Content-Type: multipart/mixed; boundary="-------------0211140457317" This is a multi-part message in MIME format. ---------------0211140457317 Content-Type: text/plain; name="02-459.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-459.keywords" Nonequilibrium satistical mechanics, entropy production, stationary state, XY-Model ---------------0211140457317 Content-Type: application/postscript; name="EntropyXY_us.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="EntropyXY_us.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: EntropyXY.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Roman Times-Bold Times-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o EntropyXY_us.ps -t letter EntropyXY %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.14:1310 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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y Fp(1)p Fz(;)578 b(x)28 b Fv(=)g(1)p Fz(;)1533 1302 y(\033)1592 1251 y FF(\()p Fr(x)p FF(\))1588 1327 y(3)1707 1302 y Fs(\001)17 b(\001)g(\001)e Fz(\033)1899 1251 y FF(\(0\))1895 1327 y(3)1994 1302 y Fz(;)173 b(x)28 b(<)g Fv(1)p Fz(:)0 1505 y FA(A)d(simple)e(calculation)i(sho)n(ws)e (that)h(the)h(elements)f(of)h Fz(T)14 b Fq(S)2083 1520 y Fy(\000)2166 1505 y FA(de\002ned)25 b(by)649 1724 y Fz(a)700 1739 y Fr(x)772 1724 y Fs(\021)j Fz(T)14 b(S)1014 1683 y FF(\()p Fr(x)p FF(\))1112 1724 y Fv(\()p Fz(\033)1209 1674 y FF(\()p Fr(x)p FF(\))1205 1749 y(1)1330 1724 y Fs(\000)23 b Fz(i\033)1522 1674 y FF(\()p Fr(x)p FF(\))1518 1749 y(2)1621 1724 y Fv(\))p Fz(=)p Fv(2)p Fz(;)215 b(a)2050 1683 y Fy(\003)2050 1749 y Fr(x)2122 1724 y Fv(=)28 b Fz(T)14 b(S)2363 1683 y FF(\()p Fr(x)p FF(\))2461 1724 y Fv(\()p Fz(\033)2558 1674 y FF(\()p Fr(x)p FF(\))2554 1749 y(1)2679 1724 y Fv(+)22 b Fz(i\033)2869 1674 y FF(\()p Fr(x)p FF(\))2865 1749 y(2)2968 1724 y Fv(\))p Fz(=)p Fv(2)p Fz(;)408 b FA(\(3.15\))0 1934 y(are)29 b(fermionic)e (annihilation)f(and)i(creation)g(operators:)36 b(The)o(y)28 b(satisfy)f(the)g(canonical)h(anticommutation)0 2054 y(relations)c(\(CAR\))736 2263 y Fs(f)p Fz(a)837 2278 y Fr(x)881 2263 y Fz(;)17 b(a)976 2278 y Fr(y)1017 2263 y Fs(g)28 b Fv(=)f(0)p Fz(;)216 b Fs(f)p Fz(a)1591 2222 y Fy(\003)1591 2288 y Fr(x)1635 2263 y Fz(;)17 b(a)1730 2222 y Fy(\003)1730 2288 y Fr(y)1771 2263 y Fs(g)28 b Fv(=)f(0)p Fz(;)216 b Fs(f)p Fz(a)2345 2278 y Fr(x)2389 2263 y Fz(;)17 b(a)2484 2222 y Fy(\003)2484 2288 y Fr(y)2525 2263 y Fs(g)28 b Fv(=)f Fz(\016)2749 2278 y Fr(x;y)2850 2263 y Fz(;)p Black 662 w FA(\(3.16\))p Black 0 2473 a(where)36 b Fs(f)p Fz(A;)17 b(B)5 b Fs(g)48 b(\021)h Fz(AB)35 b Fv(+)30 b Fz(B)5 b(A)p FA(.)65 b(W)-8 b(e)36 b(denote)f(by)h Fq(F)g FA(the)g Fz(C)2214 2437 y Fy(\003)2253 2473 y FA(-)g(subalgebra)g(of)g Fq(A)g FA(generated)g(by)g(these)0 