Content-Type: multipart/mixed; boundary="-------------0506111027508" This is a multi-part message in MIME format. ---------------0506111027508 Content-Type: text/plain; name="05-207.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-207.keywords" nonequilibrium statistical mechanics,NESS, steady-state,entropy production, Kubo formula, Onsager reciprocity relations, free Fermi gas, linear response theory ---------------0506111027508 Content-Type: application/postscript; name="QSM-intro.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="QSM-intro.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: QSM-intro.dvi %%Pages: 66 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Roman CMR8 Times-Bold CMMI10 CMSY7 Times-Italic %%+ CMSY10 MSBM10 CMMI7 CMR10 CMEX10 EUFM10 CMR7 CMR5 CMMI5 CMMI12 CMSY8 %%+ MSAM10 CMSY5 TeX-cmex7 EUFM7 MSBM7 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o QSM-intro.ps -t letter QSM-intro %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.06.11:1658 %%BeginProcSet: texc.pro %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put dup 22 /harpoonupright put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 77 /M put dup 104 /h put dup 106 /j put dup 114 /r put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163B3DA4F9700DE5807F164169FBB 1458C43CD471029C362871D2FB69E0E5E617BC06F3B8621E3528E4B47E731AE1 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 77 /circleplusdisplay put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 114 /radicalbigg put dup 115 /radicalBigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 8E4B40E7988921D551EC0867EBCA44C05657F0DC913E7B3004A5F3E1337B6987 FEBC45F989C8DC6DC0AD577E903F05D0D54208A0AE7F28C734F130C133B48422 BED48639A2B74E4C08F2E710E24A99F347E0F4394CE64EACB549576E89044E52 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Ft(C)1119 1473 y Fs(\003)1158 1504 y Fx(-scattering)19 b(and)h(NESS)70 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(11)739 1692 y Fu(4)82 b(Open)21 b(quantum)g(systems)1826 b(14)863 1795 y Fx(4.1)86 b(De\002nition)53 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) h(.)f(.)124 b(14)863 1898 y(4.2)86 b Ft(C)1119 1868 y Fs(\003)1158 1898 y Fx(-scattering)19 b(for)h(open)f(quantum)f(systems) 38 b(.)j(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(16)863 2001 y(4.3)86 b(The)20 b(\002rst)h(and)f(second)f(la)o(w)i(of) e(thermodynamics)33 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) h(.)f(.)124 b(17)863 2105 y(4.4)86 b(Linear)20 b(response)f(theory)63 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(18)863 2208 y(4.5)86 b(Fermi)20 b(Golden)g(Rule)g(\(FGR\))h(thermodynamics)54 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(21)739 2396 y Fu(5)82 b(Fr)o(ee)20 b(F)n(ermi)g(gas)g(r)o(eser)o(v)o (oir)1784 b(25)863 2499 y Fx(5.1)86 b(General)20 b(description)47 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(25)863 2602 y(5.2)86 b(Examples)62 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)124 b(30)739 2791 y Fu(6)82 b(The)22 b(simple)f(electr)o(onic)e(black-box)g(\(SEBB\))j(model)1004 b(33)863 2894 y Fx(6.1)86 b(The)20 b(model)32 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(33)863 2997 y(6.2)86 b(The)20 b(\003ux)o(es)43 b(.)e(.)g(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(35)863 3100 y(6.3)86 b(The)20 b(equi)n(v)n(alent)f(free)g(Fermi)i(gas)f(.)41 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)124 b(36)863 3203 y(6.4)86 b(Assumptions)80 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(39)739 3392 y Fu(7)82 b(Thermodynamics)21 b(of)f(the)g(SEBB)i(model) 1342 b(42)863 3495 y Fx(7.1)86 b(Non-equilibrium)17 b(steady)j(states) 79 b(.)42 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)124 b(42)863 3598 y(7.2)86 b(The)20 b(Hilbert-Schmidt)f(condition)48 b(.)42 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(43)863 3701 y(7.3)86 b(The)20 b(heat)g(and)g(char)o(ge)e(\003ux)o(es) 63 b(.)41 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(44)863 3804 y(7.4)86 b(Entrop)o(y)19 b(production)51 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f (.)124 b(45)863 3907 y(7.5)86 b(Equilibrium)18 b(correlation)h (functions)24 b(.)41 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(46)863 4011 y(7.6)86 b(Onsager)20 b(relations.)k(K)o(ubo)19 b(formulas.)48 b(.)41 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) h(.)f(.)124 b(48)739 4199 y Fu(8)82 b(FGR)20 b(thermodynamics)g(of)g (the)g(SEBB)j(model)1173 b(49)863 4302 y Fx(8.1)86 b(The)20 b(weak)g(coupling)e(limit)75 b(.)41 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(49)863 4405 y(8.2)86 b(Historical)21 b(digression\227Einstein')-5 b(s)18 b(deri)n(v)n(ation)g(of)i(the)h(Planck)e(la)o(w)82 b(.)41 b(.)h(.)f(.)124 b(51)863 4508 y(8.3)86 b(FGR)22 b(\003ux)o(es,)d(entrop)o(y)g(production)e(and)j(K)o(ubo)f(formulas)43 b(.)f(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(53)863 4612 y(8.4)86 b(From)20 b(microscopic)f(to)h(FGR)h(thermodynamics)59 b(.)42 b(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(55)739 4800 y Fu(9)82 b(A)n(ppendix)2313 b(56)863 4903 y Fx(9.1)86 b(Structural)20 b(theorems)47 b(.)41 b(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)h(.)f(.)124 b(56)863 5006 y(9.2)86 b(The)20 b(Hilbert-Schmidt)f (condition)48 b(.)42 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)124 b(58)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g (non-equilibrium)c(quantum)j(statistical)i(mechanics)939 b Fx(3)291 523 y Fv(1)119 b(Intr)n(oduction)291 713 y Fx(These)17 b(lecture)h(notes)g(are)g(an)g(e)o(xpanded)d(v)o(ersion)i (of)h(the)g(lectures)f(gi)n(v)o(en)g(by)h(the)g(second)f(and)g(the)291 813 y(fourth)j(author)i(in)g(the)h(summer)e(school)h("Open)f(Quantum)g (Systems")i(held)f(in)g(Grenoble,)g(June)291 912 y(16\226July)c(4,)i (2003.)k(W)-7 b(e)21 b(are)f(grateful)f(to)h(St\351phane)f(Attal)h(and) g(Alain)g(Jo)o(ye)g(for)f(their)h(hospitality)291 1012 y(and)f(in)m(vitation)g(to)h(speak.)415 1114 y(The)29 b(lecture)h(notes)f(ha)n(v)o(e)g(their)g(root)g(in)h(the)g(recent)f(re) n(vie)n(w)g(article)g([JP4])h(and)f(our)g(goal)291 1214 y(has)24 b(been)f(to)i(e)o(xtend)e(and)g(complement)f(certain)i(topics) g(co)o(v)o(ered)e(in)i([JP4].)37 b(In)24 b(particular)m(,)f(we)291 1313 y(will)i(discuss)h(the)f(scattering)f(theory)g(of)h (non-equilibrium)c(steady)k(states)h(\(NESS\))f(\(this)g(topic)291 1413 y(has)f(been)f(only)g(quickly)g(re)n(vie)n(wed)g(in)h([JP4]\).)36 b(On)24 b(the)g(other)f(hand,)h(we)g(will)h(not)f(discuss)g(the)291 1513 y(spectral)15 b(theory)g(of)h(NESS)h(which)e(has)h(been)g(co)o(v)o (ered)e(in)i(detail)g(in)g([JP4].)24 b(Although)14 b(the)i(lecture)291 1612 y(notes)25 b(are)h(self-contained,)f(the)g(reader)g(w)o(ould)g (bene\002t)h(from)e(reading)h(them)g(in)h(parallel)f(with)291 1712 y([JP4)o(].)415 1814 y(Concerning)17 b(preliminaries,)h(we)h(will) h(assume)e(that)i(the)e(reader)g(is)i(f)o(amiliar)f(with)g(the)g(mate-) 291 1913 y(rial)g(co)o(v)o(ered)d(in)j(the)g(lecture)f(notes)h([At,)g (Jo,)g(Pi].)25 b(On)19 b(occasion,)f(we)h(will)g(mention)f(or)h(use)g (some)291 2013 y(material)g(co)o(v)o(ered)f(in)j(the)f(lectures)g([D1)o (,)h(Ja].)415 2115 y(As)30 b(in)e([JP4],)j(we)e(will)g(w)o(ork)f(in)h (the)f(mathematical)g(frame)n(w)o(ork)e(of)j(algebraic)e(quantum)291 2215 y(statistical)33 b(mechanics.)60 b(The)32 b(basic)g(notions)g(of)g (this)g(formalism)f(are)h(re)n(vie)n(wed)f(in)i(Section)291 2314 y(3.)41 b(In)25 b(Section)g(4)h(we)g(introduce)d(open)i(quantum)e (systems)j(and)f(describe)g(their)g(basic)h(proper)n(-)291 2414 y(ties.)43 b(Linear)25 b(response)g(theory)g(\(this)h(topic)g(has) g(not)g(been)f(discussed)h(in)g([JP4]\))g(is)h(described)291 2514 y(in)j(Subsection)f(4.4.)55 b(Linear)30 b(response)f(theory)h(of)g (open)f(quantum)g(systems)h(\(K)o(ubo)f(formu-)291 2613 y(las,)e(Onsager)e(relations,)i(Central)e(Limit)h(Theorem\))e(has)i (been)f(studied)g(in)h(the)g(recent)f(papers)291 2713 y([FMU)o(,)c(FMSU,)f(AJPP)q(,)g(JPR2)q(].)415 2815 y(The)i(second)g (part)g(of)g(the)g(lecture)g(notes)g(\(Sections)g(6\2268\))f(is)j(de)n (v)n(oted)d(to)h(an)h(e)o(xample.)30 b(The)291 2914 y(model)20 b(we)i(will)g(discuss)g(is)g(the)f(simplest)h(non-tri)n(vial)d(e)o (xample)h(of)h(the)h(Electronic)e(Black)h(Box)291 3014 y(Model)27 b(studied)h(in)h([AJPP])g(and)e(we)i(will)g(refer)f(to)h(it) g(as)g(the)f Fr(Simple)g(Electr)l(onic)g(Blac)n(k)h(Box)291 3114 y(Model)22 b Fx(\(SEBB\).)h(The)g(SEBB)h(model)d(is)j(to)f(a)g (lar)o(ge)f(e)o(xtent)g(e)o(xactly)g(solv)n(able\227its)h(NESS)g(and) 291 3213 y(entrop)o(y)28 b(production)f(can)j(be)f(e)o(xactly)g (computed)f(and)h(K)o(ubo)g(formulas)g(can)g(be)h(v)o(eri\002ed)f(by) 291 3313 y(an)c(e)o(xplicit)g(computation.)38 b(F)o(or)25 b(reasons)g(of)g(space,)i(ho)n(we)n(v)o(er)m(,)d(we)i(will)g(not)f (discuss)h(tw)o(o)g(im-)291 3413 