 y FB(,)20 b(for)g(some)f Fx(i)28 b(>)f(m)20 b FB(and/or)f(some)257 1073 y Fx(j)34 b(>)28 b(m)p FB(,)20 b(which)f(are)h(conserv)n(ation)e (la)o(ws)g(of)h(the)g(perturbed)g(dynamics.)28 b(\223Leak)o(y)19 b(junctions\224)257 1194 y(are)32 b(contacts)e(where)h(the)g (interaction)f Fx(W)14 b Fs(\(\003)1863 1158 y Fn(I)1902 1194 y Fx(;)j Fs(\003)2014 1158 y Fn(I)-7 b(I)2077 1194 y Fs(\))31 b Fm(violates)f(some)g FB(of)h(the)g(conserv)n(ation)257 1314 y(la)o(ws)f Fx(Q)546 1278 y Fo(\003)595 1255 y Fe(I)546 1339 y Fn(i)660 1314 y Fv(\012)d Fs(1)-22 b(l)29 b FB(and/or)g Fs(1)-22 b(l)25 b Fv(\012)i Fx(Q)1393 1278 y Fo(\003)1442 1255 y Fe(I)-7 b(I)1393 1339 y Fn(j)1502 1314 y FB(,)31 b Fx(i;)17 b(j)43 b(>)38 b(m)p FB(.)47 b(F)o(or)30 b(con)l(v)o (enience,)g(we)h(shall)e(sometimes)257 1435 y(assume)e(that)g(the)g (operators)h Fx(Q)1380 1398 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1380 1459 y(j)1527 1435 y FB(,)28 b Fx(j)38 b Fs(=)32 b(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(m)p FB(,)28 b(are)g(the)f Fm(only)h FB(conserv)n(ation)e(la)o(ws)g(of)257 1555 y(the)d(perturbed)f(dynamics,)g(and)g Fx(M)1506 1519 y Fn(I)1575 1555 y Fs(=)27 b Fx(M)1782 1519 y Fn(I)-7 b(I)1874 1555 y Fs(=)28 b Fx(m)p FB(.)i(Let)23 b Fr(s)k Fs(=)h(\()p Fx(s)2542 1570 y Fo(1)2581 1555 y Fx(;)17 b(:)g(:)g(:)f(;)h(s)2846 1570 y Fn(m)2912 1555 y Fx(;)g Fs(0)p Fx(;)g(:)g(:)g(:)e(;)i Fs(0\))p FB(.)30 b(W)-8 b(e)257 1675 y(de\002ne)1300 1796 y Fx(')1364 1755 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1364 1820 y Fj(s)1511 1796 y Fs(\()p Fx(a)p Fs(\))55 b(:=)h Fx(')1916 1755 y Fo(\003)1965 1731 y Fe(I)1916 1820 y Fj(s)2026 1796 y Fv(\012)23 b Fx(')2190 1755 y Fo(\003)2239 1731 y Fe(I)-7 b(I)2190 1820 y Fj(s)2298 1796 y Fs(\()p Fx(a)p Fs(\))p Fx(;)803 b FB(\(2.36\))257 1952 y(for)25 b Fx(a)j Fv(2)g Fx(B)5 b Fs(\()p Fv(H)773 1916 y Fn(I)814 1952 y Fs(\))22 b Fv(\012)g Fx(B)5 b Fs(\()p Fv(H)1175 1916 y Fn(I)-7 b(I)1239 1952 y Fs(\))p FB(,)25 b(and)1291 2130 y Fx(')1355 2145 y Fj(s)1396 2130 y Fs(\()p Fx(a)p Fs(\))56 b(:=)f Fx(n)22 b Fv(\000)h Fs(lim)1782 2212 y Fo(\003)1831 2189 y Fe(I)1866 2212 y Fu(\0451)1772 2291 y Fo(\003)1821 2268 y Fe(I)-7 b(I)1877 2291 y Fu(\0451)2097 2130 y Fx(')2161 2089 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2161 2154 y Fj(s)2308 2130 y Fs(\()p Fx(a)p Fs(\))p Fx(;)793 b FB(\(2.37\))257 2448 y(for)25 b Fx(a)j Fv(2)g(F)10 b FB(;)25 b(see)g(assumptions)d(\(A1\),)j (eq.)g(\(2.18\).)404 2568 y(T)l(unnelling)d(junctions)i(can)h(then)f (be)h(characterized)h(by)f(the)f(requirement)h(that)1138 2746 y Fx(')1202 2705 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1202 2771 y Fj(s)1349 2665 y Ft(\000)1395 2746 y Fx(W)1501 2665 y Ft(\000)1546 2746 y Fs(\003)1614 2705 y Fn(I)1654 2746 y Fx(;)17 b Fs(\003)1766 2705 y Fn(I)-7 b(I)1829 2665 y Ft(\001\001)1976 2746 y Fs(=)55 b Fx(W)2213 2665 y Ft(\000)2258 2746 y Fs(\003)2326 2705 y Fn(I)2366 2746 y Fx(;)17 b Fs(\003)2478 2705 y Fn(I)-7 b(I)2541 2665 y Ft(\001)2587 2746 y Fx(;)641 b FB(\(2.38\))257 2924 y(for)27 b(arbitrary)g Fs(\003)838 2888 y Fn(I)877 2924 y Fx(;)17 b Fs(\003)989 2888 y Fn(I)-7 b(I)1053 2924 y FB(,)26 b(and)h(hence,)g(using)e(\(2.18\))i(and)f(\(2.28\),)g(we)h (\002nd)g(that,)f(in)g(the)g(ther)n(-)257 3044 y(modynamic)e(limit,) 1581 3165 y Fx(')1645 3180 y Fj(s)1687 3165 y Fs(\()p Fx(W)14 b Fs(\))55 b(=)g Fx(W)m(:)1084 b FB(\(2.39\))257 3321 y(As)19 b(an)g Fm(initial)f(state)g FB(of)h(a)h(tunnelling)d (junction)h(we)h(shall)f(usually)g(choose)h(a)h(state)e Fx(!)23 b FB(close)c(to)257 3441 y(a)j(tensor)g(product)f(state,)h Fx(!)1210 3405 y Fo(\003)1259 3382 y Fe(I)1206 3474 y Fn(\014)1249 3455 y Fe(I)1284 3474 y Fn(;)p Fj(\026)1353 3455 y Fe(I)1403 3441 y Fv(\012)10 b Fx(!)1555 3405 y Fo(\003)1604 3382 y Fe(I)-7 b(I)1551 3474 y Fn(\014)1594 3455 y Fe(I)g(I)1650 3474 y Fn(;)p Fj(\026)1720 3455 y Fe(I)g(I)1780 3441 y FB(,)22 b(of)g(tw)o(o)f(equilibrium)f(states)h (of)h(the)f(uncoupled)257 3577 y(reserv)n(oirs,)k(where)g Fx(\014)1027 3541 y Fn(I)1067 3577 y Fx(;)17 b Fr(\026)1181 3541 y Fn(I)1246 3577 y FB(and)25 b Fx(\014)1476 3541 y Fn(I)-7 b(I)1539 3577 y Fx(;)17 b Fr(\026)1653 3541 y Fn(I)-7 b(I)1742 3577 y FB(are)26 b(arbitrary)-6 b(,)24 b(\(with)g Fr(\026)2581 3541 y Fn(I)2649 3577 y Fv(2)k(M)2863 3541 y Fn(I)2902 3577 y Fx(;)17 b Fr(\026)3016 3541 y Fn(I)-7 b(I)3108 3577 y Fv(2)28 b(M)3322 3541 y Fn(I)-7 b(I)3386 3577 y FB(\).)404 3697 y(T)f(w)o(o)47 b(reserv)n(oirs)h (joined)e(by)i(a)g(tunnelling)e(junction)g(can)i(e)o(xchange)f(ener)n (gy)h(and)257 3817 y(\223char)n(ge\224)23 b(\(as)e(measured)g(by)g(the) g(conserv)n(ation)f(la)o(ws)g Fx(Q)2242 3781 y Fo(\003)2291 3758 y Fe(I)2242 3842 y Fn(j)2339 3817 y Fv(\012)9 b Fs(1)-22 b(l)o FB(,)22 b Fs(1)-22 b(l)7 b Fv(\012)i Fx(Q)2751 3781 y Fo(\003)2800 3758 y Fe(I)-7 b(I)2751 3842 y Fn(j)2860 3817 y FB(,)22 b Fx(j)34 b Fs(=)28 b(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(m)p FB(\),)257 3957 y(or)23 b(leak)h(some)e(\223char)n(ge\224)i (corresponding)e(to)h Fx(Q)1932 3921 y Fo(\003)1981 3897 y Fe(I)1932 3982 y Fn(j)2036 3957 y Fv(\012)16 b Fs(1)-22 b(l)o FB(,)23 b(or)h(to)e Fs(1)-22 b(l)14 b Fv(\012)i Fx(Q)2677 3921 y Fo(\003)2726 3897 y Fe(I)-7 b(I)2677 3982 y Fn(j)2787 3957 y FB(,)23 b(for)g(some)g Fx(j)33 b(>)28 b(m)p FB(.)404 4115 y Fz(Ener)o(gy)j(curr)n(ent)p FB(.)78 b(The)30 b(operator)g(corresponding)f(to)h(a)g(measurement)f (of)h(the)g Fm(gain)257 4235 y(of)d(internal)f(ener)l(gy)i(per)f (second)g FB(of)g(reserv)n(oir)g Fx(r)s FB(,)h(with)e Fx(r)35 b Fs(=)d Fx(I)j FB(or)27 b Fx(I)-9 b(I)8 b FB(,)28 b(at)f(time)f Fx(t)i FB(is)f(con)l(v)o(e-)257 4356 y(niently)d (de\002ned)h(in)g(the)f(Heisenber)n(g)h(picture)g(by)1270 4580 y Fx(P)1347 4538 y Fn(r)1384 4580 y Fs(\()p Fx(t)p Fs(\))83 b(:=)1792 4512 y Fx(d)p 1774 4557 86 4 v 1774 4648 a(dt)1895 4580 y(\013)1958 4538 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1957 4604 y(t)2104 4580 y Fs(\()p Fx(H)2231 4538 y Fn(r)2269 4580 y Fs(\))1592 4814 y(=)1784 4746 y Fx(i)p 1774 4791 54 4 v 1774 4882 a Fl(~)1863 4814 y Fx(\013)1926 4773 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1925 4838 y(t)2072 4814 y Fs(\([)p Fx(H)r(;)17 b(H)2353 4773 y Fn(r)2391 4814 y Fs(]\))p Fx(;)772 b FB(\(2.40\))257 5029 y(where)33 b Fx(\013)596 4993 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)595 5054 y(t)774 5029 y FB(is)32 b(as)g(in)g(\(2.29\),)h (and)f Fx(H)1662 4993 y Fn(r)1741 5029 y Fs(=)41 b Fx(H)1947 4993 y Fn(I)2014 5029 y Fv(\012)28 b Fs(1)-22 b(l)31 b FB(or)h Fs(=)41 b(1)-22 b(l)26 b Fv(\012)i Fx(H)2712 4993 y Fn(I)-7 b(I)2775 5029 y FB(,)34 b(for)e Fx(r)44 b Fs(=)d Fx(I)f FB(or)32 b Fx(I)-9 b(I)8 b FB(,)257 5150 y(respecti)n(v)o(ely)-6 b(.)29 b(By)c(\(2.25\))g(and)g(\(2.27\),)1098 5370 y Fx(P)1175 5328 y Fn(r)1213 5370 y Fs(\()p Fx(t)p Fs(\))55 b(=)1530 5302 y Fx(i)p 1520 5347 V 1520 5438 a Fl(~)1609 5370 y Fx(\013)1672 5328 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1671 5394 y(t)1818 5289 y Ft(\000\002)1905 5370 y Fx(W)14 b Fs(\(\003)2117 5328 y Fn(I)2157 5370 y Fx(;)j Fs(\003)2269 5328 y Fn(I)-7 b(I)2332 5370 y Fs(\))p Fx(;)17 b(H)2503 5328 y Fn(r)2540 5289 y Ft(\003\001)2627 5370 y Fx(:)601 b FB(\(2.41\))p eop %%Page: 13 14 13 13 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(13)257 573 y(By)35 b(\(2.17\),)h(\(2.28\))e(and)g(\(2.31\),)i(the) e(operator)g(corresponding)g(to)f(the)h(ener)n(gy)h(gain)e(per)257 693 y(second)25 b(of)g(reserv)n(oir)g Fx(r)i FB(has)e(a)g (thermodynamic)e(limit)h(gi)n(v)o(en)f(by)1218 959 y Fx(P)1295 918 y Fn(r)1332 959 y Fs(\()p Fx(t)p Fs(\))56 b(=)f Fv(\000)1757 892 y Fx(d)p 1734 936 97 4 v 1734 1028 a(ds)1866 959 y(\013)1928 974 y Fn(t)1974 879 y Ft(\000)2020 959 y Fx(\013)2083 918 y Fn(r)2082 984 y(s)2120 959 y Fs(\()p Fx(W)14 b Fs(\))2302 879 y Ft(\001)2347 875 y(\014)2347 934 y(\014)2381 998 y Fn(s)p Fo(=0)2508 959 y Fx(;)720 b FB(\(2.42\))257 1204 y(where)32 b Fx(\013)595 1167 y Fn(r)594 1228 y(s)664 1204 y FB(is)e(the)h(time)g(e)n(v)n (olution)e(of)i(reserv)n(oir)g Fx(r)j FB(in)c(the)h(thermodynamic)f (limit,)h(in)f(the)257 1324 y(absence)c(of)e(an)o(y)h(contacts.)30 b(It)25 b(follo)n(ws)e(from)i(\(2.41\))f(and)h(\(2.42\))g(that)888 1586 y Fx(P)965 1545 y Fn(I)1005 1586 y Fs(\()p Fx(t)p Fs(\))d(+)g Fx(P)1313 1545 y Fn(I)-7 b(I)1376 1586 y Fs(\()p Fx(t)p Fs(\))83 b(=)1749 1519 y Fx(i)p 1739 1563 54 4 v 1739 1654 a Fl(~)1828 1586 y Fx(\013)1891 1545 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1890 1611 y(t)2054 1505 y Ft(\000\002)2141 1586 y Fx(W)14 b Fs(\(\003)2353 1545 y Fn(I)2392 1586 y Fx(;)j Fs(\003)2504 1545 y Fn(I)-7 b(I)2568 1586 y Fs(\))p Fx(;)17 b(H)2739 1545 y Fo(0)2777 1505 y Ft(\003\001)1570 1820 y Fs(=)1749 1753 y Fx(i)p 1739 1797 V 1739 1889 a Fl(~)1828 1820 y Fx(\013)1891 1779 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1890 1845 y(t)2037 1740 y Ft(\000\002)2124 1820 y Fx(W)14 b Fs(\(\003)2336 1779 y Fn(I)2376 1820 y Fx(;)j Fs(\003)2488 1779 y Fn(I)-7 b(I)2551 1820 y Fs(\))p Fx(;)17 b(H)2722 1740 y Ft(\003)o(\001)1570 2059 y Fs(=)83 b Fv(\000)1851 1991 y Fx(d)p 1833 2036 86 4 v 1833 2127 a(dt)1954 2059 y(\013)2017 2017 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2016 2083 y(t)2163 1978 y Ft(\000)2209 2059 y Fx(W)14 b Fs(\(\003)2421 2017 y Fn(I)2460 2059 y Fx(;)j Fs(\003)2572 2017 y Fn(I)-7 b(I)2636 2059 y Fs(\))2674 1978 y Ft(\001)2719 2059 y Fx(;)509 b FB(\(2.43\))257 2314 y(where)26 b Fx(H)615 2278 y Fo(0)679 2314 y FB(and)e Fx(H)33 b FB(are)25 b(as)g(in)g(eqs.)f(\(2.25\),)h(\(2.27\),)f(and,)h (in)f(the)h(thermodynamic)e(limit,)1221 2580 y Fx(P)1298 2539 y Fn(I)1337 2580 y Fs(\()p Fx(t)p Fs(\))g(+)f Fx(P)1646 2539 y Fn(I)-7 b(I)1709 2580 y Fs(\()p Fx(t)p Fs(\))55 b(=)h Fv(\000)2128 2512 y Fx(d)p 2111 2557 V 2111 2648 a(dt)2231 2580 y(\013)2293 2595 y Fn(t)2323 2580 y Fs(\()p Fx(W)14 b Fs(\))p Fx(;)723 b FB(\(2.44\))257 2824 y(with)24 b Fx(W)42 b Fv(2)28 b(F)10 b FB(.)404 2945 y(W)-8 b(e)38 b(observ)o(e)g(that)g(if)g Fx(!)j FB(is)d(an)g Fm(arbitr)o(ary)e (time-tr)o(anslation)f(in)l(variant)i(state)g FB(of)h(the)257 3065 y(coupled)25 b(system,)e(we)i(ha)n(v)o(e)g(that)1359 3285 y Fx(!)1440 3204 y Ft(\000)1486 3285 y Fx(P)1563 3244 y Fn(I)1602 3285 y Fs(\()p Fx(t)p Fs(\))e(+)f Fx(P)1911 3244 y Fn(I)-7 b(I)1974 3285 y Fs(\()p Fx(t)p Fs(\))2085 3204 y Ft(\001)2186 3285 y Fs(=)55 b(0)p Fx(;)862 b FB(\(2.45\))257 3505 y(for)25 b(all)g(times.)404 3675 y Fz(Char)o(ge)30 b(curr)n(ent)p FB(.)79 b(The)30 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5262 y Fs(\()p Fx(t)p Fs(\))55 b(=)1507 5195 y Fx(i)p 1496 5239 V 1496 5330 a Fl(~)1585 5262 y Fx(\013)1648 5221 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1647 5287 y(t)1811 5181 y Ft(\000\002)1898 5262 y Fx(W)31 b Fs(\(\003)2127 5221 y Fn(I)2166 5262 y Fx(;)17 b Fs(\003)2278 5221 y Fn(I)-7 b(I)2342 5262 y Fs(\))p Fx(;)17 b(Q)2501 5221 y Fn(r)2501 5287 y(j)2538 5181 y Ft(\003\001)2626 5262 y Fx(:)602 b FB(\(2.47\))p eop %%Page: 14 15 14 14 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(14)257 573 y(In)25 b(the)g(thermodynamic)e(limit,)1169 839 y Fx(I)1220 798 y Fn(r)1212 864 y(j)1258 839 y Fs(\()p Fx(t)p Fs(\))55 b(=)g Fv(\000)1662 772 y Fs(1)p 1659 816 54 4 v 1659 907 a Fl(~)1808 772 y Fx(@)p 1758 816 156 4 v 1758 907 a(@)23 b(s)1878 922 y Fn(j)1949 839 y Fx(\013)2011 854 y Fn(t)2041 758 y Ft(\000)2086 839 y Fx(')2150 798 y Fn(r)2150 864 y Fj(s)2192 839 y Fs(\()p Fx(W)14 b Fs(\))2374 758 y Ft(\001)2419 754 y(\014)2419 814 y(\014)2452 878 y Fj(s)p Fo(=0)3255 839 y FB(\(2.48\))257 1111 y(with)32 b Fx(')531 1075 y Fn(r)531 1136 y Fj(s)604 1111 y FB(as)h(in)e(\(2.33\);)36 b Fx(r)44 b Fs(=)d Fx(I)8 b(;)17 b(I)-9 b(I)8 b FB(,)33 b Fx(j)48 b Fs(=)41 b(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(M)2153 1075 y Fn(r)2192 1111 y FB(.)53 b(Recall)32 b(that,)i(for)e Fx(j)48 b Fs(=)41 b(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(m)p FB(,)257 1232 y(the)24 b(operators)g Fx(Q)875 1195 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)875 1256 y(j)1049 1232 y Fs(=)28 b Fx(Q)1230 1195 y Fo(\003)1279 1172 y Fe(I)1230 1256 y Fn(j)1337 1232 y Fv(\012)20 b Fs(1)-22 b(l)17 b(+)i(1)-22 b(l)17 b Fv(\012)i Fx(Q)1846 1195 y Fo(\003)1895 1172 y Fe(I)-7 b(I)1846 1256 y Fn(j)1979 1232 y FB(are)25 b(conserv)n(ation)e(la)o(ws)g(of)h (the)f(perturbed)257 1352 y(dynamics,)h(see)h(\(2.38\),)g(and)f (therefore)730 1614 y Fx(I)781 1573 y Fn(I)773 1639 y(j)820 1614 y Fs(\()p Fx(t)p Fs(\))39 b(+)g Fx(I)1136 1573 y Fn(I)-7 b(I)1128 1639 y(j)1200 1614 y Fs(\()p Fx(t)p Fs(\))83 b(=)1573 1547 y Fx(i)p 1563 1591 54 4 v 1563 1682 a Fl(~)1651 1614 y 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Fv(H)1374 3390 y Fo(\003)1423 3366 y Fe(I)1462 3426 y Fs(\))31 b Fv(\012)g Fx(B)5 b Fs(\()p Fv(H)1841 3390 y Fo(\003)1890 3366 y Fe(I)-7 b(I)1950 3426 y Fs(\))p FB(,)39 b(where)e(the)f(equilibrium)e(states)i Fx(!)3314 3390 y Fo(\003)3363 3366 y Fe(r)3310 3452 y Fn(\014)3353 3433 y Fe(r)3387 3452 y Fn(;)p Fj(\026)3456 3433 y Fe(r)257 3546 y FB(ha)n(v)o(e)e(been)h(de\002ned)f(in)g (\(2.10\).)59 b(The)34 b(state)g Fx(!)1934 3510 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2114 3546 y FB(is)34 b(in)l(v)n(ariant)f(under)h (the)g(unperturbed)257 3667 y(time)24 b(e)n(v)n(olutions)f Fx(\013)958 3631 y Fo(\003)1007 3607 y Fe(r)957 3691 y Fn(t)1045 3667 y FB(,)i Fx(r)31 b Fs(=)c Fx(I)8 b(;)17 b(I)-9 b(I)8 b FB(,)25 b(of)f(the)h(reserv)n(oirs.)404 3787 y(At)19 b(some)g(time)g Fx(t)985 3802 y Fo(0)1024 3787 y FB(,)i(the)e(contact)g(between)h(the)f(reserv)n(oirs)h(is)f (opened,)h(and)f(we)h(are)g(inter)n(-)257 3907 y(ested)k(in)g(the)g(e)n (v)n(olution)e(of)i(the)g(state)f Fx(!)1654 3871 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1825 3907 y FB(under)24 b(the)f Fm(perturbed)h FB(time)f(e)n(v)n(olution,)f Fx(\013)3324 3871 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)3323 3932 y(t)3470 3907 y FB(,)257 4028 y(introduced)24 b(in)h(\(2.27\),)g(\(2.29\).)30 b(In)25 b(particular)l(,)g(we)g(are)h(interested)e(in)h(calculating)f (the)h(rate)257 4148 y(of)c(ener)n(gy)h(gain,)e(or)h(loss,)1203 4123 y Fs(_)1175 4148 y Fx(U)1251 4112 y Fo(\003)1300 4089 y Fe(r)1340 4148 y Fs(\()p Fx(t)p Fs(\))p FB(,)g(the)g(gain)f(or)h (loss)f(of)h(char)n(ge)h Fx(j)6 b FB(,)40 b Fs(_)-45 b Fx(q)2641 4112 y Fo(\003)2690 4089 y Fe(r)2637 4173 y Fn(j)2728 4148 y Fs(\()p Fx(t)p Fs(\))p Fx(;)17 b(j)34 b Fs(=)27 b(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(M)3432 4112 y Fn(r)3470 4148 y FB(,)257 4269 y(per)22 b(second)f(and)h(the)f (entrop)o(y)g(production)f(rate)i Fv(E)2024 4232 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2171 4269 y Fs(\()p Fx(t)p Fs(\))p FB(,)22 b(under)f(the)h(perturbed)f(time)g(e)n(v)n(o-)257 4389 y(lution,)j(in)g(the)h(state)f Fx(!)1061 4353 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1207 4389 y FB(.)31 b(By)25 b(eqs.)g(\(2.40\))g(and)f(\(2.41\),)935 4551 y Fs(_)908 4576 y Fx(U)984 4535 y Fo(\003)1033 4512 y Fe(r)1072 4576 y Fs(\()p Fx(t)p Fs(\))83 b(=)g Fx(!)1490 4535 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1636 4576 y Fs(\()p Fx(P)1751 4535 y Fn(r)1789 4576 y Fs(\()p Fx(t)p Fs(\)\))1266 4767 y(=)1445 4700 y Fx(i)p 1435 4744 54 4 v 1435 4836 a Fl(~)1524 4767 y Fx(!)1589 4726 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1735 4686 y Ft(\000)1781 4767 y Fx(\013)1844 4726 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1843 4792 y(t)1990 4686 y Ft(\000\002)2077 4767 y Fx(W)14 b Fs(\(\003)2289 4726 y Fn(I)2328 4767 y Fx(;)j Fs(\003)2440 4726 y Fn(I)-7 b(I)2504 4767 y Fs(\))p Fx(;)17 b(H)2675 4726 y Fn(r)2712 4686 y Ft(\003)o(\001\001) 3255 4767 y FB(\(2.58\))257 4979 y(and,)25 b(by)g(\(2.46\),)933 5166 y Fs(_)-45 b Fx(q)962 5125 y Fo(\003)1011 5102 y Fe(r)958 5191 y Fn(j)1049 5166 y Fs(\()p Fx(t)p Fs(\))83 b(=)g Fx(!)1467 5125 