2593 y(annihilation)23 b(and)i(creation)f(operators,)h(and)g(we)g (remark)g(that)1542 2802 y Fq(F)j Fs(\032)g Fq(S)1819 2817 y FF(+)1900 2802 y Fv(+)22 b Fz(T)14 b Fq(S)2152 2817 y Fy(\000)2210 2802 y Fz(:)1302 b FA(\(3.17\))0 3012 y(Extending)23 b(the)i Fs(\003)p FA(-)g(automorphism)e Fz(\022)28 b FA(to)c Fq(A)h FA(by)g(setting)1680 3221 y Fz(\022)s Fv(\()p Fz(T)14 b Fv(\))27 b Fs(\021)h Fz(T)8 b(;)0 3430 y FA(yields)25 b(the)g(decomposition)e Fq(F)29 b Fv(=)g Fq(F)1279 3445 y FF(+)1361 3430 y Fv(+)22 b Fq(F)1520 3445 y Fy(\000)1579 3430 y FA(.)33 b(Moreo)o(v)o(er)l(,)25 b(from)g(Equ.)32 b(\(3.15\),)26 b(we)g(obtain)e(that)h Fz(a)3445 3445 y Fr(x)3515 3430 y FA(and)g Fz(a)3735 3394 y Fy(\003)3735 3455 y Fr(x)0 3551 y FA(are)h(odd.)k(Note)24 b(that)h(the)f(relations)g(\(3.15\))h(are)h(easily)e(in)l(v)o(erted)g (to)g(gi)n(v)o(e)227 3770 y Fz(\033)286 3719 y FF(\()p Fr(x)p FF(\))282 3795 y(1)413 3770 y Fv(=)j Fz(T)14 b(S)653 3729 y FF(\()p Fr(x)p FF(\))752 3770 y Fv(\()p Fz(a)841 3785 y Fr(x)907 3770 y Fv(+)22 b Fz(a)1056 3729 y Fy(\003)1056 3795 y Fr(x)1100 3770 y Fv(\))p Fz(;)216 b(\033)1440 3719 y FF(\()p Fr(x)p FF(\))1436 3795 y(2)1566 3770 y Fv(=)28 b Fz(i)17 b(T)d(S)1857 3729 y FF(\()p Fr(x)p FF(\))1955 3770 y Fv(\()p Fz(a)2044 3785 y Fr(x)2110 3770 y Fs(\000)23 b Fz(a)2261 3729 y Fy(\003)2261 3795 y Fr(x)2305 3770 y Fv(\))p Fz(;)216 b(\033)2645 3719 y FF(\()p Fr(x)p FF(\))2641 3795 y(3)2771 3770 y Fv(=)28 b(2)p Fz(a)2975 3729 y Fy(\003)2975 3795 y Fr(x)3019 3770 y Fz(a)3070 3785 y Fr(x)3136 3770 y Fs(\000)23 b Fv(1)p Fz(;)227 b FA(\(3.18\))0 3979 y(from)25 b(which)f(we)h(conclude) g(that)1553 4100 y Fq(S)i Fs(\032)i Fq(F)1830 4115 y FF(+)1911 4100 y Fv(+)22 b Fz(T)14 b Fq(F)2141 4115 y Fy(\000)2199 4100 y Fz(:)1313 b FA(\(3.19\))0 4270 y(The)25 b(tw)o(o)f(inclusions)f(\(3.17\))i(and)g(\(3.19\))f(\002nally)h(yield) 1327 4479 y Fq(S)1410 4494 y FF(+)1496 4479 y Fv(=)i Fq(F)1660 4494 y FF(+)1719 4479 y Fz(;)216 b Fq(S)2045 4494 y Fy(\000)2132 4479 y Fv(=)27 b Fz(T)14 b Fq(F)2367 4494 y Fy(\000)2426 4479 y Fz(:)0 4688 y FA(In)37 b(particular)l(,)i (we)e(ha)n(v)o(e)f Fz(\036)p Fv(\()p Fz(X)8 b Fv(\))49 b Fs(2)h Fq(F)1385 4703 y FF(+)1444 4688 y FA(,)40 b(and)c(a)h(simple)f (calculation)f(leads)i(to)f(the)g(follo)n(wing)f(e)o(xplicit)0 4808 y(formulae)306 5112 y Fz(\036)p Fv(\()p Fz(X)8 b Fv(\))27 b(=)660 4908 y Fo(8)660 4998 y(<)660 5177 y(:)790 4991 y Fs(\000)877 4952 y FF(1)p 877 4968 36 4 v 877 5025 a(2)923 4991 y Fz(\025)p Fv(\(2)p Fz(a)1118 4955 y Fy(\003)1118 5016 y Fr(x)1161 4991 y Fz(a)1212 5006 y Fr(x)1279 4991 y Fs(\000)22 b Fv(1\))p Fz(;)1260 b(X)35 b Fv(=)28 b Fs(f)p Fz(x)p Fs(g)p Fz(;)800 5072 y FF(1)p 800 5088 V 800 5146 a(2)845 5111 y Fs(f)p Fz(a)946 5075 y Fy(\003)946 5136 y Fr(x)990 5111 y Fz(a)1041 