y(portant)e(topics)i(co)o(v)o(ered)e (in)j([AJPP]\227the)f(stability)g(theory)f(\(which)g(is)i(essentially)f (based)g(on)291 3512 y([AM)o(,)h(BM]\))g(and)f(the)h(proof)e(of)i(the)g (Central)f(Limit)h(Theorem.)43 b(The)27 b(interested)f(reader)g(may)291 3612 y(complement)21 b(Sections)j(6\2268)f(with)h(the)f(original)g (paper)f([AJPP)q(])i(and)f(the)g(recent)g(lecture)h(notes)291 3711 y([JKP].)415 3813 y(Section)30 b(5,)j(in)e(which)e(we)i(discuss)g (statistical)h(mechanics)d(of)h(a)h(free)f(Fermi)g(gas,)j(is)e(the)291 3913 y(bridge)18 b(between)i(the)g(tw)o(o)g(parts)h(of)f(the)g(lecture) f(notes.)291 4124 y Fu(Ackno)o(wledgment.)31 b Fx(The)22 b(research)g(of)g(V)-11 b(.J.)23 b(w)o(as)g(partly)f(supported)f(by)h (NSERC.)h(P)o(art)g(of)f(this)291 4224 y(w)o(ork)h(w)o(as)i(done)e (while)h(Y)-11 b(.P)i(.)25 b(w)o(as)g(a)f(CRM-ISM)h(postdoc)e(at)h (McGill)h(Uni)n(v)o(ersity)e(and)h(Centre)291 4324 y(de)c(Recherches)f (Math\351matiques)g(in)h(Montreal.)291 4617 y Fv(2)119 b(Conceptual)31 b(framew)o(ork)291 4807 y Fx(The)c(concept)g(of)h (reference)f(state)h(will)h(play)f(an)g(important)f(role)g(in)i(our)e (discussion)h(of)g(non-)291 4907 y(equilibrium)18 b(statistical)j (mechanics.)j(T)-7 b(o)20 b(clarify)f(this)h(notion,)f(let)h(us)h (consider)d(\002rst)j(a)g(classical)291 5006 y(dynamical)29 b(system)j(with)f(\002nitely)g(man)o(y)f(de)o(grees)g(of)h(freedom)e (and)i(compact)f(phase)g(space)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 739 232 a Fx(4)1697 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Ft(X)38 b Fq(\032)31 b Fp(R)1002 493 y Fo(n)1047 523 y Fx(.)39 b(The)25 b(normalized)e(Lebesgue)g(measure)h Fn(d)p Ft(x)i Fx(on)e Ft(X)32 b Fx(pro)o(vides)23 b(a)i(physically)f (natural)739 623 y(statistics)g(on)e(the)h(phase)f(space)g(in)h(the)f (sense)h(that)g(initial)g(con\002gurations)d(sampled)i(according)739 722 y(to)17 b(it)h(can)f(be)g(considered)e(typical)i(\(see)g([Ru4]\).) 23 b(Note)17 b(that)h(this)f(has)h(nothing)d(to)i(do)g(with)g(the)h(f)o (act)739 822 y(that)f Fn(d)p Ft(x)g Fx(is)h(in)m(v)n(ariant)d(under)g (the)i(\003o)n(w)f(of)h(the)f(system\227an)o(y)g(measure)f(of)i(the)f (form)g Ft(\032)p Fn(\()p Ft(x)p Fn(\)d)p Ft(x)j Fx(with)739 922 y(a)i(strictly)f(positi)n(v)o(e)g(density)g Ft(\032)h Fx(w)o(ould)f(serv)o(e)g(the)g(same)h(purpose.)j(The)c(situation)g(is)h (completely)739 1021 y(dif)n(ferent)e(if)i(the)g(system)g(has)g (in\002nitely)g(man)o(y)e(de)o(grees)h(of)h(freedom.)k(In)20 b(this)i(case,)f(there)f(is)i(no)739 1121 y(natural)c(replacement)f (for)h(the)h(Lebesgue)f Fn(d)p Ft(x)p Fx(.)26 b(In)18 b(f)o(act,)h(a)h(measure)e(on)g(an)h(in\002nite-dimensional)739 1220 y(phase)27 b(space)h(physically)e(describes)h(a)h(thermodynamic)d (state)j(of)f(the)h(system.)48 b(Suppose)26 b(for)739 1320 y(e)o(xample)d(that)i(the)g(system)g(is)g(Hamiltonian)f(and)g(is)i (in)f(thermal)e(equilibrium)g(at)i(in)m(v)o(erse)f(tem-)739 1420 y(perature)h Ft(\014)32 b 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Fx(F)o(or)d(this)h(reason)e(er)o (godicity)f(is)k(sometimes)e(called)g(return)f(to)i(equilibrium)d(in)i (mean;)g(see)h([Ro1)o(,)739 2069 y(Ro2)o(].)25 b(Similarly)-5 b(,)20 b Ft(!)j Fx(is)e(mixing)e(\(or)h(returns)f(to)h(equilibrium\))e (if)n(f)1791 2245 y Fn(lim)1750 2302 y Fs(j)p Fo(t)p Fs(j!1)1961 2245 y Ft(\026)p Fn(\()p Ft(\034)2088 2211 y Fo(t)2118 2245 y Fn(\()p Ft(A)p Fn(\)\))24 b(=)f Ft(!)s Fn(\()p Ft(A)p Fn(\))p Ft(;)739 2467 y Fx(for)c(all)i Ft(\026)i Fq(2)h(N)1180 2479 y Fo(!)1249 2467 y Fx(and)19 b Ft(A)24 b Fq(2)f(O)r Fx(.)863 2566 y(Let)g Ft(\021)j Fx(and)c Ft(\026)g Fx(be)h(tw)o(o)f(positi)n(v)o(e)f(linear)h (functionals)f(in)h Fq(O)2568 2536 y Fs(\003)2607 2566 y Fx(,)h(and)f(suppose)f(that)h Ft(\021)30 b Fq(\025)d Ft(\036)g Fq(\025)g Fn(0)739 2666 y Fx(for)c(some)h Ft(\026)p Fx(-normal)f Ft(\036)i Fx(implies)f Ft(\036)31 b Fn(=)f(0)p Fx(.)37 b(W)-7 b(e)25 b(then)f(say)g(that)g Ft(\021)k Fx(and)c Ft(\026)h Fx(are)f(mutually)f(singular)739 2766 y(\(or)18 b(orthogonal\),)d(and)j(write)h Ft(\021)26 b Fq(?)d Ft(\026)p Fx(.)i(An)18 b(equi)n(v)n(alent)f(\(more)g (symmetric\))h(de\002nition)f(is:)25 b Ft(\021)h Fq(?)d Ft(\026)739 2865 y Fx(if)n(f)d Ft(\021)26 b Fq(\025)d Ft(\036)g Fq(\025)g Fn(0)d Fx(and)g Ft(\026)j Fq(\025)g Ft(\036)g Fq(\025)g Fn(0)d Fx(imply)g Ft(\036)j Fn(=)g(0)p Fx(.)863 2965 y(T)-7 b(w)o(o)23 b(positi)n(v)o(e)e(linear)h (functionals)f Ft(\021)26 b Fx(and)21 b Ft(\026)i Fx(in)g Fq(O)2355 2935 y Fs(\003)2416 2965 y Fx(are)f(called)g(disjoint)g(if)g Fq(N)3174 2977 y Fo(\021)3235 2965 y Fq(\\)e(N)3378 2977 y Fo(\026)3450 2965 y Fn(=)26 b Fq(;)p Fx(.)739 3064 y(If)c Ft(\021)k Fx(and)c Ft(\026)g Fx(are)h(disjoint,)f(then)f Ft(\021)30 b Fq(?)d Ft(\026)p Fx(.)k(The)22 b(con)m(v)o(erse)e(does)i (not)g(hold\227)g(it)h(is)g(possible)f(that)g Ft(\021)739 3164 y Fx(and)e Ft(\026)g Fx(are)g(mutually)f(singular)h(b)n(ut)g(not)g (disjoint.)863 3264 y(T)-7 b(o)27 b(elucidate)f(further)f(these)h (important)f(notions,)i(we)g(recall)f(the)h(follo)n(wing)e(well-kno)n (wn)739 3363 y(results;)20 b(see)h(Lemmas)f(4.1.19)e(and)i(4.2.8)f(in)h ([BR1].)739 3539 y Fu(Pr)o(oposition)f(3.1)40 b Fr(Let)20 b Ft(\026)1490 3551 y Fj(1)1528 3539 y Ft(;)14 b(\026)1615 3551 y Fj(2)1675 3539 y Fq(2)23 b(O)1821 3509 y Fs(\003)1880 3539 y Fr(be)d(two)g(positive)f(linear)h(functionals)e(and)h Ft(\026)k Fn(=)f Ft(\026)3360 3551 y Fj(1)3414 3539 y Fn(+)16 b Ft(\026)3545 3551 y Fj(2)3582 3539 y Fr(.)739 3639 y(Then)k(the)g(following)f(statements)i(ar)m(e)f(equivalent:)799 3800 y(\(i\))41 b Ft(\026)969 3812 y Fj(1)1029 3800 y Fq(?)23 b Ft(\026)1167 3812 y Fj(2)1204 3800 y Fr(.)776 3963 y(\(ii\))41 b(Ther)m(e)20 b(e)n(xists)i(a)e(pr)l(ojection)f Ft(P)33 b Fr(in)20 b Ft(\031)1972 3975 y Fo(\026)2017 3963 y Fn(\()p Fq(O)r Fn(\))2149 3933 y Fs(0)2194 3963 y Fr(suc)o(h)g(that)1113 4139 y Ft(\026)1163 4151 y Fj(1)1200 4139 y Fn(\()p Ft(A)p Fn(\))k(=)1438 4072 y Fl(\000)1476 4139 y Ft(P)12 b Fn(\012)1601 4151 y Fo(\026)1645 4139 y Ft(;)i(\031)1729 4151 y Fo(\026)1774 4139 y Fn(\()p Ft(A)p Fn(\)\012)1960 4151 y Fo(\026)2005 4072 y Fl(\001)2043 4139 y Ft(;)180 b(\026)2296 4151 y Fj(2)2334 4139 y Fn(\()p Ft(A)p Fn(\))24 b(=)2571 4072 y Fl(\000)2609 4139 y Fn(\()p Ft(I)i Fq(\000)18 b Ft(P)12 b Fn(\)\012)2943 4151 y Fo(\026)2988 4139 y Ft(;)i(\031)3072 4151 y Fo(\026)3116 4139 y Fn(\()p Ft(A)p Fn(\)\012)3302 4151 y Fo(\026)3348 4072 y Fl(\001)3386 4139 y Ft(:)753 4347 y Fr(\(iii\))41 b(The)19 b(GNS)g(r)m(epr)m (esentation)f Fn(\()p Fq(H)1839 4359 y Fo(\026)1884 4347 y Ft(;)c(\031)1968 4359 y Fo(\026)2013 4347 y Ft(;)g Fn(\012)2110 4359 y Fo(\026)2154 4347 y Fn(\))20 b Fr(is)g(a)f(dir)m (ect)g(sum)g(of)h(the)f(two)g(GNS)h(r)m(epr)m(esenta-)919 4446 y(tions)g Fn(\()p Fq(H)1203 4458 y Fo(\026)1243 4466 y Fi(1)1280 4446 y Ft(;)14 b(\031)1364 4458 y Fo(\026)1404 4466 y Fi(1)1441 4446 y Ft(;)g Fn(\012)1538 4458 y Fo(\026)1578 4466 y Fi(1)1615 4446 y Fn(\))21 b Fr(and)f Fn(\()p Fq(H)1916 4458 y Fo(\026)1956 4466 y Fi(2)1993 4446 y Ft(;)14 b(\031)2077 4458 y Fo(\026)2117 4466 y Fi(2)2154 4446 y Ft(;)g Fn(\012)2251 4458 y Fo(\026)2291 4466 y Fi(2)2328 4446 y Fn(\))p Fr(,)21 b Fx(i.e.,)1165 4622 y Fq(H)1235 4634 y Fo(\026)1302 4622 y Fn(=)i Fq(H)1460 4634 y Fo(\026)1500 4642 y Fi(1)1555 4622 y Fq(\010)18 b(H)1708 4634 y Fo(\026)1748 4642 y Fi(2)1785 4622 y Ft(;)180 b(\031)2035 4634 y Fo(\026)2103 4622 y Fn(=)23 b Ft(\031)2238 4634 y Fo(\026)2278 4642 y Fi(1)2334 4622 y Fq(\010)18 b Ft(\031)2464 4634 y Fo(\026)2504 4642 y Fi(2)2541 4622 y Ft(;)180 b Fn(\012)2804 4634 y Fo(\026)2871 4622 y Fn(=)23 b(\012)3019 4634 y Fo(\026)3059 4642 y Fi(1)3114 4622 y Fq(\010)18 b Fn(\012)3257 4634 y Fo(\026)3297 4642 y Fi(2)3334 4622 y Ft(:)739 4907 y Fu(Pr)o(oposition)h(3.2)40 b Fr(Let)20 b Ft(\026)1490 4919 y Fj(1)1528 4907 y Ft(;)14 b(\026)1615 4919 y Fj(2)1675 4907 y Fq(2)23 b(O)1821 4876 y Fs(\003)1880 4907 y Fr(be)d(two)g (positive)f(linear)h(functionals)e(and)h Ft(\026)k Fn(=)f Ft(\026)3360 4919 y Fj(1)3414 4907 y Fn(+)16 b Ft(\026)3545 4919 y Fj(2)3582 4907 y Fr(.)739 5006 y(Then)k(the)g(following)f (statements)i(ar)m(e)f(equivalent:)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g (non-equilibrium)c(quantum)j(statistical)i(mechanics)939 b Fx(7)351 523 y Fr(\(i\))40 b Ft(\026)520 535 y Fj(1)578 523 y Fr(and)20 b Ft(\026)774 535 y Fj(2)832 523 y Fr(ar)m(e)g (disjoint.)327 686 y(\(ii\))41 b(Ther)m(e)21 b(e)n(xists)g(a)g(pr)l (ojection)e Ft(P)33 b Fr(in)20 b Ft(\031)1524 698 y Fo(\026)1569 686 y Fn(\()p Fq(O)r Fn(\))1701 656 y Fs(0)1743 686 y Fq(\\)f Ft(\031)1864 698 y Fo(\026)1909 686 y Fn(\()p Fq(O)r Fn(\))2041 656 y Fs(00)2105 686 y Fr(suc)o(h)h(that)665 862 y Ft(\026)715 874 y Fj(1)752 862 y Fn(\()p Ft(A)p Fn(\))k(=)990 795 y Fl(\000)1028 862 y Ft(P)12 b Fn(\012)1153 874 y Fo(\026)1197 862 y Ft(;)i(\031)1281 874 y Fo(\026)1326 862 y Fn(\()p Ft(A)p Fn(\)\012)1512 874 y Fo(\026)1557 795 y