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1613 5085 y Ft(\000)1659 5166 y Fx(I)1710 5125 y Fn(r)1702 5191 y(j)1748 5166 y Fs(\()p Fx(t)p Fs(\))1859 5085 y Ft(\001)1243 5370 y Fs(=)1423 5302 y Fx(i)p 1412 5347 V 1412 5438 a Fl(~)1501 5370 y Fx(!)1566 5328 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1712 5289 y Ft(\000)1758 5370 y Fx(\013)1821 5328 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1820 5394 y(t)1967 5289 y Ft(\000\002)2054 5370 y Fx(W)14 b Fs(\(\003)2266 5328 y Fn(I)2306 5370 y Fx(;)j Fs(\003)2418 5328 y Fn(I)-7 b(I)2481 5370 y Fs(\))p Fx(;)17 b(Q)2640 5328 y Fn(r)2640 5394 y(j)2678 5289 y Ft(\003)o(\001\001)2811 5370 y Fx(:)417 b FB(\(2.59\))p eop %%Page: 16 17 16 16 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(16)257 573 y(By)26 b(assumption)c(\(A2\),)j(see)g(\(2.20\),)g(the) g(states)f Fx(!)2016 537 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2187 573 y FB(ha)n(v)o(e)24 b(a)h(thermodynamic)f(limit)1340 747 y Fx(!)1405 706 y Fo(0)1444 747 y Fs(\()p Fx(a)p Fs(\))55 b(=)110 b(lim)1768 821 y Fo(\003)1817 798 y Fe(I)1852 821 y Fu(\0451)1757 901 y Fo(\003)1806 877 y Fe(I)-7 b(I)1862 901 y Fu(\0451)2047 747 y Fx(!)2112 706 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2259 747 y Fs(\()p Fx(a)p Fs(\))p Fx(;)842 b FB(\(2.60\))257 1067 y(for)26 b Fx(a)j Fv(2)g(F)38 b Fs(=)29 b Fv(F)871 1031 y Fn(I)933 1067 y Fv(\012)23 b(F)1115 1031 y Fn(I)-7 b(I)1178 1067 y FB(.)33 b(It)25 b(follo)n(ws)f(from)h(this)g(property)-6 b(,)24 b(from)h(assumption)f(\(A3\),)h(and)257 1188 y(from)36 b(equations)e(\(2.31\),)k(\(2.42\),)f(and)f(\(2.48\),)h(that)e(the)g (quantities)2787 1162 y Fs(_)2759 1188 y Fx(U)2835 1151 y Fo(\003)2884 1128 y Fe(r)2924 1188 y Fs(\()p Fx(t)p Fs(\))g FB(and)54 b Fs(_)-45 b Fx(q)3297 1151 y Fo(\003)3346 1128 y Fe(r)3293 1212 y Fn(j)3384 1188 y Fs(\()p Fx(t)p Fs(\))257 1308 y FB(ha)n(v)o(e)25 b(thermodynamic)e(limits)1109 1482 y Fv(P)1186 1441 y Fn(r)1224 1482 y Fs(\()p Fx(t)p Fs(\))83 b(:=)139 b(lim)1615 1556 y Fo(\003)1664 1533 y Fe(I)1699 1556 y Fu(\0451)1604 1636 y Fo(\003)1653 1612 y Fe(I)-7 b(I)1709 1636 y Fu(\0451)1922 1457 y Fs(_)1895 1482 y Fx(U)1971 1441 y 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Fx(\013)2189 5328 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2188 5394 y(t)2352 5370 y Fs(\(ln)16 b Fx(\032)2538 5328 y Fn(r)2576 5370 y Fs(\))2614 5289 y Ft(\001)2701 5370 y Fx(:)527 b FB(\(2.67\))p eop %%Page: 17 18 17 17 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(17)257 573 y(Thus,)24 b(by)h(\(2.55\),)818 768 y Fs(_)791 793 y Fx(S)857 752 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1003 793 y Fs(\()p Fx(t)p Fs(\))84 b(:=)e Fv(E)1445 752 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1592 793 y Fs(\()p Fx(t)p Fs(\))1211 988 y(=)96 b Fv(\000)1489 920 y Fx(d)p 1470 965 86 4 v 1470 1056 a(dt)1592 988 y Fs(tr)1701 907 y Ft(\000)1746 988 y Fx(\032)1796 946 y Fn(I)1859 988 y Fv(\012)22 b Fx(\032)2008 946 y Fn(I)-7 b(I)2089 988 y Fx(\013)2152 946 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2151 1012 y(t)2332 907 y Ft(\000)2378 988 y Fs(ln)16 b Fx(\032)2526 946 y Fn(I)2588 988 y Fv(\012)23 b Fx(\032)2738 946 y Fn(I)-7 b(I)2802 907 y Ft(\001\001)2935 988 y Fx(:)293 b FB(\(2.68\))257 1232 y(By)26 b(\(2.61\))e(and)h(\(2.62\))g(this)e (quantity)h(has)h(a)g(thermodynamic)e(limit)1086 1466 y Fv(E)9 b Fs(\()p Fx(t)p Fs(\))55 b(=)1447 1371 y Ft(X)1417 1583 y Fn(r)r Fo(=)p Fn(I)5 b(;I)-7 b(I)1655 1466 y Fx(\014)1716 1425 y Fn(r)1770 1466 y Fs([)p Fv(P)1874 1425 y Fn(r)1912 1466 y Fs(\()p Fx(t)p Fs(\))22 b Fv(\000)h Fr(\026)2216 1425 y Fn(r)2276 1466 y Fv(\001)f Fc(J)2422 1424 y Fn(r)2460 1466 y Fs(\()p Fx(t)p Fs(\)])42 b Fx(:)588 b FB(\(2.69\))257 1781 y(Inte)o(grating)24 b(\(2.68\))h(in)f(time,)g(we)h(\002nd)g(that) 376 2001 y Fx(S)442 1960 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)588 2001 y Fs(\()p Fx(t)p Fs(\))22 b Fv(\000)h Fx(S)887 1960 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1034 2001 y Fs(\(0\))27 b(=)55 b Fv(\000)p Fs(tr)1504 1921 y Ft(\000)1549 2001 y Fx(\032)1599 1960 y Fn(I)1662 2001 y Fv(\012)22 b Fx(\032)1811 1960 y Fn(I)-7 b(I)1892 1921 y Ft(\002)1934 2001 y Fx(\013)1997 1960 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1996 2026 y(t)2160 1921 y Ft(\000)2205 2001 y Fs(ln)16 b Fx(\032)2353 1960 y Fn(I)2416 2001 y Fv(\012)22 b Fx(\032)2565 1960 y Fn(I)-7 b(I)2630 1921 y Ft(\001)2697 2001 y Fv(\000)23 b Fs(ln)16 b Fx(\032)2945 1960 y Fn(I)3007 2001 y Fv(\012)23 b Fx(\032)3157 1960 y Fn(I)-7 b(I)3221 1921 y Ft(\003\001)3350 2001 y Fx(:)3255 2122 y FB(\(2.70\))257 2242 y(This)31 b(equation)f(sho)n (ws)g(that)g Fx(S)1364 2206 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1511 2242 y Fs(\()p Fx(t)p Fs(\))27 b Fv(\000)g Fx(S)1819 2206 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1965 2242 y Fs(\(0\))31 b FB(is)g(nothing)e(b)n(ut)i(the)g Fm(r)l(elative)f(entr)l(opy)257 2363 y FB(of)e(the)g(density)e(matrix)h Fx(\013)1184 2326 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1183 2387 y(t)1330 2363 y Fs(\()p Fx(\032)1418 2326 y Fn(I)1483 2363 y Fv(\012)24 b Fx(\032)1634 2326 y Fn(I)-7 b(I)1699 2363 y Fs(\))27 b FB(with)g(respect)h(to)f(the)h(density)e(matrix)h Fx(\032)3187 2326 y Fn(I)3252 2363 y Fv(\012)d Fx(\032)3403 2326 y Fn(I)-7 b(I)3468 2363 y FB(;)257 2483 y(see)23 b(e.g.)30 b([BR,)23 b(v)n(ol)f(II])h(for)f(a)h(de\002nition)e(of)h(relati)n(v)o (e)f(entrop)o(y)-6 b(,)22 b(which)g(dif)n(fers)g(from)g(ours)g(by)257 2603 y(the)j(sign,)f(and)h([JP1])g(for)g(similar)l(,)f(independent)g (considerations.)29 b(If)c Fx(A)g FB(is)g(a)g(non-ne)o(gati)n(v)o(e)257 2724 y(matrix)f(and)h Fx(B)30 b FB(is)25 b(a)g(strictly)e(positi)n(v)o (e)g(matrix)h(then)1092 2944 y Fv(\000)p Fs(tr)33 b(\()p Fx(A)17 b Fs(ln)f Fx(B)27 b Fv(\000)c Fx(A)17 b Fs(ln)e Fx(A)p Fs(\))56 b Fv(\025)g Fs(tr)16 b(\()p Fx(A)22 b Fv(\000)h Fx(B)5 b Fs(\))p Fx(;)594 b FB(\(2.71\))257 3164 y(see)25 b(Lemma)e(6.2.21)g(of)i([BR,)g(v)n(ol)e(II].)i(Setting)e Fx(A)28 b Fs(=)f Fx(\032)2191 3128 y Fn(I)2251 3164 y Fv(\012)20 b Fx(\032)2398 3128 y Fn(I)-7 b(I)2486 3164 y FB(and)24 b Fx(B)33 b Fs(=)27 b Fx(\013)2927 3128 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2926 3188 y(t)3074 3164 y Fs(\()p Fx(\032)3162 3128 y Fn(I)3221 3164 y Fv(\012)20 b Fx(\032)3368 3128 y Fn(I)-7 b(I)3432 3164 y Fs(\))p FB(,)257 3284 y(we)25 b(\002nd)g(that)518 3504 y Fx(S)584 3463 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)730 3504 y Fs(\()p Fx(t)p Fs(\))22 b Fv(\000)h Fx(S)1029 3463 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1176 3504 y Fs(\(0\))55 b Fv(\025)g Fs(tr)1581 3423 y Ft(\000)1627 3504 y Fx(\032)1677 3463 y Fn(I)1739 3504 y Fv(\012)23 b Fx(\032)1889 3463 y Fn(I)-7 b(I)1975 3504 y Fv(\000)23 b Fx(\013)2138 3463 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2137 3529 y(t)2284 3504 y Fs(\()p Fx(\032)2372 3463 y Fn(I)2434 3504 y Fv(\012)23 b Fx(\032)2584 3463 y Fn(I)-7 b(I)2648 3504 y Fs(\))2686 3423 y Ft(\001)2787 3504 y Fs(=)55 b(0)p Fx(;)261 b FB(\(2.72\))257 3724 y(by)25 b(the)g(unitarity)e(of)i(time)f(e)n(v)n(olution)f(and)i(the)f (c)o(yclicity)g(of)h(the)f(trace.)32 b(It)24 b(follo)n(ws)g(that)785 3941 y Fs(1)p 774 3985 71 4 v 774 4077 a Fx(T)897 3873 y Ft(Z)996 3899 y Fn(T)952 4098 y Fo(0)1068 4008 y Fv(E)1130 3967 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1276 4008 y Fs(\()p Fx(t)p Fs(\))17 b(dt)55 b(=)1703 3941 y(1)p 1692 3985 V 1692 4077 a Fx(T)1815 3928 y Ft(\000)1860 4008 y Fx(S)1926 3967 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2073 4008 y Fs(\()p Fx(T)14 b Fs(\))22 b Fv(\000)g Fx(S)2407 3967 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2554 4008 y Fs(\(0\))2679 3928 y Ft(\001)2779 4008 y Fv(\025)56 b Fs(0)p Fx(;)267 b FB(\(2.73\))257 4274 y(and)25 b(this)f(inequality)f(remains)i(ob)o(viously)d(v)n(alid)i (in)g(the)h(thermodynamic)e(limit:)1455 4491 y Fs(1)p 1444 4535 V 1444 4627 a Fx(T)1566 4423 y Ft(Z)1666 4449 y Fn(T)1621 4648 y Fo(0)1737 4558 y Fv(E)9 b Fs(\()p Fx(t)p Fs(\))28 b(dt)55 b Fv(\025)h Fs(0)25 b Fx(:)936 b FB(\(2.74\))257 4824 y(Thus,)24 b(if)h(the)g(limit)1564 4945 y Fs(lim)1548 5004 y Fn(t)p Fu(!1)1756 4945 y Fv(E)9 b Fs(\()p Fx(t)p Fs(\))55 b(=:)h Fv(E)1059 b FB(\(2.75\))257 5144 y(e)o(xists)24 b(then)1199 5309 y Fv(E)63 b Fs(=)84 b(lim)1447 5371 y Fn(T)10 b Fu(!1)1701 5241 y Fs(1)p 1690 5286 V 1690 5377 a Fx(T)1812 5173 y Ft(Z)1912 5199 y Fn(T)1868 5399 y Fo(0)1984 5309 y Fv(E)f Fs(\()p Fx(t)p Fs(\))17 b(dt)54 b Fv(\025)i Fs(0)25 b Fx(;)701 b FB(\(2.76\))p eop %%Page: 18 19 18 18 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(18)257 573 y(i.e.,)33 b(the)e Fm(entr)l(opy)f(pr)l(oduction)f(r)o (ate)h Fv(E)9 b FB(,)32 b(in)f(the)g(thermodynamic)e(limit,)i(is)g (non-ne)o(gati)n(v)o(e,)257 693 y(as)f(time)f Fx(t)h FB(tends)e(to)i Fv(1)p FB(;)h(see)f([Ru1].)45 b(In)30 b(Sect.)g(5,)g(we)g(shall)f(study)f(e)o(xamples)h(where)h Fv(E)38 b FB(is)257 814 y Fm(strictly)24 b(positive)p FB(.)404 934 y(Let)29 b(us)h(assume)f(that)g(the)h(entrop)o(y)f (production)g(rate)h Fv(E)9 b Fs(\()p Fx(t)p Fs(\))30 b FB(con)l(v)o(er)n(ges)f(as)h Fx(t)37 b Fv(!)g(1)p FB(.)45 b(It)257 1054 y(follo)n(ws)24 b(from)g(\(2.74\))h(and)g(\(2.76\))f (that)h(it)f(is)g Fm(nonne)l(gative)p FB(.)404 1175 y(Let)g(us)h(set) 1469 1295 y Fv(P)64 b Fs(:=)55 b Fv(P)1837 1254 y Fn(I)1933 1295 y Fs(=)g Fv(\000P)2218 1254 y Fn(I)-7 b(I)3255 1295 y FB(\(2.77\))257 1469 y(and,)25 b(for)g Fx(j)34 b Fs(=)27 b(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(m)p FB(,)1434 1590 y Fv(J)1502 1605 y Fn(j)1593 1590 y Fs(:=)56 b Fv(J)1838 1549 y Fn(I)1820 1615 y(j)1933 1590 y Fs(=)f 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Fx(\014)1416 2184 y Fn(I)-7 b(I)1479 2144 y Ft(\001)1558 2225 y Fv(P)31 b(\000)1757 2144 y Ft(\000)1803 2225 y Fx(\014)1864 2184 y Fn(I)1903 2225 y Fr(\026)1974 2184 y Fn(I)2036 2225 y Fv(\000)23 b Fx(\014)2197 2184 y Fn(I)-7 b(I)2260 2225 y Fr(\026)2331 2184 y Fn(I)g(I)2395 2144 y Ft(\001)2479 2225 y Fv(\001)22 b Fc(J)73 b Fv(\025)56 b Fs(0)25 b Fx(:)341 b FB(\(2.79\))257 2445 y(The)21 b(currents)f Fv(J)860 2409 y Fn(r)842 2470 y(j)918 2445 y FB(v)n(anish)f(for)i (thermal)e(contacts,)i(and)f(\(2.79\))g(sho)n(ws)f(that)h(ener)n(gy)h (is)e(trans-)257 2565 y(ferred)26 b(from)f(the)f(hotter)h(to)f(the)h (colder)g(reserv)n(oir)g(\227)f(as)h(e)o(xpected.)404 2686 y(The)42 b(thermoelectric)f(situation)g(corresponds)g(to)h Fx(M)2351 2650 y Fn(I)2451 2686 y Fs(=)60 b Fx(M)2691 2650 y Fn(I)-7 b(I)2815 2686 y Fs(=)60 b Fx(m)g Fs(=)f(1)p FB(,)46 b(and)257 2806 y Fx(Q)334 2770 y Fn(r)334 2831 y Fo(1)402 2806 y Fs(=)29 b Fx(N)595 2770 y Fn(r)658 2806 y FB(is)c(the)g(particle)g(number)g(operator)-5 b(.)31 b(F)o(or)25 b(identical)f(temperatures)h(b)n(ut)g(dif)n(ferent) 257 2927 y(chemical)i(potentials,)f(\(2.79\))h(sho)n(ws)e(that)i (particles)f(are)i(transferred)g(from)e(the)h(reserv)n(oir)257 3047 y(with)c(the)h(higher)g(chemical)f(potential)g(to)g(the)h(reserv)n (oir)g(with)f(the)h(lo)n(wer)f(chemical)h(poten-)257 3167 y(tial.)30 b(Notice)24 b(that)g(ener)n(gy)h(may)f(\003o)n(w)g (from)g(the)g(colder)g(reserv)n(oir)g(to)g(the)g(hotter)g(one)g(when) 257 3288 y(the)h(chemical)g(potentials)e(are)i(dif)n(ferent)g (\(consider)g(e.g.)f Fx(\014)2338 3252 y Fn(I)2378 3288 y Fx(\026)2437 3252 y Fn(I)2504 3288 y Fv(\035)j Fx(\014)2692 3252 y Fn(I)-7 b(I)2756 3288 y Fx(\026)2815 3252 y Fn(I)g(I)2903 3288 y FB(b)n(ut)25 b Fx(\014)3115 3252 y Fn(I)3182 3288 y Fx(>)i(\014)3346 3252 y Fn(I)-7 b(I)3410 3288 y FB(\).)404 3408 y(Also)29 b(interesting)f(is)h(the)h(case)g(of)g Fm(adiabatic)e FB(thermal)h(contacts)g(between)h(tw)o(o)f(reser)n(-)257 3528 y(v)n(oirs,)39 b(i.e.,)f(without)d(heat)h(e)o(xchange.)65 b(A)36 b(general)g(discussion)f(of)h(systems)f(with)g(time-)257 3649 y(dependent)f(interactions)f(con\002ned)h(to)g(time-dependent)f (re)o(gions)g(is)g(gi)n(v)o(en)g(in)h([FMSU],)257 3769 y(and)25 b(will)f(be)h(elaborated)g(upon)f(in)g(a)i(forthcoming)d (paper)-5 b(.)257 4064 y Fw(2.4)119 b(Existence)30 b(of)f(stationary)g (states)g(in)h(the)h(thermodynamic)f(limit)257 4252 y FB(The)g(abo)o(v)o(e)e(considerations,)h(and)g(in)g(particular)g (\(2.61\),)h(\(2.62\),)g(and)f(\(2.75\),)h(suggest)f(to)257 4372 y(study)24 b(the)h(question)e(whether)i(the)g(in\002nite-v)n (olume)e(states)1245 4592 y Fx(!)1306 4607 y Fn(t)1335 4592 y Fs(\()p Fx(a)p Fs(\))56 b(:=)f Fx(!)1741 4551 y Fo(0)1796 4592 y Fs(\()p Fx(\013)1896 4607 y Fn(t)1926 4592 y Fs(\()p Fx(a)p Fs(\)\))17 b Fx(;)66 b(a)28 b Fv(2)g(F)35 b Fx(;)747 b FB(\(2.80\))257 4812 y(ha)n(v)o(e)25 b(a)g(limit,)e(as)i Fx(t)j Fv(!)f(1)p FB(.)404 4933 y(The)k(state)h Fx(!)870 4897 y Fo(0)909 4933 y FB(,)h(de\002ned)f(in)f(\(2.60\),)j(is)d(ob)o (viously)e(in)l(v)n(ariant)i(under)g(the)h(unperturbed)257 5053 y(time)24 b(e)n(v)n(olution)f Fx(\013)919 5017 y Fo(0)918 5078 y Fn(t)983 5053 y FB(de\002ned)i(in)g(\(2.30\).)30 b(Thus)1156 5273 y Fx(!)1217 5288 y Fn(t)1247 5273 y Fs(\()p Fx(a)p Fs(\))55 b(=)g Fx(!)1625 5232 y Fo(0)1681 5192 y Ft(\000)1726 5273 y Fx(\013)1789 5232 y Fo(0)1788 5298 y Fu(\000)p Fn(t)1889 5273 y Fs(\()p Fx(\013)1989 5288 y Fn(t)2019 5273 y Fs(\()p Fx(a)p Fs(\)\))2184 5192 y Ft(\001)2246 5273 y Fx(;)42 b(a)28 b Fv(2)g(F)10 b Fx(:)658 b FB(\(2.81\))p eop %%Page: 19 20 19 19 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(19)257 573 y(A)33 b(suf)n(\002cient)f(condition)f(for)i(the)f(e)o (xistence)g(of)h(a)g Fm(stationary)d FB(\(i.e.,)35 b(time-translation)c (in-)257 693 y(v)n(ariant\))25 b Fm(limiting)e(state)p FB(,)1214 899 y Fx(!)1275 914 y Fo(stat)1397 899 y Fs(\()p Fx(a)p Fs(\))55 b(=)71 b(lim)1710 958 y Fn(t)p Fu(!1)1921 899 y Fx(!)1982 914 y Fn(t)2012 899 y Fs(\()p Fx(a)p Fs(\))p Fx(;)66 b(a)28 b Fv(2)g(F)35 b Fx(;)716 b FB(\(2.82\))257 1129 y(is)25 b(gi)n(v)o(en)e(in)404 1370 y Fz(\(A4\))50 b(Existence)25 b(of)g(a)g(scattering)h(endomor)o(phism)p FB(.)404 1491 y Fm(The)f(limits)1253 1611 y Fx(\033)1308 1626 y Fu(\006)1368 1611 y Fs(\()p Fx(a)p Fs(\))55 b(=)h Fx(n)22 b Fv(\000)h Fs(lim)1729 1679 y Fn(t)p Fu(!