5126 y Fr(x)p FF(+1)1197 5111 y Fv(+)22 b Fz(a)1346 5075 y Fy(\003)1346 5136 y Fr(x)p FF(+1)1481 5111 y Fz(a)1532 5126 y Fr(x)1598 5111 y Fv(+)g Fz(\015)5 b Fv(\()p Fz(a)1841 5075 y Fy(\003)1841 5136 y Fr(x)1885 5111 y Fz(a)1936 5075 y Fy(\003)1936 5136 y Fr(x)p FF(+1)2092 5111 y Fv(+)22 b Fz(a)2241 5126 y Fr(x)p FF(+1)2376 5111 y Fz(a)2427 5126 y Fr(x)2471 5111 y Fv(\))p Fs(g)p Fz(;)166 b(X)35 b Fv(=)28 b Fs(f)p Fz(x;)17 b(x)22 b Fv(+)g(1)p Fs(g)p Fz(;)790 5232 y Fv(0)p Fz(;)1886 b FA(otherwise)o Fz(:)p Black Black eop end %%Page: 12 12 TeXDict begin 12 11 bop Black 0 100 a FA(12)2327 b Fx(W)-9 b(.H.)25 b(Aschbacher)g(and)g(C.-A.)g(Pillet)p Black 0 407 a Fd(3.4)119 b(The)30 b(Bogoliubo)o(v)g(A)-6 b(utomor)o(phism)0 643 y FA(Let)25 b Fq(h)i Fs(\021)h Fz(`)383 607 y FF(2)423 643 y Fv(\()p Ft(Z)p Fv(\))19 b Fs(\012)k Ft(C)753 607 y FF(2)826 643 y Fs(')28 b Fz(`)972 607 y FF(2)1011 643 y Fv(\()p Ft(Z)p Fv(\))20 b Fs(\010)j Fz(`)1317 607 y FF(2)1356 643 y Fv(\()p Ft(Z)p Fv(\))f FA(and)j(de\002ne)g(the)g (linear)g(map)627 875 y Fq(h)i Fs(3)h Fz(f)39 b Fs(\021)28 b Fv(\()p Fz(f)1078 890 y FF(+)1137 875 y Fz(;)17 b(f)1229 890 y Fy(\000)1288 875 y Fv(\))27 b Fs(7!)h Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\))27 b Fs(\021)1827 780 y Fo(X)1834 992 y Fr(x)p Fy(2)p Fn(Z)1988 875 y Fv(\()p Fz(f)2074 890 y FF(+)2133 875 y Fv(\()p Fz(x)p Fv(\))17 b Fz(a)2332 834 y Fy(\003)2332 900 y Fr(x)2398 875 y Fv(+)22 b Fz(f)2544 890 y Fy(\000)2603 875 y Fv(\()p Fz(x)p Fv(\))17 b Fz(a)2802 890 y Fr(x)2846 875 y Fv(\))28 b Fs(2)g Fq(F)3067 890 y Fy(\000)3126 875 y Fz(:)0 1178 y FA(It)d(follo)n(ws)e(from)i(the)f (CAR)i(\(3.16\))f(that)f(this)g(sum)g(con)l(v)o(er)n(ges)g(in)h(the)f Fz(C)2549 1142 y Fy(\003)2589 1178 y FA(-)h(norm)f(of)h Fq(F)p FA(,)g(and)f(that)1362 1393 y Fs(f)p Fz(B)1491 1352 y Fy(\003)1530 1393 y Fv(\()p Fz(f)11 b Fv(\))p Fz(;)17 b(B)5 b Fv(\()p Fz(g)t Fv(\))p Fs(g)26 b Fv(=)i(\()p Fz(f)5 b(;)17 b(g)t Fv(\))g Fp(1)p Fz(;)1120 b FA(\(3.20\))0 1607 y(where)25 b(we)g(ha)n(v)o(e)g(set)f Fz(B)833 1571 y Fy(\003)873 1607 y Fv(\()p Fz(f)11 b Fv(\))27 b Fs(\021)h Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\))1354 1571 y Fy(\003)1394 1607 y FA(.)30 b(Moreo)o(v)o(er)l(,)1546 1822 y Fz(B)1625 1781 y Fy(\003)1664 1822 y Fv(\()p Fz(f)11 b Fv(\))27 b(=)h Fz(B)5 b Fv(\()p Fz(J)k(f)i Fv(\))p Fz(;)0 2037 y FA(where)25 b Fz(J)34 b FA(is)25 b(the)f(antiunitary)g(in)l(v)n (olution)f(on)h Fq(h)h FA(de\002ned)g(by)1398 2252 y Fz(J)20 b Fv(:)33 b(\()p Fz(f)1618 2267 y FF(+)1677 2252 y Fz(;)17 b(f)1769 2267 y Fy(\000)1828 2252 y Fv(\))28 b Fs(7!)f Fv(\()2088 2226 y(\026)2059 2252 y Fz(f)2107 2267 y Fy(\000)2166 2252 y Fz(;)2239 2226 y Fv(\026)2210 2252 y Fz(f)2258 2267 y FF(+)2317 2252 y Fv(\))p Fz(:)0 2467 y FA(Clearly)-6 b(,)24 b