Fl(\001)1595 862 y Ft(;)180 b(\026)1848 874 y Fj(2)1885 862 y Fn(\()p Ft(A)p Fn(\))24 b(=)2123 795 y Fl(\000)2161 862 y Fn(\()p Ft(I)i Fq(\000)18 b Ft(P)12 b Fn(\)\012)2495 874 y Fo(\026)2539 862 y Ft(;)i(\031)2623 874 y Fo(\026)2668 862 y Fn(\()p Ft(A)p Fn(\)\012)2854 874 y Fo(\026)2899 795 y Fl(\001)2937 862 y Ft(:)415 1147 y Fx(Let)27 b Ft(\021)s(;)14 b(\026)35 b Fq(2)g(O)877 1116 y Fs(\003)943 1147 y Fx(be)26 b(tw)o(o)h(positi)n(v)o(e)e(linear)i(functionals.)42 b(The)26 b(functional)f Ft(\021)30 b Fx(has)d(a)g(unique)291 1246 y(decomposition)d Ft(\021)39 b Fn(=)c Ft(\021)1028 1258 y Fo(n)1097 1246 y Fn(+)23 b Ft(\021)1226 1258 y Fo(s)1262 1246 y Fx(,)29 b(where)d Ft(\021)1583 1258 y Fo(n)1628 1246 y Ft(;)14 b(\021)1706 1258 y Fo(s)1770 1246 y Fx(are)27 b(positi)n(v)o(e,)g Ft(\021)2249 1258 y Fo(n)2330 1246 y Fq(\034)36 b Ft(\026)p Fx(,)29 b(and)d Ft(\021)2737 1258 y Fo(s)2809 1246 y Fq(?)35 b Ft(\026)p Fx(.)46 b(The)291 1346 y(uniqueness)18 b(of)i(the)g(decomposition)e (implies)i(that)g(if)h Ft(\021)j Fx(is)d Ft(\034)9 b Fx(-in)m(v)n(ariant,)18 b(then)i(so)h(are)f Ft(\021)2851 1358 y Fo(n)2917 1346 y Fx(and)f Ft(\021)3098 1358 y Fo(s)3134 1346 y Fx(.)415 1446 y(T)-7 b(o)25 b(elucidate)e(the)i (nature)e(of)h(this)h(decomposition)d(we)i(need)g(to)g(recall)h(the)f (notions)f(of)h(the)291 1545 y(uni)n(v)o(ersal)c(representation)g(and)i (the)g(uni)n(v)o(ersal)e(en)m(v)o(eloping)f(v)n(on)j(Neumann)e(algebra) h(of)h Fq(O)r Fx(;)i(see)291 1645 y(Section)19 b(III.2)g(in)i([T)-7 b(a)o(])21 b(and)e(Section)h(10.1)f(in)i([KR].)415 1744 y(Set)606 1920 y Fq(H)676 1932 y Fj(un)777 1920 y Fq(\021)925 1841 y Fl(M)864 2023 y Fo(!)r Fs(2)p Fo(E)s Fj(\()p Fs(O)r Fj(\))1125 1920 y Fq(H)1195 1932 y Fo(!)1243 1920 y Ft(;)180 b(\031)1493 1932 y Fj(un)1594 1920 y Fq(\021)1742 1841 y Fl(M)1682 2023 y Fo(!)r Fs(2)p Fo(E)s Fj(\()p Fs(O)r Fj(\))1942 1920 y Ft(\031)1989 1932 y Fo(!)2038 1920 y Ft(;)g Fk(M)2328 1932 y Fj(un)2428 1920 y Fq(\021)23 b Ft(\031)2563 1932 y Fj(un)2641 1920 y Fn(\()p Fq(O)r Fn(\))2773 1886 y Fs(00)2816 1920 y Ft(:)291 2193 y Fn(\()p Fq(H)393 2205 y Fj(un)470 2193 y Ft(;)14 b(\031)554 2205 y Fj(un)632 2193 y Fn(\))25 b Fx(is)h(a)f(f)o(aithful)e (representation.)36 b(It)24 b(is)i(called)e Fr(the)g(univer)o(sal)g(r)m (epr)m(esentation)g Fx(of)g Fq(O)r Fx(.)291 2293 y Fk(M)378 2305 y Fj(un)479 2293 y Fq(\032)g(B)s Fn(\()p Fq(H)728 2305 y Fj(un)805 2293 y Fn(\))d Fx(is)h(its)f(uni)n(v)o(ersal)f(en)m(v) o(eloping)d(v)n(on)j(Neumann)f(algebra.)26 b(F)o(or)20 b(an)o(y)g Ft(!)26 b Fq(2)f Ft(E)5 b Fn(\()p Fq(O)r Fn(\))291 2392 y Fx(the)20 b(map)1347 2568 y Ft(\031)1394 2580 y Fj(un)1471 2568 y Fn(\()p Fq(O)r Fn(\))84 b Fq(!)f Ft(\031)1900 2580 y Fo(!)1949 2568 y Fn(\()p Fq(O)r Fn(\))1353 2693 y Ft(\031)1400 2705 y Fj(un)1478 2693 y Fn(\()p Ft(A)p Fn(\))g Fq(7!)g Ft(\031)1900 2705 y Fo(!)1949 2693 y Fn(\()p Ft(A)p Fn(\))p Ft(;)291 2869 y Fx(e)o(xtends)29 b(to)h(a)g(surjecti)n(v)o(e)f Fq(\003)p Fx(-morphism)j Fn(~)-46 b Ft(\031)1570 2881 y Fo(!)1659 2869 y Fn(:)42 b Fk(M)1811 2881 y Fj(un)1930 2869 y Fq(!)f Fk(M)2141 2881 y Fo(!)2189 2869 y Fx(.)55 b(It)30 b(follo)n(ws)g(that)g Ft(!)j Fx(uniquely)291 2969 y(e)o(xtends)19 b(to)i(a)g(normal)e(state) 28 b Fn(~)-49 b Ft(!)s Fn(\()p Fq(\001)p Fn(\))25 b Fq(\021)e Fn(\(\012)1484 2981 y Fo(!)1532 2969 y Ft(;)18 b Fn(~)-46 b Ft(\031)1616 2981 y Fo(!)1664 2969 y Fn(\()p Fq(\001)p Fn(\)\012)1811 2981 y Fo(!)1860 2969 y Fn(\))22 b Fx(on)e Fk(M)2105 2981 y Fj(un)2182 2969 y Fx(.)27 b(Moreo)o(v)o(er)m(,)18 b(one)i(easily)g(sho)n(ws)291 3068 y(that)863 3168 y Fn(Ker)d(~)-47 b Ft(\031)1057 3180 y Fo(!)1129 3168 y Fn(=)22 b Fq(f)p Ft(A)h Fq(2)h Fk(M)1509 3180 y Fj(un)1600 3168 y Fq(j)19 b Fn(~)-47 b Ft(\027)5 b Fn(\()p Ft(A)p Fn(\))24 b(=)f(0)d(for)g(an)n(y)h Ft(\027)29 b Fq(2)23 b(N)2470 3180 y Fo(!)2518 3168 y Fq(g)p Ft(:)413 b Fx(\(3.1\))291 3313 y(Since)21 b Fn(Ker)d(~)-47 b Ft(\031)692 3325 y Fo(!)763 3313 y Fx(is)22 b(a)h Ft(\033)s Fx(-weakly)e(closed)g(tw)o(o)h (sided)g(ideal)f(in)h Fk(M)2175 3325 y Fj(un)2253 3313 y Fx(,)g(there)f(e)o(xists)h(an)g(orthogonal)291 3413 y(projection)e Ft(p)692 3425 y Fo(!)766 3413 y Fq(2)27 b Fk(M)935 3425 y Fj(un)1032 3413 y Fq(\\)21 b Fk(M)1195 3383 y Fs(0)1195 3433 y Fj(un)1295 3413 y Fx(such)h(that)g Fn(Ker)17 b(~)-46 b Ft(\031)1812 3425 y Fo(!)1887 3413 y Fn(=)26 b Ft(p)2020 3425 y Fo(!)2067 3413 y Fk(M)2154 3425 y Fj(un)2232 3413 y Fx(.)31 b(The)22 b(orthogonal)d(projection)291 3513 y Ft(z)330 3525 y Fo(!)419 3513 y Fq(\021)41 b Ft(I)33 b Fq(\000)25 b Ft(p)726 3525 y Fo(!)816 3513 y Fq(2)42 b Fk(M)1000 3525 y Fj(un)1103 3513 y Fq(\\)26 b Fk(M)1271 3482 y Fs(0)1271 3533 y Fj(un)1380 3513 y Fx(is)31 b(called)f(the)g Fr(support)f(pr)l(ojection)h Fx(of)g(the)g(state)h Ft(!)s Fx(.)55 b(The)291 3612 y(restriction)30 b(of)35 b Fn(~)-46 b Ft(\031)807 3624 y Fo(!)887 3612 y Fx(to)31 b Ft(z)1022 3624 y Fo(!)1070 3612 y Fk(M)1157 3624 y Fj(un)1266 3612 y Fx(is)h(an)f(isomorphism)e(between)h(the)h(v)n(on)g(Neumann)e (algebras)291 3712 y Ft(z)330 3724 y Fo(!)377 3712 y Fk(M)464 3724 y Fj(un)563 3712 y Fx(and)19 b Fk(M)790 3724 y Fo(!)838 3712 y Fx(.)26 b(W)-7 b(e)21 b(shall)g(denote)e(by)h Ft(\036)1587 3724 y Fo(!)1656 3712 y Fx(the)g(in)m(v)o(erse)f (isomorphism.)415 3811 y(Let)k(no)n(w)g Ft(\021)s(;)14 b(\026)28 b Fq(2)h(O)1025 3781 y Fs(\003)1087 3811 y Fx(be)23 b(tw)o(o)g(positi)n(v)o(e)f(linear)g(functionals.)32 b(By)24 b(scaling,)f(without)f(loss)i(of)291 3911 y(generality)i(we)i (may)f(assume)h(that)g(the)o(y)f(are)h(states.)48 b(Since)34 b Fn(~)-48 b Ft(\021)32 b Fx(is)c(a)g(normal)f(state)h(on)g Fk(M)3003 3923 y Fj(un)3109 3911 y Fx(it)291 4011 y(follo)n(ws)19 b(that)26 b Fn(~)-48 b Ft(\021)22 b Fq(\016)c Ft(\036)875 4023 y Fo(\026)940 4011 y Fx(is)j(a)g(normal)e(state)h(on)g Fk(M)1693 4023 y Fo(\026)1759 4011 y Fx(and)f(hence)g(that)i Ft(\021)2300 4023 y Fo(n)2368 4011 y Fq(\021)29 b Fn(~)-48 b Ft(\021)21 b Fq(\016)d Ft(\036)2627 4023 y Fo(\026)2690 4011 y Fq(\016)g Ft(\031)2797 4023 y Fo(\026)2862 4011 y Fx(de\002nes)i(a)291 4110 y Ft(\026)p Fx(-normal)h(positi)n(v)o(e)h (linear)h(functional)e(on)i Fq(O)r Fx(.)34 b(Moreo)o(v)o(er)m(,)21 b(from)h(the)h(relation)f Ft(\036)2716 4122 y Fo(\026)2781 4110 y Fq(\016)e Ft(\031)2890 4122 y Fo(\026)2935 4110 y Fn(\()p Ft(A)p Fn(\))29 b(=)291 4210 y Ft(z)330 4222 y Fo(\026)374 4210 y Ft(\031)421 4222 y Fj(un)499 4210 y Fn(\()p Ft(A)p Fn(\))21 b Fx(it)g(follo)n(ws)f(that)1173 4386 y Ft(\021)1214 4398 y Fo(n)1259 4386 y Fn(\()p Ft(A)p Fn(\))k(=)f(\(\012)1589 4398 y Fo(\021)1630 4386 y Ft(;)18 b Fn(~)-46 b Ft(\031)1714 4398 y Fo(\021)1754 4386 y Fn(\()p Ft(z)1825 4398 y Fo(\026)1870 4386 y Fn(\))p Ft(\031)1949 4398 y Fo(\021)1990 4386 y Fn(\()p Ft(A)p Fn(\)\012)2176 4398 y Fo(\021)2217 4386 y Fn(\))p Ft(:)291 4562 y Fx(Setting)1176 4662 y Ft(\021)1217 4674 y Fo(s)1253 4662 y Fn(\()p Ft(A)p Fn(\))24 b Fq(\021)f Fn(\(\012)1583 4674 y Fo(\021)1623 4662 y Ft(;)c Fn(~)-47 b Ft(\031)1707 4674 y Fo(\021)1748 4662 y Fn(\()p Ft(p)1822 4674 y Fo(\026)1866 4662 y Fn(\))p Ft(\031)1945 4674 y Fo(\021)1987 4662 y Fn(\()p Ft(A)p Fn(\)\012)2173 4674 y Fo(\021)2214 4662 y Fn(\))p Ft(;)291 4807 y Fx(we)26 b(obtain)f(a)h(decomposition)d Ft(\021)36 b Fn(=)d Ft(\021)1442 4819 y Fo(n)1510 4807 y Fn(+)22 b Ft(\021)1638 4819 y Fo(s)1674 4807 y Fx(.)42 b(T)-7 b(o)26 b(sho)n(w)g(that)g Ft(\021)2241 4819 y Fo(s)2310 4807 y Fq(?)33 b Ft(\026)26 b Fx(let)g Ft(!)j Fx(be)d(a)g Ft(\026)p Fx(-normal)291 4907 y(positi)n(v)o(e)14 b(linear)i(functional)e(on)h Fq(O)k Fx(such)c(that)h Ft(\021)1659 4919 y Fo(s)1718 4907 y Fq(\025)23 b Ft(!)s Fx(.)g(By)16 b(the)g(unicity)f(of)h(the)f(normal)g(e)o(xtension)297 5006 y Fn(~)-48 b Ft(\021)332 5018 y Fo(s)398 5006 y Fx(one)30 b(has)36 b Fn(~)-48 b Ft(\021)731 5018 y Fo(s)767 5006 y Fn(\()p Ft(A)p Fn(\))42 b(=)48 b(~)-48 b Ft(\021)s Fn(\()p Ft(p)1160 5018 y Fo(\026)1204 5006 y Ft(A)p Fn(\))32 b Fx(for)d Ft(A)42 b Fq(2)g Fk(M)1745 5018 y Fj(un)1823 5006 y Fx(.)55 b(Since)31 b Ft(\031)2162 5018 y Fj(un)2239 5006 y Fn(\()p Fq(O)r Fn(\))h Fx(is)f Ft(\033)s Fx(-strongly)e(dense)h (in)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 739 232 a Fx(8)1697 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fk(M)826 535 y Fj(un)935 523 y Fx(it)31 b(follo)n(ws)g(from)e (the)i(inequality)36 b Fn(~)-48 b Ft(\021)2021 535 y Fo(s)2083 523 y Fq(\016)26 b Ft(\031)2198 535 y Fj(un)2318 523 y Fq(\025)49 b Fn(~)-49 b Ft(!)29 b Fq(\016)d Ft(\031)2621 535 y Fj(un)2730 523 y Fx(that)37 b Fn(~)-48 b Ft(\021)s Fn(\()p Ft(p)3004 535 y Fo(\026)3049 523 y Ft(A)p Fn(\))43 b Fq(\025)49 b Fn(~)-49 b Ft(!)s Fn(\()p Ft(A)p Fn(\))32 b Fx(for)739 623 y(an)o(y)25 b(positi)n(v)o(e)f Ft(A)33 b Fq(2)g Fk(M)1439 635 y Fj(un)1517 623 y Fx(.)41 b(Since)26 b Ft(!)j Fx(is)d Ft(\026)p Fx(-normal,)f(it)h(further)e(follo)n(ws)h (from)f(Equ.)41 b(\(3.1\))24 b(that)739 722 y Ft(!)s Fn(\()p Ft(A)p