\0061)2041 1611 y Fx(\013)2104 1570 y Fo(0)2103 1636 y Fu(\000)p Fn(t)2205 1611 y Fs(\()o Fx(\013)2304 1626 y Fn(t)2334 1611 y Fs(\()p Fx(a)p Fs(\)\))756 b FB(\(2.83\))257 1820 y Fm(e)n(xist,)21 b(for)e(all)g Fx(a)28 b Fv(2)g(F)10 b Fm(,)20 b(and)g(de\002ne)1472 1784 y Fu(\003)1512 1820 y Fm(endomorphisms)e(of)h Fv(F)10 b Fm(,)20 b(i.e)o(.,)g Fx(\033)2608 1835 y Fu(\006)2688 1820 y Fm(ar)l(e)g(homomorphisms)257 1941 y(of)25 b(the)f Fx(C)583 1904 y Fu(\003)623 1941 y Fm(-alg)o(ebr)o(a)f Fv(F)35 b Fm(with)24 b(the)h(pr)l(operty)e(that)h Fx(\033)2036 1956 y Fu(\006)2096 1941 y Fs(\()p Fx(a)p Fs(\))2223 1904 y Fu(\003)2290 1941 y Fs(=)k Fx(\033)2449 1956 y Fu(\006)2508 1941 y Fs(\()p Fx(a)2597 1904 y Fu(\003)2637 1941 y Fs(\))p Fm(,)c(for)h(all)f Fx(a)k Fv(2)g(F)10 b Fm(.)257 2181 y FB(The)28 b(usefulness)f(of)g(these)h(so-called)f Fm(scattering)g(\(or)g(M\370ller\))g(endomorphisms)f FB(has)h(\002rst)257 2302 y(been)42 b(recognized)g(in)f([He,)46 b(Rob];)k(interesting)40 b(e)o(xamples)h(ha)n(v)o(e)g(been)h (constructed)f(in)257 2422 y([BM].)28 b(In)g(the)g(conte)o(xt)e(of)i (thermal)f(contacts)g(and)h(tunnelling)d(junctions,)i(the)o(y)g(ha)n(v) o(e)g(\002rst)257 2542 y(been)e(used)g(in)f([DFG];)i(see)f(also)f ([Ma].)404 2663 y(It)38 b(is)g(important)f(to)i(note)f(that)g (scattering)g(endomorphisms)e(do)i(not)g(e)o(xist)g(in)g Fm(\002nite)257 2783 y FB(v)n(olume,)29 b(because)g(the)f(free)i(and)e (the)h(perturbed)f(time)g(e)n(v)n(olutions)e(of)j(the)f(tw)o(o)g (reserv)n(oirs)257 2904 y(are)e(generated)f(by)g(Hamiltonians)1293 3109 y Fx(H)1382 3068 y Fo(0)1504 3109 y Fs(=)83 b Fx(H)1752 3068 y Fn(I)1814 3109 y Fv(\012)23 b Fs(1)-22 b(l)20 b(+)i(1)-22 b(l)21 b Fv(\012)i Fx(H)2352 3068 y Fn(I)-7 b(I)2415 3109 y Fx(;)1333 3254 y(H)90 b Fs(=)83 b Fx(H)1752 3213 y Fo(0)1813 3254 y Fs(+)22 b Fx(W)14 b Fs(\(\003)2123 3213 y Fn(I)2163 3254 y Fx(;)j Fs(\003)2275 3213 y Fn(I)-7 b(I)2338 3254 y Fs(\))p Fx(;)257 3459 y FB(see)27 b(\(2.25\))e(and)h (\(2.27\),)g(with)g Fm(pur)l(e-point)f FB(spectra)h(when)g Fs(\003)2404 3423 y Fn(I)2470 3459 y FB(and)f Fs(\003)2707 3423 y Fn(I)-7 b(I)2797 3459 y FB(are)27 b(compact.)34 b(It)26 b(is)257 3580 y(thus)i(natural)g(to)g(w)o(onder)g(about)g(the)g (meaning)g(of)g(scattering)g(endomorphisms)e(for)j(lar)n(ge)257 3700 y(b)n(ut)20 b(\002nite)g(reserv)n(oirs.)29 b(Let)20 b(us)g(sk)o(etch)g(the)g(answer)g(to)g(this)f(question.)28 b(W)-8 b(e)20 b(\002x)h(an)f(arbitrarily)257 3821 y(small,)40 b(b)n(ut)d(positi)n(v)o(e)e(number)h Fx(")p FB(.)69 b(F)o(or)37 b(e)n(v)o(ery)g(operator)g Fx(a)51 b Fv(2)h(F)10 b FB(,)40 b(there)e(e)o(xist)e(compact)257 3941 y(re)o(gions)25 b Fs(\003)643 3905 y Fn(r)681 3941 y Fs(\()p Fx(";)17 b(a)p Fs(\))p FB(,)26 b Fx(r)33 b Fs(=)d Fx(I)8 b(;)17 b(I)-9 b(I)8 b FB(,)26 b(and)h(an)f(operator)g Fx(a)2063 3956 y Fn(")2130 3941 y Fv(2)31 b Fx(B)5 b Fs(\()p Fv(H)2429 3905 y Fo(\003)2478 3881 y Fe(I)2518 3941 y Fs(\))23 b Fv(\012)g Fx(B)5 b Fs(\()p Fv(H)2881 3905 y Fo(\003)2930 3881 y Fe(I)-7 b(I)2990 3941 y Fs(\))p FB(,)27 b(with)e Fs(\003)3351 3905 y Fn(r)3419 3941 y Fs(=)257 4061 y(\003)325 4025 y Fn(r)363 4061 y Fs(\()p Fx(";)17 b(a)p Fs(\))p FB(,)24 b Fx(r)31 b Fs(=)c Fx(I)8 b(;)17 b(I)-9 b(I)8 b FB(,)25 b(such)f(that)1542 4290 y Fv(k)p Fx(a)f Fv(\000)f Fx(a)1816 4305 y Fn(")1853 4290 y Fv(k)55 b Fx(<)2101 4222 y(")p 2099 4267 49 4 v 2099 4358 a Fs(4)2183 4290 y Fx(:)257 4526 y FB(Then,)25 b(by)f(\(2.82\))h(and)g(\(2.83\),)f (there)h(is)g(some)f Fx(T)14 b Fs(\()p Fx(";)j(a)p Fs(\))27 b Fx(<)g Fv(1)e FB(such)f(that)1346 4670 y Ft(\014)1346 4730 y(\014)1380 4755 y Fx(!)1441 4770 y Fo(stat)1563 4755 y Fs(\()p 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Fn(I)-7 b(I)1787 5394 y(t)1934 5370 y Fs(\()p Fx(a)2023 5385 y Fn(")2060 5370 y Fs(\))p Fv(k)55 b Fx(<)2346 5302 y(")p 2344 5347 V 2344 5438 a Fs(4)2428 5370 y Fx(;)p eop %%Page: 20 21 20 20 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(20)257 573 y(for)23 b(all)e Fx(t)28 b Fv(2)g Fs([0)p Fx(;)17 b(T)d Fs(])p FB(.)30 b(Finally)-6 b(,)21 b(by)h(assumption)e (\(A2\),)i(one)g(can)h(choose)f Fs(\003)2797 537 y Fn(r)2834 573 y Fs(\()p Fx(";)17 b(a;)g(T)d Fs(\))21 b FB(so)h(lar)n(ge)257 693 y(that)982 729 y Ft(\014)982 789 y(\014)1015 814 y Fx(!)1080 772 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1243 733 y Ft(\000)1288 814 y Fx(\013)1351 772 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1350 838 y(t)1498 814 y Fs(\()p Fx(a)1587 829 y Fn(")1624 814 y Fs(\))1662 733 y Ft(\001)1729 814 y Fv(\000)23 b Fx(!)1894 772 y Fo(0)1949 733 y Ft(\000)1995 814 y Fx(\013)2058 772 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)2057 838 y(t)2204 814 y Fs(\()p Fx(a)2293 829 y Fn(")2330 814 y Fs(\))2368 733 y Ft(\001)2430 729 y(\014)2430 789 y(\014)2519 814 y Fx(<)2662 746 y(")p 2660 791 49 4 v 2660 882 a Fs(4)2744 814 y Fx(;)257 1016 y FB(pro)o(vided)27 b Fs(\003)706 980 y Fn(r)776 1016 y Fv(\033)33 b Fs(\003)954 980 y Fn(r)992 1016 y Fs(\()p Fx(";)17 b(a;)g(T)d Fs(\))p FB(,)27 b Fx(r)35 b Fs(=)d Fx(I)8 b(;)17 b(I)-9 b(I)8 b FB(,)28 b(for)g Fm(all)f FB(times)f Fx(t)33 b Fv(2)g Fs([0)p Fx(;)17 b(T)d Fs(])p FB(.)38 b(It)28 b(follo)n(ws)d(that,)j (for)257 1137 y(an)o(y)d Fx(T)14 b FB(,)24 b(with)g Fs(0)55 b Fx(<)h(T)14 b Fs(\()p Fx(";)j(a)p Fs(\))54 b Fx(<)h(T)69 b(<)56 b Fv(1)24 b Fx(;)h FB(and)g(for)g Fs(\003)2268 1100 y Fn(r)2361 1137 y Fv(\033)56 b Fs(\003)2562 1100 y Fn(r)2599 1137 y Fs(\()p Fx(";)17 b(a;)g(T)d Fs(\))p Fx(;)66 b(r)58 b Fs(=)d Fx(I)8 b(;)17 b(I)-9 b(I)8 b(;)1131 1266 y Ft(\014)1131 1325 y(\014)1164 1350 y Fx(!)1225 1365 y Fo(stat)1347 1350 y Fs(\()p Fx(a)p Fs(\))22 b Fv(\000)h Fx(!)1661 1309 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1823 1270 y Ft(\000)1869 1350 y Fx(\013)1932 1309 y Fn(I)5 b Fu([)p Fn(I)-7 b(I)1931 1375 y(t)2078 1350 y Fs(\()p Fx(a)2167 1365 y Fn(")2204 1350 y Fs(\))2242 1270 y Ft(\001)2305 1266 y(\014)2305 1325 y(\014)2393 1350 y Fx(<)55 b(")25 b(;)633 b FB(\(2.84\))257 1564 y(for)25 b(all)g(times)f Fx(t)p FB(,)h(with)f Fx(T)14 b Fs(\()p Fx(";)j(a)p Fs(\))27 b Fx(<)g(t)h(<)g(T)14 b FB(.)404 1684 y(These)28 b(simple)g (considerations,)g(combined)f(with)h(\(2.44\))g(and)h(\(2.48\),)g(sho)n (w)e(that)h(the)257 1805 y(ener)n(gy-gain)22 b(rates)990 1780 y Fs(_)963 1805 y Fx(U)1039 1769 y Fo(\003)1088 1745 y Fe(r)1127 1805 y Fs(\()p Fx(t)p Fs(\))g FB(and)g(the)f(currents) 40 b Fs(_)-45 b Fx(q)1959 1769 y Fo(\003)2008 1745 y Fe(r)1955 1829 y Fn(j)2047 1805 y Fs(\()p Fx(t)p Fs(\))21 b FB(of)h(tw)o(o)g(v)o(ery)f(lar)n(ge,)i(b)n(ut)e(\002nite)g(reser)n(-) 257 1925 y(v)n(oirs,)j Fx(r)31 b Fs(=)c Fx(I)8 b(;)17 b(I)-9 b(I)8 b FB(,)25 b(are)g(well)g(approximated)f(by)g(the)h(ener)n (gy-gain)f(rates)923 2185 y Fv(P)1000 2144 y Fn(r)1094 2185 y Fs(:=)71 b(lim)1252 2245 y Fn(t)p Fu(!1)1463 2185 y Fv(P)1540 2144 y Fn(r)1579 2185 y Fs(\()p Fx(t)p Fs(\))55 b(=)g Fv(\000)1987 2117 y Fx(d)p 1963 2162 97 4 v 1963 2253 a(ds)2095 2185 y(!)2156 2200 y Fo(stat)2295 2185 y Fs(\()p Fx(\013)2396 2144 y Fn(r)2395 2210 y(s)2433 2185 y Fs(\()p Fx(W)14 b Fs(\)\))2669 2100 y Ft(\014)2669 2160 y(\014)2703 2224 y Fn(s)p Fo(=0)3255 2185 y FB(\(2.85\))257 2424 y(and)25 b(the)g(currents)813 2663 y Fv(J)899 2622 y Fn(r)881 2688 y(j)992 2663 y Fs(:=)71 b(lim)1151 2723 y Fn(t)p Fu(!1)1362 2663 y Fv(J)1448 2622 y Fn(r)1430 2688 y(j)1486 2663 y Fs(\()p Fx(t)p Fs(\))55 b(=)g Fv(\000)1872 2596 y Fs(1)p 1870 2641 54 4 v 1870 2732 a Fl(~)2011 2596 y Fx(@)p 1969 2641 140 4 v 1969 2732 a(@)5 b(s)2071 2747 y Fn(j)2144 2663 y Fx(!)2205 2678 y Fo(stat)2343 2663 y Fs(\()p Fx(')2445 2622 y Fn(r)2445 2688 y Fj(s)2486 2663 y Fs(\()p Fx(W)14 b Fs(\)\))2722 2579 y Ft(\014)2722 2639 y(\014)2756 2703 y Fj(s)p Fo(=0)2912 2663 y Fx(;)316 b FB(\(2.86\))257 2929 y(respecti)n(v)o(ely)-6 b(,)22 b(for)h(a)g(lar)n(ge)g(range)h(of)f Fm(suf)n(\002ciently)f(lar)l(g)o(e) p FB(,)h(b)n(ut)f Fm(not)g(e)n(xceedingly)i(lar)l(g)o(e)e FB(times)257 3050 y Fx(t)p FB(;)j(\(see)g(\(2.61\),)g(\(2.62\)\).)404 3288 y Fz(Remark)p FB(.)56 b(It)25 b(is)f(usually)g(much)g(easier)h(to) g(pro)o(v)o(e)f(that)g(the)h(limits)1227 3502 y Fx(\033)1286 3461 y Fu(0)1282 3527 y(\006)1341 3502 y Fs(\()p Fx(a)p Fs(\))56 b(:=)f Fx(n)23 b Fv(\000)f Fs(lim)1729 3570 y Fn(t)p Fu(!\0061)2042 3502 y Fx(\013)2104 3517 y Fu(\000)p Fn(t)2205 3421 y Ft(\000)2251 3502 y Fx(\013)2314 3461 y Fo(0)2313 3527 y Fn(t)2353 3502 y Fs(\()p Fx(a)p Fs(\))2480 3421 y Ft(\001)3255 3502 y FB(\(2.87\))257 3757 y(e)o(xist)i(and)h(are) g(operators)g(in)f Fv(F)10 b FB(,)25 b(for)g(arbitrary)f Fx(a)k Fv(2)g(F)10 b FB(,)25 b(rather)g(than)f(to)h(establish)e(the)i (e)o(xis-)257 3877 y(tence)h(of)g(the)f(scattering)g(endomorphisms)f Fx(\033)1880 3892 y Fu(\006)1965 3877 y FB(in)h(\(2.83\).)33 b(If)26 b(the)f(unperturbed)g(dynamics)257 3997 y(of)30 b(the)f(reserv)n(oirs)g(is)g(dispersi)n(v)o(e,)f(\(as)i(for)f (non-interacting,)g(non-relati)n(vistic)e(electrons\),)257 4118 y(one)34 b(may)f(hope)g(to)g(pro)o(v)o(e)g(\(2.87\))g(by)g(using)g (a)h(simple)e(Cook)h(ar)n(gument;)k(see)d(e.g.)57 b([He,)257 4238 y(Rob,)25 b(CFKS].)h(If)g(both)e(limits)f(\(2.83\))h(and)h (\(2.87\))g(e)o(xist)e(then)1182 4452 y Fx(\033)1237 4467 y Fu(\006)1313 4371 y Ft(\000)1359 4452 y Fx(\033)1418 4411 y Fu(0)1414 4477 y(\006)1473 4452 y Fs(\()p Fx(a)p Fs(\))1600 4371 y Ft(\001)1701 4452 y Fs(=)55 b Fx(\033)1891 4411 y Fu(0)1887 4477 y(\006)1964 4452 y Fs(\()o Fx(\033)2056 4467 y Fu(\006)2116 4452 y Fs(\()p Fx(a)p Fs(\)\))g(=)h Fx(a)25 b(;)684 b FB(\(2.88\))257 4666 y(i.e.,)30 b Fx(\033)493 4629 y Fu(0)489 4690 y(\006)577 4666 y FB(is)e(a)h(left)g(and)f(right)g (in)l(v)o(erse)g(of)h Fx(\033)1774 4681 y Fu(\006)1834 4666 y FB(,)g(and)g(hence)g Fx(\033)2377 4681 y Fu(\006)2465 4666 y FB(is)f(a)2633 4629 y Fu(\003)2673 4666 y Fm(automorphism)e FB(of)j Fv(F)10 b FB(.)257 4786 y(This)24 b(will)g(turn)h(out)f(to)g (hold)g(in)h(the)f(e)o(xamples)g(discussed)g(in)g(subsequent)g (sections.)257 5080 y Fw(2.5)119 b(Uniqueness)32 b(and)e(stability)g (pr)n(operties)h(of)e(stationary)g(states)257 5268 y FB(W)-8 b(e)30 b(\002rst)f(describe)g(the)g(property)g(of)h(return)f (to)g(equilibrium)e(for)i(a)h(single)e(reserv)n(oir)-5 b(.)43 b(Let)257 5388 y Fx(!)d FB(be)c(a)g(state)g(on)f(the)h(\002eld)g (algebra)g(of)g(a)g(single)f(reserv)n(oir)l(,)k Fv(F)2589 5352 y Fn(r)2627 5388 y FB(,)f(i.e.,)h Fx(!)g FB(is)d(a)g(positi)n(v)o (e,)p eop %%Page: 21 22 21 21 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(21)257 573 y(linear)28 b(functional)f(on)g Fv(F)1153 537 y Fn(r)1218 573 y FB(normalized)g(such)h(that)f Fx(!)t Fs(\(1)-22 b(l)n(\))33 b(=)g(1)p FB(.)39 b(From)27 b Fv(F)2856 537 y Fn(r)2922 573 y FB(and)g Fx(!)k FB(one)d(can)257 693 y(construct)e(a)h(Hilbert)f(space)h Fv(H)1361 708 y Fn(!)1411 693 y FB(,)g(a)g(representation)f Fx(\031)2180 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5249 y Ff(0)2638 5338 y Fn(n)2709 5233 y Ft(\001)2755 5313 y Fs(\012)2825 5233 y Ft(\013)2897 5313 y Fx(:)p eop %%Page: 23 24 23 23 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(23)257 573 y(Let)1380 693 y Fx(b)1421 708 y Fo(1)1517 693 y Fs(:=)55 b Fx(u)1731 652 y Fn(\013)1731 718 y(n)1805 693 y Fx(;)66 b(b)1939 708 y Fo(2)2035 693 y Fs(:=)55 b Fx(u)2249 652 y Fn(\013)2294 629 y Ff(0)2249 718 y Fn(n)2345 693 y Fx(;)257 865 y FB(for)25 b(some)g(\002x)o(ed)f Fx(n;)17 b(\013)q(;)g(\013)1128 828 y Fu(0)1151 865 y FB(.)30 b(Then)873 997 y Ft(\012)921 1077 y Fx(\031)t Fs(\()p Fx(b)1059 1092 y Fo(1)1098 1077 y Fs(\)\012)p Fx(;)k(\031)1326 997 y Ft(\000)1372 1077 y Fx(\013)1434 1092 y Fn(t)1463 1077 y Fs(\()p Fx(a)p Fs(\))1590 997 y Ft(\001)1653 1077 y Fx(\031)t Fs(\()p Fx(b)1791 1092 y Fo(2)1830 1077 y Fs(\)\012)1938 997 y Ft(\013)2041 1077 y Fs(=)55 b Fx(!)2233 1092 y Fo(stat)2355 997 y Ft(\000)2401 1077 y Fx(b)2442 1036 y Fu(\003)2442 1102 y Fo(1)2482 1077 y Fx(\013)2544 1092 y Fn(t)2573 1077 y Fs(\()p Fx(a)p Fs(\))p Fx(b)2741 1092 y Fo(2)2781 997 y Ft(\001)2852 1077 y Fx(:)376 b FB(\(2.99\))257 1290 y(Since)26 b Fx(!)565 1305 y Fo(stat)687 1290 y Fs(\()p Fx(a)p Fs(\))55 b(=)g Fx(!)1065 1254 y Fo(0)1104 1210 y Ft(\000)1150 1290 y Fx(\033)1205 1305 y Fo(+)1264 1290 y Fs(\()p Fx(a)p Fs(\))1391 1210 y Ft(\001)1462 1290 y FB(,)25 b(and)f(since,)h(by)f(eq.)h(\(2.83\),)1327 1503 y Fx(\033)1382 1518 y Fo(+)1442 1422 y Ft(\000)1487 1503 y Fx(\013)1549 1518 y Fn(t)1579 1503 y Fs(\()p Fx(a)p Fs(\))1706 1422 y Ft(\001)1807 1503 y Fs(=)55 b Fx(\013)2001 1462 y Fo(0)2000 1528 y Fn(t)2040 1422 y Ft(\000)2086 1503 y Fx(\033)2141 1518 y Fo(+)2201 1503 y Fs(\()p Fx(a)p Fs(\))2328 1422 y Ft(\001)2398 1503 y Fx(;)780 b FB(\(2.100\))257 1716 y(for)25 b(arbitrary)g Fx(a)j Fv(2)g(F)10 b FB(,)25 b(the)f(R.S.)i(of)f(\(2.99\))f(is)h(gi)n(v)o(en)e(by)827 1929 y Fx(!)888 1944 y Fo(stat)1010 1848 y Ft(\000)1055 1929 y Fx(b)1096 1888 y Fu(\003)1096 1953 y Fo(1)1136 1929 y Fx(\013)1198 1944 y Fn(t)1228 1929 y Fs(\()p Fx(a)p Fs(\))p Fx(b)1396 1944 y Fo(2)1436 1848 y Ft(\001)1537 1929 y Fs(=)55 b Fx(!)1733 1888 y Fo(0)1772 1848 y Ft(\000)1817 1929 y Fx(\033)1872 1944 y Fo(+)1932 1929 y Fs(\()p Fx(b)2011 1888 y Fu(\003)2011 1953 y Fo(1)2051 1929 y Fs(\))17 b Fx(\013)2169 1888 y Fo(0)2168 1953 y Fn(t)2207 1848 y Ft(\000)2253 1929 y Fx(\033)2308 1944 y Fo(+)2368 1929 y Fs(\()p Fx(a)p Fs(\))2495 1848 y Ft(\001)2557 1929 y Fx(\033)2612 1944 y Fo(+)2671 1929 y Fs(\()p Fx(b)2750 1944 y Fo(2)2790 1929 y Fs(\))2828 1848 y Ft(\001)2899 1929 y Fx(:)257 2142 y FB(It)25 b(follo)n(ws)e(from)i(\(2.96\))g(by)f (polarization)g(that)299 2355 y Fs(lim)284 2414 y Fn(t)p Fu(!1)495 2355 y Fx(!)560 2313 y Fo(0)598 2274 y Ft(\000)644 2355 y Fx(\033)699 2370 y Fo(+)759 2355 y Fs(\()p Fx(b)838 2313 y Fu(\003)838 2379 y Fo(1)877 2355 y Fs(\))17 b Fx(\013)995 2313 y Fo(0)994 2379 y Fn(t)1034 2274 y Ft(\000)1080 2355 y Fx(\033)1135 2370 y Fo(+)1194 2355 y Fs(\()p Fx(a)p Fs(\))1321 2274 y Ft(\001)1384 2355 y Fx(\033)1439 2370 y Fo(+)1498 2355 y Fs(\()p Fx(b)1577 2370 y Fo(2)1617 2355 y Fs(\))1655 2274 y Ft(\001)1784 2355 y Fs(=)82 b Fx(!)2007 2313 y Fo(0)2046 2274 y Ft(\000)2092 2355 y Fx(\033)2147 2370 y Fo(+)2206 2355 y Fs(\()p Fx(b)2285 2313 y Fu(\003)2285 2379 y Fo(1)2325 2355 y Fs(\))17 b Fx(\033)2435 2370 y Fo(+)2494 2355 y Fs(\()p Fx(b)2573 2370 y Fo(2)2613 2355 y Fs(\))2651 2274 y Ft(\001)2713 2355 y Fx(!)2778 2313 y Fo(0)2817 2274 y Ft(\000)2863 2355 y Fx(\033)2918 2370 y Fo(+)2977 2355 y Fs(\()p Fx(a)p Fs(\))3104 2274 y Ft(\001)1784 2542 y Fs(=)82 b Fx(!)2003 2557 y Fo(stat)2125 2462 y Ft(\000)2171 2542 y Fx(b)2212 2501 y Fu(\003)2212 2567 y Fo(1)2269 2542 y Fx(b)2310 2557 y Fo(2)2349 2462 y Ft(\001)2423 2542 y Fx(!)2484 2557 y Fo(stat)2606 2542 y Fs(\()p Fx(a)p Fs(\))1784 2688 y(=)1942 2607 y Ft(\012)1989 2688 y Fx(\031)t Fs(\()p Fx(b)2127 2703 y Fo(1)2167 2688 y Fs(\)\012)p Fx(;)17 b(\031)t Fs(\()p Fx(b)2457 2703 y Fo(2)2497 2688 y Fs(\)\012)2605 2607 y Ft(\013)2680 2688 y Fx(!)2741 2703 y Fo(stat)2863 2688 y Fs(\()p Fx(a)p Fs(\))25 b Fx(:)163 b FB(\(2.101\))257 2900 y(Our)25 b(contention,)f(eq.)g(\(2.98\),)h(follo)n(ws)e(from)i (this.)257 3194 y Fw(2.6)119 b(Cluster)31 b(pr)n(operties)f(and)h(pr)n (o\002les)f(in)h Fb(d)p Fa(>)o Fb(2)g Fw(dimensional)526 3344 y(systems)257 3531 y FB(The)26 b(last)f(question)g(we)h(wish)e(to) i(address,)f(in)h(this)e(summary)h(of)h(the)f(general)h(theory)-6 b(,)25 b(con-)257 3652 y(cerns)36 b Fm(cluster)e(pr)l(operties)f FB(of)j(the)e(stationary)h(state)f Fx(!)2231 3667 y Fo(stat)2353 3652 y FB(,)k(which)d(will)f(sho)n(w)g(that)g Fx(!)3373 3667 y Fo(stat)257 3772 y Fm(cannot)26 b FB(be)g(an)h(equilibrium)d (\(KMS\))i(state)g(for)h(the)f(dynamics,)f Fx(\013)2590 3787 y Fn(t)2620 3772 y FB(,)h(of)h(the)f(coupled)f(reser)n(-)257 3893 y(v)n(oirs)g(and)h(is,)f(in)g(general,)h Fm(not)f FB(normal)f(relati)n(v)o(e)h(to)g(the)g(product)g(state,)g Fx(!)2873 3856 y Fo(0)2912 3893 y FB(,)h(of)f(the)h(uncou-)257 4013 y(pled)f(reserv)n(oirs.)404 4133 y(W)-8 b(e)33 b(consider)g(tw)o (o)g(increasing)g(f)o(amilies)f(of)h(reserv)n(oirs)g(con\002ned)h(to)f (re)o(gions)f Fs(\003)3337 4097 y Fn(r)3418 4133 y Fv(\032)257 4254 y Fl(R)323 4218 y Fo(3)369 4254 y FB(,)25 b(with)1399 4374 y Fs(\003)1467 4333 y Fn(r)1532 4374 y Fv(\045)i Fl(R)1725 4333 y Fo(3)1796 4374 y Fx(;)66 b(r)58 b Fs(=)d Fx(I)8 b(;)17 b(I)-9 b(I)33 b(;)851 b FB(\(2.102\))257 4545 y(joined)34 b(together)g(by)g(a)h(thermal)f(contact)h(or)f(a)h (tunnelling)e(junction)g(localized)h(near)h(the)257 4666 y(origin,)23 b Fr(x)28 b Fs(=)f(0)p FB(,)d(of)g(physical)e(space.)31 b(The)24 b(more)f(realistic)h(situation)e(where)i(the)f(reserv)n(oirs) 257 4786 y(are)29 b(con\002ned)f(to)g(tw)o(o)f(complementary)g(half)h (spaces,)h Fl(R)2271 4750 y Fo(3)2271 4811 y(+)2364 4786 y FB(and)f Fl(R)2602 4750 y Fo(3)2602 4811 y Fu(\000)2667 4786 y FB(,)h(respecti)n(v)o(ely)-6 b(,)27 b(with)g(a)257 4907 y(junction)i(localized)g(near)i(the)e(origin,)i(has)e(been)h (considered)g(in)f([DFG];)h(see)h(also)e([Ru1,)257 5027 y(Ru2].)65 b(It)37 b(will)e(be)h(studied)f(in)h(more)g(detail)g(else)n (where.)64 b(In)37 b(order)f(to)g(describe)g(spatial)257 5147 y(properties)25 b(of)g(the)f(system,)g(we)h(mak)o(e)g(the)f(follo) n(wing)f(assumption.)404 5388 y Fz(\(A5\))50 b(Existence)25 b(of)g(space)h(translations)p FB(.)p eop %%Page: 24 25 24 24 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(24)404 573 y Fm(F)-10 b(or)24 b(eac)o(h)g(r)l(eservoir)h Fx(r)30 b Fs(=)d Fx(I)8 b(;)17 b(I)8 b(I)g Fm(,)25 b(ther)l(e)f(e)n (xists)h(a)2134 537 y Fu(\003)2174 573 y Fm(automorphism)d(\(semi-\))j (gr)l(oup)1541 708 y Ft(\010)1599 789 y Fx(\034)1652 747 y Fn(r)1641 813 y Fj(x)1720 704 