Fq(F)h FA(is)f(the)h Fz(C)737 2431 y Fy(\003)776 2467 y FA(-)g(algebra)g(generated)g(by)f (polynomials)f(in)h Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\))p FA(,)24 b(and,)h(since)f Fz(\022)s Fv(\()p Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\)\))28 b(=)f Fs(\000)p Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\))p FA(,)0 2587 y(the)34 b(e)n(v)o(en)f(part)h Fq(F)624 2602 y FF(+)718 2587 y FA(is)f(generated)i(by)f(e)n(v)o(en)f (polynomials)f(in)i(the)g Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\))p FA(.)58 b(Thus)34 b Fq(F)g FA(is)f(a)i(self-dual)f(CAR)0 2707 y(algebra)25 b(as)g(introduced)f(by)h(Araki)f(in)h([3])g(\(see)g (also)f([5])i(and)e([14]\).)0 2876 y(T)-8 b(o)25 b(a)g(\002nite)f(rank) h(operator)g Fz(k)31 b Fs(\021)1172 2801 y Fo(P)1277 2828 y Fr(n)1277 2905 y(j)t FF(=1)1421 2876 y Fz(f)1469 2891 y Fr(j)1522 2876 y Fv(\()p Fz(g)1607 2891 y Fr(j)1643 2876 y Fz(;)i Fs(\001)17 b Fv(\))24 b FA(on)h Fq(h)p FA(,)g Fz(f)2085 2891 y Fr(j)2121 2876 y Fz(;)17 b(g)2212 2891 y Fr(j)2276 2876 y Fs(2)28 b Fq(h)p FA(,)d(we)g(associate)1237 3184 y Fp(B)p Fv(\()p Fz(k)s Fv(\))i Fs(\021)1630 3059 y Fr(n)1579 3089 y Fo(X)1590 3299 y Fr(j)t FF(=1)1740 3184 y Fz(B)5 b Fv(\()p Fz(f)1905 3199 y Fr(j)1942 3184 y Fv(\))17 b Fz(B)2076 3143 y Fy(\003)2115 3184 y Fv(\()p Fz(g)2200 3199 y Fr(j)2236 3184 y Fv(\))27 b Fs(2)i Fq(F)2457 3199 y FF(+)2515 3184 y Fz(;)997 b FA(\(3.21\))0 3494 y(which)22 b(is)g(easily)g(seen)h(to)f(depend)h(only)f(on)g Fz(k)k FA(and)c(not)g(on)h(its)e(representation)i(in)f(terms)g(of)g Fz(g)3259 3509 y Fr(j)3318 3494 y FA(and)h Fz(f)3533 3509 y Fr(j)3570 3494 y FA(.)30 b(The)0 3614 y(follo)n(wing)23 b(properties)h(are)i(immediate)d(consequences)i(of)g(this)f (de\002nition)g(and)g(of)h(the)g(CAR)h(\(3.20\):)1226 3829 y Fp(B)p Fv(\()p Fz(k)1398 3788 y Fy(\003)1437 3829 y Fv(\))83 b(=)g Fp(B)p Fv(\()p Fz(k)s Fv(\))1927 3788 y Fy(\003)1966 3829 y Fz(;)969 3974 y Fp(B)p Fv(\()p Fz(k)25 b Fv(+)d Fz(j)6 b Fv(\()p Fz(k)s Fv(\)\))83 b(=)g(tr\()p Fz(k)s Fv(\))17 b Fp(1)p Fz(;)954 4120 y Fv([)p Fp(B)p Fv(\()p Fz(k)s Fv(\))p Fz(;)g(B)5 b Fv(\()p Fz(f)11 b Fv(\)])82 b(=)h Fz(B)5 b Fv(\(\()p Fz(k)25 b Fs(\000)e Fz(j)6 b Fv(\()p Fz(k)s Fv(\)\))p Fz(f)11 b Fv(\))p Fz(;)934 4265 y Fv([)p Fp(B)p Fv(\()p Fz(k)s Fv(\))p Fz(;)17 b Fp(B)p Fv(\()p Fz(k)1387 4224 y Fy(0)1410 4265 y Fv(\)])83 b(=)g Fp(B)p Fv(\([)p Fz(k)25 b Fs(\000)d Fz(j)6 b Fv(\()p Fz(k)s Fv(\))p Fz(;)17 b(k)2311 4224 y Fy(0)2357 4265 y Fs(\000)22 b Fz(j)6 b Fv(\()p Fz(k)2594 4224 y Fy(0)2617 4265 y Fv(\)]\))p Fz(=)p Fv(2)p Fz(;)0 4480 y FA(where)25 b Fz(j)6 b Fv(\()p Fz(k)s Fv(\))28 b Fs(\021)g Fz(J)9 b(k)694 4444 y Fy(\003)734 4480 y Fz(J)g FA(.)31 b(In)25 b(particular)l(,)f(if)h Fz(k)g Fv(+)d Fz(j)6 b Fv(\()p Fz(k)s Fv(\))28 b(=)f(0)p