Fn(\))i(=)32 b(~)-49 b Ft(!)s Fn(\()p Ft(\031)1170 734 y Fj(un)1248 722 y Fn(\()p Ft(A)p Fn(\)\))27 b(=)32 b(~)-49 b Ft(!)s Fn(\()p Ft(z)1649 734 y Fo(\026)1694 722 y Ft(\031)1741 734 y Fj(un)1818 722 y Fn(\()p Ft(A)p Fn(\)\))27 b Fq(\024)k Fn(~)-47 b Ft(\021)s Fn(\()p Ft(p)2212 734 y Fo(\026)2256 722 y Ft(z)2295 734 y Fo(\026)2339 722 y Ft(\031)2386 734 y Fj(un)2464 722 y Fn(\()p Ft(A)p Fn(\)\))27 b(=)e(0)d Fx(for)f(an)o(y)g(positi)n(v)o(e)g Ft(A)26 b Fq(2)g(O)r Fx(,)739 822 y Fr(i.e)o(.,)f Ft(!)33 b Fn(=)d(0)p Fx(.)38 b(Since)28 b Fn(~)-46 b Ft(\031)1423 834 y Fo(\021)1489 822 y Fx(is)25 b(surjecti)n(v)o(e,)f(one)g(has)29 b Fn(~)-47 b Ft(\031)2262 834 y Fo(\021)2303 822 y Fn(\()p Ft(z)2374 834 y Fo(\026)2418 822 y Fn(\))31 b Fq(2)g Fk(M)2654 834 y Fo(\021)2717 822 y Fq(\\)21 b Fk(M)2880 792 y Fs(0)2880 842 y Fo(\021)2946 822 y Fx(and,)j(by)g(Proposition)739 922 y(3.2,)19 b(the)h(functionals)f Ft(\021)1437 934 y Fo(n)1503 922 y Fx(and)h Ft(\021)1685 934 y Fo(s)1741 922 y Fx(are)g(disjoint.)863 1026 y(T)-7 b(w)o(o)24 b(states)h Ft(!)1293 1038 y Fj(1)1354 1026 y Fx(and)e Ft(!)1550 1038 y Fj(2)1612 1026 y Fx(are)g(called)h Fr(quasi-equivalent)c Fx(if)k Fq(N)2680 1038 y Fo(!)2722 1046 y Fi(1)2788 1026 y Fn(=)29 b Fq(N)2950 1038 y Fo(!)2992 1046 y Fi(2)3029 1026 y Fx(.)36 b(The)o(y)22 b(are)i(called)739 1126 y(unitarily)g(equi) n(v)n(alent)f(if)i(their)g(GNS)h(representations)d Fn(\()p Fq(H)2489 1138 y Fo(!)2531 1146 y Fh(j)2566 1126 y Ft(;)14 b(\031)2650 1138 y Fo(!)2692 1146 y Fh(j)2728 1126 y Ft(;)g Fn(\012)2825 1138 y Fo(!)2867 1146 y Fh(j)2901 1126 y Fn(\))26 b Fx(are)f(unitarily)f(equi)n(v-)739 1225 y(alent,)34 b(namely)d(if)g(there)h(is)g(a)g(unitary)e Ft(U)53 b Fn(:)44 b Fq(H)2184 1237 y Fo(!)2226 1245 y Fi(1)2307 1225 y Fq(!)g(H)2504 1237 y Fo(!)2546 1245 y Fi(2)2615 1225 y Fx(such)31 b(that)h Ft(U)9 b Fn(\012)3082 1237 y Fo(!)3124 1245 y Fi(1)3204 1225 y Fn(=)44 b(\012)3373 1237 y Fo(!)3415 1245 y Fi(2)3483 1225 y Fx(and)739 1325 y Ft(U)9 b(\031)852 1337 y Fo(!)894 1345 y Fi(1)930 1325 y Fn(\()p Fq(\001)p Fn(\))24 b(=)f Ft(\031)1176 1337 y Fo(!)1218 1345 y Fi(2)1254 1325 y Fn(\()p Fq(\001)p Fn(\))p Ft(U)9 b Fx(.)26 b(Clearly)-5 b(,)20 b(unitarily)f(equi)n(v)n (alent)f(states)k(are)e(quasi-equi)n(v)n(alent.)863 1429 y(If)26 b Ft(!)j Fx(is)d Ft(\034)9 b Fx(-in)m(v)n(ariant,)25 b(then)g(there)g(e)o(xists)h(a)g(unique)e(self-adjoint)g(operator)g Ft(L)i Fx(on)f Fq(H)3377 1441 y Fo(!)3451 1429 y Fx(such)739 1529 y(that)1396 1643 y Ft(L)p Fn(\012)1513 1655 y Fo(!)1584 1643 y Fn(=)d(0)p Ft(;)180 b(\031)1963 1655 y Fo(!)2011 1643 y Fn(\()p Ft(\034)2088 1609 y Fo(t)2118 1643 y Fn(\()p Ft(A)p Fn(\)\))24 b(=)f(e)2425 1609 y Fj(i)p Fo(tL)2518 1643 y Ft(\031)2565 1655 y Fo(!)2614 1643 y Fn(\()p Ft(A)p Fn(\)e)2777 1609 y Fs(\000)p Fj(i)p Fo(tL)2923 1643 y Ft(:)739 1807 y Fx(W)-7 b(e)21 b(will)g(call)g Ft(L)f Fx(the)g Ft(!)s Fx(-Liouvillean)e(of)i Ft(\034)9 b Fx(.)863 1912 y(The)23 b(state)g Ft(!)j Fx(is)e(called)f(f)o(actor)f(state)h (\(or)f(primary)f(state\))j(if)f(its)g(en)m(v)o(eloping)d(v)n(on)i (Neumann)739 2012 y(algebra)g Fk(M)1094 2024 y Fo(!)1167 2012 y Fx(is)i(a)g(f)o(actor)m(,)f(namely)g(if)g Fk(M)1972 2024 y Fo(!)2042 2012 y Fq(\\)e Fk(M)2205 1981 y Fs(0)2205 2032 y Fo(!)2282 2012 y Fn(=)29 b Fp(C)p Ft(I)7 b Fx(.)35 b(By)24 b(Proposition)e(3.2)h Ft(!)k Fx(is)d(a)g(f)o(actor)739 2111 y(state)e(if)n(f)e(it)i(cannot)e(be)h(written)g(as)g(a)h(nontri)n (vial)d(con)m(v)o(e)o(x)f(combination)h(of)i(disjoint)f(states.)29 b(This)739 2211 y(implies)21 b(that)f(if)h Ft(!)j Fx(is)e(a)f(f)o (actor)f(state)h(and)f Ft(\026)h Fx(is)h(a)f(positi)n(v)o(e)f(linear)g (functional)f(in)h Fq(O)3169 2181 y Fs(\003)3208 2211 y Fx(,)h(then)f(either)739 2310 y Ft(!)26 b Fq(\034)d Ft(\026)d Fx(or)g Ft(!)26 b Fq(?)d Ft(\026)p Fx(.)863 2415 y(T)-7 b(w)o(o)21 b(f)o(actor)e(states)i Ft(!)1499 2427 y Fj(1)1557 2415 y Fx(and)e Ft(!)1749 2427 y Fj(2)1806 2415 y Fx(are)h(either)g(quasi-equi)n(v)n(alent)d(or)j(disjoint.)k(The) o(y)19 b(are)h(quasi-)739 2515 y(equi)n(v)n(alent)i(if)n(f)h Fn(\()p Ft(!)1289 2527 y Fj(1)1348 2515 y Fn(+)d Ft(!)1485 2527 y Fj(2)1522 2515 y Fn(\))p Ft(=)p Fn(2)k Fx(is)g(also)g(a)h(f)o (actor)e(state)h(\(this)g(follo)n(ws)f(from)g(Theorem)f(4.3.19)g(in)739 2614 y([BR1]\).)863 2719 y(The)j(state)g Ft(!)j Fx(is)e(called)e (modular)f(if)i(there)g(e)o(xists)g(a)g Ft(C)2489 2689 y Fs(\003)2527 2719 y Fx(-dynamics)e Ft(\033)2945 2731 y Fo(!)3019 2719 y Fx(on)i Fq(O)i Fx(such)e(that)f Ft(!)739 2818 y Fx(is)f(a)f Fn(\()p Ft(\033)955 2830 y Fo(!)1004 2818 y Ft(;)14 b Fq(\000)p Fn(1\))p Fx(-KMS)21 b(state.)30 b(If)22 b Ft(!)j Fx(is)d(modular)m(,)e(then)h Fn(\012)2386 2830 y Fo(!)2457 2818 y Fx(is)h(a)g(separating)f(v)o(ector)g(for)g Fk(M)3392 2830 y Fo(!)3440 2818 y Fx(,)h(and)739 2918 y(we)30 b(denote)e(by)h Fn(\001)1299 2930 y Fo(!)1347 2918 y Fx(,)j Ft(J)38 b Fx(and)29 b Fq(P)36 b Fx(the)30 b(modular)d(operator)m(,)i(the)h(modular)d(conjugation)g(and)i(the)739 3018 y(natural)18 b(cone)g(associated)g(to)h Fn(\012)1669 3030 y Fo(!)1717 3018 y Fx(.)25 b(T)-7 b(o)19 b(an)o(y)e Ft(C)2070 2987 y Fs(\003)2109 3018 y Fx(-dynamics)g Ft(\034)29 b Fx(on)18 b Fq(O)k Fx(one)c(can)g(associate)h(a)g(unique)739 3117 y(self-adjoint)g(operator)f Ft(L)i Fx(on)g Fq(H)1686 3129 y Fo(!)1755 3117 y Fx(such)g(that)g(for)g(all)h Ft(t)1343 3310 y(\031)1390 3322 y Fo(!)1439 3310 y Fn(\()p Ft(\034)1516 3275 y Fo(t)1546 3310 y Fn(\()p Ft(A)p Fn(\)\))j(=)f(e) 1853 3275 y Fj(i)p Fo(tL)1946 3310 y Ft(\031)1993 3322 y Fo(!)2041 3310 y Fn(\()p Ft(A)p Fn(\)e)2204 3275 y Fs(\000)p Fj(i)p Fo(tL)2351 3310 y Ft(;)180 b Fn(e)2591 3275 y Fs(\000)p Fj(i)p Fo(tL)2736 3310 y Fq(P)30 b Fn(=)22 b Fq(P)7 b Ft(:)739 3502 y Fx(The)21 b(operator)f Ft(L)h Fx(is)i(called)e(standard)g(Liouvillean)f(of)h Ft(\034)32 b Fx(associated)21 b(to)h Ft(!)s Fx(.)29 b(If)21 b Ft(!)k Fx(is)d Ft(\034)9 b Fx(-in)m(v)n(ariant,)739 3602 y(then)20 b Ft(L)p Fn(\012)1020 3614 y Fo(!)1090 3602 y Fn(=)j(0)p Fx(,)d(and)f(the)i(standard)e(Liouvillean)f(is)j(equal)f(to)g(the)h Ft(!)s Fx(-Liouvillean)c(of)j Ft(\034)9 b Fx(.)863 3706 y(The)24 b(importance)d(of)j(the)f(standard)g(Liouvillean)e Ft(L)j Fx(stems)g(from)f(the)g(f)o(act)h(that)g(if)g(a)g(state)g Ft(\021)739 3806 y Fx(is)i Ft(!)s Fx(-normal)d(and)h Ft(\034)9 b Fx(-in)m(v)n(ariant,)25 b(then)f(there)h(e)o(xists)g(a)g (unique)f(v)o(ector)g Fn(\012)2898 3818 y Fo(\021)2970 3806 y Fq(2)32 b Fn(Ker)13 b Ft(L)22 b Fq(\\)g(P)32 b Fx(such)739 3905 y(that)19 b Ft(\021)s Fn(\()p Fq(\001)p Fn(\))24 b(=)f(\(\012)1218 3917 y Fo(\021)1259 3905 y Ft(;)14 b(\031)1343 3917 y Fo(!)1391 3905 y Fn(\()p Fq(\001)p Fn(\)\012)1538 3917 y Fo(\021)1579 3905 y Fn(\))p Fx(.)26 b(This)19 b(f)o(act)h(has)f(tw)o(o)h(important)e(consequences.)23 b(On)c(one)g(hand,)f(if)739 4005 y Ft(\021)23 b Fx(is)d Ft(!)s Fx(-normal)e(and)g Ft(\034)9 b Fx(-in)m(v)n(ariant,)18 b(then)h(some)g(er)o(godic)f(properties)f(of)j(the)f(quantum)e (dynamical)739 4105 y(system)h Fn(\()p Fq(O)r Ft(;)c(\034)5 b(;)14 b(\021)s Fn(\))20 b Fx(can)e(be)f(described)g(in)h(terms)g(of)g (the)g(spectral)g(properties)e(of)i Ft(L)p Fx(;)h(see)g([JP2)o(,)g (Pi].)739 4204 y(On)g(the)g(other)g(hand,)f(if)h Fn(Ker)13 b Ft(L)23 b Fn(=)f Fq(f)p Fn(0)p Fq(g)p Fx(,)d(then)f(the)h Ft(C)2273 4174 y Fs(\003)2312 4204 y Fx(-dynamics)f Ft(\034)29 b Fx(has)19 b(no)g Ft(!)s Fx(-normal)e(in)m(v)n(ariant)739 4304 y(states.)56 b(The)29 b(papers)h([BFS,)h(DJ,)f(FM1,)g(FM2,)g(FMS,) h(JP1,)f(JP2,)g(JP3,)h(Me1)o(,)f(Me2,)g(Og)o(])h(are)739 4404 y(centered)19 b(around)f(this)j(set)g(of)f(ideas.)863 4508 y(In)33 b(quantum)e(statistical)j(mechanics)e(one)g(also)h (encounters)e Ft(L)2791 4478 y Fo(p)2829 4508 y Fx(-Liouvilleans,)j (for)f Ft(p)46 b Fq(2)739 4608 y Fn([1)p Ft(;)14 b Fq(1)p Fn(])23 b Fx(\(the)f(standard)g(Liouvillean)f(is)i(equal)g(to)f(the)h Ft(L)2384 4578 y Fj(2)2421 4608 y Fx(-Liouvillean\).)30 b(The)23 b Ft(L)3123 4578 y Fo(p)3160 4608 y Fx(-Liouvilleans)739 4707 y(are)k(closely)g(related)g(to)g(the)g(Araki-Masuda)e Ft(L)2170 4677 y Fo(p)2208 4707 y Fx(-spaces)i([ArM)o(].)46 b Ft(L)2821 4677 y Fj(1)2886 4707 y Fx(and)27 b Ft(L)3091 4677 y Fs(1)3160 4707 y Fx(-Liouvilleans)739 4807 y(ha)n(v)o(e)22 b(played)f(a)h(central)g(role)g(in)g(the)g(spectral)g(theory)f(of)h (NESS)h(de)n(v)o(eloped)c(in)k([JP5)o(].)32 b(The)21 b(use)739 4907 y(of)g(other)g Ft(L)1080 4876 y Fo(p)1118 4907 y Fx(-Liouvilleans)e(is)k(more)d(recent)h(\(see)h([JPR2]\))f(and)g (the)o(y)g(will)h(not)f(be)g(discussed)h(in)739 5006 y(this)f(lecture.