y Ft(\014)1720 764 y(\014)1780 789 y Fr(x)j Fv(2)g Fl(R)2034 747 y Fo(3)2034 813 y(\()p Fu(\006)p Fo(\))2154 708 y Ft(\011)3205 789 y FB(\(2.103\))257 1004 y Fm(of)i(the)g(\002eld)g(alg)o(ebr)o(a)e Fv(F)1138 968 y Fn(r)1176 1004 y Fm(,)j(r)l(epr)l(esenting)e(space)h (tr)o(anslations)d(of)j Fl(R)2694 968 y Fo(3)2769 1004 y Fm(\()p Fl(R)2869 968 y Fo(3)2869 1029 y Fu(\006)2934 1004 y Fm(,)h(r)l(espectively\))257 1125 y(on)25 b Fv(F)464 1089 y Fn(r)501 1125 y Fm(.)257 1366 y FB(F)o(or)g(the)g(system)e(of)i (tw)o(o)g(coupled)f(reserv)n(oirs,)1322 1581 y Fx(\034)1364 1596 y Fj(x)1470 1581 y Fs(:=)56 b Fx(\034)1682 1540 y Fn(I)1671 1606 y Fj(x)1744 1581 y Fv(\012)23 b Fx(\034)1897 1540 y Fn(I)-7 b(I)1886 1606 y Fj(x)1986 1581 y Fx(;)67 b Fr(x)27 b Fv(2)h Fl(R)2333 1540 y Fo(3)2403 1581 y Fx(;)775 b FB(\(2.104\))257 1797 y(de\002nes)38 b(a)g(representation)f (of)h(space)g(translations)e(as)i(a)g(3-parameter)g(group)f(of)3251 1761 y Fu(\003)3290 1797 y FB(auto-)257 1917 y(morphisms)24 b(on)h(the)h(\002eld)g(algebra)52 b Fv(F)38 b Fs(=)p 1763 1831 389 4 v 30 w Fv(F)1845 1889 y Fn(I)1906 1917 y Fv(\012)23 b(F)2088 1889 y Fn(I)-7 b(I)2151 1917 y FB(.)34 b(It)25 b(is)h(plausible)e(that)h(space)h(transla-)257 2038 y(tions)e(satisfy)g(the)h(follo)n(wing)e(assumption.)404 2276 y Fz(\(A6\))50 b(Asymptotic)24 b(abelianness)i(of)e(space)i (translations,)e(and)i(homogeneity)655 2397 y(of)f(r)n(eser)o(v)o(oirs) p FB(.)404 2517 y Fm(The)k(action)e(of)h Fx(\034)1007 2532 y Fj(x)1086 2517 y Fm(on)g Fv(F)38 b Fm(is)28 b(norm-continuous)e (in)i Fr(x)34 b Fv(2)h Fl(R)2501 2481 y Fo(3)2574 2517 y Fm(and)28 b(for)g(all)g(oper)o(ator)o(s)e Fx(a)257 2637 y Fm(and)f Fx(b)g Fm(in)g Fv(F)10 b Fm(,)1397 2758 y Fs(lim)1351 2824 y Fu(j)p Fj(x)p Fu(j!1)1622 2758 y Fv(k)17 b Fs([)p Fx(\034)1758 2773 y Fj(x)1808 2758 y Fs(\()p Fx(a)p Fs(\))p Fx(;)g(b)p Fs(])g Fv(k)56 b Fs(=)f(0)25 b Fx(:)803 b FB(\(2.105\))257 2979 y Fm(Furthermor)l(e)o(,)22 b(the)h(dynamics)f(and)g(the)h(equilibrium)e(states)g(of)i(the)f (uncoupled)g(r)l(eservoir)o(s)257 3100 y(ar)l(e)j(homo)o(g)o(eneous,)f (in)g(the)h(sense)g(that)1370 3316 y Fx(\013)1433 3274 y Fo(0)1432 3340 y Fn(t)1472 3235 y Ft(\000)1517 3316 y Fx(\034)1559 3331 y Fj(x)1610 3316 y Fs(\()p Fx(a)p Fs(\))1737 3235 y Ft(\001)1838 3316 y Fs(=)56 b Fx(\034)2012 3331 y Fj(x)2063 3235 y Ft(\000)2108 3316 y Fx(\013)2171 3274 y Fo(0)2170 3340 y Fn(t)2210 3316 y Fs(\()p Fx(a)p Fs(\))2337 3235 y Ft(\001)3205 3316 y FB(\(2.106\))257 3531 y Fm(and)1434 3652 y Fx(!)1499 3611 y Fo(0)1538 3571 y Ft(\000)1583 3652 y Fx(\034)1625 3667 y Fj(x)1676 3652 y Fs(\()p Fx(a)p Fs(\))1803 3571 y Ft(\001)1904 3652 y Fs(=)g Fx(!)2101 3611 y Fo(0)2139 3652 y Fs(\()p Fx(a)p Fs(\))25 b Fx(;)887 b FB(\(2.107\))257 3824 y Fm(for)25 b(all)49 b Fx(a)27 b Fv(2)i(F)10 b Fm(.)257 4065 y FB(The)26 b(local)f(nature)g(of)h(the)f(perturbation,)f Fx(W)14 b FB(,)25 b(of)h(the)f(dynamics)f(of)i(the)f(system)f(due)i(to) e(the)257 4185 y(contact)i(or)g(junction,)f(see)i(Assumption)c(\(A3\),) k(eqs.)f(\(2.27\))f(and)h(\(2.28\),)g(and)g(assumption)257 4306 y(\(A6\),)f(then)g(imply)e(that,)h(for)h(all)g Fx(a)j Fv(2)g(F)10 b FB(,)1124 4522 y Fs(lim)1078 4588 y Fu(j)p Fj(x)p Fu(j!1)1349 4522 y Fv(k)p Fx(\013)1461 4537 y Fn(t)1491 4441 y Ft(\000)1536 4522 y Fx(\034)1578 4537 y Fj(x)1629 4522 y Fs(\()p Fx(a)p Fs(\))1756 4441 y Ft(\001)1824 4522 y Fv(\000)23 b Fx(\013)1987 4480 y Fo(0)1986 4546 y Fn(t)2026 4441 y Ft(\000)2072 4522 y Fx(\034)2114 4537 y Fj(x)2165 4522 y Fs(\()p Fx(a)p Fs(\))2292 4441 y Ft(\001)2337 4522 y Fv(k)55 b Fs(=)h(0)24 b Fx(;)531 b FB(\(2.108\))257 4786 y(for)39 b(all)g(times)e Fx(t)p FB(.)73 b(A)39 b(proof)f(of)h (\(2.108\))f(follo)n(ws)g(from)g(the)h(Lie-Schwinger)f(series)h(for)257 4907 y Fx(\013)319 4922 y Fu(\000)p Fn(t)404 4907 y Fs(\()p Fx(\013)505 4870 y Fo(0)504 4931 y Fn(t)544 4907 y Fs(\()p Fx(a)p Fs(\)\))21 b FB(and)f(use)g(of)h(\(2.28\))f(and)g(\(2.105\).)29 b(Relation)20 b(\(2.108\))g(sho)n(ws)f(that)h(observ)n(ables)257 5027 y(localized)25 b(f)o(ar)g(from)g(the)g(junction)e(e)n(v)n(olv)o(e) h(according)g(to)h(the)f(non-interacting)g(dynamics.)404 5147 y(It)e(is)h(tempting,)e(and)i(can)g(be)g(justi\002ed)f(in)h(e)o (xamples,)f(to)g(strengthen)g(assumption)f(\(A4\))257 5268 y(\(e)o(xistence)k(of)g(scattering)f(endomorphism\))f(as)i(follo)n (ws.)p eop %%Page: 25 26 25 25 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(25)404 573 y Fz(\(A7\))50 b(Cluster)25 b(pr)n(operties)h(of)f(the) h(scattering)f(endomor)o(phism)p FB(.)404 693 y Fm(The)g(limits)1070 814 y Fx(n)d Fv(\000)h Fs(lim)1117 882 y Fn(t)p Fu(!\0061)1430 814 y Fx(\013)1493 772 y Fo(0)1492 838 y Fu(\000)p Fn(t)1576 733 y Ft(\000)1622 814 y Fx(\013)1684 829 y Fn(t)1713 733 y Ft(\000)1759 814 y Fx(\034)1801 829 y Fj(x)1852 814 y Fs(\()p Fx(a)p Fs(\))1979 733 y Ft(\001\001)2126 814 y Fs(=)55 b Fx(\033)2312 829 y Fu(\006)2371 733 y Ft(\000)2417 814 y Fx(\034)2459 829 y Fj(x)2510 814 y Fs(\()p Fx(a)p Fs(\))2637 733 y Ft(\001)3205 814 y FB(\(2.109\))257 1039 y Fm(ar)l(e)25 b(uniform)f(in)g Fr(x)k Fv(2)g Fl(R)1108 1003 y Fo(3)1153 1039 y Fm(,)d(for)f(e)o(very)h Fx(a)j Fv(2)g(F)10 b Fm(.)257 1280 y FB(Equations)24 b(\(2.108\))g(and)h (\(2.109\))f(imply)g(that)1209 1497 y Fs(lim)1163 1564 y Fu(j)p Fj(x)q Fu(j!1)1435 1497 y Fv(k)p Fx(\033)1540 1512 y Fu(\006)1599 1417 y Ft(\000)1644 1497 y Fx(\034)1686 1512 y Fj(x)1737 1497 y Fs(\()p Fx(a)p Fs(\))1864 1417 y Ft(\001)1932 1497 y Fv(\000)f Fx(\034)2074 1512 y Fj(x)2125 1497 y Fs(\()p Fx(a)p Fs(\))p Fv(k)55 b Fs(=)g(0)25 b Fx(;)616 b FB(\(2.110\))257 1764 y(for)25 b(e)n(v)o(ery)g Fx(a)j Fv(2)g(F)10 b FB(.)30 b(From)25 b(this)f(property)g(we)h (conclude)g(that)972 1982 y Fs(lim)927 2049 y Fu(j)p Fj(x)p Fu(j!1)1198 1982 y Fx(!)1259 1997 y Fo(stat)1381 1901 y Ft(\000)1426 1982 y Fx(\034)1468 1997 y Fj(x)1519 1982 y Fs(\()p Fx(a)p Fs(\))1646 1901 y Ft(\001)1775 1982 y Fs(=)129 b(lim)1934 2049 y Fu(j)p Fj(x)p Fu(j!1)2205 1982 y Fx(!)2270 1941 y Fo(0)2309 1901 y Ft(\000)2354 1982 y Fx(\033)2409 1997 y Fo(+)2469 1901 y Ft(\000)2515 1982 y Fx(\034)2557 1997 y Fj(x)2607 1982 y Fs(\()p Fx(a)p Fs(\))2734 1901 y 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b(hold,)i(while)d(it)h(usually) f(f)o(ails)g(in)h(dimension)e Fx(d)47 b Fs(=)f(1)p Fx(;)17 b Fs(2)p FB(.)61 b(F)o(or)35 b(people)g(f)o(amiliar)f(with)257 934 y(elementary)25 b(f)o(acts)g(of)g(scattering)f(theory)h(this)e (will)h(not)h(come)f(as)h(a)g(surprise.)404 1054 y(In)30 b(a)h(subsequent)e(paper)l(,)j(we)f(will)f(sho)n(w)f(that,)i(for)g(a)g (lar)n(ge)g(class)f(of)h(reserv)n(oirs,)g(one)257 1175 y(can)g(construct)f Fm(\223observables)g(at)g(in\002nity\224)p FB(,)h(see)g(e.g.)48 b([BR,)32 b(V)-13 b(ol.)47 b(II],)31 b(corresponding)f(to)257 1295 y(the)d(operators)g Fx(P)881 1259 y Fn(r)918 1295 y Fs(\()p Fx(t)p Fs(\))g FB(and)g Fx(I)1278 1259 y Fn(r)1270 1320 y(j)1316 1295 y Fs(\()p Fx(t)p Fs(\))g FB(de\002ned)h(in)e(eqs.)h(\(2.42\),)g(\(2.48\),)g (respecti)n(v)o(ely)-6 b(.)35 b(Clearly)-6 b(,)257 1416 y(the)38 b(e)o(xpectation)f(v)n(alues)g(of)h(these)f(operators)h Fm(vanish)f FB(in)g(the)h(product)f(state)h Fx(!)3176 1379 y Fo(0)3253 1416 y FB(of)f(the)257 1536 y(uncoupled)26 b(reserv)n(oirs)h(and)f(are)i(gi)n(v)o(en)d(by)h Fv(P)1883 1500 y Fn(r)1949 1536 y FB(and)g Fv(J)2205 1500 y Fn(r)2187 1561 y(j)2270 1536 y FB(in)g(the)h(stationary)e(state,)i Fx(!)3238 1551 y Fo(stat)3360 1536 y FB(,)g(of)257 1656 y(the)k(coupled)f(reserv)n(oirs.)47 b(If)31 b(we)g(can)g(sho)n(w)e (that)h Fv(P)2127 1620 y Fn(r)2204 1656 y Fv(6)p Fs(=)38 b(0)p FB(,)31 b(or)g(that)f Fv(J)2803 1620 y Fn(r)2785 1681 y(j)2879 1656 y Fv(6)p Fs(=)38 b(0)p FB(,)32 b(for)f Fx(r)40 b Fs(=)e Fx(I)257 1777 y FB(or)d Fx(I)-9 b(I)42 b FB(and)34 b(some)f Fx(j)6 b FB(,)37 b(then)c(it)h(follo)n(ws)e(that)i Fx(!)1894 1792 y Fo(stat)2050 1777 y FB(is)g Fm(not)f FB(normal)g(relati)n(v)o(e)g(to)h Fx(!)3134 1741 y Fo(0)3173 1777 y FB(.)58 b(In)34 b(the)257 1897 y(e)o(xamples)e(studied)g(in)g (Sects.)h(4)g(and)f(5,)j(we)e(shall)f(encounter)h(instances)f(where)h Fv(P)3280 1861 y Fn(r)3351 1897 y FB(and)257 2017 y Fc(J)354 1975 y Fn(r)417 2017 y FB(do)24 b Fm(not)g FB(v)n(anish.)257 2357 y FA(3)144 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4424 y Fo(\003)1508 4465 y Fs(:=)55 b Fx(L)1732 4424 y Fo(2)1772 4384 y Ft(\000)1817 4465 y Fs(\003)p Fx(;)17 b(d)1980 4424 y Fn(d)2020 4465 y Fx(x)2075 4384 y Ft(\001)2143 4465 y Fv(\012)23 b Fl(C)2309 4424 y Fn(k)2382 4465 y Fx(;)895 b FB(\(3.2\))257 4657 y(where)32 b Fx(k)41 b Fs(=)738 4582 y Ft(P)844 4608 y Fn(l)844 4686 y(\013)p Fo(=1)983 4657 y Fs(\(2)p Fx(S)1130 4672 y Fn(\013)1206 4657 y Fs(+)26 b(1\))p FB(,)32 b Fx(S)1512 4672 y Fn(\013)1592 4657 y FB(is)e(the)h(spin)f(of)h(species)f Fx(\013)q FB(,)i(and)e Fx(l)j FB(is)e(the)f(number)g(of)257 4777 y(species.)404 4898 y(The)d(one-particle)g(dynamics)f(is)h(generated)h (by)f(the)g(follo)n(wing)e(self)o(adjoint)h(operator)l(,)257 5018 y Fx(t)292 4982 y Fo(\003)346 5018 y FB(,)e(acting)h(on)f Fx(h)844 4982 y Fo(\003)898 5018 y FB(,)1447 5194 y Fx(t)1482 5153 y Fo(\003)1590 5194 y Fs(=)55 b Fv(\000)1839 5127 y Fl(~)1893 5091 y Fo(2)p 1808 5171 154 4 v 1808 5262 a Fs(2)p Fx(M)1997 5194 y Fs(\001)23 b Fv(\012)f Fs(1)-22 b(l)24 b Fx(;)998 b FB(\(3.3\))p eop %%Page: 27 28 27 27 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(27)257 573 y(where)27 b Fx(M)38 b FB(is)26 b(the)g(mass)g(of)g (the)h(particle)f(and)h Fs(\001)g FB(is)f(the)g(Laplace)h(operator)f (on)h Fx(L)3122 537 y Fo(2)3161 573 y Fs(\(\003)p Fx(;)17 b(d)3362 537 y Fn(d)3402 573 y Fx(x)p Fs(\))257 693 y FB(with)31 b(self)o(adjoint)e(boundary)h(conditions)g(\(e.g.)49 b(Dirichlet,)31 b(Neumann,)h(or)f(periodic\))g(im-)257 814 y(posed)25 b(at)g(the)f(boundary)-6 b(,)24 b Fx(@)5 b Fs(\003)p FB(,)25 b(of)g Fs(\003)p FB(.)404 934 y(In)g(the)f(follo)n (wing,)f(we)i(choose)g(units)e(in)i(which)f Fl(~)k Fs(=)f(1)e FB(and)g Fx(M)38 b Fs(=)2799 895 y Fo(1)p 2799 911 36 4 v 2799 968 a(2)2844 934 y FB(.)404 1054 y(Other)18 b(operators)h(are)h(physically)d(interesting.)27 b(Electrons)18 b(in)h(semi-conductors)e(w)o(ould)257 1175 y(in)l(v)n(olv)o(e)29 b(a)h(potential)f(operator)h(that)f(is)h(diagonal)f(in)g(the)h(space)g (representation.)46 b(A)29 b(mag-)257 1295 y(netic)f(\002eld)h(could)f (also)g(be)g(considered;)i(the)e(Laplacian)g(should)f(be)i(replaced)g (by)f(the)g(co-)257 1416 y(v)n(ariant)19 b(Laplacian,)h(and)g(a)g (coupling)e(between)h(the)h(spin)e(of)i(the)f(particle)g(and)h(the)f (magnetic)257 1536 y(\002eld)28 b(should)e(be)h(introduced.)36 b(In)28 b(this)e(paper)l(,)i(we)f(restrict)g(our)g(attention)e(to)i (the)g(situation)257 1656 y(\(3.3\).)404 1777 y(Ne)o(xt,)f(we)h (consider)f(a)h(system)f(consisting)e(of)j Fx(n)g FB(identical)f (particles)g(of)h(the)f(kind)g(just)257 1897 y(considered,)i(all)f (con\002ned)g(to)g(the)g(re)o(gion)f Fs(\003)p FB(.)37 b(Its)27 b(state)g(space)h(is)e(gi)n(v)o(en)g(by)h(a)g(subspace)g(of) 257 2017 y(the)e Fx(n)p FB(-fold)g(tensor)f(product)g(of)h Fx(h)1442 1981 y Fo(\003)1521 2017 y FB(of)f(\002x)o(ed)h(symmetry)e (type,)1263 2237 y Fx(h)1319 2196 y Fo(\003)1319 2262 y Fn(n)1428 2237 y Fs(:=)55 b Fx(P)14 b Fs(\()p Fx(h)1757 2196 y Fo(\003)1809 2237 y Fs(\))1847 2196 y Fu(\012)p Fn(n)1949 2237 y Fx(;)67 b(h)2099 2196 y Fo(\003)2099 2262 y(0)2207 2237 y Fs(:=)56 b Fl(C)51 b Fx(;)814 b FB(\(3.4\))257 2457 y(where)29 b Fx(P)42 b FB(is)28 b(the)h(orthogonal) e(projection)g(onto)h(the)g(subspace)h(of)f(w)o(a)n(v)o(e)h(functions)e (of)h(the)257 2578 y(selected)40 b(symmetry)d(type)i(under)h (permutations)d(of)j(the)f Fx(n)g FB(particle)g(v)n(ariables.)74 b(If)39 b(the)257 2698 y(particles)29 b(are)g Fm(bosons)f FB(then)h Fx(P)48 b Fv(\021)36 b Fx(P)1568 2713 y Fo(+)1656 2698 y FB(projects)28 b(onto)g Fm(completely)h(symmetric)f Fx(n)p FB(-particle)257 2819 y(w)o(a)n(v)o(e)d(functions;)e(while,)h (for)h Fm(fermions)p FB(,)e Fx(P)41 b Fv(\021)28 b Fx(P)1993 2834 y Fu(\000)2077 2819 y FB(projects)c(onto)f Fm(totally)h (anti-symmetric)257 2939 y FB(w)o(a)n(v)o(e)h(functions.)30 b(In)24 b(this)g(paper)l(,)h(we)h(focus)e(our)h(attention)f(on)g (fermions.)404 3059 y(If)38 b(the)h(particles)e(do)i(not)e(interact)h (with)g(each)h(other)f(the)g(Hamiltonian,)i Fx(T)3161 3023 y Fo(\003)3147 3084 y Fn(n)3252 3059 y FB(of)e(the)257 3180 y Fx(n)p FB(-particle)25 b(system)f(is)g(gi)n(v)o(en)g(by)1145 3472 y Fx(T)1216 3431 y Fo(\003)1202 3497 y Fn(n)1325 3472 y Fs(:=)1533 3348 y Fn(n)1483 3378 y Ft(X)1494 3588 y Fn(j)t Fo(=1)1643 3472 y Fs(1)-22 b(l)21 b Fv(\012)i Fx(:)17 b(:)g(:)22 b Fv(\012)g Fx(t)2090 3431 y Fo(\003)2090 3497 y Fn(j)2166 3472 y Fv(\012)g Fx(:)17 b(:)g(:)22 b Fv(\012)g Fs(1)-22 b(l)24 b Fx(;)697 b FB(\(3.5\))257 3801 y(where)26 b Fx(t)561 3765 y Fo(\003)561 3825 y Fn(j)639 3801 y FB(acts)f(on)f(the)h Fx(j)1136 3765 y Fo(th)1232 3801 y FB(f)o(actor)g(in)f(the)h Fx(n)p FB(-fold)g(tensor)f (product)g(in)h(\(3.4\).)404 3921 y(If)g(the)f(number)g(of)h(particles) g(can)g(\003uctuate)f(\(e.g.,)h(because)g(the)g(system)e(is)i(coupled)f (to)257 4041 y(a)h(particle)g(reserv)n(oir)g(such)g(as)g(a)g(battery\)) f(then)h(it)f(is)g(con)l(v)o(enient)g(to)g(use)h(the)g(formalism)e(of) 257 4162 y(\223second)i(quantization\224,)f(which)g(we)h(brie\003y)g (recall.)404 4282 y(The)g Fm(F)-10 b(oc)n(k)24 b(space)h FB(is)f(de\002ned)h(by)1522 4575 y Fv(H)1607 4534 y Fo(\003)1716 4575 y Fs(:=)1914 4450 y Fu(1)1874 4480 y Ft(M)1883 4689 y Fn(n)p Fo(=0)2069 4575 y Fx(h)2125 4534 y Fo(\003)2125 4599 y Fn(n)2203 4575 y Fx(:)1074 b FB(\(3.6\))257 4889 y(The)25 b(free)h(dynamics)e(on)g Fv(H)1233 4853 y Fo(\003)1311 4889 y FB(is)h(generated)g(by)f(the)h Fm(Hamiltonian)1513 5181 y Fx(H)1602 5140 y Fo(\003)1710 5181 y Fs(:=)1909 5057 y Fu(1)1869 5087 y Ft(M)1877 5296 y Fn(n)p Fo(=0)2063 5181 y Fx(T)2134 5140 y Fo(\003)2120 5206 y Fn(n)2212 5181 y Fx(;)1065 b FB(\(3.7\))p eop %%Page: 28 29 28 28 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(28)257 573 y(with)24 b Fx(T)530 537 y Fo(\003)516 597 y Fn(n)608 573 y FB(as)h(in)f(\(3.5\).)31 b(The)25 b Fm(particle)f(number)g(oper)o(ator)p FB(,)f Fx(N)2394 537 y Fo(\003)2448 573 y FB(,)h(is)h(de\002ned)g(by)1421 831 y Fx(N)1509 790 y Fo(\003)1618 831 y Fs(:=)1817 706 y Fu(1)1777 736 y Ft(M)1786 945 y Fn(n)p Fo(=0)1972 831 y Fx(n)d Fv(\001)g Fs(1)-22 b(l)2156 746 y Ft(\014)2156 806 y(\014)2189 870 y Fn(h)2230 851 y Fp(\003)2230 886 y Fe(n)2304 831 y Fx(:)973 b FB(\(3.8\))257 1108 y(Let)25 b Fx(\024)g FB(be)g(a)g(symmetric)f Fx(k)h Fv(\002)e Fx(k)28 b FB(matrix)c(acting)g(on)h Fl(C)2128 1072 y Fn(k)2177 1108 y FB(.)31 b(W)-8 b(e)25 b(set)1000 1362 y Fx(K)1090 1321 y Fo(\003)1083 1387 y Fn(n)1199 1362 y Fs(:=)1408 1238 y Fn(n)1357 1268 y Ft(X)1368 1478 y Fn(j)t Fo(=1)1517 1362 y Fs(1)-22 b(l)21 b Fv(\012)i Fx(:)17 b(:)g(:)22 b Fv(\012)1929 1282 y Ft(\000)1975 1362 y Fs(1)-22 b(l)21 b Fv(\012)h Fx(\024)2206 1377 y Fn(j)2243 1282 y Ft(\001)2311 1362 y Fv(\012)h Fx(:)17 b(:)g(:)k Fv(\012)i Fs(1)-22 b(l)23 b Fx(;)257 1653 y FB(where)e Fs(1)-22 b(l)t Fv(\012)5 b Fx(\024)718 1668 y Fn(j)775 1653 y FB(acts)20 b(on)g(the)g Fx(j)1258 1617 y Fo(th)1349 1653 y FB(f)o(actor)l(,)h Fv(')29 b Fx(L)1794 1617 y Fo(2)1833 1653 y Fs(\(\003)p Fx(;)17 b(d)2034 1617 y Fn(d)2074 1653 y Fx(x)p Fs(\))5 b Fv(\012)g Fl(C)2320 1617 y Fn(k)2369 1653 y FB(,)21 b(in)f(the)g Fx(n)p FB(-fold)g(tensor)g (product)257 1773 y(de\002ning)28 b Fx(h)668 1737 y Fo(\003)668 1798 y Fn(n)721 1773 y FB(.)41 b(F)o(or)28 b Fx(t)987 1737 y Fo(\003)1069 1773 y FB(as)g(in)g(\(3.3\),)g(a)h(typical)e(char)n (ge)i(operator)l(,)g Fx(Q)2653 1737 y Fo(\003)2740 1773 y Fv(\021)34 b Fx(Q)2928 1737 y Fo(\003)2982 1773 y Fs(\()p Fx(\024)p Fs(\))p FB(,)29 b(is)e(of)h(the)257 1894 y(form)1509 2066 y Fx(Q)1586 2025 y Fo(\003)1695 2066 y Fs(:=)1893 1942 y Fu(1)1853 1972 y Ft(M)1862 2181 y Fn(n)p Fo(=0)2048 