FA(,)e(one)g(has)1202 4695 y Fv(e)1245 4654 y Fr(t)p Fb(B)p FF(\()p Fr(k)r FF(\))p Fr(=)p FF(2)1514 4695 y Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\))17 b(e)1788 4654 y Fy(\000)p Fr(t)p Fb(B)p FF(\()p Fr(k)r FF(\))p Fr(=)p FF(2)2121 4695 y Fv(=)28 b Fz(B)5 b Fv(\(e)2385 4654 y Fr(tk)2453 4695 y Fz(f)11 b Fv(\))p Fz(:)962 b FA(\(3.22\))0 4958 y(The)25 b(local)f(Hamiltonian)f Fz(H)1002 4973 y FF(\003)1080 4958 y FA(can)i(be)g(e)o(xpressed)g(as)1115 5187 y Fz(H)1196 5202 y FF(\003)1276 5187 y Fv(=)j Fp(B)p Fv(\()p Fz(h)1554 5202 y FF(\003)1607 5187 y Fv(\))p Fz(=)p Fv(2)p Fz(;)215 b(h)2041 5202 y FF(\003)2122 5187 y Fv(=)2237 5092 y Fo(X)2226 5303 y Fr(X)5 b Fy(\032)p FF(\003)2409 5187 y Fz(')p Fv(\()p Fz(X)j Fv(\))p Fz(;)p Black Black eop end %%Page: 13 13 TeXDict begin 13 12 bop Black 0 100 a Fx(Non-Equilibrium)22 b(Steady)j(States)g(of)g(the)g Fz(X)8 b(Y)46 b Fx(Chain)1725 b 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Fz(')p Fv(\()p Fz(X)i Fv(\)\))27 b(=)h(0)p FA(,)23 b(the)g(local)g(dynamics)f Fz(\034)1970 1324 y FF(\003)2047 1309 y FA(e)o(xtends)g(from)h Fq(S)2673 1324 y FF(+)2755 1309 y FA(to)g(a)h(Bogoliubo)o(v)d(automor)n(-)0 1429 y(phism)i(of)i(the)g(self-dual)g(CAR)g(algebra)g Fq(F)1420 1654 y Fz(\034)1473 1613 y Fr(t)1462 1679 y FF(\003)1515 1654 y Fv(\()p Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\)\))27 b(=)h Fz(B)5 b Fv(\(e)2096 1613 y Fr(ith)2186 1624 y Ff(\003)2236 1654 y Fz(f)11 b Fv(\))p Fz(:)0 1880 y FA(The)25 b(limit)e Fv(\003)k Fs(")h Ft(Z)22 b FA(leads)j(to)1449 2006 y Fz(\034)1502 1965 y Fr(t)1532 2006 y Fv(\()p Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\)\))27 b(=)g Fz(B)5 b Fv(\(e)2112 1965 y Fr(ith)2207 2006 y Fz(f)11 b Fv(\))p Fz(;)0 2186 y FA(where)25 b(the)g(translation)e(in)l(v)n(ariant)g(Hamiltonian)g Fz(h)28 b Fs(\021)1944 2112 y Fo(P)2049 2215 y Fr(X)5 b Fy(\032)p Fn(Z)2232 2186 y Fz(')p Fv(\()p Fz(X)j Fv(\))25 b FA(is)f(gi)n(v)o(en,)f(in)h(the)g(F)o(ourier)h(represen-)0 2318 y(tation)f Fq(h)j Fv(=)h Fz(L)501 2282 y FF(2)541 2318 y Fv(\()p Fz(S)645 2282 y FF(1)684 2318 y Fz(;)742 2274 y Fr(d\030)p 738 2295 79 4 v 738 2352 a FF(2)p Fr(\031)826 2318 y Fv(\))22 b Fs(\012)g Ft(C)1051 2282 y FF(2)1097 2318 y FA(,)j(by)f(the)h(formula)1151 2543 y Fz(h)j Fs(\021)g Fv(\(cos)18 b Fz(\030)26 b Fs(\000)d Fz(\025)p Fv(\))f Fs(\012)g Fz(\033)1966 2558 y FF(3)2028 2543 y Fs(\000)h Fz(\015)f Fv(sin)16 b Fz(\030)27 b Fs(\012)22 b Fz(\033)2561 2558 y FF(2)2601 2543 y Fz(:)911 b FA(\(3.23\))0 2769 y(The)25 b(decoupled)f(dynamics)f Fz(\034)1062 2784 y FF(0)1127 2769 y FA(is)h(implemented)f(in)h(a)h(similar)e(w)o(ay)i(in)f (the)g(self-dual)g(CAR)i(algebra.)31 b(The)0 2889 y(corresponding)24 b(Hamiltonian)f Fz(h)1174 2904 y FF(0)1238 2889 y FA(decouples)i (according)f(to)h(the)f(decomposition)895 3115 y Fq(h)j Fv(=)h Fz(`)1119 3073 y FF(2)1158 3115 y Fv(\()p Ft(Z)1265 3130 y Fr(L)1315 3115 y Fv(\))22 b Fs(\012)g Ft(C)1540 3073 y FF(2)1608 3115 y Fs(\010)h Fz(`)1749 3073 y FF(2)1788 3115 y Fv(\()p Ft(Z)1895 3130 y Fj(\003)1951 3115 y Fv(\))f Fs(\012)h Ft(C)2177 3073 y FF(2)2245 3115 y Fs(\010)f Fz(`)2385 3073 y FF(2)2425 3115 y Fv(\()p Ft(Z)2532 3130 y Fr(R)2587 3115 y Fv(\))g Fs(\012)g Ft(C)2812 3073 y FF(2)2858 3115 y Fz(:)0 3340 y FA(It)j(is)f(gi)n(v)o(en)f(by)1292 3466 y Fz(h)1348 3481 y FF(0)1415 3466 y Fs(\021)28 b Fz(h)22 b Fs(\000)h Fz(v)31 b Fv(=)d Fz(h)1936 3481 y Fr(L)2010 3466 y Fs(\010)23 b Fz(h)2166 3481 y Fj(\003)2247 3466 y Fs(\010)g Fz(h)2403 3481 y Fr(R)2461 3466 y Fz(;)1051 b FA(\(3.24\))0 3647 y(where)25 b(the)g(\002nite)g(rank)g(perturbation) e Fz(v)29 b FA(looks)24 b(lik)o(e)987 3872 y Fz(v)32 b Fs(\021)c Fz(')p Fv(\()p Fs(f\000)p Fz(M)33 b Fs(\000)22 b Fv(1)p Fz(;)17 b Fs(\000)p Fz(M)10 b Fs(g)p Fv(\))23 b(+)f Fz(')p Fv(\()p Fs(f)p Fz(M)5 b(;)17 b(M)33 b Fv(+)22 b(1)p Fs(g)p Fv(\))p Fz(:)746 b FA(\(3.25\))0 4230 y Fd(3.5)119 b(Quasi-fr)n(ee)30 b(States)0 4473 y FA(A)36 b(quasi-free)g(state)f(on)h Fq(F)g FA(is)f(a)h(state)f Fz(!)40 b FA(which)35 b(v)n(anishes)f(on)i Fq(F)2346 4488 y Fy(\000)2441 4473 y FA(and)f(satis\002es)h(the)f(W)l(ick)h(e)o (xpansion)0 4594 y(formula)587 4772 y Fz(!)t Fv(\()p Fz(B)5 b Fv(\()p Fz(f)855 4787 y FF(1)894 4772 y Fv(\))17 b Fs(\001)g(\001)g(\001)d Fz(B)5 b Fv(\()p Fz(f)1246 4787 y FF(2)p Fr(n)1328 4772 y Fv(\)\))28 b(=)1535 4677 y Fo(X)1586 4886 y Fr(\031)1696 4772 y Fv(sign)o(\()p Fz(\031)t Fv(\))2075 4647 y Fr(n)2033 4677 y Fo(Y)2032 4889 y Fr(k)r FF(=1)2178 4772 y Fz(!)t Fv(\()p Fz(B)5 b Fv(\()p Fz(f)2446 4787 y Fr(\031)r FF(\(2)p Fr(k)r Fy(\000)p FF(1\))2711 4772 y Fv(\))p Fz(B)g Fv(\()p Fz(f)2914 4787 y Fr(\031)r FF(\(2)p Fr(k)r FF(\))3090 4772 y Fv(\)\))p Fz(;)346 b FA(\(3.26\))0 5042 y(where)25 b(the)g(sum)f(runs)g(o)o(v)o (er)g(all)h(permutations)e Fz(\031)32 b Fs(2)c Fz(S)1912 5057 y FF(2)p Fr(n)1994 5042 y FA(,)d(with)f(signature)g Fv(sign)o(\()p Fz(\031)t Fv(\))p FA(,)h(such)g(that)1263 5268 y Fz(\031)t Fv(\(2)p Fz(k)s Fv(\))p Fz(;)17 b(\031)t Fv(\(2)p Fz(k)24 b Fv(+)e(1\))27 b Fz(>)h(\031)t Fv(\(2)p Fz(k)d Fs(\000)d Fv(1\))p Fz(:)p Black Black eop end %%Page: 14 14 TeXDict begin 14 13 bop Black 0 100 a FA(14)2327 b Fx(W)-9 b(.H.)25 b(Aschbacher)g(and)g(C.