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g (non-equilibrium)c(quantum)j(statistical)i(mechanics)939 b Fx(9)291 523 y Fm(3.2)99 b(Non-equilibrium)24 b(steady)g(states)g (\(NESS\))h(and)f(entr)n(opy)h(pr)n(oduction)291 679 y Fx(The)20 b(central)h(notions)f(of)h(non-equilibrium)c(statistical)22 b(mechanics)e(are)h(non-equilibrium)c(stea-)291 778 y(dy)k(states)j (\(NESS\))e(and)f(entrop)o(y)g(production.)28 b(Our)22 b(de\002nition)f(of)h(NESS)h(follo)n(ws)e(closely)h(the)291 878 y(idea)c(of)h(Ruelle)g(that)g(a)g(\223natural\224)e(steady)i(state) 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b Fn(\))25 b Fq(\032)f Ft(E)5 b Fn(\()p Fq(O)r Ft(;)14 b(\034)9 b Fn(\))p Fx(.)30 b(Moreo)o(v)o(er)m(,)18 b(since)j Ft(E)5 b Fn(\()p Fq(O)r Fn(\))23 b Fx(is)f(weak-)p Fq(\003)e Fx(compact,)291 2897 y Fn(\006)351 2909 y Fj(+)406 2897 y Fn(\()p Ft(!)s(;)14 b(\034)9 b Fn(\))21 b Fx(is)g(non-empty)-5 b(.)415 2997 y(As)20 b(already)e(mentioned,)f(our)h(de\002nition)g(of)g (entrop)o(y)g(production)e(is)k(based)e(on)h(the)g(concept)291 3096 y(of)g(relati)n(v)o(e)g(entrop)o(y.)24 b(The)19 b(relati)n(v)o(e)h(entrop)o(y)e(of)i(tw)o(o)g(density)f(matrices)h Ft(\032)h Fx(and)e Ft(!)k Fx(is)e(de\002ned,)e(by)291 3196 y(analogy)f(with)j(the)f(relati)n(v)o(e)f(entrop)o(y)g(of)h(tw)o (o)g(measures,)g(by)f(the)i(formula)1148 3356 y Fn(En)n(t)o(\()p Ft(\032)p Fq(j)p Ft(!)s Fn(\))i Fq(\021)g Fn(T)-7 b(r\()p Ft(\032)p Fn(\(log)14 b Ft(!)21 b Fq(\000)d Fn(log)c Ft(\032)p Fn(\)\))p Ft(:)699 b Fx(\(3.2\))291 3517 y(It)20 b(is)i(easy)f(to)f(sho)n(w)h(that)f Fn(En)n(t\()p Ft(\032)p Fq(j)p Ft(!)s Fn(\))k 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(entrop)o(y)e(produc-)739 4201 y(tion)f(follo)n(wing)e([JP4].)25 b(The)20 b(proofs)f(are)h(gi)n(v)o(en)f(in)h(Appendix)e(9.1.)863 4300 y(First,)27 b(we)f(will)g(discuss)g(the)f(dependence)d(of)j Fn(\006)2306 4312 y Fj(+)2361 4300 y Fn(\()p Ft(!)s(;)14 b(\034)2521 4312 y Fo(V)2579 4300 y Fn(\))26 b Fx(on)f(the)g(reference) f(state)i Ft(!)s Fx(.)40 b(On)739 4400 y(physical)25 b(grounds,)g(one)g(may)h(e)o(xpect)f(that)h(if)g Ft(!)j Fx(is)e(suf)n(\002ciently)e(re)o(gular)g(and)g Ft(\021)30 b Fx(is)d Ft(!)s Fx(-normal,)739 4499 y(then)20 b Fn(\006)963 4511 y Fj(+)1018 4499 y Fn(\()p Ft(\021)s(;)14 b(\034)1167 4511 y Fo(V)1225 4499 y Fn(\))23 b(=)g(\006)1428 4511 y Fj(+)1483 4499 y Fn(\()p Ft(!)s(;)14 b(\034)1643 4511 y Fo(V)1701 4499 y Fn(\))p Fx(.)739 4657 y Fu(Theor)o(em)20 b(3.3)40 b Fr(Assume)27 b(that)g Ft(!)k Fr(is)d(a)f(factor)g(state)g (on)g(the)g Ft(C)2614 4627 y Fs(\003)2652 4657 y Fr(-alg)o(ebr)o(a)f Fq(O)k Fr(and)c(that,)i(for)f(all)739 4757 y Ft(\021)f Fq(2)e(N)953 4769 y Fo(!)1022 4757 y Fr(and)19 b Ft(A;)14 b(B)27 b Fq(2)d(O)r Fr(,)1613 4972 y Fn(lim)1580 5026 y Fo(T)9 b Fs(!1)1793 4916 y Fn(1)p 1784 4953 61 4 v 1784 5029 a Ft(T)1868 4859 y Fl(Z)1951 4880 y Fo(T)1914 5048 y Fj(0)2017 4972 y Ft(\021)s Fn(\([)p Ft(\034)2161 4938 y Fo(t)2152 4993 y(V)2211 4972 y Fn(\()p Ft(A)p Fn(\))p Ft(;)14 b(B)t Fn(]\))g(d)p Ft(t)24 b Fn(=)f(0)p Ft(;)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(11)291 523 y Fr(holds)24 b(\(weak)h(asymptotic)f(abelianness)f(in) i(mean\).)38 b(Then)25 b Fn(\006)2132 535 y Fj(+)2187 523 y Fn(\()p Ft(\021)s(;)14 b(\034)2336 535 y Fo(V)2394 523 y Fn(\))32 b(=)f(\006)2614 535 y Fj(+)2669 523 y Fn(\()p Ft(!)s(;)14 b(\034)2829 535 y Fo(V)2887 523 y Fn(\))26 b Fr(for)f(all)291 623 y Ft(\021)h Fq(2)d(N)504 635 y Fo(!)552 623 y Fr(.)291 800 y Fx(The)c(second)h(structural)f (property)f(we)j(w)o(ould)e(lik)o(e)i(to)f(mention)f(is:)291 977 y Fu(Theor)o(em)h(3.4)40 b Fr(Let)21 b Ft(\021)27 b Fq(2)d(O)1120 947 y Fs(\003)1179 977 y Fr(be)d Ft(!)s Fr(-normal)e(and)g Ft(\034)1803 989 y Fo(V)1861 977 y Fr(-in)m(variant.)24 b(Then)c Ft(\021)s Fn(\()p Ft(\033)2547 989 y Fo(V)2606 977 y Fn(\))k(=)f(0)p Fr(.)j(In)20 b(partic-)291 1077 y(ular)-9 b(,)19 b(the)h(entr)l(opy)g(pr)l(oduction)f(of)h(the)g (normal)g(part)g(of)g(any)g(NESS)g(is)h(equal)e(to)h(zer)l(o.)415 1254 y Fx(If)e Fn(En)n(t\()p Ft(\021)s Fq(j)p Ft(!)s Fn(\))23 b Ft(>)g Fq(\0001)p Fx(,)c(then)f(Theorem)e(3.4)i(is)h(an)f (immediate)g(consequence)e(of)i(the)g(entrop)o(y)291 1354 y(balance)24 b(equation)f(\(3.3\).)39 b(The)24 b(case)i Fn(En)n(t\()p Ft(\021)s Fq(j)p Ft(!)s Fn(\))32 b(=)g Fq(\0001)25 b Fx(has)h(been)e(treated)h(in)g([JP7])g(and)f(the)291 1453 y(proof)16 b(requires)h(the)g(full)h(machinery)e(of)h(Araki')-5 b(s)18 b(perturbation)d(theory)-5 b(.)23 b(W)-7 b(e)19 b(will)f(not)g(reproduce)291 1553 y(it)i(here.)415 1652 y(If)f Ft(!)542 1664 y Fj(+)616 1652 y Fx(is)h(a)f(f)o(actor)f(state,)i (then)e(either)g Ft(!)1574 1664 y Fj(+)1652 1652 y Fq(\034)23 b Ft(!)f Fx(or)d Ft(!)1973 1664 y Fj(+)2051 1652 y Fq(?)j Ft(!)s Fx(.)j(Hence,)18 b(Theorem)f(3.4)h(yields:)291 1814 y Fu(Cor)o(ollary)g(3.5)40 b Fr(If)25 b Ft(!)929 1826 y Fj(+)1009 1814 y Fr(is)g(a)f(factor)h(state)g(and)e Fn(Ep\()p Ft(!)1897 1826 y Fj(+)1952 1814 y Fn(\))31 b Ft(>)g Fn(0)p Fr(,)25 b(then)f Ft(!)2419 1826 y Fj(+)2505 1814 y Fq(?)30 b Ft(!)s Fr(.)38 b(If)25 b Ft(!)j Fr(is)d(also)f(a)291 1914 y(factor)c(state)o(,)g(then)g Ft(!)923 1926 y Fj(+)998 1914 y Fr(and)g Ft(!)j Fr(ar)m(e)d(disjoint.)415 2076 y Fx(Certain)29 b(structural)g(properties)f(can)h(be)g(characterized)e (in)j(terms)f(of)g(the)g(standard)f(Liou-)291 2175 y(villean.)51 b(Let)29 b Ft(L)g Fx(be)g(the)g(standard)f(Liouvillean)f(associated)i (to)g Ft(\034)39 b Fx(and)29 b Ft(L)2525 2187 y Fo(V)2611 2175 y Fx(the)h(standard)d(Li-)291 2275 y(ouvillean)22 b(associated)h(to)h Ft(\034)1114 2287 y Fo(V)1172 2275 y Fx(.)35 b(By)24 b(the)g(well-kno)n(wn)d(Araki')-5 b(s)24 b(perturbation)d(formula,)h(one)h(has)291 2374 y Ft(L)348 2386 y Fo(V)428 2374 y Fn(=)f Ft(L)c Fn(+)g Ft(V)38 b Fq(\000)18 b Ft(J)8 b(V)19 b(J)28 b Fx(\(see)21 b([DJP,)f(Pi)q(]\).)291 2552 y Fu(Theor)o(em)g(3.6)40 b Fr(Assume)20 b(that)g Ft(!)k Fr(is)d(modular)-9 b(.)351 2713 y(\(i\))40 b(Under)24 b(the)h(assumptions)e(of)i(Theor)m(em)f(3.3,)h(if)g Fn(Ker)13 b Ft(L)2103 2725 y Fo(V)2191 2713 y Fq(6)p Fn(=)30 b Fq(f)p Fn(0)p Fq(g)p Fr(,)24 b(then)g(it)h(is)h(one-dimen-)470 2813 y(sional)20 b(and)f(ther)m(e)i(e)n(xists)g(a)g(unique)d(normal,)i Ft(\034)1858 2825 y Fo(V)1916 2813 y Fr(-in)m(variant)e(state)j Ft(!)2495 2825 y Fo(V)2573 2813 y Fr(suc)o(h)e(that)1467 2990 y Fn(\006)1527 3002 y Fj(+)1582 2990 y Fn(\()p Ft(!)s(;)14 b(\034)1742 3002 y Fo(V)1800 2990 y Fn(\))23 b(=)g Fq(f)p Ft(!)2037 3002 y Fo(V)2094 2990 y Fq(g)p Ft(:)327 3200 y Fr(\(ii\))41 b(If)21 b Fn(Ker)13 b Ft(L)747 3212 y Fo(V)827 3200 y Fn(=)23 b Fq(f)p Fn(0)p Fq(g)p Fr(,)c(then)g(any)h (NESS)g(in)g Fn(\006)1740 3212 y Fj(+)1795 3200 y Fn(\()p Ft(!)s(;)14 b(\034)1955 3212 y Fo(V)2013 3200 y Fn(\))21 b Fr(is)g(pur)m(ely)f(singular)-9 b(.)304 3363 y(\(iii\))41 b(If)36 b Fn(Ker)13 b Ft(L)762 3375 y Fo(V)856 3363 y Fr(contains)34 b(a)i(separ)o(ating)f(vector)g(for)i Fk(M)2106 3375 y Fo(!)2154 3363 y Fr(,)j(then)35 b Fn(\006)2454 3375 y Fj(+)2509 3363 y Fn(\()p Ft(!)s(;)14 b(\034)2669 3375 y Fo(V)2727 3363 y Fn(\))37 b Fr(contains)d(a)470 3463 y(unique)19 b(state)i Ft(!)947 3475 y Fj(+)1022 3463 y Fr(and)f(this)g(state)h(is)g Ft(!)s Fr(-normal.)291 3700 y Fm(3.4)99 b Fg(C)592 3664 y Ff(\003)631 3700 y Fm(-scattering)26 b(and)f(NESS)291 3855 y Fx(Let)i Fn(\()p Fq(O)r Ft(;)14 b(\034)9 b Fn(\))30 b Fx(be)e(a)g Ft(C)910 3825 y Fs(\003)948 3855 y Fx(-dynamical)e(system)h(and)g Ft(V)47 b Fx(a)28 b(local)g(perturbation.)44 b(The)27 b(abstract)h Ft(C)3089 3825 y Fs(\003)3127 3855 y Fx(-)291 3955 y(scattering)19 b(approach)f(to)i(the)h(study)e(of)h(NESS)h(is)g (based)f(on)g(the)g(follo)n(wing)e(assumption:)291 4150 y Fu(Assumption)j(\(S\))f Fx(The)g(strong)f(limit)1319 4327 y Ft(\013)1372 4292 y Fj(+)1372 4352 y Fo(V)1453 4327 y Fq(\021)k Fn(s)18 b Fq(\000)g Fn(lim)1587 4384 y Fo(t)p Fs(!1)1804 4327 y Ft(\034)1849 4293 y Fs(\000)p Fo(t)1949 4327 y Fq(\016)g Ft(\034)2054 4293 y Fo(t)2045 4348 y(V)2103 4327 y Ft(;)291 4534 y Fx(e)o(xists.)415 4729 y(The)31 b(map)g Ft(\013)804 4694 y Fj(+)804 4754 y Fo(V)894 4729 y Fx(is)h(an)g(isometric)f Fq(\003)p Fx(-endomorphism)c(of)k Fq(O)r Fx(,)k(and)c(is)h(often)f(called)g (M\370ller)291 4829 y(morphism.)23 b Ft(\013)727 4793 y Fj(+)727 4853 y Fo(V)805 4829 y Fx(is)e(one-to-one)d(b)n(ut)i(it)h (is)g(generally)e(not)h(onto,)f(namely)1369 5006 y Fq(O)1435 5018 y Fj(+)1513 5006 y Fq(\021)k Fn(Ran)13 b Ft(\013)1816 4971 y Fj(+)1816 5031 y Fo(V)1897 5006 y Fq(6)p Fn(=)23 b Fq(O)r Ft(:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 739 232 a Fx(12)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx(Since)25 b Ft(\013)1002 488 y Fj(+)1002 547 y Fo(V)1082 523 y Fq(\016)c Ft(\034)1190 493 y Fo(t)1181 546 y(V)1271 523 y Fn(=)31 b Ft(\034)1412 493 y Fo(t)1464 523 y Fq(\016)21 b Ft(\013)1580 488 y Fj(+)1580 547 y Fo(V)1638 523 y Fx(,)26 b(the)f(pair)g Fn(\()p Fq(O)2065 535 y Fj(+)2120 523 y Ft(;)14 b(\034)9 b Fn(\))27 b Fx(is)f(a)f Ft(C)2469 493 y Fs(\003)2507 523 y Fx(-dynamical)e(system)i(and)g Ft(\013)3361 488 y Fj(+)3361 547 y Fo(V)3444 523 y Fx(is)h(an)739 623 y(isomorphism)18 b(between)i(the)g(dynamical)e(systems)j Fn(\()p Fq(O)r Ft(;)14 b(\034)2436 635 y Fo(V)2495 623 y Fn(\))21 b Fx(and)e Fn(\()p Fq(O)2786 635 y Fj(+)2842 623 y Ft(;)14 