2066 y Fx(K)2138 2025 y Fo(\003)2131 2091 y Fn(n)2216 2066 y Fx(:)1061 b FB(\(3.9\))404 2319 y(The)39 b(operators)f Fx(H)1096 2282 y Fo(\003)1149 2319 y Fx(;)17 b(N)1281 2282 y Fo(\003)1373 2319 y FB(and)39 b Fx(Q)1633 2282 y Fo(\003)1686 2319 y Fs(\()p Fx(\024)p Fs(\))g FB(are)g(unbounded,)i (self)o(adjoint)d(operators)h(on)257 2439 y Fv(H)342 2403 y Fo(\003)396 2439 y FB(;)24 b(see)h(e.g.)31 b([RS].)404 2559 y(Ne)o(xt,)26 b(we)h(describe)f(the)g(structure)h(of)f Fv(H)1867 2523 y Fo(\003)1947 2559 y FB(in)g(some)g(more)g(detail)g (and)h(introduce)e(cre-)257 2680 y(ation)g(and)g(annihilation)e (operators.)31 b(Let)25 b Fr(x)p Fx(;)17 b Fr(y)s Fx(;)g(:)g(:)g(:)25 b FB(denote)f(points)g(in)h(physical)e(space)j Fl(R)3424 2643 y Fn(d)3470 2680 y FB(,)257 2800 y(and)e(let)f Fx(s)k Fs(=)h(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(k)26 b FB(label)d(on)g (orthonormal)f(basis)h(in)g Fl(C)2315 2764 y Fn(k)2363 2800 y FB(.)31 b(V)-11 b(ectors)23 b Fx(f)2789 2815 y Fn(n)2859 2800 y FB(in)g(the)g Fx(n)p FB(-particle)257 2920 y(space)j Fx(h)560 2884 y Fo(\003)560 2945 y Fn(n)638 2920 y FB(can)f(be)g(represented)g(as)g(square-inte)o(grable)f(w)o(a)n (v)o(e)g(functions,)1397 3103 y Fx(f)1445 3118 y Fn(n)1509 3103 y Fs(\()p Fr(x)1613 3118 y Fo(1)1652 3103 y Fx(;)17 b(s)1742 3118 y Fo(1)1781 3103 y Fx(;)g(:)g(:)g(:)f(;)h Fr(x)2065 3118 y Fn(n)2112 3103 y Fx(;)g(s)2202 3118 y Fn(n)2249 3103 y Fs(\))41 b Fx(;)257 3285 y FB(with)24 b(support)g(in)h Fs(\003)954 3249 y Fn(dn)1065 3285 y Fv(\032)j Fl(R)1236 3249 y Fn(dn)1325 3285 y FB(,)d(which,)g(for)g (fermions,)f(are)i(totally)d(anti-symmetric)g(under)257 3405 y(permutations)j(of)i(their)f Fx(n)g FB(ar)n(guments.)38 b(V)-11 b(ectors)27 b Fx( )t(;)17 b(\036;)g(:)g(:)g(:)27 b FB(in)g(F)o(ock)g(space)h(correspond)f(to)257 3526 y(sequences,)1157 3646 y Fx( )59 b Fs(=)c(\()p Fx(f)1496 3661 y Fn(n)1543 3646 y Fs(\))1581 3605 y Fu(1)1581 3671 y Fn(n)p Fo(=0)1768 3646 y Fx(;)67 b(\036)55 b Fs(=)g(\()p Fx(g)2191 3661 y Fn(n)2238 3646 y Fs(\))2276 3605 y Fu(1)2276 3671 y Fn(n)p Fo(=0)2438 3646 y Fx(;)17 b(:)g(:)g(:)658 b FB(\(3.10\))257 3804 y(of)25 b Fx(n)p FB(-particle)g(w)o(a)n(v)o(e)g (functions)f(in)g Fx(h)1565 3768 y Fo(\003)1565 3829 y Fn(n)1618 3804 y FB(.)31 b(The)25 b(scalar)g(product)f(on)h Fv(H)2650 3768 y Fo(\003)2728 3804 y FB(is)f(de\002ned)i(by)334 4062 y Fv(h)p Fx( )t(;)17 b(\036)p Fv(i)54 b Fs(:=)803 3937 y Fu(1)766 3967 y Ft(X)772 4177 y Fn(n)p Fo(=0)975 3967 y Ft(X)927 4177 y Fn(s)960 4186 y Fp(1)994 4177 y Fn(;:::)n(;s)1125 4185 y Fe(n)1184 3926 y Ft(Z)1239 4152 y Fo(\003)1351 3937 y Fn(n)1309 3967 y Ft(Y)1311 4177 y Fn(j)t Fo(=1)1453 4062 y Fx(d)p Fr(x)1569 4077 y Fn(j)p 1605 3975 873 4 v 1605 4062 a Fx(f)1653 4077 y Fn(n)1700 4062 y Fs(\()p Fr(x)1804 4077 y Fo(1)1843 4062 y Fx(;)17 b(s)1933 4077 y Fo(1)1973 4062 y Fx(;)g(:)g(:)g(:)e(;)i Fr(x)2257 4077 y Fn(n)2304 4062 y Fx(;)g(s)2394 4077 y Fn(n)2440 4062 y Fs(\))g Fx(g)2542 4077 y Fn(n)2589 4062 y Fs(\()p Fr(x)2692 4077 y Fo(1)2732 4062 y Fx(;)g(s)2822 4077 y Fo(1)2861 4062 y Fx(;)g(:)g(:)g(:)e(;)i Fr(x)3145 4077 y Fn(n)3192 4062 y Fx(;)g(s)3282 4077 y Fn(n)3328 4062 y Fs(\))25 b Fx(:)3255 4276 y FB(\(3.11\))257 4397 y(The)i(v)o(ector)e(represented)i(by)f(the)g(sequence)g Fs(\()p Fx(f)1949 4412 y Fn(n)1996 4397 y Fs(\))2034 4361 y Fu(1)2034 4421 y Fn(n)p Fo(=0)2171 4397 y FB(,)h(with)e Fx(f)2474 4412 y Fo(0)2544 4397 y Fs(=)30 b(1)p FB(,)c Fx(f)2798 4412 y Fn(n)2875 4397 y Fv(\021)31 b Fs(0)p FB(,)26 b(for)g Fx(n)31 b Fv(\025)g Fs(1)p FB(,)257 4517 y(is)25 b(denoted)f(by)h Fs(\012)g FB(and)g(is)f(called)h(the)g Fm(vacuum)f(\(vector\))p FB(.)404 4638 y(Let)34 b Fv(D)48 b Fs(=)d Fv(D)897 4601 y Fo(\003)984 4638 y FB(be)35 b(the)f(linear)g(domain)f(of)i(v)o(ectors)e Fx( )50 b Fs(=)45 b(\()p Fx(f)2621 4653 y Fn(n)2668 4638 y Fs(\))2706 4601 y Fu(1)2706 4662 y Fn(n)p Fo(=0)2877 4638 y FB(in)34 b Fv(H)3074 4601 y Fo(\003)3162 4638 y FB(with)f(the)257 4758 y(property)h(that)f Fm(all)g FB(b)n(ut)g Fm(\002nitely)h(many)f Fx(f)1713 4773 y Fn(n)1760 4758 y FB(')-5 b(s)34 b Fm(vanish)p FB(.)57 b(Clearly)-6 b(,)35 b Fv(D)i FB(is)c(dense)h(in)f Fv(H)3222 4722 y Fo(\003)3276 4758 y FB(.)57 b(F)o(or)257 4878 y Fx(f)39 b Fv(2)28 b Fx(h)494 4842 y Fo(\003)547 4878 y FB(,)d(we)g(de\002ne)g(an)g Fm(annihilation)d(oper)o(ator)p FB(,)h Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p FB(,)25 b(by)714 4980 y Ft(\000)759 5061 y Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p Fx( )1012 4980 y Ft(\001)1058 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573 y Fv(2)g(D)s FB(,)d(and)1591 787 y Fx(a)p Fs(\()p Fx(f)11 b Fs(\)\012)56 b(:=)f(0)25 b Fx(:)1093 b FB(\(3.13\))257 1000 y(The)32 b Fm(cr)l(eation)g(oper)o (ator)p FB(,)g Fx(a)1262 964 y Fu(\003)1302 1000 y Fs(\()p Fx(f)11 b Fs(\))p FB(,)33 b(is)f(de\002ned)g(to)g(be)g(the)g(adjoint)f (of)h Fx(a)p Fs(\()p Fx(f)11 b Fs(\))32 b FB(on)f Fv(H)3167 964 y Fo(\003)3253 1000 y FB(and)h(is)257 1121 y(easily)25 b(seen)g(to)f(be)h(well)f(de\002ned)i(on)e Fv(D)s FB(.)404 1241 y(It)g(is)g(well)g(kno)n(wn,)g(see)h(e.g.)30 b([RS,)c(BR],)f (that,)f(for)h Fm(fermions)p FB(,)f(the)g(follo)n(wing)e Fm(\223canoni-)257 1362 y(cal)j(anti-commutation)d(r)l(elations\224)i FB(\(CAR\))i(hold:)1396 1495 y Ft(\010)1454 1575 y Fx(a)1505 1534 y Fo(#)1568 1575 y Fs(\()p Fx(f)11 b Fs(\))p Fx(;)41 b(a)1822 1534 y Fo(#)1885 1575 y Fs(\()p Fx(g)t Fs(\))2012 1495 y Ft(\011)2125 1575 y Fs(=)55 b(0)25 b Fx(;)898 b FB(\(3.14\))257 1789 y(for)35 b(arbitrary)f Fx(f)5 b(;)17 b(g)37 b FB(in)d Fx(h)1134 1753 y Fo(\003)1222 1789 y FB(where)g Fx(a)1550 1753 y Fo(#)1658 1789 y Fs(=)45 b Fx(a)34 b FB(or)h Fx(a)2033 1753 y Fu(\003)2072 1789 y FB(,)i(and)d Fv(f)p Fx(A;)17 b(B)5 b Fv(g)44 b Fs(:=)h Fx(AB)34 b Fs(+)29 b Fx(B)5 b(A)35 b FB(is)e(the)257 1910 y(anti-commutator)23 b(of)i(tw)o(o)g(operators)f Fx(A)h FB(and)g Fx(B)5 b FB(;)1297 2123 y Fv(f)p Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p Fx(;)41 b(a)1652 2082 y Fu(\003)1692 2123 y Fs(\()p Fx(g)t Fs(\))p Fv(g)54 b Fs(=)i(\()p Fx(f)5 b(;)17 b(g)t Fs(\))k Fv(\001)h Fs(1)-22 b(l)23 b Fx(;)800 b FB(\(3.15\))257 2337 y(where)24 b Fs(\()p Fx(f)5 b(;)17 b(g)t Fs(\))27 b(:=)906 2262 y Ft(P)1011 2366 y Fn(s)1064 2257 y Ft(R)1111 2372 y Fo(\003)1181 2337 y Fx(d)p Fr(x)p 1314 2250 290 4 v 16 w Fx(f)11 b Fs(\()p Fr(x)o Fx(;)17 b(s)p Fs(\))g Fx(g)t Fs(\()p Fr(x)n Fx(;)g(s)p Fs(\))24 b FB(is)f(the)g(scalar)h(product)f(on)g Fx(h)2924 2301 y Fo(\003)2978 2337 y FB(.)30 b(F)o(or)24 b(bosons,)257 2458 y(\(3.14\))30 b(and)g(\(3.15\))g(hold)g(if)g(anti-commutators)e (are)j(replaced)f(by)g(commutators)f(\(CCR\).)257 2578 y(F)o(ormally)-6 b(,)1176 2743 y Fx(a)p Fs(\()p Fx(f)11 b Fs(\))55 b(=)1520 2649 y Ft(X)1576 2858 y Fn(s)1681 2608 y Ft(Z)1736 2833 y Fo(\003)1806 2743 y Fx(d)p Fr(x)p 1950 2657 V 27 w Fx(f)11 b Fs(\()p Fr(x)o Fx(;)17 b(s)p Fs(\))27 b Fx(a)p Fs(\()p Fr(x)p Fx(;)17 b(s)p Fs(\))p Fx(;)257 2995 y FB(and)1129 3141 y Fx(a)1180 3099 y Fu(\003)1220 3141 y Fs(\()p Fx(f)11 b Fs(\))55 b(=)1513 3046 y Ft(X)1569 3255 y Fn(s)1674 3005 y Ft(Z)1729 3230 y Fo(\003)1799 3141 y Fx(d)p Fr(x)27 b Fx(a)1994 3099 y Fu(\003)2034 3141 y Fs(\()p Fr(x)o Fx(;)17 b(s)p Fs(\))g Fx(f)11 b Fs(\()p Fr(x)o Fx(;)17 b(s)p Fs(\))24 b Fx(;)257 3393 y FB(with)1029 3513 y Fv(f)o Fx(a)p Fs(\()p Fr(x)p Fx(;)17 b(s)p Fs(\))p Fx(;)33 b(a)1472 3472 y Fu(\003)1512 3513 y Fs(\()p Fr(x)1615 3472 y Fu(0)1639 3513 y Fx(;)17 b(s)1729 3472 y Fu(0)1752 3513 y Fs(\))p Fv(g)55 b Fs(=)g Fx(\016)2069 3528 y Fn(ss)2135 3509 y Ff(0)2177 3513 y Fx(\016)2224 3472 y Fo(\()p Fn(d)p Fo(\))2320 3513 y Fs(\()p Fr(x)21 b Fv(\000)i Fr(x)2611 3472 y Fu(0)2634 3513 y Fs(\))i Fx(:)531 b FB(\(3.16\))404 3685 y(A)32 b(remarkable)h(consequence)f(of) h(the)f(CAR)h(is)f(that)g(the)g(operators)g Fx(a)p Fs(\()p Fx(f)11 b Fs(\))33 b FB(and)f Fx(a)3321 3648 y Fu(\003)3361 3685 y Fs(\()p Fx(f)11 b Fs(\))257 3805 y FB(are)26 b Fm(bounded)e FB(in)g(norm)h(by)1007 4019 y Fv(k)p Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p Fv(k)55 b Fs(=)g Fv(k)p Fx(a)1580 3978 y Fu(\003)1619 4019 y Fs(\()p Fx(f)11 b Fs(\))p Fv(k)55 b Fs(=)g Fv(k)p Fx(f)11 b Fv(k)55 b Fs(:=)2362 3929 y Ft(p)p 2462 3929 232 4 v 90 x Fs(\()p Fx(f)5 b(;)17 b(f)11 b Fs(\))25 b Fx(:)509 b FB(\(3.17\))257 4233 y(T)-8 b(o)25 b(see)g(this,)f(we)h(choose)g(an)f(arbitrary)h Fx( )32 b Fv(2)c(D)g FB(and)c(note)h(that)976 4446 y Fv(k)p Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p Fx( )t Fv(k)1329 4405 y Fo(2)1406 4446 y Fs(+)39 b Fv(k)p Fx(a)1622 4405 y Fu(\003)1661 4446 y Fs(\()p Fx(f)11 b Fs(\))p Fx( )t Fv(k)1913 4405 y Fo(2)1259 4592 y Fs(=)1391 4511 y Ft(\012)1438 4592 y Fx(a)p Fs(\()p Fx(f)g Fs(\))p Fx( )t(;)33 b(a)p Fs(\()p Fx(f)11 b Fs(\))p Fx( )2004 4511 y Ft(\013)2089 4592 y Fs(+)2204 4511 y Ft(\012)2251 4592 y Fx(a)2302 4551 y Fu(\003)2342 4592 y Fs(\()p Fx(f)g Fs(\))p Fx( )t(;)33 b(a)2655 4551 y Fu(\003)2694 4592 y Fs(\()p Fx(f)11 b Fs(\))p Fx( )2896 4511 y Ft(\013)1259 4737 y Fs(=)1391 4656 y Ft(\012)1438 4737 y Fx( )t(;)17 b Fv(f)o Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p Fx(;)33 b(a)1895 4696 y Fu(\003)1935 4737 y Fs(\()p Fx(f)11 b Fs(\))p Fv(g)16 b Fx( )2203 4656 y Ft(\013)1259 4882 y Fs(=)56 b(\()p Fx(f)5 b(;)17 b(f)11 b Fs(\))17 b Fv(h)p Fx( )t(;)g( )t Fv(i)23 b Fx(;)257 5096 y FB(so)i(that)1350 5216 y Fv(k)p Fx(a)1451 5175 y Fo(#)1514 5216 y Fs(\()p Fx(f)11 b Fs(\))p Fx( )t Fv(k)55 b(\024)h(k)p Fx(f)11 b Fv(k)21 b(\001)h(k)p Fx( )t Fv(k)i Fx(:)853 b FB(\(3.18\))257 5388 y(Equality)24 b(in)g(\(3.18\))h(is)f (seen)h(from)g(e)o(xamples.)p eop %%Page: 30 31 30 30 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(30)404 573 y(Eq.)24 b(\(3.17\))h(is)f(f)o(alse)h(for)g Fm(bosons)p FB(,)f Fx(a)p Fs(\()p Fx(f)11 b Fs(\))25 b FB(and)f Fx(a)2031 537 y Fu(\003)2071 573 y Fs(\()p Fx(f)11 b Fs(\))24 b FB(being)h Fm(unbounded)e FB(operators.)404 693 y(F)o(or)28 b(fermions,)f(polynomials)f(in)h Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p FB(,)29 b Fx(a)1893 657 y Fu(\003)1933 693 y Fs(\()p Fx(f)11 b Fs(\))p FB(,)28 b Fx(f)44 b Fv(2)34 b Fx(h)2369 657 y Fo(\003)2422 693 y FB(,)28 b(form)g(a)2769 657 y Fu(\003)2809 693 y FB(algebra)g(of)g(opera-)257 814 y(tors)c(on)g Fv(H)640 777 y Fo(\003)717 814 y FB(which)g(is)f (weakly)h(dense)g(in)f Fx(B)5 b Fs(\()p Fv(H)1941 777 y Fo(\003)1995 814 y Fs(\))p FB(.)30 b(The)24 b(\223observ)n(able)g (algebra\224)g Fv(A)3206 777 y Fo(\003)3283 814 y FB(is)f(the)257 934 y(norm)29 b(closure)h(of)f(the)h(algebra)g(of)f(these)h (polynomials)d(in)i Fx(a)p Fs(\()p Fx(f)11 b Fs(\))p Fx(;)17 b(a)2659 898 y Fu(\003)2698 934 y Fs(\()p Fx(f)11 b Fs(\))p FB(,)31 b Fx(f)47 b Fv(2)37 b Fx(h)3143 898 y Fo(\003)3196 934 y FB(,)31 b(which)257 1054 y(commute)g(with)f(the)i (number)f(operator)g Fx(N)1809 1018 y Fo(\003)1894 1054 y FB(and,)i(possibly)-6 b(,)31 b(with)g(further)h(char)n(ge)g(oper)n(-) 257 1175 y(ators)d Fx(Q)557 1139 y Fo(\003)611 1175 y Fs(\()p Fx(\024)p Fs(\))p FB(,)h(for)g(certain)f(choices)g(of)h Fx(\024)p FB(.)44 b(Ev)o(ery)29 b(monomial)e(in)i Fx(a)h FB(and)f Fx(a)2921 1139 y Fu(\003)2990 1175 y FB(belonging)f(to)257 1295 y Fv(A)337 1259 y Fo(\003)416 1295 y FB(has)e(equally)f(man)o(y)g (f)o(actors)h(of)h Fx(a)f FB(and)g Fx(a)1844 1259 y Fu(\003)1883 1295 y FB(,)h(since)e(it)h(must)f(conserv)o(e)g(the)h(total)f(particle) 257 1416 y(number)-5 b(.)31 b(A)25 b(general)h(monomial)d(in)i Fx(a)g FB(and)h Fx(a)1858 1379 y Fu(\003)1922 1416 y FB(is)f Fm(W)-5 b(ic)n(k-or)l(der)l(ed)25 b FB(if)g(all)g Fx(a)2834 1379 y Fu(\003)2874 1416 y FB(')-5 b(s)25 b(are)h(to)e(the)h (left)257 1536 y(of)g(all)g Fx(a)p FB(')-5 b(s.)404 1656 y(In)25 b(terms)f(of)h(creation)g(and)f(annihilation)f(operators,)i (the)g(operators)f Fx(H)2933 1620 y Fo(\003)2986 1656 y Fx(;)17 b(N)3118 1620 y Fo(\003)3196 1656 y FB(and)25 b Fx(Q)3442 1620 y Fo(\003)257 1777 y FB(can)h(be)f(e)o(xpressed)f(as)h (follo)n(ws.)1107 2034 y Fx(H)1196 1993 y Fo(\003)1304 2034 y Fs(=)1407 1939 y Ft(X)1463 2148 y Fn(s)1568 1898 y Ft(Z)1623 2124 y Fo(\003)1693 2034 y Fx(d)p Fr(x)16 b Fx(a)1877 1993 y Fu(\003)1916 2034 y Fs(\()p Fr(x)p Fx(;)h(s)p Fs(\)\()p Fx(t)2221 1993 y Fo(\003)2274 2034 y Fx(a)p Fs(\)\()p Fr(x)o Fx(;)g(s)p Fs(\))25 b Fx(;)609 b FB(\(3.19\))1180 2394 y Fx(N)1268 2353 y Fo(\003)1377 2394 y Fs(=)1481 2299 y Ft(X)1537 2509 y Fn(s)1641 2258 y Ft(Z)1697 2484 y Fo(\003)1766 2394 y Fx(d)p Fr(x)16 b Fx(a)1950 2353 y Fu(\003)1990 2394 y Fs(\()p Fr(x)p Fx(;)h(s)p Fs(\))g Fx(a)p Fs(\()p Fr(x)o Fx(;)g(s)p Fs(\))24 b Fx(;)683 b FB(\(3.20\))257 2645 y(and)1026 2791 y Fx(Q)1103 2749 y Fo(\003)1156 2791 y Fs(\()p Fx(\024)p Fs(\))56 b(=)1447 2696 y Ft(X)1466 2909 y Fn(s;s)1552 2890 y Ff(0)1608 2655 y Ft(Z)1663 2880 y Fo(\003)1733 2791 y Fx(d)p Fr(x)16 b Fx(a)1917 2749 y Fu(\003)1956 2791 y Fs(\()p Fr(x)p Fx(;)h(s)p Fs(\))g Fx(\024)2261 2806 y Fn(ss)2327 2787 y Ff(0)2369 2791 y Fx(a)p Fs(\()p Fr(x)p Fx(;)g(s)2614 2749 y Fu(0)2637 2791 y Fs(\))24 b Fx(:)529 b FB(\(3.21\))404 3058 y(In)38 b(the)h(e)o(xamples)f(discussed)f(belo)n(w)h(and)h(in)f (Sects.)h(4)g(and)f(5,)k(we)d(usually)f(re)o(gard)257 3179 y Fx(N)345 3142 y Fo(\003)435 3179 y Fs(=)d Fx(Q)623 3142 y Fo(\003)677 3179 y Fs(\()p Fx(\024)g Fs(=)h(1)-22 b(l)o(\))29 b FB(to)g(be)g(the)g Fm(only)g(conservation)e(law)p FB(,)k(besides)d Fx(H)2775 3142 y Fo(\003)2828 3179 y FB(,)i(rele)n(v)n(ant)e(for)h(the)257 3299 y(description)21 b(of)h(the)f(reserv)n(oirs.)30 b(In)22 b(a)g(general)g(discussion,)f (we)h(consider)f Fx(M)33 b FB(conserv)n(ation)257 3419 y(la)o(ws,)24 b Fx(Q)565 3383 y Fo(\003)565 3444 y Fn(j)646 3419 y Fs(=)k Fx(Q)827 3383 y Fo(\003)880 3419 y Fs(\()p Fx(\024)974 3434 y Fn(j)1011 3419 y Fs(\))p FB(,)d Fx(j)33 b Fs(=)28 b(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(M)10 b FB(,)26 b(and)e(choose)h Fx(t)2203 3383 y Fo(\003)2281 3419 y FB(as)g(in)f(\(3.3\).)404 3540 y(The)h(main)f(result)g(of)h(this)f (section)g(is)g(the)h(follo)n(wing)e(theorem.)257 3736 y Fz(Theor)n(em)e(3.1.)32 b Fm(F)-10 b(or)18 b Fx(t)1037 3700 y Fo(\003)1110 3736 y Fm(as)g(in)h(\(3.3\),)g(and)g Fx(Q)1795 3700 y Fo(\003)1795 3761 y Fn(j)1876 3736 y Fs(=)27 b Fx(Q)2056 3700 y Fo(\003)2056 3761 y Fn(j)2110 3736 y Fs(\()p Fx(\024)2204 3751 y Fn(j)2240 3736 y Fs(\))p Fx(;)17 b(j)34 b Fs(=)27 b(1)p Fx(;)17 b(:)g(:)g(:)f(;)h(M)10 b Fm(,)20 b(with)f Fx(\024)3163 3751 y Fo(1)3202 3736 y Fx(;)e(:)g(:)g(:)f(;)h(\024)3477 3751 y Fn(M)257 3857 y Fm(arbitr)o(ary)-5 b(,)21 b(commuting)g(symmetric)i Fx(k)17 b Fv(\002)d Fx(k)27 b Fm(matrices,)22 b(the)h(equilibrium)e (states)h Fx(!)3138 3821 y Fo(\003)3134 3883 y Fn(\014)s(;)p Fj(\026)3273 3857 y Fm(intr)l(o-)257 3977 y(duced)j(in)g(eq.)f (\(2.10\))h(e)n(xist,)f(for)h(arbitr)o(ary)d Fx(\014)34 b Fv(\025)28 b Fs(0)d Fm(and)f Fr(\026)j Fv(2)h Fl(R)2492 3941 y Fn(M)2577 3977 y Fm(.)404 4098 y(Assumptions)c(\(A1\),)k (\(A2\),)g(\(A5\))f(and)f(\(A6\))h(of)g(Sect.)g(2,)g(concerning)g(the)f (e)n(xistence)i(of)257 4218 y(the)d(thermodynamic)f(limit,)f Fs(\003)28 b Fv(\045)f Fl(R)1570 4182 y Fn(d)1616 4218 y Fm(,)e(hold.)404 4415 y FB(The)31 b(proof)f(of)h(Theorem)g(3.1)f(is)h (standard.)48 b(A)31 b(careful)h(e)o(xposition)c(can)j(be)g(found)g(in) 257 4535 y([BR],)26 b(Sect.)g(5.2.)404 4655 y(In)f(Section)g(4)g(we)g (shall)g(consider)g(a)g(system)f(consisting)f(of)j(tw)o(o)e(identical)g (reserv)n(oirs,)257 4776 y Fx(I)45 b FB(and)37 b Fx(I)-9 b(I)8 b FB(,)40 b(both)c(composed)f(of)i(non-interacting,)i(non-relati) n(vistic)34 b(fermions)i(con\002ned)257 4896 y(to)f(some)g(re)o(gion)g Fs(\003)47 b(=)g(\003)1211 4860 y Fn(I)1298 4896 y Fs(=)g(\003)1489 4860 y Fn(I)-7 b(I)1588 4896 y FB(of)36 b Fl(R)1773 4860 y Fn(d)1819 4896 y FB(.)63 b(A)35 b(con)l(v)o(enient)f(notation)g(for)i (creation)f(and)257 5017 y(annihilation)23 b(operators)i(for)g(the)g (tw)o(o)f(reserv)n(oirs)h(is)f(the)h(follo)n(wing)e(one.)1056 