-A.)g(Pillet)p Black 0 407 a FA(Such)e(a)g(state)f(is)g(completely)g(characterized)i(by)e (its)g(tw)o(o-point)f(function)h Fz(!)t Fv(\()p Fz(B)2788 371 y Fy(\003)2827 407 y Fv(\()p Fz(f)11 b Fv(\))p Fz(B)5 b Fv(\()p Fz(g)t Fv(\)\))p FA(,)22 b(which)g(in)g(turn)0 527 y(determines)i(a)h(bounded)f(operator)h Fz(T)39 b FA(on)24 b Fq(h)h FA(such)g(that)1364 764 y Fv(\()p Fz(f)5 b(;)17 b(T)d(g)t Fv(\))26 b Fs(\021)i Fz(!)t Fv(\()p Fz(B)1972 723 y Fy(\003)2011 764 y Fv(\()p Fz(f)11 b Fv(\))p Fz(B)5 b Fv(\()p Fz(g)t Fv(\)\))p Fz(:)0 1000 y FA(An)o(y)24 b(self-adjoint)g(operator)h Fz(T)38 b FA(on)25 b Fq(h)g FA(such)f(that)1254 1237 y Fv(0)j Fs(\024)i Fz(T)41 b Fs(\024)28 b Fz(I)8 b(;)216 b(T)36 b Fv(+)22 b Fz(j)6 b Fv(\()p Fz(T)14 b Fv(\))27 b(=)h Fz(I)8 b(;)1013 b FA(\(3.27\))0 1474 y(determines)25 b(in)h(this)f(w)o(ay)h(a)h(unique) e(quasi-free)i(state)e(on)h Fq(F)g FA(\(see)h([3]\).)35 b(Let)26 b Fz(k)j FA(be)d(a)h(self-adjoint)e(operator)0 1594 y(on)c Fq(h)h FA(such)f(that)g Fz(k)13 b Fv(+)d Fz(j)c Fv(\()p Fz(k)s Fv(\))27 b(=)g(0)p FA(,)22 b(and)g Fz(\034)1330 1609 y Fr(k)1394 1594 y FA(the)g(corresponding)e(group)h (of)h(Bogoliubo)o(v)d(automorphisms)g(of)j Fq(F)0 1714 y FA(\()p Fw(i.e)o(.,)h Fz(\034)255 1678 y Fr(t)244 1740 y(k)287 1714 y Fv(\()p Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\)\))27 b(=)h Fz(B)5 b Fv(\(e)868 1678 y Fr(itk)960 1714 y Fz(f)11 b Fv(\))p FA(\).)30 b(Then,)23 b Fv(\(1)15 b(+)g(e)1634 1678 y Fr(\014)s(k)1719 1714 y Fv(\))1757 1678 y Fy(\000)p FF(1)1874 1714 y FA(satis\002es)23 b(Condition)e(\(3.27\))i(and)g(the)g (corresponding)0 1835 y(quasi-free)34 b(state)e(is)h Fv(\()p Fz(\034)827 1850 y Fr(k)870 1835 y Fz(;)17 b(\014)6 b Fv(\))p FA(-)33 b(KMS.)g(It)g(follo)n(ws)e(that)i(the)g(state)g Fz(!)2395 1788 y Fr(M)s(;\014)2526 1799 y Fi(L)2570 1788 y Fr(;\014)2630 1799 y Fi(R)2391 1859 y FF(0)2717 1835 y FA(de\002ned)h(in)e(Equ.)56 b(\(3.13\),)35 b(or)0 1955 y(more)25 b(precisely)f(its)g(restriction)g(to)h Fq(F)1326 1970 y FF(+)1384 1955 y FA(,)g(e)o(xtends)f(to)g(the)h(quasi-free)g (state)g(on)f Fq(F)h FA(determined)f(by)1608 2233 y Fz(T)1665 2248 y FF(0)1732 2233 y Fs(\021)1967 2166 y Fv(1)p 1847 2210 288 4 v 1847 2301 a(1)e(+)g(e)2059 2273 y Fr(k)2096 2282 y Ff(0)2145 2233 y Fz(;)1367 b FA(\(3.28\))0 2501 y(where)1381 2638 y Fz(k)1432 2653 y FF(0)1499 2638 y Fs(\021)28 b Fz(\014)1659 2653 y Fr(L)1728 2638 y Fz(h)1784 2653 y Fr(L)1858 2638 y Fs(\010)23 b Fv(0)f Fs(\010)g Fz(\014)2183 2653 y Fr(R)2258 2638 y Fz(h)2314 2653 y Fr(R)2372 2638 y Fz(:)1140 b FA(\(3.29\))0 3068 y FB(4)143 b(Scattering)34 b(Theory)0 3359 y FA(In)i(this)g(section,)i(we)f(apply) e(Ruelle')-5 b(s)36 b(scattering)g(approach)g(\(see)h([25]\))g(to)f (the)g(construction)f(of)h(non-)0 3479 y(equilibrium)22 b(steady)i(states)f(of)h(the)g Fz(C)1343 3443 y Fy(\003)1382 3479 y FA(-)g(dynamical)f(system)g Fv(\()p Fq(F)p Fz(;)17 b(\034)11 b Fv(\))p FA(.)31 b(Due)24 b(to)f(the)h(f)o(act)g(that)g(the) g(dynamics)0 3600 y(is)37 b(implemented)f(by)h(a)h(group)f(of)g (Bogoliubo)o(v)f(automorphisms,)i(the)f(analysis)f(reduces)i(to)f(a)h (simple)0 3720 y(Hilbert)24 b(space)h(scattering)g(problem.)p Black 