b(\034)9 b Fn(\))p Fx(.)863 722 y(If)31 b(the)g(reference)f(state)h Ft(!)k Fx(is)d Ft(\034)9 b Fx(-in)m(v)n(ariant,)32 b(then)e Ft(!)2425 734 y Fj(+)2523 722 y Fn(=)43 b Ft(!)29 b Fq(\016)d Ft(\013)2833 687 y Fj(+)2833 747 y Fo(V)2923 722 y Fx(is)32 b(the)f(unique)e(NESS)739 822 y(associated)20 b(to)g Ft(!)k Fx(and)19 b Ft(\034)1438 834 y Fo(V)1517 822 y Fx(and)1772 980 y Fn(w)1833 946 y Fs(\003)1890 980 y Fq(\000)f Fn(lim)1852 1037 y Fo(t)p Fs(!1)2102 980 y Ft(!)j Fq(\016)d Ft(\034)2280 946 y Fo(t)2271 1001 y(V)2352 980 y Fn(=)23 b Ft(!)2492 992 y Fj(+)2547 980 y Ft(:)739 1175 y Fx(Note)d(in)g(particular)f(that)i (if)f Ft(!)j Fx(is)f(a)e Fn(\()p Ft(\034)5 b(;)14 b(\014)t Fn(\))p Fx(-KMS)21 b(state,)g(then)f Ft(!)2603 1187 y Fj(+)2678 1175 y Fx(is)h(a)g Fn(\()p Ft(\034)2880 1187 y Fo(V)2938 1175 y Ft(;)14 b(\014)t Fn(\))p Fx(-KMS)21 b(state.)863 1274 y(The)i(map)f Ft(\013)1235 1239 y Fj(+)1235 1299 y Fo(V)1317 1274 y Fx(is)i(the)f(algebraic)e(analog)h(of)h(the)g (w)o(a)n(v)o(e)g(operator)e(in)i(Hilbert)g(space)f(scatter)n(-)739 1374 y(ing)j(theory)-5 b(.)41 b(A)26 b(simple)g(and)f(useful)g(result)h (in)g(Hilbert)g(space)g(scattering)f(theory)g(is)h(the)g(Cook)739 1474 y(criterion)19 b(for)g(the)i(e)o(xistence)e(of)h(the)g(w)o(a)n(v)o (e)g(operator)-5 b(.)24 b(Its)d(algebraic)e(analog)g(is:)739 1632 y Fu(Pr)o(oposition)g(3.7)100 b Fr(\(i\))41 b(Assume)25 b(that)h(ther)m(e)f(e)n(xists)i(a)e(dense)g(subset)h Fq(O)2913 1644 y Fj(0)2983 1632 y Fq(\032)32 b(O)c Fr(suc)o(h)d(that)g (for)919 1732 y(all)20 b Ft(A)j Fq(2)h(O)1257 1744 y Fj(0)1294 1732 y Fr(,)1804 1757 y Fl(Z)1887 1777 y Fs(1)1850 1945 y Fj(0)1971 1870 y Fq(k)p Fn([)p Ft(V)5 b(;)14 b(\034)2171 1835 y Fo(t)2162 1890 y(V)2220 1870 y Fn(\()p Ft(A)p Fn(\)])p Fq(k)g Fn(d)p Ft(t)23 b(<)g Fq(1)p Ft(:)726 b Fx(\(3.5\))919 2045 y Fr(Then)20 b(Assumption)f Fx(\(S\))h Fr(holds.)776 2202 y(\(ii\))41 b(Assume)20 b(that)g(ther)m(e)g(e)n (xists)i(a)e(dense)g(subset)g Fq(O)2301 2214 y Fj(1)2362 2202 y Fq(\032)i(O)h Fr(suc)o(h)d(that)g(for)h(all)f Ft(A)j Fq(2)h(O)3316 2214 y Fj(1)3353 2202 y Fr(,)1814 2290 y Fl(Z)1897 2311 y Fs(1)1860 2479 y Fj(0)1981 2403 y Fq(k)p Fn([)p Ft(V)5 b(;)14 b(\034)2181 2369 y Fo(t)2210 2403 y Fn(\()p Ft(A)p Fn(\)])p Fq(k)g Fn(d)p Ft(t)24 b(<)e Fq(1)p Ft(:)736 b Fx(\(3.6\))919 2617 y Fr(Then)20 b Fq(O)1172 2629 y Fj(+)1250 2617 y Fn(=)i Fq(O)i Fr(and)19 b Ft(\013)1625 2582 y Fj(+)1625 2642 y Fo(V)1703 2617 y Fr(is)j(a)e Fq(\003)p Fr(-automorphism)d(of)k Fq(O)r Fr(.)739 2776 y Fu(Pr)o(oof)o(.)d Fx(F)o(or)i(all)h Ft(A)i Fq(2)g(O)g Fx(we)e(ha)n(v)o(e)1142 2974 y Ft(\034)1187 2940 y Fs(\000)p Fo(t)1264 2948 y Fi(2)1320 2974 y Fq(\016)d Ft(\034)1425 2937 y Fo(t)1450 2945 y Fi(2)1416 2999 y Fo(V)1487 2974 y Fn(\()p Ft(A)p Fn(\))h Fq(\000)f Ft(\034)1760 2940 y Fs(\000)p Fo(t)1837 2948 y Fi(1)1893 2974 y Fq(\016)g Ft(\034)1998 2937 y Fo(t)2023 2945 y Fi(1)1989 2999 y Fo(V)2060 2974 y Fn(\()p Ft(A)p Fn(\))24 b(=)f(i)2335 2861 y Fl(Z)2418 2882 y Fo(t)2443 2890 y Fi(2)2381 3050 y Fo(t)2406 3058 y Fi(1)2493 2974 y Ft(\034)2538 2940 y Fs(\000)p Fo(t)2620 2974 y Fn(\([)p Ft(V)5 b(;)14 b(\034)2810 2940 y Fo(t)2801 2995 y(V)2859 2974 y Fn(\()p Ft(A)p Fn(\)]\))g(d)p Ft(t;)1142 3314 y(\034)1187 3277 y Fs(\000)p Fo(t)1264 3285 y Fi(2)1178 3338 y Fo(V)1320 3314 y Fq(\016)k Ft(\034)1425 3280 y Fo(t)1450 3288 y Fi(2)1487 3314 y Fn(\()p Ft(A)p Fn(\))h Fq(\000)f Ft(\034)1760 3277 y Fs(\000)p Fo(t)1837 3285 y Fi(1)1751 3338 y Fo(V)1893 3314 y Fq(\016)g Ft(\034)1998 3280 y Fo(t)2023 3288 y Fi(1)2060 3314 y Fn(\()p Ft(A)p Fn(\))24 b(=)f Fq(\000)p Fn(i)2400 3201 y Fl(Z)2482 3222 y Fo(t)2507 3230 y Fi(2)2445 3390 y Fo(t)2470 3398 y Fi(1)2558 3314 y Ft(\034)2603 3279 y Fs(\000)p Fo(t)2594 3338 y(V)2685 3314 y Fn(\([)p Ft(V)5 b(;)14 b(\034)2875 3280 y Fo(t)2905 3314 y Fn(\()p Ft(A)p Fn(\)]\))g(d)p Ft(t;)3444 3143 y Fx(\(3.7\))739 3521 y(and)20 b(so)1186 3717 y Fq(k)p Ft(\034)1273 3683 y Fs(\000)p Fo(t)1350 3691 y Fi(2)1405 3717 y Fq(\016)e Ft(\034)1510 3680 y Fo(t)1535 3688 y Fi(2)1501 3742 y Fo(V)1572 3717 y Fn(\()p Ft(A)p Fn(\))h Fq(\000)f Ft(\034)1845 3683 y Fs(\000)p Fo(t)1922 3691 y Fi(1)1978 3717 y Fq(\016)g Ft(\034)2083 3680 y Fo(t)2108 3688 y Fi(1)2074 3742 y Fo(V)2145 3717 y Fn(\()p Ft(A)p Fn(\))p Fq(k)24 b(\024)2445 3604 y Fl(Z)2528 3625 y Fo(t)2553 3633 y Fi(2)2491 3793 y Fo(t)2516 3801 y Fi(1)2604 3717 y Fq(k)p Fn([)p Ft(V)5 b(;)14 b(\034)2804 3683 y Fo(t)2795 3738 y(V)2852 3717 y Fn(\()p Ft(A)p Fn(\)])p Fq(k)g Fn(d)p Ft(t;)1186 4057 y Fq(k)p Ft(\034)1273 4020 y Fs(\000)p Fo(t)1350 4028 y Fi(2)1264 4082 y Fo(V)1405 4057 y Fq(\016)k Ft(\034)1510 4023 y Fo(t)1535 4031 y Fi(2)1572 4057 y Fn(\()p Ft(A)p Fn(\))h Fq(\000)f Ft(\034)1845 4020 y Fs(\000)p Fo(t)1922 4028 y Fi(1)1836 4082 y Fo(V)1978 4057 y Fq(\016)g Ft(\034)2083 4023 y Fo(t)2108 4031 y Fi(1)2145 4057 y Fn(\()p Ft(A)p Fn(\))p Fq(k)24 b(\024)2424 3944 y Fl(Z)2507 3965 y Fo(t)2532 3973 y Fi(2)2470 4133 y Fo(t)2495 4141 y Fi(1)2583 4057 y Fq(k)p Fn([)p Ft(V)5 b(;)14 b(\034)2783 4023 y Fo(t)2812 4057 y Fn(\()p Ft(A)p Fn(\)])p Fq(k)g Fn(d)p Ft(t:)3444 3886 y Fx(\(3.8\))739 4265 y(T)-7 b(o)23 b(pro)o(v)o(e)d(P)o(art)46 b(\(i\),)22 b(note)g(that)h(\(3.5\))e(and)h(the)h(\002rst)g(estimate)g (in)f(\(3.8\))f(imply)h(that)h(for)f Ft(A)27 b Fq(2)h(O)3566 4277 y Fj(0)739 4365 y Fx(the)20 b(norm)f(limit)1687 4464 y Ft(\013)1740 4429 y Fj(+)1740 4489 y Fo(V)1797 4464 y Fn(\()p Ft(A)p Fn(\))24 b Fq(\021)44 b Fn(lim)2035 4514 y Fo(t)p Fs(!1)2206 4464 y Ft(\034)2251 4430 y Fs(\000)p Fo(t)2352 4464 y Fq(\016)18 b Ft(\034)2457 4430 y Fo(t)2448 4485 y(V)2505 4464 y Fn(\()p Ft(A)p Fn(\))p Ft(;)739 4632 y Fx(e)o(xists.)26 b(Since)20 b Fq(O)1245 4644 y Fj(0)1303 4632 y Fx(is)i(dense)e(and)g Ft(\034)1776 4602 y Fs(\000)p Fo(t)1876 4632 y Fq(\016)e Ft(\034)1981 4602 y Fo(t)1972 4655 y(V)2051 4632 y Fx(is)j(isometric,)f(the)g(limit)h(e)o (xists)g(for)f(all)h Ft(A)i Fq(2)h(O)r Fx(,)d(and)739 4732 y Ft(\013)792 4696 y Fj(+)792 4756 y Fo(V)870 4732 y Fx(is)g(a)f Fq(\003)p Fx(-morphism)d(of)j Fq(O)r Fx(.)25 b(T)-7 b(o)20 b(pro)o(v)o(e)e(P)o(art)40 b(\(ii\))20 b(note)f(that)h(the)g(second)f(estimate)h(in)g(\(3.8\))e(and)739 4831 y(\(3.6\))h(imply)g(that)i(the)f(norm)f(limit)1698 4990 y Ft(\014)1749 4954 y Fj(+)1745 5014 y Fo(V)1805 4990 y Fn(\()p Ft(A)p Fn(\))24 b Fq(\021)43 b Fn(lim)2042 5039 y Fo(t)p Fs(!1)2214 4990 y Ft(\034)2259 4954 y Fs(\000)p Fo(t)2250 5014 y(V)2359 4990 y Fq(\016)18 b Ft(\034)2464 4955 y Fo(t)2494 4990 y Fn(\()p Ft(A)p Fn(\))p Ft(;)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(13)291 523 y(also)20 b(e)o(xists)h(for)e(all)i Ft(A)i Fq(2)h(O)r Fx(.)i(Since)20 b Ft(\013)1412 488 y Fj(+)1412 547 y Fo(V)1488 523 y Fq(\016)e Ft(\014)1599 488 y Fj(+)1595 547 y Fo(V)1655 523 y Fn(\()p Ft(A)p Fn(\))23 b(=)g Ft(A)p Fx(,)e Ft(\013)2049 488 y Fj(+)2049 547 y Fo(V)2127 523 y Fx(is)g(a)g Fq(\003)p Fx(-automorphism)c(of)j Fq(O)r Fx(.)p Fe(\003)415 668 y Fx(Until)i(the)f(end)g(of)h(this)g(subsection) e(we)i(will)g(assume)g(that)f(the)h(Assumption)e(\(S\))i(holds)f(and) 291 767 y(that)f Ft(!)j Fx(is)e Ft(\034)9 b Fx(-in)m(v)n(ariant.)415 867 y(Let)39 b Fn(~)-49 b Ft(!)48 b Fq(\021)c Ft(!)k Fe(\026)d Fq(O)1013 879 y Fj(+)1100 867 y Fx(and)32 b(let)g Fn(\()p Fq(H)1475 879 y Fj(~)-38 b Fo(!)1519 867 y Ft(;)14 b(\031)1608 879 y Fj(~)-38 b Fo(!)1651 867 y Ft(;)14 b Fn(\012)1753 879 y Fj(~)-38 b Fo(!)1796 867 y Fn(\))33 b Fx(be)f(the)g(GNS-representation)e(of)h Fq(O)2970 879 y Fj(+)3058 867 y Fx(as-)291 967 y(sociated)f(to)37 b Fn(~)-49 b Ft(!)s Fx(.)57 b(Ob)o(viously)-5 b(,)31 b(if)f Ft(\013)1350 931 y Fj(+)1350 991 y Fo(V)1439 967 y Fx(is)i(an)e (automorphism,)g(then)37 b Fn(~)-48 b Ft(!)45 b Fn(=)c Ft(!)s Fx(.)57 b(W)-7 b(e)31 b(denote)f(by)291 1066 y Fn(\()p Fq(H)393 1078 y Fo(!)435 1086 y Fi(+)486 1066 y Ft(;)14 b(\031)570 1078 y Fo(!)612 1086 y Fi(+)663 1066 y Ft(;)g Fn(\012)760 1078 y Fo(!)802 1086 y Fi(+)852 1066 y Fn(\))30 b Fx(the)f(GNS)h(representation)d(of)h Fq(O)k Fx(associated)d(to)g Ft(!)2454 1078 y Fj(+)2509 1066 y Fx(.)52 b(Let)29 b Ft(L)2784 1078 y Fj(~)-38 b Fo(!)2856 1066 y Fx(and)29 b Ft(L)3063 1078 y Fo(!)3105 1086 y Fi(+)291 1166 y Fx(be)d(the)g(standard)f(Liouvilleans)g (associated,)j(respecti)n(v)o(ely)-5 b(,)25 b(to)i Fn(\()p Fq(O)2296 1178 y Fj(+)2351 1166 y Ft(;)14 b(\034)5 b(;)21 b Fn(~)-49 b Ft(!)s Fn(\))27 b Fx(and)f Fn(\()p Fq(O)r Ft(;)14 b(\034)2900 1178 y Fo(V)2958 1166 y Ft(;)g(!)3047 1178 y Fj(+)3102 1166 y Fn(\))p Fx(.)291 1266 y(Recall)20 b(that)h Ft(L)731 1278 y Fj(~)-38 b Fo(!)794 1266 y Fx(is)21 b(the)f(unique)f(self-adjoint)g(operator)f(on)i Fq(H)2114 1278 y Fj(~)-38 b Fo(!)2178 1266 y Fx(such)20 b(that)g(for)g Ft(A)j Fq(2)h(O)2844 1278 y Fj(+)2899 1266 y Fx(,)881 1437 y Ft(L)943 1449 y Fj(~)-38 b Fo(!)986 1437 y Fn(\012)1051 1449 y Fj(~)g Fo(!)1117 1437 y Fn(=)22 b(0)p Ft(;)180 b(\031)1501 1449 y Fj(~)-38 b Fo(!)1544 1437 y Fn(\()p Ft(\034)1621 