5229 y Fx(a)1107 5188 y Fo(#)1170 5229 y Fs(\()p Fr(x)o Fx(;)17 b(s;)g(I)8 b Fs(\))83 b(:=)g Fx(a)1816 5188 y Fo(#)1879 5229 y Fs(\()p Fr(x)o Fx(;)17 b(s)p Fs(\))2110 5144 y Ft(\014)2110 5204 y(\014)2143 5268 y Fu(H)2203 5249 y Fp(\003)2244 5235 y Fe(I)2310 5229 y Fv(\012)23 b Fs(1)-22 b(l)2464 5144 y Ft(\014)2464 5204 y(\014)2497 5268 y Fu(H)2557 5249 y Fp(\003)2598 5235 y Fe(I)-7 b(I)2688 5229 y Fx(;)1021 5388 y(a)1072 5347 y Fo(#)1136 5388 y Fs(\()p Fr(x)o Fx(;)17 b(s;)g(I)-9 b(I)8 b Fs(\))83 b(:=)g(1)-22 b(l)1818 5303 y Ft(\014)1818 5363 y(\014)1851 5427 y Fu(H)1911 5408 y Fp(\003)1952 5394 y Fe(I)2019 5388 y Fv(\012)22 b Fx(a)2169 5347 y Fo(#)2233 5388 y Fs(\()p Fr(x)o Fx(;)17 b(s)p Fs(\))2464 5303 y Ft(\014)2464 5363 y(\014)2497 5427 y Fu(H)2557 5408 y Fp(\003)2598 5394 y Fe(I)-7 b(I)2688 5388 y Fx(:)540 b FB(\(3.22\))p eop %%Page: 31 32 31 31 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(31)257 573 y(W)-8 b(e)20 b(note)f(that)g(all)g(the)g(operators)g Fx(a)1468 537 y Fo(#)1531 573 y Fs(\()p Fx(f)5 b(;)17 b(I)8 b Fs(\))28 b(=)f Fx(a)1937 537 y Fo(#)2000 573 y Fs(\()p Fx(f)11 b Fs(\))r Fv(\012)r Fs(1)-22 b(l)18 b Fm(commute)h FB(with)f(all)h(the)h(operators)257 693 y Fx(a)308 657 y Fo(#)372 693 y Fs(\()p Fx(f)5 b(;)17 b(I)-9 b(I)8 b Fs(\))27 b(=)h(1)-22 b(l)18 b Fv(\012)h Fx(a)981 657 y Fo(#)1044 693 y Fs(\()p Fx(f)11 b Fs(\))p FB(.)30 b(If,)25 b(for)f(con)l(v)o(enience,)g(we)g(prefer)h(that)e(the) o(y)h(anti-commute)e(we)257 814 y(can)k(accomplish)d(this)h(feature)i (by)e(a)h(standard)g Fm(Klein-J)n(or)l(dan-W)-5 b(igner)23 b(tr)o(ansformation)p FB(:)1162 1002 y Fx(a)1213 961 y Fo(#)1276 1002 y Fs(\()p Fx(f)5 b(;)17 b(I)8 b Fs(\))83 b Fv(7!)g Fx(a)1817 961 y Fo(#)1880 1002 y Fs(\()p Fx(f)5 b(;)17 b(I)8 b Fs(\))24 b Fx(;)1128 1179 y(a)1179 1138 y Fo(#)1242 1179 y Fs(\()p Fx(f)5 b(;)17 b(I)-9 b(I)8 b Fs(\))83 b Fv(7!)g Fx(a)1817 1138 y Fo(#)1880 1179 y Fs(\()p Fx(f)5 b(;)17 b(I)-9 b(I)8 b Fs(\))27 b Fx(e)2210 1138 y Fn(i\031)r Fo(\()p Fn(N)2367 1115 y Fp(\003)2408 1094 y Fe(I)2449 1138 y Fu(\012)p Fo(1)-18 b(l\))2598 1179 y Fx(:)630 b FB(\(3.23\))257 1367 y(The)28 b(operators)g(on)f(the) h(R.S.)h(of)f(\(3.23\))f(will)g(again)g(be)h(denoted)g(by)f Fx(a)2765 1331 y Fo(#)2828 1367 y Fs(\()p Fx(f)5 b(;)17 b(r)s Fs(\))p FB(,)29 b Fx(r)36 b Fs(=)d Fx(I)8 b(;)17 b(I)-9 b(I)8 b FB(.)257 1488 y(W)-8 b(e)26 b(introduce)e(the)g(follo)n (wing)f(notation.)1104 1676 y Fx(X)90 b Fs(:=)83 b(\()p Fr(x)p Fx(;)17 b(s;)g(r)s Fs(\))27 b Fv(2)h Fl(R)1971 1635 y Fn(d)2039 1676 y Fv(\002)23 b(f)p Fs(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(k)s Fv(g)22 b(\002)h(f)p Fx(I)8 b(;)17 b(I)-9 b(I)8 b Fv(g)p Fx(;)266 b FB(\(3.24\))981 1821 y Fx(X)1070 1780 y Fo(\()p Fn(N)7 b Fo(\))1275 1821 y Fs(:=)83 b(\()p Fx(X)1580 1836 y Fo(1)1620 1821 y Fx(;)17 b(:)g(:)g(:)e(;)i(X)1919 1836 y Fn(N)1986 1821 y Fs(\))42 b Fx(;)1162 b FB(\(3.25\))1015 1966 y Fx(x)1070 1925 y Fo(\()p Fn(N)7 b Fo(\))1275 1966 y Fs(:=)83 b(\()p Fr(x)1565 1981 y Fo(1)1604 1966 y Fx(;)17 b(:)g(:)g(:)f(;)h Fr(x)1889 1981 y Fn(n)1936 1966 y Fs(\))27 b Fv(2)h Fl(R)2161 1925 y Fn(dN)2271 1966 y Fx(;)957 b FB(\(3.26\))1024 2112 y Fx(s)1070 2071 y Fo(\()p Fn(N)7 b Fo(\))1275 2112 y Fs(:=)83 b(\()p Fx(s)1545 2127 y Fo(1)1585 2112 y Fx(;)17 b(:)g(:)g(:)e(;)i(s)1849 2127 y Fn(N)1916 2112 y Fs(\))28 b Fv(2)g(f)p Fs(1)p Fx(;)17 b(:)g(:)g(:)e(;)i(k)s Fv(g)2497 2071 y Fn(N)2564 2112 y Fx(;)664 b FB(\(3.27\))1024 2257 y Fx(r)1071 2216 y Fo(\()p Fn(N)7 b Fo(\))1275 2257 y Fs(:=)83 b(\()p Fx(r)1543 2272 y Fo(1)1583 2257 y Fx(;)17 b(:)g(:)g(:)e(;)i(r)1845 2272 y Fn(N)1912 2257 y Fs(\))28 b Fv(2)g(f)p Fx(I)8 b(;)17 b(I)8 b(I)g Fv(g)2369 2216 y Fn(N)2435 2257 y Fx(;)793 b FB(\(3.28\))928 2360 y Ft(Z)983 2586 y Fo(\003)1053 2496 y Fx(dX)90 b Fs(:=)1491 2401 y Ft(X)1461 2613 y Fn(r)r Fo(=)p Fn(I)5 b(;I)-7 b(I)1762 2371 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b(is)h(understood)e(that)i Fx(D)1977 5252 y Fo(0)2016 5237 y Fx(a)j Fs(=)g Fx(a)p FB(.)p eop %%Page: 34 35 34 34 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(34)404 573 y(W)-8 b(e)25 b(de\002ne)g(the)g(norm)f(of)h(an)g (interaction)f Fx(W)39 b FB(by)24 b(setting)340 875 y Fv(k)p Fx(W)14 b Fv(k)546 834 y Fu(0)597 875 y Fs(=)705 781 y Ft(X)700 992 y Fn(N)7 b Fu(\025)p Fo(1)870 875 y Fx(N)j Fs(2)1007 834 y Fo(\()p Fn(d)p Fo(+2\))p Fn(N)1426 751 y(k)1373 781 y Ft(X)1273 990 y Fn(s)1306 999 y Fp(1)1340 990 y Fn(;:::)n(;s)1471 1001 y Fe(N)1528 990 y Fo(=1)1789 751 y Fn(k)1736 781 y Ft(X)1635 997 y Fn(s)1668 974 y Ff(0)1668 1018 y Fp(1)1702 997 y Fn(;:::)o(;s)1834 974 y Ff(0)1834 1020 y Fe(N)1890 997 y Fo(=1)823 1073 y Ft(X)677 1285 y Fn(r)709 1294 y Fp(1)743 1285 y Fn(;:::)o(;r)874 1296 y Fe(N)931 1285 y Fo(=)p Fn(I)5 b(;I)g(I)1275 1073 y Ft(X)1129 1290 y Fn(r)1163 1267 y Ff(0)1161 1311 y Fp(1)1196 1290 y Fn(;:::)n(;r)1328 1267 y Ff(0)1326 1313 y Fe(N)1383 1290 y Fo(=)p 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col0 sh gr /Times-Italic-iso ff 210.00 scf sf 5685 2040 m gs 1 -1 sc (r = 1) col0 sh gr /Times-Italic-iso ff 150.00 scf sf 5760 2115 m gs 1 -1 sc (3) col0 sh gr /Times-Italic-iso ff 210.00 scf sf 6930 2190 m gs 1 -1 sc (r = 0) col0 sh gr /Times-Italic-iso ff 150.00 scf sf 6990 2280 m gs 1 -1 sc (2) col0 sh gr /Times-Italic-iso ff 210.00 scf sf 3525 1875 m gs 1 -1 sc (r = 3) col0 sh gr /Times-Italic-iso ff 150.00 scf sf 3600 1935 m gs 1 -1 sc (m-2) col0 sh gr % Polyline 7.500 slw n 975 1200 m 975 1350 l 3525 1350 l 3525 1200 l gs col0 s gr % Polyline n 2325 1200 m 2325 1500 l 8100 1500 l 8100 1200 l gs col0 s gr % Polyline n 3600 1200 m 3600 1650 l 5700 1650 l 5700 1200 l gs col0 s gr % Polyline n 5775 1200 m 5775 1800 l 8175 1800 l 8175 1200 l gs col0 s gr % Polyline n 6975 1200 m 6975 1950 l 9375 1950 l 9375 1200 l gs col0 s gr % Polyline n 8250 1200 m 8250 1350 l 9300 1350 l 9300 1200 l gs col0 s gr /Times-Italic-iso ff 270.00 scf sf 4500 900 m gs 1 -1 sc (...) col0 sh gr /Times-Italic-iso ff 270.00 scf sf 900 900 m gs 1 -1 sc (m) col0 sh gr /Times-Italic-iso ff 270.00 scf sf 2100 900 m gs 1 -1 sc (m-1) col0 sh gr /Times-Italic-iso ff 270.00 scf sf 3300 900 m gs 1 -1 sc (m-2) col0 sh gr /Times-Italic-iso ff 270.00 scf sf 5700 900 m gs 1 -1 sc (3) col0 sh gr /Times-Italic-iso ff 270.00 scf sf 6900 900 m gs 1 -1 sc (2) col0 sh gr /Times-Italic-iso ff 270.00 scf sf 8100 900 m gs 1 -1 sc (1) col0 sh gr /Times-Italic-iso ff 270.00 scf sf 9300 900 m gs 1 -1 sc (0) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 257 1376 a(Figure)29 b(2:)37 b(Illustration)27 b(for)i(the)f(numbers)g Fx(r)1838 1391 y Fn(m)1904 1376 y Fx(;)17 b(:)g(:)g(:)f(;)h(r)2167 1391 y Fo(2)2234 1376 y FB(that)28 b(occur)h(in)f(the)g(choice)h(of)f(con-)257 1496 y(tractions.)i(W)-8 b(e)25 b(see)h(that)e(the)o(y)g(de\002ne)h(a)g (tree.)257 1783 y(an)33 b(operator)g(at)g(time)f Fx(t)1099 1798 y Fo(0)1182 1783 y Fs(=)42 b(0)p FB(;)37 b(an)c(operator)g(at)g (time)f Fx(t)2256 1798 y Fo(2)2329 1783 y FB(contracts)g(with)g(an)h (operator)g(at)257 1904 y(time)i Fx(t)505 1919 y Fn(r)537 1928 y Fp(2)611 1904 y FB(with)f Fx(r)867 1919 y Fo(2)954 1904 y Fs(=)46 b(0)p Fx(;)17 b Fs(1)p FB(;)40 b(.)15 b(.)g(.)g(;)39 b(an)d(operator)f(at)g(time)g Fx(t)2325 1919 y Fn(m)2427 1904 y FB(contracts)g(with)f(an)h(operator)257 2024 y(at)j(time)f Fx(t)617 2039 y Fn(r)649 2047 y Fe(m)749 2024 y FB(with)g Fx(r)1008 2039 y Fn(m)1125 2024 y Fs(=)51 b(0)p Fx(;)17 b(:)g(:)g(:)f(;)h(m)31 b Fv(\000)h Fs(1)p FB(.)69 b(See)38 b(Fig.)f(2)h(for)f(an)h(illustration.)67 b(T)-8 b(o)37 b(a)h(set)257 2144 y(of)e(contractions)e(corresponds)h(a) h(monomial)d(of)j(creation)f(and)g(annihilation)f(operators,)257 2265 y(multiplied)23 b(by)h(anticommutators)f(of)i(contracted)g (operators.)404 2385 y(The)32 b(monomial)e(of)h(creation)h(and)g (annihilation)e(operators)h(is)h(bounded)f(in)g(operator)257 2505 y(norm)d(by)g(1.)42 b(Contracted)29 b(operators)f(are)h(estimated) f(using)f(\(4.15\).)42 b(This)27 b(yields)h(a)h(f)o(actor)257 2626 y(in)l(v)n(olving)23 b(times,)h(namely)1539 2694 y Fn(m)1506 2724 y Ft(Y)1509 2934 y Fn(j)t Fo(=1)1634 2738 y Ft(\000)1679 2819 y Fs(1)e Fv(^)1912 2780 y Fo(4)p Fn(\031)p 1849 2796 204 4 v 1849 2853 a(t)1874 2863 y Fe(j)1907 2853 y Fu(\000)p Fn(t)1987 2861 y Fe(r)2016 2877 y(j)2063 2738 y Ft(\001)2108 2760 y Fn(d=)p Fo(2)2219 2819 y Fx(:)257 3087 y FB(Second,)32 b(one)d(obtains)g(a)h(f)o(actor)g (in)l(v)n(olving)e(indices)h(of)h(Hermite)f(functions)g(for)h(the)f (con-)257 3208 y(tracted)e(operators.)35 b(An)26 b(upper)h(bound)e(on)h (this)g(f)o(actor)g(is)g(obtained)g(by)g(writing)f(a)i(product)257 3328 y(o)o(v)o(er)d(all)h(indices,)f(namely)1245 3524 y Fn(m)1213 3554 y Ft(Y)1215 3764 y Fn(j)t Fo(=0)1377 3513 y Fn(N)1433 3523 y Fe(j)1357 3554 y Ft(Y)1357 3766 y Fn(k)r Fo(=1)1548 3524 y Fn(d)1502 3554 y Ft(Y)1509 3764 y Fn(i)p Fo(=1)1629 3648 y Fs(\()p Fx(q)1710 3663 y Fn(j)t(k)r(i)1832 3648 y Fs(+)e(1\))2027 3580 y Fp(1)p 2026 3592 31 3 v 2026 3633 a(2)2071 3648 y Fs(\()p Fx(q)2156 3607 y Fu(0)2152 3673 y Fn(j)t(k)r(i)2273 3648 y Fs(+)g(1\))2468 3580 y Fp(1)p 2468 3592 V 2468 3633 a(2)2513 3648 y Fx(:)257 3977 y FB(Here,)41 b Fr(q)574 4000 y Fn(j)t(k)649 3977 y Fx(;)17 b Fr(q)751 3941 y Fu(0)751 4003 y Fn(j)t(k)877 3977 y Fv(2)51 b Fl(N)1060 3941 y Fn(d)1144 3977 y FB(are)38 b(indices)f(for)g(Hermite)g(functions)f(determined)h(by)g(the)g Fx(k)s FB(-th)257 4097 y(element)25 b(of)g Fr(Q)792 4121 y Fn(j)829 4097 y FB(.)404 4217 y(It)32 b(remains)h(to)f(estimate)g (the)h(number)f(of)h(contraction)f(schemes,)j(gi)n(v)o(en)c Fx(r)3102 4232 y Fo(2)3141 4217 y Fx(;)17 b(:)g(:)g(:)f(;)h(r)3404 4232 y Fn(m)3470 4217 y FB(.)257 4338 y(W)-8 b(e)26 b(de\002ne)976 4458 y Fx(e)1021 4473 y Fn(j)1085 4458 y Fs(=)1189 4373 y Ft(\014)1189 4433 y(\014)1222 4458 y Fv(f)p Fx(k)k Fs(:)e Fx(r)1452 4473 y Fn(k)1523 4458 y Fs(=)f Fx(j)6 b Fv(g)1722 4373 y Ft(\014)1722 4433 y(\014)1777 4458 y Fs(+)22 b(1)g Fv(\000)h Fx(\016)2089 4473 y Fn(j)t Fo(0)2161 4458 y Fx(;)116 b Fs(0)28 b Fv(\024)g Fx(j)33 b Fv(\024)28 b Fx(m:)479 b FB(\(4.19\))257 4633 y(Notice)29 b(that)f Fs(1)34 b Fv(\024)h Fx(e)970 4648 y Fo(0)1044 4633 y Fv(\024)g Fx(m)29 b FB(and)f Fs(1)35 b Fv(\024)g Fx(e)1683 4648 y Fn(j)1754 4633 y Fv(\024)g Fx(m)25 b Fv(\000)h Fx(j)k Fs(+)25 b(1)k FB(if)f Fx(j)41 b Fv(6)p Fs(=)34 b(0)p FB(.)42 b Fx(e)2769 4648 y Fn(j)2834 4633 y FB(is)28 b(the)g(number)g(of)257 4753 y(operators)f(at)h(time)e Fx(t)994 4768 y Fn(j)1058 4753 y FB(that)h(belong)f(to)h(a)h (contraction)e(and)i(it)e(is)h(necessarily)g(smaller)g(than)257 4873 y Fs(2)p Fx(N)384 4888 y Fn(j)421 4873 y FB(.)j(Since)23 b(there)g(are)h Fs(2)p Fx(N)1214 4888 y Fn(j)1273 4873 y FB(operators)e(at)h(time)f Fx(t)1996 4888 y Fn(j)2033 4873 y FB(,)h(the)g(number)f(of)h(possible)e(contractions)257 4994 y(is)1578 5042 y Fn(m)1546 5072 y Ft(Y)1548 5282 y Fn(j)t Fo(=0)1802 5099 y Fs(\(2)p Fx(N)1967 5114 y Fn(j)2003 5099 y Fs(\)!)p 1700 5143 470 4 v 1700 5235 a(\(2)p Fx(N)1865 5250 y Fn(j)1923 5235 y Fv(\000)i Fx(e)2068 5250 y Fn(j)2105 5235 y Fs(\)!)2180 5166 y Fx(:)p eop %%Page: 39 40 39 39 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(39)257 573 y(The)19 b(abo)o(v)o(e)f(estimates)g(could)g(be)h (impro)o(v)o(ed)e(by)h(observing)g(that)h(man)o(y)e(contraction)i (schemes)257 693 y(yield)30 b(zero;)i(namely)-6 b(,)30 b(in)g(the)g(case)g(where)h(both)e(operators)h(are)g(creation)g(or)g (annihilation)257 814 y(operators;)f(or)e(if)h(the)f(spins)f(are)i(dif) n(ferent;)g(or)g(if)f(the)o(y)g(belong)g(to)g(dif)n(ferent)g(reserv)n (oirs.)38 b(It)257 934 y(is)25 b(not)f(easy)h(to)f(tak)o(e)h(adv)n (antage)f(of)h(this,)f(ho)n(we)n(v)o(er)-5 b(.)404 1054 y(W)d(e)25 b(no)n(w)f(gather)g(the)h(abo)o(v)o(e)f(estimates)f(to)i (obtain)f(the)g(bound)357 1211 y Ft(Z)457 1237 y Fu(1)412 1437 y Fo(0)548 1347 y Fv(k)p Fs([)p Fx(W)14 b Fs(\()p Fx(t)p Fs(\))p Fx(;)j(D)967 1362 y Fn(m)p Fu(\000)p Fo(1)1123 1347 y Fs(\()p Fx(t)p Fs(\))p Fx(A)p Fs(])p Fv(k)p Fx(dt)28 b Fv(\024)894 1538 y Ft(X)809 1750 y Fj(Q)871 1767 y Fp(0)905 1750 y Fn(;:::)o(;)p Fj(Q)1064 1767 y Fe(m)1140 1633 y Fv(j)1193 1608 y Fs(~)1168 1633 y Fx(A)1241 1648 y Fn(N)1297 1657 y Fp(0)1335 1633 y Fs(\()p Fx(Q)1450 1582 y Fo(\()p Fn(N)1533 1591 y Fp(0)1568 1582 y Fo(\))1450 1657 y(0)1600 1633 y Fx(;)17 b(Q)1721 1592 y Fu(0)1721 1658 y Fo(0)1760 1581 y(\()p Fn(N)1843 1590 y Fp(0)1878 1581 y Fo(\))1910 1633 y Fs(\))p Fv(j)2025 1508 y Fn(m)1992 1538 y Ft(Y)1995 1748 y Fn(j)t Fo(=1)2136 1633 y Fv(j)j Fs(~)-69 b Fx(w)2234 1648 y Fn(N)2290 1658 y Fe(j)2326 1633 y Fs(\()p Fx(Q)2441 1579 y Fo(\()p Fn(N)2524 1589 y Fe(j)2557 1579 y Fo(\))2441 1658 y Fn(j)2589 1633 y Fx(;)17 b(Q)2710 1592 y Fu(0)2710 1658 y Fn(j)2747 1581 y Fo(\()p Fn(N)2830 1591 y Fe(j)2863 1581 y Fo(\))2894 1633 y Fs(\))p Fv(j)664 2001 y(\002)796 1876 y Fn(m)764 1906 y Ft(Y)766 2116 y Fn(j)t Fo(=0)928 1865 y Fn(N)984 1875 y Fe(j)909 1906 y Ft(Y)908 2119 y Fn(k)r Fo(=1)1099 1876 y Fn(d)1053 1906 y Ft(Y)1060 2116 y Fn(i)p Fo(=1)1180 2001 y Fs(\()p Fx(q)1261 2016 y Fn(j)t(k)r(i)1383 2001 y Fs(+)22 b(1\))1578 1933 y Fp(1)p 1578 1945 31 3 v 1578 1986 a(2)1622 2001 y Fs(\()p Fx(q)1707 1960 y Fu(0)1703 2026 y Fn(j)t(k)r(i)1824 2001 y Fs(+)g(1\))2019 1933 y Fp(1)p 2019 1945 V 2019 1986 a(2)2080 1865 y Ft(Z)2136 2091 y Fu(1)p Fn(>t)2287 2099 y Fe(m)2345 2091 y Fn(>)p Fu(\001\001\001)o Fn(>t)2539 2100 y Fp(1)2574 2091 y Fn(>)p Fo(0)2685 2001 y Fx(dt)2771 2016 y Fo(1)2827 2001 y Fx(:)17 b(:)g(:)f(dt)3044 2016 y Fn(m)613 2350 y Fv(\002)726 2225 y Fn(m)p Fu(\000)p Fo(1)731 2255 y Ft(X)712 2464 y Fn(r)744 2472 y Fe(m)803 2464 y Fo(=0)963 2225 y Fn(m)p Fu(\000)p Fo(2)967 2255 y Ft(X)910 2464 y Fn(r)942 2473 y Fe(m)p Ff(\000)p Fp(1)1079 2464 y Fo(=0)1185 2350 y Fv(\001)h(\001)g(\001)1379 2225 y Fo(1)1324 2255 y Ft(X)1318 2464 y Fn(r)1350 2473 y Fp(2)1385 2464 y Fo(=0)1524 2225 y Fn(m)1491 2255 y Ft(Y)1494 2465 y Fn(j)t Fo(=1)1619 2269 y Ft(\000)1664 2350 y Fs(1)22 b Fv(^)1897 2310 y Fo(4)p Fn(\031)p 1834 2327 204 4 v 1834 2384 a(t)1859 2394 y Fe(j)1892 2384 y Fu(\000)p Fn(t)1972 2392 y Fe(r)2001 2408 y(j)2048 2269 y Ft(\001)2093 2291 y Fn(d=)p Fo(2)2253 2225 y Fn(m)2221 2255 y Ft(Y)2223 2465 y Fn(j)t Fo(=0)2375 2282 y Fx(\037)p Fs([)p Fx(e)2508 2297 y Fn(j)2572 2282 y Fv(\024)29 b Fs(2)p Fx(N)2805 2297 y Fn(j)2841 2282 y Fs(]\(2)p Fx(N)3033 2297 y Fn(j)3069 2282 y Fs(\)!)p 2375 2327 760 4 v 2520 2418 a(\(2)p Fx(N)2685 2433 y Fn(j)2743 2418 y Fv(\000)23 b Fx(e)2888 2433 y Fn(j)2925 2418 y Fs(\)!)3144 2350 y Fx(:)84 b FB(\(4.20\))404 2666 y(A)29 b(sequence)h(of)f(numbers)g Fx(r)1429 2681 y Fo(2)1468 2666 y Fx(;)17 b(:)g(:)g(:)f(;)h(r)1731 2681 y Fn(m)1827 2666 y FB(can)29 b(be)h(represented)f(by)h(a)f(graph)h(with)e(set)h(of) 257 2787 y(v)o(ertices)d Fv(f)p Fs(0)p Fx(;)17 b(:)g(:)g(:)e(;)i(m)p Fv(g)p FB(,)26 b(and)g(an)f(edge)h(between)g Fx(i)g FB(and)g Fx(j)32 b FB(whene)n(v)o(er)25 b Fx(r)2710 2802 y Fn(j)2776 2787 y Fs(=)k Fx(i)p FB(.)34 b(This)25 b(graph)h(is)257 2907 y(a)h(tree:)35 b(there)27 b(are)h Fx(m)f FB(edges,)g(and)g(each)g (v)o(erte)o(x)f Fx(j)37 b Fv(6)p Fs(=)31 b(0)c FB(is)f(directly)g (connected)h(to)f(a)h(v)o(erte)o(x)257 3027 y Fx(i)42 b(<)f(j)6 b FB(,)34 b(hence)e(each)h(v)o(erte)o(x)e(is)g(e)n(v)o (entually)g(connected)h(to)f(0.)53 b(The)32 b(numbers)f Fx(e)3133 3042 y Fn(j)3202 3027 y FB(de\002ned)257 3148 y(in)d(\(4.19\))f(are)h(then)f(the)h(incidence)f(numbers)g(of)g(the)h (tree)g(\227)f Fx(e)2518 3163 y Fn(j)2582 3148 y FB(is)g(the)h(number)f (of)g(edges)257 3268 y(containing)21 b(the)g(v)o(erte)o(x)g Fx(j)6 b FB(.)30 b(This)21 b(is)g(illustrated)g(in)g(Fig.)h(2.)29 b(W)-8 b(e)23 b(can)f(symmetrize)f(the)g(bound)257 3389 y(by)j(summing)f(o)o(v)o(er)g Fm(all)h FB(trees)g Fx(T)38 b FB(with)24 b Fx(m)d Fs(+)f(1)k FB(v)o(ertices;)g(this)f(step)h(will)f (allo)n(w)h(to)g(deal)g(with)257 3509 y(the)h(time)f(inte)o(grals.)29 