0 3968 a Fu(Lemma)g(4.1)p Black 49 w Fw(F)-10 b(or)24 b(any)g Fz(A)k Fs(2)g Fq(F)p Fw(,)d(the)g(norm)f(limit)1343 4204 y Fz(\015)1394 4219 y FF(+)1452 4204 y Fv(\()p Fz(A)p Fv(\))k Fs(\021)71 b Fv(lim)1734 4264 y Fr(t)p Fy(!)p FF(+)p Fy(1)1972 4204 y Fz(\034)2025 4163 y Fy(\000)p Fr(t)2014 4229 y FF(0)2133 4204 y Fs(\016)22 b Fz(\034)2258 4163 y Fr(t)2288 4204 y Fv(\()p Fz(A)p Fv(\))0 4473 y Fw(e)n(xists.)30 b(The)c(M\370ller)e(morphism)f Fz(\015)1220 4488 y FF(+)1304 4473 y Fw(is)h(completely)g(c)o(har)o(acterized)f(by) 1439 4710 y Fz(\015)1490 4725 y FF(+)1549 4710 y Fv(\()p Fz(B)5 b Fv(\()p Fz(f)11 b Fv(\)\))27 b(=)h Fz(B)5 b Fv(\(\012)2157 4725 y Fy(\000)2216 4710 y Fz(f)11 b Fv(\))p Fz(;)0 4946 y Fw(for)24 b Fz(f)39 b Fs(2)28 b Fq(h)p Fw(,)c(wher)l(e)i(the)f(wave)g(oper)o(ator)e Fv(\012)1506 4961 y Fy(\000)1591 4946 y Fw(is)h(given)h(by)1398 5183 y Fv(\012)1468 5198 y Fy(\000)1555 5183 y Fs(\021)j Fv(s)23 b Fs(\000)g Fv(lim)1698 5251 y Fr(t)p Fy(!)p FF(+)p Fy(1)1989 5183 y Fv(e)2032 5142 y Fy(\000)p Fr(ith)2177 5151 y Ff(0)2217 5183 y Fv(e)2260 5142 y Fr(ith)2354 5183 y Fz(:)p Black Black eop end %%Page: 15 15 TeXDict begin 15 14 bop Black 0 100 a Fx(Non-Equilibrium)22 b(Steady)j(States)g(of)g(the)g Fz(X)8 b(Y)46 b Fx(Chain)1725 b FA(15)p Black 0 407 a Fu(Pr)n(oof)o(.)54 b FA(It)33 b(follo)n(ws)e(from)h(Equ.)54 b(\(3.23\))33 b(that)f Fz(h)h FA(has)g(purely)f(absolutely)f(continuous)g(spectrum.)54 b(Since)0 527 y(the)25 b(perturbation)g Fz(v)k FA(is)c(\002nite)g (rank,)h(it)f(follo)n(ws)f(from)h(Kato-Birman)g(theory)g(that)g(the)g (w)o(a)n(v)o(e)h(operator)f Fv(\012)3720 542 y Fy(\000)0 648 y FA(e)o(xists)d(and)g(is)h(complete)f(\()p 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b(in)g(norm)g(to)g Fz(B)5 b Fv(\(\012)1055 1355 y Fy(\000)1115 1340 y Fz(f)11 b Fv(\))p FA(.)35 b(The)27 b(norm)e(con)l(v)o(er)n(gence)i(of)g Fz(\034)2376 1299 y Fy(\000)p Fr(t)2365 1364 y FF(0)2484 1340 y Fs(\016)c Fz(\034)2610 1304 y Fr(t)2640 1340 y Fv(\()p Fz(A)p Fv(\))k FA(e)o(xtends)e(by)h(continuity)f(to)0 1460 y(all)g Fz(A)i Fs(2)h Fq(F)p FA(.)3297 b Fa(\003)0 1635 y FA(Since)31 b(the)f(state)g Fz(!)682 1587 y Fr(M)s(;\014)813 1598 y Fi(L)858 1587 y Fr(;\014)918 1598 y Fi(R)678 1659 y FF(0)1001 1635 y FA(de\002ned)h(in)f(Equ.)47 b(\(3.13\))30 b(is)f(in)l(v)n(ariant)h(under)g(the)g(decoupled)g(dynamics)f Fz(\034)3715 1650 y FF(0)3755 1635 y FA(,)0 1755 y(we)f(ha)n(v)o(e)g Fz(!)422 1708 y Fr(M)s(;\014)553 1719 y Fi(L)597 1708 y Fr(;\014)657 1719 y Fi(R)418 1779 y FF(0)735 1755 y Fs(\016)d Fz(\034)863 1719 y Fr(t)926 1755 y Fv(=)33 b Fz(!)1100 1708 y Fr(M)s(;\014)1231 1719 y Fi(L)1276 1708 y Fr(;\014)1336 1719 y Fi(R)1096 1779 y FF(0)1414 1755 y Fs(\016)24 b Fz(\034)1541 1714 y 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