1402 y Fo(t)1651 1437 y Fn(\()p Ft(A)p Fn(\)\))25 b(=)d(e)1958 1402 y Fj(i)p Fo(tL)2053 1410 y Fi(~)-33 b Fh(!)2094 1437 y Ft(\031)2146 1449 y Fj(~)-38 b Fo(!)2190 1437 y Fn(\()p Ft(A)p Fn(\)e)2353 1402 y Fs(\000)p Fj(i)p Fo(tL)2500 1410 y Fi(~)-33 b Fh(!)2541 1437 y Ft(;)291 1608 y Fx(and)19 b(similarly)h(for)g Ft(L)922 1620 y Fo(!)964 1628 y Fi(+)1014 1608 y Fx(.)291 1764 y Fu(Pr)o(oposition)e (3.8)41 b Fr(The)20 b(map)1160 1935 y Ft(U)9 b(\031)1278 1947 y Fj(~)-38 b Fo(!)1321 1935 y Fn(\()p Ft(\013)1406 1900 y Fj(+)1406 1960 y Fo(V)1464 1935 y Fn(\()p Ft(A)p Fn(\)\)\012)1687 1947 y Fj(~)g Fo(!)1755 1935 y Fn(=)22 b Ft(\031)1889 1947 y Fo(!)1931 1955 y Fi(+)1982 1935 y Fn(\()p Ft(A)p Fn(\)\012)2168 1947 y Fo(!)2210 1955 y Fi(+)2262 1935 y Ft(;)291 2106 y Fr(e)n(xtends)e(to)g(a)g(unitary)g Ft(U)32 b Fn(:)23 b Fq(H)1177 2118 y Fj(~)-38 b Fo(!)1243 2106 y Fq(!)23 b(H)1419 2118 y Fo(!)1461 2126 y Fi(+)1533 2106 y Fr(whic)o(h)d(intertwines)g Ft(L)2202 2118 y Fj(~)-38 b Fo(!)2266 2106 y Fr(and)19 b Ft(L)2468 2118 y Fo(!)2510 2126 y Fi(+)2560 2106 y Fr(,)i(i.e)o(.,)1468 2277 y Ft(U)9 b(L)1596 2289 y Fj(~)-38 b Fo(!)1661 2277 y Fn(=)23 b Ft(L)1806 2289 y Fo(!)1848 2297 y Fi(+)1898 2277 y Ft(U:)291 2605 y Fu(Pr)o(oof)o(.)17 b Fx(Set)k Ft(\031)704 2575 y Fs(0)706 2627 y Fj(~)-38 b Fo(!)749 2605 y Fn(\()p Ft(A)p Fn(\))24 b Fq(\021)f Ft(\031)1039 2617 y Fj(~)-38 b Fo(!)1082 2605 y Fn(\()p Ft(\013)1167 2570 y Fj(+)1167 2630 y Fo(V)1225 2605 y Fn(\()p Ft(A)p Fn(\)\))22 b Fx(and)d(note)h (that)g Ft(\031)1904 2575 y Fs(0)1906 2627 y Fj(~)-38 b Fo(!)1949 2605 y Fn(\()p Fq(O)r Fn(\)\012)2146 2617 y Fj(~)g Fo(!)2213 2605 y Fn(=)23 b Ft(\031)2353 2617 y Fj(~)-38 b Fo(!)2396 2605 y Fn(\()p Fq(O)2494 2617 y Fj(+)2550 2605 y Fn(\)\012)2647 2617 y Fj(~)g Fo(!)2690 2605 y Fx(,)21 b(so)f(that)g Fn(\012)3036 2617 y Fj(~)-38 b Fo(!)3099 2605 y Fx(is)291 2705 y(c)o(yclic)19 b(for)h Ft(\031)676 2675 y Fs(0)678 2727 y Fj(~)-38 b Fo(!)721 2705 y Fn(\()p Fq(O)r Fn(\))p Fx(.)27 b(Since)389 2876 y Ft(!)441 2888 y Fj(+)496 2876 y Fn(\()p Ft(A)p Fn(\))c(=)g Ft(!)s Fn(\()p Ft(\013)873 2840 y Fj(+)873 2900 y Fo(V)931 2876 y Fn(\()p Ft(A)p Fn(\)\))h(=)29 b(~)-48 b Ft(!)r Fn(\()p Ft(\013)1340 2840 y Fj(+)1340 2900 y Fo(V)1398 2876 y Fn(\()p Ft(A)p Fn(\)\))24 b(=)f(\(\012)1765 2888 y Fj(~)-38 b Fo(!)1808 2876 y Ft(;)14 b(\031)1897 2888 y Fj(~)-38 b Fo(!)1941 2876 y Fn(\()p Ft(\013)2026 2840 y Fj(+)2026 2900 y Fo(V)2084 2876 y Fn(\()p Ft(A)p Fn(\)\)\012)2307 2888 y Fj(~)g Fo(!)2351 2876 y Fn(\))23 b(=)g(\(\012)2591 2888 y Fj(~)-38 b Fo(!)2634 2876 y Ft(;)14 b(\031)2721 2841 y Fs(0)2723 2896 y Fj(~)-38 b Fo(!)2766 2876 y Fn(\()p Ft(A)p Fn(\)\012)2957 2888 y Fj(~)g Fo(!)3001 2876 y Fn(\))p Ft(;)291 3047 y Fn(\()p Fq(H)398 3059 y Fj(~)g Fo(!)441 3047 y Ft(;)14 b(\031)528 3017 y Fs(0)530 3069 y Fj(~)-38 b Fo(!)573 3047 y Ft(;)14 b Fn(\012)675 3059 y Fj(~)-38 b Fo(!)718 3047 y Fn(\))29 b Fx(is)f(also)g(a)g(GNS)h (representation)c(of)j Fq(O)i Fx(associated)e(to)g Ft(!)2491 3059 y Fj(+)2545 3047 y Fx(.)48 b(Since)28 b(GNS)g(rep-)291 3146 y(resentations)j(associated)h(to)h(the)f(same)h(state)g(are)g (unitarily)e(equi)n(v)n(alent,)i(there)f(is)i(a)f(unitary)291 3246 y Ft(U)e Fn(:)23 b Fq(H)500 3258 y Fj(~)-38 b Fo(!)567 3246 y Fq(!)23 b(H)743 3258 y Fo(!)785 3266 y Fi(+)856 3246 y Fx(such)d(that)h Ft(U)9 b Fn(\012)1306 3258 y Fj(~)-38 b Fo(!)1372 3246 y Fn(=)22 b(\012)1519 3258 y Fo(!)1561 3266 y Fi(+)1633 3246 y Fx(and)1350 3417 y Ft(U)9 b(\031)1466 3383 y Fs(0)1468 3438 y Fj(~)-38 b Fo(!)1511 3417 y Fn(\()p Ft(A)p Fn(\))24 b(=)f Ft(\031)1796 3429 y Fo(!)1838 3437 y Fi(+)1889 3417 y Fn(\()p Ft(A)p Fn(\))p Ft(U:)291 3588 y Fx(Finally)-5 b(,)19 b(the)h(identities)607 3760 y Ft(U)9 b Fn(e)710 3725 y Fj(i)p Fo(tL)805 3733 y Fi(~)-33 b Fh(!)846 3760 y Ft(\031)896 3725 y Fs(0)898 3780 y Fj(~)-38 b Fo(!)941 3760 y Fn(\()p Ft(A)p Fn(\)\012)1132 3772 y Fj(~)g Fo(!)1199 3760 y Fn(=)23 b Ft(U)9 b(\031)1405 3772 y Fj(~)-38 b Fo(!)1448 3760 y Fn(\()p Ft(\034)1525 3725 y Fo(t)1555 3760 y Fn(\()p Ft(\013)1640 3724 y Fj(+)1640 3784 y Fo(V)1698 3760 y Fn(\()p Ft(A)p Fn(\)\)\)\012)1953 3772 y Fj(~)g Fo(!)2021 3760 y Fn(=)22 b Ft(U)9 b(\031)2226 3772 y Fj(~)-38 b Fo(!)2269 3760 y Fn(\()p Ft(\013)2354 3724 y Fj(+)2354 3784 y Fo(V)2413 3760 y Fn(\()p Ft(\034)2490 3725 y Fo(t)2481 3780 y(V)2539 3760 y Fn(\()p Ft(A)p Fn(\)\)\)\012)2794 3772 y Fj(~)g Fo(!)1199 3971 y Fn(=)23 b Ft(\031)1334 3983 y Fo(!)1376 3991 y Fi(+)1427 3971 y Fn(\()p Ft(\034)1504 3937 y Fo(t)1495 3991 y(V)1553 3971 y Fn(\()p Ft(A)p Fn(\)\)\012)1771 3983 y Fo(!)1813 3991 y Fi(+)1888 3971 y Fn(=)g(e)2013 3933 y Fj(i)p Fo(tL)2103 3941 y Fh(!)2140 3954 y Fi(+)2194 3971 y Ft(\031)2241 3983 y Fo(!)2283 3991 y Fi(+)2334 3971 y Fn(\()p Ft(A)p Fn(\)\012)2520 3983 y Fo(!)2562 3991 y Fi(+)1199 4182 y Fn(=)g(e)1324 4145 y Fj(i)p Fo(tL)1414 4153 y Fh(!)1451 4166 y Fi(+)1505 4182 y Ft(U)9 b(\031)1621 4148 y Fs(0)1623 4203 y Fj(~)-38 b Fo(!)1666 4182 y Fn(\()p Ft(A)p Fn(\)\012)1857 4194 y Fj(~)g Fo(!)1901 4182 y Ft(;)291 4348 y Fx(yield)19 b(that)i Ft(U)29 b Fx(intertwines)20 b Ft(L)1161 4360 y Fj(~)-38 b Fo(!)1224 4348 y Fx(and)20 b Ft(L)1422 4364 y Fo(!)1466 4347 y Fi(+)1516 4348 y Fx(.)26 b Fe(\003)415 4493 y Fx(W)-7 b(e)21 b(\002nish)g(this)f(subsection)g(with:)291 4664 y Fu(Pr)o(oposition)e(3.9)101 b Fr(\(i\))40 b(Assume)21 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%%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin end %%EndSetup TeXDict begin 1 0 bop Black Black Black 515 3893 a @beginspecial 0 @llx 0 @lly 232 @urx 292 @ury 2320 @rwi @setspecial %%BeginDocument: pre.ps %!PS-Adobe-2.0 EPSF-2.0 %%Title: pre.fig %%Creator: fig2dev Version 3.2 Patchlevel 4 %%CreationDate: Fri Feb 18 21:35:00 2005 %%For: cap@weyl.cap.lnet (Claude-Alain Pillet,,,) %%BoundingBox: 0 0 232 292 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def 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338.1 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proc char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0 slj 0 slc 0.06000 0.06000 sc % % Fig objects follow % % % here starts figure with depth 70 % Polyline 15.000 slw n 1926 1759 m 1701 1759 1701 3002 225 arcto 4 {pop} repeat 1701 3227 2944 3227 225 arcto 4 {pop} repeat 3169 3227 3169 1984 225 arcto 4 {pop} repeat 3169 1759 1926 1759 225 arcto 4 {pop} repeat cp gs 0.80 setgray ef gr gs col0 s gr % Polyline 0.000 slw n 646 4099 m 4408 4099 l 4408 5613 l 646 5613 l cp gs 0.80 setgray ef gr % Polyline n 646 796 m 4454 796 l 4454 1209 l 646 1209 l cp gs 0.80 setgray ef gr % here ends figure; % % here starts figure with depth 60 % Polyline 15.000 slw n 646 796 m 4454 796 l gs col0 s gr % Polyline n 646 1209 m 4454 1209 l gs col0 s gr % Polyline n 646 5613 m 646 4099 l 4408 4099 l 4408 5613 l gs col0 s gr % here ends figure; % % here starts figure with depth 50 % Ellipse 7.500 slw gs 2099 1530 tr -90.000 rot n 0 0 658 106 0 360 DrawEllipse 90.000 rot gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 96.07 94.93] PATmp PATsp ef gr PATusp gs col0 s gr gr % Ellipse gs 2756 3664 tr -90.000 rot n 0 0 986 160 0 360 DrawEllipse 90.000 rot gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 118.00 233.60] PATmp PATsp ef gr PATusp gs col0 s gr gr % here ends figure; $F2psEnd rs end showpage %%EndDocument @endspecial 0 0 0 TeXcolorrgb 1375 2368 a Fc(S)p Black 0 0 0 TeXcolorrgb 1375 3538 a(R)1476 3556 y Fb(1)p Black 0 0 0 TeXcolorrgb 1375 1611 a Fc(R)1476 1629 y Fb(2)p Black 0 0 0 TeXcolorrgb 1053 1863 a Fa(V)1121 1881 y Fb(2)p Black 0 0 0 TeXcolorrgb 1673 2964 a Fa(V)1741 2982 y Fb(1)p Black Black Black Black eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF %%EndDocument @endspecial 599 4717 a(Figure)20 b(1:)25 b(Junctions)20 b Ft(V)1310 4729 y Fj(1)1347 4717 y Fx(,)h Ft(V)1437 4729 y Fj(2)1495 4717 y Fx(between)f(the)g(system)g Fq(S)28 b Fx(and)19 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1251 y Fn(\()p Ft(V)c Fn(\)\))24 b(=)2412 1148 y Fo(M)2386 1173 y Fl(X)2389 1349 y Fo(j)s Fj(=1)2520 1251 y Ft(\034)2565 1217 y Fo(t)2556 1272 y(V)2614 1251 y Fn(\()p Ft(\016)2683 1263 y Fo(j)2718 1251 y Fn(\()p Ft(V)2798 1263 y Fo(j)2834 1251 y Fn(\)\))p Ft(:)291 1523 y Fx(Thus,)f(we)g(can)g(identify)f(the)h(observ)n(able)e (describing)h(the)h(heat)g(\003ux)g(out)g(of)f(the)i Ft(j)5 b Fx(-th)22 b(reserv)n(oir)291 1623 y(as)1163 1732 y Fn(\010)1223 1744 y Fo(j)1281 1732 y Fn(=)h Ft(\016)1406 1744 y Fo(j)1441 1732 y Fn(\()p Ft(V)c Fn(\))k(=)g Ft(\016)1720 1744 y Fo(j)1755 1732 y Fn(\()p Ft(V)1835 1744 y Fo(j)1870 1732 y Fn(\))h(=)e Ft(\016)2050 1744 y Fs(R)2111 1732 y Fn(\()p Ft(V)2191 1744 y Fo(j)2227 1732 y Fn(\))p Ft(:)291 1892 y Fx(W)-7 b(e)21 b(note)f(that)g(if)g Fk(r)h Fx(is)h(a)e(time-re)n (v)o(ersal,)e(then)i Fk(r)p Fn(\(\010)1751 1904 y Fo(j)1787 1892 y Fn(\))j(=)g Fq(\000)p Fn(\010)2055 1904 y Fo(j)2089 