b(Reor)n(ganizing,)c(we)g(obtain)357 3666 y Ft(Z)457 3692 y Fu(1)412 3891 y Fo(0)548 3801 y Fv(k)p Fs([)p Fx(W)14 b Fs(\()p Fx(t)p Fs(\))p Fx(;)j(D)967 3816 y Fn(m)p Fu(\000)p Fo(1)1123 3801 y Fs(\()p Fx(t)p Fs(\))p Fx(a)p Fs(])p Fv(k)p Fx(dt)28 b Fv(\024)808 4010 y Ft(X)832 4222 y Fj(Q)893 4239 y Fp(0)969 4105 y Fv(j)q Fs(~)-50 b Fx(a)1048 4120 y Fn(N)1104 4129 y Fp(0)1142 4105 y Fs(\()p Fx(Q)1257 4054 y Fo(\()p Fn(N)1340 4063 y Fp(0)1376 4054 y Fo(\))1257 4129 y(0)1407 4105 y Fx(;)17 b(Q)1528 4064 y Fu(0)1528 4130 y Fo(0)1568 4053 y(\()p Fn(N)1651 4062 y Fp(0)1686 4053 y Fo(\))1717 4105 y Fs(\))p Fv(j)1818 3980 y Fn(N)1874 3989 y Fp(0)1800 4010 y Ft(Y)1800 4223 y Fn(k)r Fo(=1)1990 3980 y Fn(d)1945 4010 y Ft(Y)1951 4220 y Fn(i)p Fo(=1)2072 4105 y Fs(\()p Fx(q)2153 4120 y Fo(0)p Fn(k)r(i)2277 4105 y Fs(+)22 b(1\))2472 4037 y Fp(1)p 2472 4049 31 3 v 2472 4090 a(2)2517 4105 y Fs(\()p Fx(q)2602 4064 y Fu(0)2598 4130 y Fo(0)p Fn(k)r(i)2722 4105 y Fs(+)g(1\))2917 4037 y Fp(1)p 2917 4049 V 2917 4090 a(2)433 4424 y Fv(\002)532 4289 y Ft(Z)588 4514 y Fu(1)p Fn(>t)739 4522 y Fe(m)797 4514 y Fn(>)p Fu(\001\001\001)o Fn(>t)991 4523 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Fn(;)p Fo(2)1919 1724 y Fs(=)43 b Fv(P)2107 1739 y Fo(2)p Fn(;)p Fo(2)2246 1724 y Fs(=)h(0)p FB(.)57 b(The)33 b(\003o)n(ws)g(v)n(anish)g(if)g (both)501 1845 y(reserv)n(oirs)25 b(are)h(at)e(the)h(same)g (temperature)f(and)h(chemical)g(potential.)419 2048 y(-)49 b(If)25 b Fx(\026)651 2012 y Fn(I)718 2048 y Fs(=)i Fx(\026)880 2012 y Fn(I)5 b(I)980 2048 y FB(and)24 b Fx(\014)1209 2012 y Fn(I)1276 2048 y Fx(>)j(\014)1440 2012 y Fn(I)5 b(I)1540 2048 y FB(then)23 b Fx(\032)1785 2063 y Fn(I)5 b(I)1861 2048 y Fs(\()p Fx(k)s Fs(\))19 b Fv(\000)g Fx(\032)2156 2063 y Fn(I)2196 2048 y Fs(\()p Fx(k)s Fs(\))28 b Fx(>)g Fs(0)p FB(,)c(for)g(all)f Fx(k)s FB(.)31 b(Consequently)-6 b(,)501 2169 y Fv(J)569 2184 y Fo(2)p Fn(;)p Fo(2)663 2169 y Fx(;)17 b Fv(P)776 2184 y Fo(2)p Fn(;)p Fo(2)898 2169 y Fx(>)28 b Fs(0)p FB(.)h(At)23 b(constant)e(chemical)h (potential,)g(there)h(is)f(a)g(particle)h(and)f(ener)n(gy)501 2289 y(\003o)n(w)j(from)f(the)h(hotter)f(to)h(the)f(colder)h(reserv)n (oir)-5 b(.)419 2492 y(-)49 b(If)40 b Fx(\026)666 2456 y Fn(I)761 2492 y Fx(>)55 b(\026)951 2456 y Fn(I)5 b(I)1066 2492 y FB(and)39 b Fx(\014)1310 2456 y Fn(I)1405 2492 y Fs(=)55 b Fx(\014)1597 2456 y Fn(I)5 b(I)1727 2492 y Fs(=)55 b Fx(\014)6 b FB(,)43 b(then)c Fx(\032)2248 2507 y Fn(I)5 b(I)2324 2492 y Fs(\()p Fx(k)s Fs(\))33 b Fv(\000)h Fx(\032)2648 2507 y Fn(I)2688 2492 y Fs(\()p Fx(k)s Fs(\))55 b Fx(<)g Fs(0)p FB(,)43 b(for)d(all)f Fx(k)s FB(.)501 2613 y(Consequently)-6 b(,)33 b Fv(J)1170 2628 y Fo(2)p Fn(;)p Fo(2)1264 2613 y Fx(;)17 b Fv(P)1377 2628 y Fo(2)p Fn(;)p Fo(2)1513 2613 y Fx(<)41 b Fs(0)p FB(.)54 b(At)32 b(constant)f(temperature,)j(there)f(is)f(a)g(particle) 501 2733 y(and)k(ener)n(gy)h(\003o)n(w)f(from)g(the)g(reserv)n(oir)g (with)g(higher)g(chemical)g(potential)f(to)g(the)501 2854 y(reserv)n(oir)25 b(with)f(lo)n(wer)h(chemical)f(potential.)404 3057 y Fm(Ohm')l(s)33 b(law)h(and)g(the)f(r)l(esistance)g(of)h(the)g (junction.)90 b FB(Suppose)34 b(that)f Fx(\014)2977 3021 y Fn(I)3061 3057 y Fs(=)44 b Fx(\014)3242 3021 y Fn(I)5 b(I)3351 3057 y FB(and)257 3177 y Fx(\026)316 3141 y Fn(I)390 3177 y Fs(=)35 b Fx(\026;)17 b(\026)663 3141 y Fn(I)5 b(I)772 3177 y Fs(=)34 b Fx(\026)24 b Fs(+)h(\001)p Fx(\026)p FB(,)k(with)f Fs(\001)p Fx(\026)g FB(small.)41 b(Retaining)28 b(only)f(the)i(leading)f(order)g(in)g Fs(\001)p Fx(\026)257 3298 y FB(in)d(the)f(e)o(xpression)g(of)h(the)f (particle)h(\003o)n(w)g(yields)1548 3563 y Fv(J)1616 3578 y Fo(2)p Fn(;)p Fo(2)1738 3563 y Fv(\031)1940 3495 y Fs(\001)p Fx(\026)p 1853 3540 314 4 v 1853 3631 a(R)q Fs(\()p Fx(\026;)17 b(\014)6 b Fs(\))2177 3563 y Fx(;)1100 b FB(\(5.6\))257 3839 y(where)26 b(the)e Fm(r)l(esistance)h Fx(R)q Fs(\()p Fx(\026;)17 b(\014)6 b Fs(\))23 b FB(is)i(determined)f (by)366 4130 y Fx(R)q Fs(\()p Fx(\026;)17 b(\014)6 b Fs(\))681 4089 y Fu(\000)p Fo(1)802 4130 y Fs(=)28 b(2)p Fx(\031)t(\014)1091 3995 y Ft(Z)1146 4220 y Fq(R)1194 4202 y Fp(2)p Fe(d)1282 4130 y Fx(d)p Fr(k)e Fx(d)p Fr(l)f Fx(\016)t Fs(\()p Fv(j)p Fr(k)r Fv(j)1733 4089 y Fo(2)1794 4130 y Fv(\000)e(j)p Fr(l)p Fv(j)1985 4089 y Fo(2)2024 4130 y Fs(\))2072 4063 y Fv(j)16 b Ft(b)-71 b Fx(w)2170 4078 y Fo(1)2209 4063 y Fs(\(\()p Fv(\000)p Fr(k)r Fx(;)17 b(I)8 b(I)g Fs(\))p Fx(;)17 b Fs(\()p Fr(l)p Fx(;)g(I)8 b Fs(\)\))p Fv(j)2924 4027 y Fo(2)2987 4063 y Fx(e)3032 4027 y Fn(\014)s Fo(\()p Fu(j)p Fj(k)q Fu(j)3186 4003 y Fp(2)3220 4027 y Fu(\000)p Fn(\026)p Fo(\))p 2072 4108 1278 4 v 2387 4199 a Fs(\()p Fx(e)2470 4171 y Fn(\014)s Fo(\()p Fu(j)p Fj(k)q Fu(j)2624 4152 y Fp(2)2658 4171 y Fu(\000)p Fn(\026)p Fo(\))2809 4199 y Fs(+)22 b(1\))2994 4171 y Fo(2)3359 4130 y Fx(:)3304 4300 y FB(\(5.7\))257 4421 y(W)-8 b(e)31 b(refer)f(to)g(Subsection)f(5.2)g(for)h(a)g (qualitati)n(v)o(e)e(discussion)g(of)h(the)h(resistance,)h(in)e(three) 257 4541 y(dimensions,)23 b Fx(d)k Fs(=)h(3)p FB(.)404 4782 y Fm(Onsa)o(g)o(er)h(r)l(ecipr)l(ocity)g(r)l(elations.)44 b FB(Let)30 b(us)g(study)f(the)g(interdependence)h(of)g(the)g(\003o)n (ws)257 4902 y(near)21 b(equilibrium.)27 b(The)20 b(rele)n(v)n(ant)f (parameters)h(are)g(the)g(dif)n(ference)g(of)g(the)g(in)l(v)o(erse)f (temper)n(-)257 5023 y(atures,)32 b(and)f(the)f(dif)n(ference)h(of)f (the)h(chemical)f(potentials)f(di)n(vided)g(by)h(the)h(temperature.)257 5143 y(Precisely)-6 b(,)28 b(we)g(set)g Fx(\014)1013 5107 y Fn(I)1085 5143 y Fs(=)33 b Fx(\014)6 b FB(;)29 b Fx(\014)1373 5107 y Fn(I)5 b(I)1481 5143 y Fs(=)33 b Fx(\014)c Fv(\000)c Fs(\001)p Fx(\014)6 b FB(;)29 b Fx(\027)39 b Fs(=)33 b Fx(\014)2232 5107 y Fn(I)2272 5143 y Fx(\026)2331 5107 y Fn(I)2370 5143 y FB(;)c Fs(\001)p Fx(\027)40 b Fs(=)33 b Fx(\014)2766 5107 y Fn(I)2805 5143 y Fx(\026)2864 5107 y Fn(I)2928 5143 y Fv(\000)25 b Fx(\014)3091 5107 y Fn(I)5 b(I)3166 5143 y Fx(\026)3225 5107 y Fn(I)g(I)3301 5143 y FB(.)39 b(W)-8 b(e)257 5264 y(consider)25 b(the)f(\003o)n(ws)h(to)f(depend)h(on)f Fx(\014)6 b FB(,)25 b Fx(\027)6 b FB(,)25 b Fs(\001)p Fx(\014)6 b FB(,)25 b(and)g Fs(\001)p Fx(\027)6 b FB(.)p eop %%Page: 44 45 44 44 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(44)404 573 y(One)25 b(easily)f(checks)h(that)818 794 y Fx(@)p 747 839 199 4 v 747 930 a(@)5 b Fs(\001)p Fx(\014)956 781 y Ft(\002)997 861 y Fx(\032)1047 876 y Fn(I)g(I)1123 861 y Fs(\()p Fr(k)r Fs(\))22 b Fv(\000)h Fx(\032)1433 876 y Fn(I)1473 861 y Fs(\()p Fr(k)r Fs(\))1611 781 y Ft(\003)1652 747 y(\014)1652 807 y(\014)1652 866 y(\014)1686 930 y Fo(\001)p Fn(\014)s Fo(=\001)p Fn(\027)t Fo(=0)2062 861 y Fs(=)2318 794 y Fx(e)2363 758 y Fn(\014)s Fo(\()p Fu(j)p Fj(k)q Fu(j)2517 734 y Fp(2)2551 758 y Fu(\000)p Fn(\026)p Fo(\))p 2176 839 647 4 v 2176 930 a Fs(\()p Fx(e)2259 901 y Fn(\014)s Fo(\()p Fu(j)p Fj(k)p Fu(j)2412 883 y Fp(2)2447 901 y Fu(\000)p Fn(\026)p Fo(\))2598 930 y Fs(+)f(1\))2783 901 y Fo(2)2832 861 y Fv(j)p Fr(k)q Fv(j)2949 820 y Fo(2)2989 861 y Fx(;)815 1080 y(@)p 747 1125 193 4 v 747 1216 a(@)5 b Fs(\001)p Fx(\027)950 1067 y Ft(\002)991 1147 y Fx(\032)1041 1162 y Fn(I)g(I)1117 1147 y Fs(\()p Fr(k)r Fs(\))22 b Fv(\000)h Fx(\032)1427 1162 y Fn(I)1467 1147 y Fs(\()p Fr(k)r Fs(\))1605 1067 y Ft(\003)1646 1033 y(\014)1646 1093 y(\014)1646 1152 y(\014)1680 1216 y Fo(\001)p Fn(\014)s Fo(=\001)p Fn(\027)t Fo(=0)2056 1147 y Fs(=)28 b Fv(\000)2389 1080 y Fx(e)2434 1044 y Fn(\014)s Fo(\()p Fu(j)p Fj(k)q Fu(j)2588 1020 y Fp(2)2622 1044 y Fu(\000)p Fn(\026)p Fo(\))p 2247 1125 647 4 v 2247 1216 a Fs(\()p Fx(e)2330 1188 y Fn(\014)s Fo(\()p Fu(j)p Fj(k)q Fu(j)2484 1169 y Fp(2)2518 1188 y Fu(\000)p Fn(\026)p Fo(\))2669 1216 y Fs(+)22 b(1\))2854 1188 y Fo(2)2903 1147 y Fx(:)374 b FB(\(5.8\))257 1418 y(The)34 b(\002rst)f(partial)g(deri)n(v)n(ati)n(v)o(e)d(is)j(tak)o(en)g (at)g(constant)f Fx(\014)6 b FB(,)35 b Fx(\027)6 b FB(,)36 b(and)d Fs(\001)p Fx(\027)6 b FB(;)38 b(the)33 b(second)g(partial)257 1539 y(deri)n(v)n(ati)n(v)o(e)23 b(is)h(at)h(constant)f Fx(\014)6 b FB(,)24 b Fx(\027)6 b FB(,)26 b(and)e Fs(\001)p Fx(\014)6 b FB(.)31 b(Then)25 b(from)f(\(5.3\))h(and)g(\(5.5\))g(we)g (observ)o(e)f(that)1147 1735 y Fx(@)5 b Fv(P)1272 1750 y Fo(2)p Fn(;)p Fo(2)p 1147 1780 221 4 v 1161 1871 a Fx(@)g Fs(\001)p Fx(\027)1377 1688 y Ft(\014)1377 1748 y(\014)1377 1808 y(\014)1410 1872 y Fo(\001)p Fn(\014)s Fo(=\001)p Fn(\027)t Fo(=0)1787 1803 y Fs(=)28 b Fv(\000)1978 1735 y Fx(@)5 b Fv(J)2102 1750 y Fo(2)p Fn(;)p Fo(2)p 1978 1780 219 4 v 1988 1871 a Fx(@)g Fs(\001)p Fx(\014)2207 1688 y Ft(\014)2207 1748 y(\014)2207 1808 y(\014)2240 1872 y Fo(\001)p Fn(\014)s Fo(=\001)p Fn(\027)t Fo(=0)2589 1803 y Fx(:)688 b FB(\(5.9\))257 2066 y(This)24 b(is)h(an)g(Onsager)g (reciprocity)f(relation)g(and)h(we)g(see)g(that)g(it)f(holds)g(at)h(lo) n(west)e(order)-5 b(.)404 2307 y Fm(Entr)l(opy)19 b(pr)l(oduction)g(r)o (ate)o(.)49 b FB(Recall)21 b(that)f Fv(P)1950 2271 y Fn(I)5 b(I)2054 2307 y Fs(=)28 b Fv(\000P)2312 2271 y Fn(I)2373 2307 y FB(\(equation)21 b(\(2.77\)\))f(and)h Fv(J)3316 2271 y Fn(I)5 b(I)3419 2307 y Fs(=)257 2427 y Fv(\000J)421 2391 y Fn(I)486 2427 y FB(\(equation)24 b(\(2.78\)\),)g(hence)1073 2647 y Fv(E)36 b Fs(=)28 b(\()p Fx(\014)1365 2606 y Fn(I)1426 2647 y Fv(\000)23 b Fx(\014)1587 2606 y Fn(I)5 b(I)1662 2647 y Fs(\))p Fv(P)31 b(\000)22 b Fs(\()p Fx(\014)1998 2606 y Fn(I)2038 2647 y Fx(\026)2097 2606 y Fn(I)2158 2647 y Fv(\000)h Fx(\014)2319 2606 y Fn(I)5 b(I)2394 2647 y Fx(\026)2453 2606 y Fn(I)g(I)2529 2647 y Fs(\))p Fv(J)18 b Fx(:)257 2867 y FB(Using)24 b(the)h(abo)o(v)o(e)f(e)o(xpressions)f(for)i Fv(J)1624 2882 y Fn(k)r(;l)1733 2867 y FB(and)f Fv(P)1970 2882 y Fn(k)r(;l)2055 2867 y FB(,)h(we)g(obtain)305 3087 y Fv(E)358 3102 y Fo(1)p Fn(;)p Fo(1)535 3087 y Fs(=)83 b Fv(E)747 3102 y Fo(1)p Fn(;)p Fo(2)869 3087 y Fs(=)27 b(0)p Fx(;)305 3233 y Fv(E)358 3248 y Fo(2)p Fn(;)p Fo(2)535 3233 y Fs(=)83 b(\()p Fx(\014)793 3192 y Fn(I)855 3233 y Fv(\000)22 b Fx(\014)1015 3192 y Fn(I)5 b(I)1091 3233 y Fs(\))p Fv(P)1198 3248 y Fo(2)p Fn(;)p Fo(2)1314 3233 y Fv(\000)23 b Fs(\()p Fx(\014)1513 3192 y Fn(I)1552 3233 y Fx(\026)1611 3192 y Fn(I)1673 3233 y Fv(\000)g Fx(\014)1834 3192 y Fn(I)5 b(I)1909 3233 y Fx(\026)1968 3192 y Fn(I)g(I)2043 3233 y Fs(\))p Fv(J)2149 3248 y Fo(2)p Fn(;)p Fo(2)535 3429 y Fs(=)83 b(2)p Fx(\031)818 3294 y Ft(Z)874 3519 y Fq(R)922 3500 y Fp(2)p Fe(d)1009 3429 y Fx(d)p Fr(k)27 b Fx(d)p Fr(l)e Fx(\016)t Fs(\()p Fv(j)p Fr(k)q Fv(j)1460 3388 y Fo(2)1522 3429 y Fv(\000)d(j)p Fr(l)p Fv(j)1712 3388 y Fo(2)1752 3429 y Fs(\))p Fv(j)16 b Ft(b)-71 b Fx(w)1888 3444 y Fo(1)1926 3429 y Fs(\(\()p Fv(\000)p Fr(k)s Fx(;)17 b(I)8 b(I)g Fs(\))p Fx(;)17 b Fs(\()p Fr(l)o Fx(;)g(I)8 b Fs(\)\))p Fv(j)2641 3388 y Fo(2)694 3713 y Fv(\002)781 3646 y(f)p Fs(\()p Fx(\014)930 3610 y Fn(I)992 3646 y Fv(\000)23 b Fx(\014)1153 3610 y Fn(I)5 b(I)1228 3646 y Fs(\))p Fv(j)p Fr(k)q Fv(j)1383 3610 y Fo(2)1445 3646 y Fv(\000)22 b Fs(\()p Fx(\014)1643 3610 y Fn(I)1683 3646 y Fx(\026)1742 3610 y Fn(I)1804 3646 y Fv(\000)g Fx(\014)1964 3610 y Fn(I)5 b(I)2039 3646 y Fx(\026)2098 3610 y Fn(I)g(I)2174 3646 y Fs(\))p Fv(gf)p Fx(e)2357 3610 y Fn(\014)2400 3587 y Fe(I)2435 3610 y Fo(\()p Fu(j)p Fj(k)q Fu(j)2546 3587 y Fp(2)2580 3610 y Fu(\000)p Fn(\026)2677 3587 y Fe(I)2712 3610 y Fo(\))2766 3646 y Fv(\000)23 b Fx(e)2911 3610 y Fn(\014)2954 3587 y Fe(I)t(I)3020 3610 y Fo(\()p Fu(j)p Fj(k)q Fu(j)3131 3587 y Fp(2)3165 3610 y Fu(\000)p Fn(\026)3262 3587 y Fe(I)t(I)3329 3610 y Fo(\))3360 3646 y Fv(g)p 781 3691 2629 4 v 1387 3784 a Fs(\()p Fx(e)1470 3755 y Fn(\014)1513 3737 y Fe(I)1549 3755 y Fo(\()p Fu(j)p Fj(k)p Fu(j)1659 3737 y Fp(2)1694 3755 y Fu(\000)p Fn(\026)1791 3737 y Fe(I)1826 3755 y Fo(\))1880 3784 y Fs(+)f(1\)\()p Fx(e)2148 3755 y Fn(\014)2191 3737 y Fe(I)t(I)2257 3755 y Fo(\()p Fu(j)p Fj(k)q Fu(j)2368 3737 y Fp(2)2402 3755 y Fu(\000)p Fn(\026)2499 3737 y Fe(I)t(I)2566 3755 y Fo(\))2620 3784 y Fs(+)g(1\))3420 3713 y Fx(:)257 4011 y FB(The)33 b(numerator)e(of)i (the)f(fraction)g(is)g(of)g(the)g(form)g Fs(\()p Fx(x)2185 3975 y Fn(I)2253 4011 y Fv(\000)c Fx(x)2413 3975 y Fn(I)5 b(I)2489 4011 y Fs(\)\()p Fx(e)2610 3975 y Fn(x)2650 3951 y Fe(I)2717 4011 y Fv(\000)28 b 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5345 y Fu(0)3045 5388 y Fx(;)g(s)p Fs(\))p Fx(a)p Fs(\()p Fr(l)3297 5345 y Fu(0)3320 5388 y Fx(;)g(s)3410 5347 y Fu(0)3433 5388 y Fs(\)])p Fx(:)p eop %%Page: 48 49 48 48 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(48)257 573 y(W)-8 b(e)26 b(e)o(xpand)e(the)g(commutator)g(on)g (the)h(right)f(side)g(and)h(apply)f(the)h(state)g Fx(!)2881 537 y Fo(0)2944 573 y FB(to)g(obtain)731 811 y Fv(J)799 826 y Fo(2)p Fn(;)p Fo(2)921 811 y Fs(=)1024 675 y Ft(Z)1124 702 y Fu(1)1080 901 y Fo(0)1215 811 y Fx(dt)1318 675 y Ft(Z)1373 901 y Fq(R)1421 882 y Fp(2)p Fe(d)1509 811 y Fx(d)p Fr(k)h Fx(d)p Fr(l)42 b Ft(b)-72 b Fx(w)1827 826 y Fo(1)1866 811 y Fs(\(\()p Fv(\000)p Fr(k)s Fx(;)17 b(I)8 b(I)g Fs(\))p Fx(;)17 b Fs(\()p Fr(l)o Fx(;)g(I)8 b Fs(\)\))41 b Ft(b)-71 b Fx(w)2648 826 y Fo(1)2687 811 y Fs(\(\()p Fv(\000)p Fr(l)q Fx(;)17 b(I)8 b Fs(\))p Fx(;)17 b Fs(\()p Fr(k)q Fx(;)g(I)8 b(I)g Fs(\)\))1047 1049 y Fv(\002)1141 938 y Ft(n)1207 1049 y Fx(e)1252 1008 y Fu(\000)p Fn(i)p Fo(\()p Fu(j)p 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1677 y(\0001)1540 1642 y Fx(dt)g(e)1701 1606 y Fn(i\034)8 b(t)1830 1642 y Fs(=)37 b(2)p Fx(\031)t(\016)t Fs(\()p Fx(\034)11 b Fs(\))p FB(,)31 b(one)e(sees)h(that)f(\(5.3\))h(holds.)44 b(It)30 b(is)257 1762 y(useful)25 b(to)f(k)o(eep)h(in)g(mind)e(that)748 2024 y Fx(\032)798 2039 y Fn(I)5 b(I)874 2024 y Fs(\()p Fr(k)r Fs(\))22 b Fv(\000)h Fx(\032)1184 2039 y Fn(I)1224 2024 y Fs(\()p Fr(k)r Fs(\))28 b(=)1711 1956 y Fx(e)1756 1920 y Fn(\014)1799 1897 y Fe(I)1834 1920 y Fo(\()p Fu(j)p Fj(k)q Fu(j)1945 1897 y Fp(2)1979 1920 y Fu(\000)p Fn(\026)2076 1897 y Fe(I)2112 1920 y Fo(\))2165 1956 y Fv(\000)23 b Fx(e)2310 1920 y Fn(\014)2353 1897 y Fe(I)t(I)2420 1920 y Fo(\()p Fu(j)p Fj(k)p Fu(j)2530 1897 y Fp(2)2564 1920 y Fu(\000)p Fn(\026)2661 1897 y Fe(I)t(I)2728 1920 y Fo(\))p 1503 2001 1465 4 v 1503 2017 a Ft(\000)1549 2097 y Fx(e)1594 2069 y Fn(\014)1637 2050 y Fe(I)1672 2069 y Fo(\()p Fu(j)p Fj(k)q Fu(j)1783 2050 y Fp(2)1817 2069 y Fu(\000)p Fn(\026)1914 2050 y Fe(I)1950 2069 y Fo(\))2003 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y(1)p 1799 5161 503 4 v 1799 5252 a(1)h(+)g Fx(e)2013 5224 y Fu(\000)p Fn(\014)s Fo(\()p Fn(r)r Fu(\000)p Fn(\026)p Fo(\))2333 5184 y Fs(+)2431 5049 y Ft(Z)2531 5075 y Fu(1)2487 5274 y Fn(\026)2622 5184 y Fx(f)2681 5143 y Fu(0)2704 5184 y Fs(\()p Fx(r)s Fs(\))3036 5117 y(1)p 2837 5161 448 4 v 2837 5252 a Fx(e)2882 5224 y Fn(\014)s Fo(\()p Fn(r)r Fu(\000)p Fn(\026)p Fo(\))3137 5252 y Fs(+)g(1)3294 5184 y Fx(:)3255 5388 y FB(\(5.22\))p eop %%Page: 49 50 49 49 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(49)257 573 y(Apply)29 b(the)g(change)g(of)g(v)n(ariables)f Fx(t)36 b Fs(=)g Fv(\000)p Fx(\014)6 b Fs(\()p Fx(r)28 b Fv(\000)d Fx(\026)p Fs(\))k FB(and)g Fx(t)36 b Fs(=)f Fx(\014)6 b Fs(\()p Fx(r)28 b Fv(\000)e Fx(\026)p Fs(\))j FB(in)f(the)h(\002rst)g(and)257 693 y(second)c(inte)o(gral)f(on)g(the)h (right)f(side)g(of)h(\(5.22\),)g(repecti)n(v)o(ely)-6 b(.)29 b(Then)24 b(one)h(has)489 880 y Fx(R)q Fs(\()p Fx(\026;)17 b(\014)6 b Fs(\))804 839 y Fu(\000)p Fo(1)925 880 y Fs(=)28 b 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244 4 v 2271 1617 a(e)2316 1588 y Fn(t)2368 1617 y Fs(+)d(1)2547 1549 y(+)g Fx(O)s Fs(\()p Fx(e)2806 1508 y Fu(\000)p Fn(\014)s(\026)2950 1549 y Fs(\))2988 1408 y Ft(\023)3078 1549 y Fx(;)257 1793 y FB(for)27 b(some)f Fx(\030)680 1808 y Fn(t)740 1793 y Fv(2)32 b Fs([)p Fv(\000)p Fx(t=\014)e Fs(+)23 b Fx(\026;)17 b(t=\014)28 b Fs(+)c Fx(\026)p Fs(])i FB(and)h(where)g (the)f(e)o(xponentially)f(small)g(remainder)257 1913 y(term)33 b(comes)f(from)h(remo)o(ving)e(the)i(cutof)n(f)f(function)g Fx(\037)p Fs(\()p Fx(t)42 b Fv(\024)h Fx(\014)6 b(\026)p Fs(\))p FB(.)54 b(Retaining)32 b(the)h(main)257 2034 y(term)25 b(\()p Fx(\014)33 b Fv(!)28 b(1)p FB(\))c(yields)1247 2276 y Fx(R)q Fs(\()p Fx(\026;)17 b(\014)6 b Fs(\))1562 2235 y Fu(\000)p Fo(1)1683 2276 y Fv(\031)28 b Fx(f)11 b Fs(\()p Fx(\026)p Fs(\))21 b(+)2137 2209 y Fx(\031)2196 2173 y Fo(2)p 2111 2253 149 4 v 2111 2344 a Fs(6)p Fx(\014)2221 2316 y Fo(2)2270 2276 y Fx(f)2329 2235 y Fu(00)2371 2276 y Fs(\()p Fx(\026)p Fs(\))257 2506 y FB(and)k(consequently)-6 b(,)1005 2734 y Fx(R)q Fs(\()p Fx(\026;)17 b(\014)6 b Fs(\))27 b Fv(\031)1865 2667 y Fs(1)p 1462 2711 856 4 v 1462 2802 a Fx(f)11 b Fs(\()p Fx(\026)p Fs(\))22 b(+)g Fx(\031)1835 2774 y Fo(2)1874 2802 y Fx(T)1945 2774 y Fo(2)1984 2802 y Fx(f)2043 2774 y Fu(00)2085 2802 y Fs(\()p Fx(\026)p Fs(\))p Fx(=)p Fs(6)2327 2734 y Fx(;)92 b(T)41 b Fv(!)27 b Fs(0)p Fx(:)257 2981 y FB(At)33 b(zero)h(temperature,)i (the)d(resistance)g(has)g(the)h(v)n(alue)e Fx(R)q Fs(\()p Fx(\026;)17 b Fv(1)p Fs(\))43 b(=)g Fx(f)11 b Fs(\()p Fx(\026)p Fs(\))2991 2945 y Fu(\000)p Fo(1)3118 2981 y FB(and)34 b(it)f(in-)257 3101 y(creases)20 b(or)f(decreases)g(with)f (increasing)h Fx(T)33 b FB(according)18 b(to)h(whether)g Fx(f)2688 3065 y Fu(00)2730 3101 y Fs(\()p Fx(\026)p Fs(\))27 b Fx(<)g Fs(0)19 b FB(or)g Fx(f)3224 3065 y Fu(00)3266 3101 y Fs(\()p Fx(\026)p Fs(\))27 b Fx(>)257 3222 y Fs(0)p FB(.)404 3462 y Fz(Ener)o(gy)f(curr)n(ent.)57 b FB(The)25 b(ener)n(gy)g(current)814 3649 y Fv(P)36 b Fs(=)27 b Fx(!)1083 3664 y Fo(stat)1205 3649 y Fs(\()p Fv(\000)p Fx(i)p Fs([)p Fx(H)k Fv(\012)22 b Fs(1)-22 b(l)o Fx(;)17 b(W)d Fs(]\))27 b(=)h Fx(!)2056 3608 y Fo(0)2094 3649 y Fs(\()p Fx(\033)2187 3664 y Fo(+)2247 3649 y Fs(\()p Fv(\000)p Fx(i)p Fs([)p Fx(H)i Fv(\012)23 b Fs(1)-22 b(l)o Fx(;)17 b(W)d Fs(]\)\))257 3836 y FB(has)33 b(been)g(introduced)f(in)g(equations)f(\(2.85\),)k(\(2.77\),)f(see)f (also)f(\(2.82\).)54 b(W)-8 b(e)33 b(set)g Fl(~)42 b Fs(=)g(1)p FB(.)257 3956 y(Using)24 b(the)h(CAR)h(and)e(e)o(xpression)g (\(5.15\))h(for)g Fx(H)8 b FB(,)24 b(one)h(obtains)659 4232 y Fs([)p Fx(H)30 b Fv(\012)23 b Fs(1)-22 b(l)n Fx(;)17 b(W)1086 4247 y Fn(N)1153 4232 y Fs(])84 b(=)1422 4096 y Ft(Z)1539 4232 y Fx(dK)1680 4191 y Fo(\()p Fn(N)7 b Fo(\))1802 4232 y Fx(dL)1919 4191 y Fo(\()p Fn(N)g Fo(\))2098 4107 y Fn(N)2057 4137 y Ft(X)2068 4347 y Fn(j)t Fo(=1)2201 4232 y Fs(\()p Fv(j)p Fr(k)2329 4247 y Fn(j)2365 4232 y Fv(j)2393 4191 y Fo(2)2433 4232 y Fx(\016)2476 4247 y Fn(r)2508 4257 y Fe(j)2540 4247 y Fn(;I)2622 4232 y Fv(\000)22 b(j)p Fr(l)2785 4247 y Fn(j)2821 4232 y Fv(j)2849 4191 y Fo(2)2888 4232 y Fx(\016)2931 4248 y Fn(r)2965 4225 y Ff(0)2963 4270 y Fe(j)2996 4248 y Fn(;I)3055 4232 y Fs(\))1447 4494 y Fv(\002)p Fy(a)1578 4453 y Fu(\003)1619 4494 y Fs(\()p Fx(K)1747 4453 y Fo(\()p Fn(N)7 b Fo(\))1869 4494 y Fs(\))17 b Ft(b)-72 b Fx(w)1977 4509 y Fn(N)2044 4494 y Fs(\()p Fv(\000)p Fx(K)2249 4453 y Fo(\()p Fn(N)7 b Fo(\))2372 4494 y Fx(;)17 b(L)2482 4453 y Fo(\()p Fn(N)7 b Fo(\))2604 4494 y Fs(\))p Fy(a)p Fs(\()p Fx(L)2800 4453 y Fo(\()p Fn(N)g Fo(\))2923 4494 y Fs(\))p Fx(;)257 4681 y FB(and)27 b(it)f(is)g(readily)g(v)o(eri\002ed)g(that)g Fv(P)1480 4696 y Fo(1)p Fn(;)p Fo(1)1605 4681 y Fs(=)31 b Fv(P)1781 4696 y Fo(1)p Fn(;)p Fo(2)1906 4681 y Fs(=)g(0)p FB(,)26 b(and)h(a)f(similar)g(calculation)f(as)i(for)f(the)257 4801 y(particle)f(current)g(sho)n(ws)f(that)1071 5035 y Fv(P)1140 5050 y Fo(2)p Fn(;)p Fo(2)1262 5035 y Fs(=)1366 4900 y Ft(Z)1466 4926 y Fu(1)1421 5125 y Fo(0)1557 5035 y Fx(dt)h(!)1733 4994 y Fo(0)1772 5035 y Fs(\([)p Fx(W)1929 5050 y Fo(1)1968 5035 y Fs(\()p Fx(t)p Fs(\))p Fx(;)17 b Fs([)p Fx(H)30 b Fv(\012)22 b Fs(1)-22 b(l)o Fx(;)17 b(W)2550 5050 y Fo(1)2589 5035 y Fs(]]\))574 b FB(\(5.23\))257 5268 y(is)25 b(gi)n(v)o(en)e(by)i(\(5.5\).)p eop %%Page: 50 51 50 50 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(50)257 573 y FA(Refer)m(ences)257 773 y FB([Af])50 b(J.)39 b(Af)n(\003eck:)60 b Fm(F)l(ields,)42 b(strings)c(and)h (critical)f(phenomena)p FB(,)43 b(\(Les)d(Houches)f(1988\),)472 893 y(Amsterdam)24 b(North)g(Holland,)g(1990)257 1091 y([A)l(CF])50 b(A.)38 b(Y)-11 b(u.)38 b(Aleksee)n(v)-6 b(,)40 b(V)-13 b(.V)38 b(Cheiano)o(v)-6 b(,)41 b(J.)c(Fr)8 b(\250)-41 b(ohlich:)57 b Fm(Univer)o(sality)37 b(of)h(T)-5 b(r)o(ansport)472 1211 y(Pr)l(operties)38 b(in)i(Equilibrium,)i(the)e (Goldstone)e(Theor)l(em,)45 b(and)39 b(Chir)o(al)g(Anomaly)p FB(,)472 1331 y(Phys.)24 b(Re)n(v)-6 b(.)25 b(Letters)f Fz(81)h FB(\(1998\),)f(3503\2263506)257 1529 y([AP])50 b(W)-9 b(.)45 b(Aschbacher)l(,)50 b(C.-A.)45 b(Pillet:)70 b Fm(Non-Equilibrium)43 b(Steady)h(States)g(of)g(the)h(XY)472 1649 y(Chain)p FB(,)25 b(preprint,)f(mp-arc,)h(2002)257 1847 y([ASS])50 b(J.)26 b(E.)g(A)-7 b(vron,)25 b(R.)h(Seiler)l(,)h(B.)f (Simon:)32 b Fm(Char)l(g)o(e)26 b(de\002ciency)-5 b(,)26 b(c)o(har)l(g)o(e)f(tr)o(ansport)f(and)472 1967 y(comparison)f(of)i (dimensions)p FB(,)e(Comm.)h(Math.)g(Phys.)g Fz(159)h FB(\(1994\),)f(no.)h(2,)f(399\226422)257 2165 y([Bee])50 b(C.)64 b(W)-9 b(.)63 b(J.)g(Beenakk)o(er:)108 b Fm(Random-matrix)62 b(theory)g(of)h(quantum)f(tr)o(ansport)p FB(,)472 2285 y(arXi)n(v:cond-mat/9612179v1,)21 b(19)k(Dec.)g(1996)257 2483 y([Ber])50 b(C.)26 b(Ber)n(ge,)g Fm(Principes)d(de)i(Combinatoir)l (e)p FB(,)f(Dunod,)g(P)o(aris,)g(1968)257 2681 y([BES])50 b(J.)29 b(Bellissard,)i(A.)e(v)n(an)g(Elst,)g(H.)h(Schulz-Baldes:)40 b Fm(The)30 b(noncommutative)e(g)o(eom-)472 2801 y(etry)36 b(of)f(the)g(quantum)f(Hall)h(ef)n(fect.)g(T)-9 b(opolo)o(gy)35 b(and)g(physics.)p FB(,)i(J.)e(Math.)g(Phys.)g Fz(35)472 2921 y FB(\(1994\),)25 b(no.)f(10,)g(5373\2265451)257 3119 y([BFS])51 b(V)-13 b(.)25 b(Bach,)g(J.)g(Fr)8 b(\250)-41 b(ohlich,)24 b(I.)h(M.)g(Sigal:)30 b Fm(Return)24 b(to)g(Equilibrium)p FB(,)f(J.)h(Math.)g(Phys.)h Fz(41)472 3239 y FB(\(2000\),)g(No.)f(6,)h (3985\2264060)257 3437 y([BLR])50 b(F)-8 b(.)29 b(Bonetto,)g(J.L.)f (Lebo)n(witz,)g(L.)g(Re)o(y-Bellet:)38 b Fm(F)-10 b(ourier')l(s)27 b(Law:)38 b(A)29 b(Challeng)o(e)f(to)472 3557 y(Theorists)p FB(,)g(Mathematical)g(physics)e(2000,)j(128\226150,)f(Imp.)g(Coll.)g (Press,)h(London,)472 3678 y(2000)257 3875 y([BM])50 b(D.)36 b(D.)g(Botvich,)j(V)-13 b(.)36 b(A.)g(Malishe)n(v:)52 b Fm(Unitary)36 b(equivalence)g(of)g(temper)o(atur)l(e)f(dy-)472 3996 y(namics)d(for)g(ideal)g(and)h(locally)f(perturbed)f(F)-7 b(ermi-gas)p FB(,)34 b(Comm.)e(Math.)g(Phys.)g Fz(91)472 4116 y FB(\(1983\),)25 b(no.)f(3,)h(301\226312)257 4314 y([BR])50 b(O.)28 b(Bratteli,)g(D.)f(Robinson:)35 b Fm(Oper)o(ator)26 b(Alg)o(ebr)o(as)f(and)i(Quantum)g(Statistical)e(Me-)472 4434 y(c)o(hanics)d FB(I,)h(II,)g(T)-7 b(e)o(xts)22 b(and)g(Monographs) g(in)g(Physics,)g(Springer)h(V)-11 b(erlag,)24 b(1987,)e(1997)257 4632 y([Ca])50 b(H.)34 b(B.)h(Callen:)49 b Fm(Thermodynamics)33 b(and)h(intr)l(oduction)d(to)j(thermostatistics)p FB(,)f(John)472 4752 y(W)l(ile)o(y)24 b(&)g(Sons,)h(second)g(edition,)e(1985)257 4950 y([CFKS])51 b(H.L.)24 b(Cycon,)g(R.G.)h(Froese,)g(W)-9 b(.)24 b(Kirsch,)g(B.)g(Simon:)30 b Fm(Sc)o(hr)2739 4951 y(\250)2731 4950 y(oding)o(er)22 b(Oper)o(ator)o(s)p FB(,)472 5070 y(T)-7 b(e)o(xts)24 b(and)h(Monographs)e(in)i(Physics,)e (Springer)i(V)-11 b(erlag,)25 b(1987)257 5268 y([CNP])50 b(C.)39 b(Connaughton,)i(A.)d(C.)h(Ne)n(well,)i(Y)-13 b(.)38 b(Pomeau:)58 b Fm(Non-stationary)36 b(spectr)o(a)i(of)472 5388 y(local)24 b(wave)i(turb)n(ulence)p FB(,)e(preprint)g(\(2003\))p eop %%Page: 51 52 51 51 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(51)257 573 y([CPW])50 b(A.)23 b(M.)e(Chang,)i(L.)f(N.)h(Pfeif)n (fer)l(,)g(K.)f(W)-9 b(.)22 b(W)-8 b(est:)29 b Fm(Observation)21 b(of)h(Chir)o(al)f(Lutting)o(er)472 693 y(Behaviour)f(in)g(Electr)l(on) f(T)-5 b(unneling)19 b(into)h(F)-5 b(r)o(actional)17 b(Quantum)j(Hall)f(Edg)o(es)p FB(,)i(Phys.)472 814 y(Re)n(v)-6 b(.)24 b(Letters,)h Fz(77)f FB(\(1996\),)h(no.)f(12,)h(2538\2262541)257 1014 y([CWCPW])51 b(A.)41 b(M.)f(Chang,)45 b(M.)40 b(K.)h(W)-5 b(u,)45 b(C.)c(C.)g(Chi,)k(L.)c(N.)g(Pfeif)n(fer)l(,)k(K.)c(W)-9 b(.)41 b(W)-8 b(est:)472 1135 y Fm(Plateau)24 b(Behaviour)g(in)g(the)h (Chir)o(al)e(Lutting)o(er)h(Liquid)g(Exponent)p FB(,)g(Phys.)g(Re)n(v) -6 b(.)24 b(Let-)472 1255 y(ters,)h Fz(86)f FB(\(2001\),)h(no.)f(1,)h (143\226146)257 1456 y([DFG])50 b(Dirren,)27 b(S.,)g(ETH)f(diploma)f (thesis)g(winter)i(1998/99,)e(chapter)i(5;)f(\(written)g(under)472 1576 y(the)f(supervision)e(of)i(J.)g(Fr)8 b(\250)-41 b(ohlich)24 b(and)h(G.M.)f(Graf\))257 1777 y([EPR])50 b(J.-P)-11 b(.)28 b(Eckmann,)f(C.-A.)h(Pillet,)g(L.)f(Re)o(y-Bellet:)36 b Fm(Entr)l(opy)26 b(pr)l(oduction)f(in)i(nonlin-)472 1897 y(ear)-11 b(,)25 b(thermally)e(driven)i(Hamiltonian)d(systems)p FB(,)i(J.)h(Stat.)f(Phys.)g Fz(95)h FB(\(1999\),)f(no.)h(1-2,)472 2017 y(305\226331)257 2218 y([FMSU])50 b(J.)29 b(Fr)8 b(\250)-41 b(ohlich,)31 b(M.)e(Merkli,)h(S.)h(Schw)o(arz,)h(D.)d (Ueltschi:)40 b Fm(Statistical)27 b(Mec)o(hanics)472 2338 y(of)e(Thermodynamic)f(Pr)l(ocesses)p FB(,)g(submitted)f(to)h(W)-8 b(orld)24 b(Scienti\002c)i(Press)257 2539 y([FP1])50 b(J.)21 b(Fr)8 b(\250)-41 b(ohlich,)21 b(B.)h(Pedrini:)29 b Fm(Ne)o(w)21 b(applications)e(of)i(the)g(c)o(hir)o(al)e(anomaly)p FB(,)i(Mathemat-)472 2659 y(ical)k(physics)e(2000,)h(9\22647,)g(Imp.)h (Coll.)f(Press,)h(London,)f(2000.)257 2860 y([FP2])50 b(J.)55 b(Fr)8 b(\250)-41 b(ohlich,)63 b(B.)56 b(Pedrini:)92 b Fm(Axions,)62 b(quantum)55 b(mec)o(hanical)f(pumping)o(,)62 b(and)472 2980 y(prime)o(val)29 b(ma)o(gnetic)h(\002elds)p FB(,)h(in)f(A.)h(Cappelli)f(and)g(G)h(Mussardo)e(\(eds.\),)j (Statistical)472 3101 y(Field)25 b(Theories,)f(289\226303,)g(Kluwer)g (Academic)h(Publishers,)f(2002)257 3302 y([FST])50 b(J.)38 b(Fr)8 b(\250)-41 b(ohlich,)40 b(U.)e(M.)f(Studer)l(,)k(E.)d(Thiran:)56 b Fm(Quantum)37 b(theory)g(of)g(lar)l(g)o(e)h(systems)472 3422 y(of)33 b(non-r)l(elativistic)e(matter)p FB(,)k(G)6 b(\264)-39 b(eom)6 b(\264)-39 b(etries)32 b(\003uctuantes)h(en)g(m)6 b(\264)-39 b(ecanique)33 b(statistique)472 3542 y(et)38 b(en)g(th)6 b(\264)-39 b(eorie)37 b(des)h(champs)f(\(Les)h(Houches,)j (1994\),)f(771\226912,)g(North-Holland,)472 3663 y(Amsterdam,)24 b(1996.)257 3863 y([Ha])50 b(R.)g(Haag:)80 b Fm(Local)49 b(Quantum)g(Physics)p FB(,)55 b(T)-7 b(e)o(xts)48 b(and)i(Monographs)e (in)h(Physics,)472 3984 y(Springer)25 b(V)-11 b(erlag,)25 b(1992)257 4184 y([He])50 b(Hepp,)34 b(K.:)47 b Fm(Rigor)l(ous)31 b(r)l(esults)h(on)h(the)g Fx(s)28 b Fv(\000)g Fx(d)33 b Fm(model)g(of)f(the)h(K)l(ondo)g(ef)n(fect.)g FB(Solid)472 4305 y(State)25 b(Communications,)e Fz(8)i FB(\(1970\),)f(2087\2262090) 257 4505 y([JP1])49 b(V)-13 b(.)34 b(Jak)s(\013)-36 b(si)6 b(\264)-39 b(c,)35 b(C.-A.)e(Pillet:)47 b Fm(On)33 b(entr)l(opy)f(pr)l (oduction)f(in)i(quantum)f(statistical)f(me-)472 4626 y(c)o(hanics)p FB(,)24 b(Comm.)g(Math.)g(Phys.)g Fz(217)g FB(\(2001\),)h(no.)f(2,)h(285\226293)257 4826 y([JP2])49 b(V)-13 b(.)21 b(Jak)s(\013)-36 b(si)6 b(\264)-39 b(c,)21 b(C.-A.)f(Pillet:)28 b Fm(Non-equilibrium)19 b(steady)h(states)f(of)h (\002nite)g(quantum)f(stys-)472 4947 y(tems)25 b(coupled)g(to)f (thermal)g(r)l(eservoir)o(s)p FB(,)h(Comm.)f(Math.)h(Phys.,)f Fz(226)h FB(\(2002\),)g(No.)g(1,)472 5067 y(131\226162)257 5268 y([La])50 b(R.)21 b(B.)h(Laughlin:)27 b Fm(Quantized)20 b(Hall)g(Conductivity)g(in)h(T)-7 b(wo)21 b(Dimensions)p FB(,)f(Phys.)h(Re)n(v)-6 b(.)472 5388 y Fz(B23)25 b FB(\(1981\))g (5632\2265633)p eop %%Page: 52 53 52 52 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(52)257 573 y([LS])50 b(L.)h(S.)h(Le)n(vito)o(v)-6 b(,)56 b(A.)c(V)-13 b(.)52 b(Shyto)o(v:)82 b Fm(T)-5 b(unneling)51 b(in)g(a)h(bosonized)e(F)-7 b(ermi)51 b(liquid)p FB(,)472 693 y(arXi)n(v:cond-mat/9510006,)22 b(1995,)i(release)h(of)g (September)g(2002)257 897 y([LSH])50 b(L.)23 b(S.)g(Le)n(vito)o(v)-6 b(,)21 b(A.)i(V)-13 b(.)24 b(Shyto)o(v)-6 b(,)21 b(B.)j(I.)f(Halperin:) 29 b Fm(Ef)n(fective)23 b(action)g(of)f(a)h(compr)l(ess-)472 1017 y(ible)34 b(QH)g(state)g(edg)o(e:)50 b(application)33 b(to)h(tunneling)p FB(,)h(arXi)n(v:cond-mat/0005016v2,)472 1137 y(July)24 b(2001)257 1341 y([Ma])49 b(G.)25 b(D.)g(Mahan:)30 b Fm(Many)25 b(P)-8 b(article)24 b(Physics)p FB(,)g(Plenum)g(Press,)h (Ne)n(w)g(Y)-11 b(ork,)24 b(1981)257 1544 y([MO])49 b(T)-7 b(.)46 b(Matsui,)51 b(Y)-13 b(.)46 b(Ogata:)74 b Fm(V)-11 b(ariational)44 b(principle)i(for)f(non-equilibrium)g(steady)472 1665 y(states)24 b(of)g(the)h Fx(X)8 b(X)33 b Fm(model)p FB(,)24 b(preprint)g(\(2003\))257 1868 y([Rob])50 b(D.)24 b(Robinson:)30 b Fm(Return)24 b(to)h(Equilibrium)p FB(,)d(Comm.)i (Math.)h(Phys.)f Fz(31)g FB(\(1973\),)h(171\226)472 1988 y(189)257 2192 y([RSII,)i(III])49 b(Reed,)26 b(M.,)e(Simon,)h(B.:)31 b Fm(Methods)24 b(of)h(Modern)g(Mathematical)e(Physics)p FB(,)i(V)-13 b(ol.)472 2312 y(II)34 b(\(F)o(ourier)g(Analysis,)h (Self-Adjointness\),)g(III)f(\(Scattering)g(Theory\),)i(Academic)472 2433 y(Press,)25 b(Ne)n(w)g(Y)-11 b(ork)24 b(1975,)g(1979)257 2636 y([Ru1])50 b(Ruelle,)29 b(D.:)39 b Fm(Natur)o(al)27 b(nonequilibrium)g(states)g(in)i(quantum)e(statistical)g(mec)o(han-)472 2756 y(ics)p FB(,)e(J.)f(Stat.)h(Phys.)f Fz(98)h FB(\(2000\),)f(no.)g (1-2,)h(57\22675)257 2960 y([Ru2])50 b(Ruelle,)38 b(D.:)52 b Fm(Entr)l(opy)35 b(pr)l(oduction)e(in)j(quantum)e(spin)h(systems)p FB(,)j(Comm.)d(Math.)472 3080 y(Phys.)24 b Fz(224)h FB(\(2001\),)f(no.) g(1,)h(3\22616.)257 3284 y([Ru3])50 b(Ruelle,)38 b(D.:)52 b Fm(Statistical)33 b(Mec)o(hanics.)i(Rigor)l(ous)e(r)l(esults)p FB(,)k(Reprint)f(of)g(the)f(1989)472 3404 y(edition.)22 b(W)-8 b(orld)22 b(Scienti\002c)h(Publishing)e(Co.,)i(Inc.,)h(Ri)n(v)o (er)e(Edge,)h(NJ;)f(Imperial)g(Col-)472 3524 y(le)o(ge)j(Press,)g (London,)e(1999)257 3728 y([SGJE])49 b(L.)26 b(Saminadayar)l(,)g(D.)g (C.)g(Glattli,)f(Y)-13 b(.)26 b(Jin,)f(B.)h(Etienne:)32 b Fm(Observation)24 b(of)i(the)f Fx(e=)p Fs(3)472 3848 y Fm(F)-5 b(r)o(actionally)33 b(Char)l(g)o(ed)i(Laughlin)f (Quasiparticle)p FB(,)i(Phys.)e(Re)n(v)-6 b(.)35 b(Lett.)g Fz(79)g FB(\(1997\),)472 3968 y(2526\2262529)257 4172 y([SR1])50 b(M.)31 b(Sigrist,)h(T)-7 b(.)31 b(M.)f(Rice:)44 b Fm(P)-8 b(ar)o(ama)o(gnetic)29 b(Ef)n(fect)i(in)g(High)g Fx(T)2759 4187 y Fn(c)2825 4172 y Fm(Super)l(conductor)o(s)472 4292 y(\226)f(A)g(Hint)f(for)h Fx(d)p Fm(-W)-9 b(ave)30 b(Super)l(conductivity)p FB(,)g(Journal)f(of)h(The)g(Physical)f (Society)h(of)472 4413 y(Japan,)25 b Fz(61)f FB(\(1992\),)h(no.)f(12,)h (December)l(,)g(4283\2264286)257 4616 y([SR2])50 b(M.)38 b(Sigrist,)k(T)-7 b(.)38 b(M.)g(Rice:)59 b Fm(Unusual)37 b(par)o(ama)o(gnetic)g(phenomena)h(in)g(gr)o(anular)472 4736 y(high-temper)o(atur)l(e)44 b(super)l(conductor)o(s)f(\226)j(A)f (consequence)g(of)g Fx(d)p Fm(-wave)h(pairing?)p FB(,)472 4857 y(Re)n(v)-6 b(.)24 b(Mod.)g(Phys.,)g Fz(67)h FB(\(1995\),)f(no.2,) h(April,)f(503\226513)257 5060 y([TKNN])49 b(D.)22 b(J.)g(Thouless,)f (M.)g(K)m(ohmoto,)g(M.)g(P)-11 b(.)23 b(Nightingale,)d(M.)i(den)f (Nijs:)29 b Fm(Quantized)472 5181 y(Hall)20 b(Conductance)h(in)f(a)h(T) -7 b(wo-Dimensional)19 b(P)-8 b(eriodic)20 b(P)-8 b(otential)p FB(,)20 b(Phys.)g(Re)n(v)-6 b(.)21 b(Lett.)472 5301 y Fz(49)k FB(\(1982\),)f(405\226408)p eop %%Page: 53 54 53 53 bop 257 266 a FC(FMU)25 b(April)f(10,)h(2003)2313 b FB(53)257 573 y([vW])49 b(B.)26 b(J.)e(v)n(an)h(W)-8 b(ees,)25 b(H.)g(v)n(an)f(Houten,)h(C.)g(W)-9 b(.)25 b(J.)g(Beenakk)o(er)l(,)h(J.)e(G.)h(W)l(illiamson,)e(L.)i(P)-11 b(.)472 693 y(K)m(ouwenho)o(v)o(en,)21 b(D.)j(v)n(an)e(der)i(Marel:)30 b Fm(Quantized)22 b(Conductance)h(of)g(P)-8 b(oint)22 b(Contacts)472 814 y(in)j(a)f(two-dimensional)f(Electr)l(on)h(Gas)p FB(,)g(Phys.)g(Re)n(v)-6 b(.)25 b(Lett.)f Fz(60)g FB(\(1988\),)h (848\226850)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0304150253571--