1892 y Fx(.)j(The)20 b(ener)o(gy)e(balance)h(equation)1344 2061 y Fo(M)1319 2086 y Fl(X)1321 2263 y Fo(j)s Fj(=1)1452 2165 y Fn(\010)1512 2177 y Fo(j)1570 2165 y Fn(=)k Ft(\016)1695 2177 y Fo(V)1753 2165 y Fn(\()p Ft(H)1854 2177 y Fs(S)1922 2165 y Fn(+)18 b Ft(V)g Fn(\))p Ft(;)291 2437 y Fx(yields)23 b(the)g(conserv)n(ation)e(of)i(ener)o(gy)e(\(the)i(\002rst)h(la)o(w)f (of)g(thermodynamics\):)28 b(for)22 b(an)o(y)h Ft(\034)2901 2449 y Fo(V)2959 2437 y Fx(-inv)n(a-)291 2536 y(riant)d(state)g Ft(\021)s Fx(,)1492 2619 y Fo(M)1466 2644 y Fl(X)1469 2820 y Fo(j)s Fj(=1)1600 2722 y Ft(\021)s Fn(\(\010)1736 2734 y Fo(j)1772 2722 y Fn(\))j(=)g(0)p Ft(:)974 b Fx(\(4.12\))415 2968 y(Besides)25 b(heat)e(\003ux)o(es,)h(there)g(might)f(be)g(other)g (\003ux)o(es)h(across)g(the)f(system)h Fq(S)k Fn(+)21 b Fq(R)j Fx(\(for)f(e)o(x-)291 3068 y(ample,)17 b(matter)h(and)f(char)o (ge)g(currents\).)23 b(W)-7 b(e)19 b(will)g(not)e(discuss)i(here)e(the) h(general)f(theory)g(of)h(such)291 3167 y(\003ux)o(es)h(\(the)h (related)f(information)f(can)i(be)f(found)g(in)h([FMU)o(,)h(FMSU,)f(TM) o(]\).)25 b(In)20 b(the)g(rest)g(of)g(this)291 3267 y(section)k(we)i (will)f(focus)g(on)f(the)h(thermodynamics)d(of)j(heat)g(\003ux)o(es.)39 b(Char)o(ge)24 b(currents)g(will)h(be)291 3366 y(discussed)20 b(in)g(the)g(conte)o(xt)f(of)h(a)h(concrete)e(model)g(in)h(the)g (second)g(part)g(of)f(this)i(lecture.)415 3469 y(W)-7 b(e)27 b(no)n(w)d(turn)h(to)g(the)h(entrop)o(y)d(production.)38 b(Assume)25 b(that)g(there)g(e)o(xists)h(a)g Ft(C)2771 3439 y Fs(\003)2809 3469 y Fx(-dynamics)291 3569 y Ft(\033)341 3539 y Fo(t)338 3592 y Fs(R)432 3569 y Fx(on)31 b Fq(O)613 3581 y Fs(R)707 3569 y Fx(such)h(that)g Ft(!)1101 3581 y Fs(R)1194 3569 y Fx(is)h Fn(\()p Ft(\033)1361 3581 y Fs(R)1423 3569 y Ft(;)14 b Fq(\000)p Fn(1\))p Fx(-KMS)31 b(state)i(and)e(such)h(that)g Ft(\033)2564 3581 y Fs(R)2658 3569 y Fx(preserv)o(es)f(each)291 3678 y(subalgebra)22 b Fq(O)741 3690 y Fs(R)798 3698 y Fh(j)833 3678 y Fx(.)36 b(Let)1029 3656 y Fn(~)1025 3678 y Ft(\016)1062 3690 y Fo(j)1121 3678 y Fx(be)24 b(the)g(generator)e(of)h(the)h(restriction) f(of)h Ft(\033)2409 3690 y Fs(R)2495 3678 y Fx(to)g Fq(O)2650 3690 y Fs(R)2707 3698 y Fh(j)2766 3678 y Fx(and)f(assume)291 3792 y(that)d Ft(V)484 3804 y Fo(j)543 3792 y Fq(2)25 b Fn(Dom)14 b(\()847 3770 y(~)843 3792 y Ft(\016)880 3804 y Fo(j)915 3792 y Fn(\))p Fx(.)27 b(The)20 b(entrop)o(y)f (production)f(observ)n(able)g(associated)j(to)f(the)h(perturbation)291 3892 y Ft(V)39 b Fx(and)20 b(the)g(reference)f(state)h Ft(!)26 b Fn(=)d Ft(!)1361 3904 y Fs(S)1428 3892 y Fq(\012)18 b Ft(!)1563 3904 y Fs(R)1624 3892 y Fx(,)i(where)g Ft(!)1941 3904 y Fs(S)1990 3892 y Fn(\()p Fq(\001)p Fn(\))k(=)e(T)-7 b(r\()p Fq(\001)p Fn(\))p Ft(=)14 b Fn(dim)g Fq(H)2639 3904 y Fs(S)2688 3892 y Fx(,)21 b(is)1426 4162 y Ft(\033)1473 4174 y Fo(V)1555 4162 y Fn(=)1667 4058 y Fo(M)1642 4083 y Fl(X)1645 4260 y Fo(j)s Fj(=1)1780 4140 y Fn(~)1776 4162 y Ft(\016)1813 4174 y Fo(j)1848 4162 y Fn(\()p Ft(V)1928 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Ft(\014)1884 4942 y Fo(j)1919 4930 y Fn(\010)1979 4942 y Fo(j)2014 4930 y Ft(:)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 739 232 a Fx(18)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx(In)32 b(particular)m(,)g(for)g(an)o(y)f(NESS)h Ft(!)1779 535 y Fj(+)1879 523 y Fq(2)45 b Fn(\006)2039 535 y Fj(+)2094 523 y Fn(\()p Ft(!)s(;)14 b(\034)2254 535 y Fo(V)2312 523 y Fn(\))p Fx(,)35 b(the)d(second)f(la)o(w)h(of)g (thermodynamics)739 623 y(holds:)1636 690 y Fo(M)1611 715 y Fl(X)1613 892 y Fo(j)s Fj(=1)1745 794 y Ft(\014)1792 806 y Fo(j)1840 794 y Ft(!)1892 806 y Fj(+)1947 794 y Fn(\(\010)2039 806 y Fo(j)2074 794 y Fn(\))24 b(=)e Fq(\000)p Fn(Ep\()p Ft(!)2469 806 y Fj(+)2524 794 y Fn(\))h Fq(\024)g Fn(0)p Ft(:)671 b Fx(\(4.13\))739 1042 y(In)32 b(f)o(act,)j(it)e(is)g (not)f(dif)n(\002cult)f(to)i(sho)n(w)f(that)g Fn(Ep\()p Ft(!)2268 1054 y Fj(+)2323 1042 y Fn(\))h Fx(is)g(independent)c(of)j (the)g(choice)g(of)g(the)739 1142 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(at)j(the)f(macroscopic)f(le)n(v)o(el.)863 2839 y(A)f(less)g(ambitious) e(application)g(of)g(linear)h(response)f(theory)g(concerns)f(transport) h(properties)739 2939 y(of)29 b(microscopic)e(and)i(mesoscopic)f (quantum)f(de)n(vices)i(\(the)g(adv)n(ances)f(in)h(nanotechnologies)739 3039 y(during)f(the)i(last)h(decade)e(ha)n(v)o(e)g(triggered)f(a)i (strong)f(interest)h(in)g(the)g(transport)f(properties)f(of)739 3138 y(such)d(de)n(vices\).)39 b(Linear)24 b(response)h(theory)f(of)g (such)h(systems)h(is)g(much)e(better)h(understood,)f(as)739 3238 y(we)c(shall)h(try)f(to)g(illustrate.)863 3341 y(In)27 b(our)e(current)h(setting,)i(the)e(forces)g(that)h(dri)n(v)o(e)e(the)i (system)f Fq(S)k Fn(+)23 b Fq(R)k Fx(out)f(of)h(equilibrium)739 3441 y(are)f(the)h(dif)n(ferent)d(in)m(v)o(erse)h(temperatures)g Ft(\014)2071 3453 y Fj(1)2108 3441 y Ft(;)14 b Fq(\001)g(\001)g(\001)28 b Ft(;)14 b(\014)2354 3453 y Fo(M)2454 3441 y Fx(of)26 b(the)h(reserv)n(oirs)e(attached)h(to)g Fq(S)6 b Fx(.)45 b(If)739 3540 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Fx(.)863 4344 y(In)32 b(phenomenological)d(non-equilibrium)f (thermodynamics,)k(the)h(duality)e(between)h(the)739 4444 y(driving)22 b(forces)h Ft(F)1283 4456 y Fo(\013)1331 4444 y Fx(,)i(also)f(called)f Fr(af)o(\002nities,)h Fx(and)f(the)h (steady)f(currents)g Ft(\036)2935 4456 y Fo(\013)3007 4444 y Fx(the)o(y)g(induce)g(is)h(e)o(x-)739 4543 y(pressed)c(by)f(the) i(entrop)o(y)d(production)g(formula)1880 4747 y Fn(Ep)23 b(=)2094 4668 y Fl(X)2132 4842 y Fo(\013)2227 4747 y Ft(F)2280 4759 y Fo(\013)2342 4747 y Ft(\036)2391 4759 y Fo(\013)2439 4747 y Ft(;)739 5006 y Fx(\(see)32 b([DGM)o(]\).)59 b(The)31 b(steady)h(currents)e(are)i(themselv)o(es)f(functions)f(of)h (the)h(af)n(\002nities)g Ft(\036)3447 5018 y Fo(\013)3539 5006 y Fn(=)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(19)291 523 y Ft(\036)340 535 y Fo(\013)387 523 y Fn(\()p Ft(F)472 535 y Fj(1)510 523 y Ft(;)14 b Fq(\001)g(\001)g(\001)g Fn(\))p Fx(.)26 b(In)20 b(the)g(linear)g(response)f(re)o(gime,)g(these) h(functions)f(are)h(gi)n(v)o(en)f(by)h(the)g(relations)1421 727 y Ft(\036)1470 739 y Fo(\013)1541 727 y Fn(=)1629 648 y Fl(X)1670 823 y Fo(\015)1763 727 y Ft(L)1820 739 y Fo(\013\015)1905 727 y Ft(F)1958 739 y Fo(\015)2001 727 y Ft(;)291 995 y Fx(which)f(de\002ne)h(the)g Fr(kinetic)g(coef)o (\002cients)f Ft(L)1552 1007 y Fo(\013\015)1637 995 y Fx(.)415 1097 y(Comparing)e(with)i(Equ.)24 b(\(4.13\))16 b(and)i(using)h(ener)o(gy)d(conserv)n(ation)h(\(4.12\))f(we)k(obtain)d (in)i(our)291 1197 y(case)1201 1365 y Fn(Ep\()p Ft(!)1388 1377 y Fo(X)5 b Fj(+)1502 1365 y Fn(\))23 b(=)1670 1261 y Fo(M)1645 1286 y Fl(X)1647 1463 y Fo(j)s Fj(=1)1778 1365 y Ft(X)1847 1377 y Fo(j)1896 1365 y Ft(!)1948 1377 y Fo(X)5 b Fj(+)2062 1365 y Fn(\(\010)2154 1377 y Fo(j)2189 1365 y Fn(\))p Ft(:)291 1610 y Fx(Thus)24 b Ft(X)551 1622 y Fo(j)612 1610 y Fx(is)i(the)g(af)n(\002nity)e(conjugated)f(to)i 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b(it)g(is)g(possible)g(to)f (pro)o(v)o(e)f(that)i(if)f(the)h(microscopic)e(thermodynamics)e(e)o (xists)k(and)f(is)291 1965 y(suf)n(\002ciently)e(re)o(gular)m(,)i(then) g(\(4.28\))e(hold.)50 b(On)28 b(the)h(other)f(hand,)i(establishing)e(e) o(xistence)g(and)291 2064 y(re)o(gularity)20 b(of)j(the)f(microscopic)f (thermodynamics)f(is)k(a)f(formidable)d(task)j(which)f(has)h(been)f(so) 291 2164 y(f)o(ar)28 b(carried)f(out)i(only)e(for)h(a)h(fe)n(w)f (models.)50 b(FGR)29 b(thermodynamics)d(is)j(v)o(ery)e(rob)n(ust)h(and) g(the)291 2263 y(weak)22 b(coupling)e(limit)j(is)g(an)f(ef)n(fecti)n(v) o(e)f(tool)h(in)g(the)g(study)g(of)g(the)g(models)g(whose)g (microscopic)291 2363 y(thermodynamics)17 b(appears)i(be)o(yond)f (reach)h(of)h(the)h(e)o(xisting)e(techniques.)415 2465 y(W)-7 b(e)22 b(will)g(return)d(to)i(this)h(topic)e(in)h(Section)f(8)h (where)f(we)i(will)f(discuss)g(the)g(FGR)h(thermody-)291 2564 y(namics)e(of)f(the)i(SEBB)g(model.)291 2854 y Fv(5)119 b(Fr)n(ee)30 b(F)m(ermi)g(gas)f(r)n(eser)o(v)o(oir)291 3043 y Fx(In)15 b(the)h(SEBB)h(model,)f(which)f(we)h(shall)h(study)e (in)h(the)g(second)f(part)h(of)f(this)i(lecture,)f(the)f(reserv)n(oir) 291 3143 y(will)22 b(be)g(described)f(by)g(an)h(in\002nitely)f(e)o (xtended)f(free)i(Fermi)f(gas.)30 b(Our)22 b(description)e(of)i(the)g (free)291 3242 y(Fermi)e(gas)g(in)g(this)h(section)f(is)h(suited)f(to)g (this)h(application.)415 3344 y(The)e(basic)g(properties)f(of)h(the)g (free)g(Fermi)g(gas)g(are)g(discussed)g(in)h(the)f(lecture)f([Me3)o(])i (and)e(in)291 3443 y(Examples)k(18)i(and)f(51)g(of)h(the)f(lecture)h ([Pi)o(])g(and)g(we)g(will)g(assume)g(that)g(the)f(reader)g(is)i(f)o (amiliar)291 3543 y(with)h(the)g(terminology)d(and)j(results)g (described)f(there.)42 b(A)27 b(more)e(detailed)g(e)o(xposition)g(can)g (be)291 3642 y(found)18 b(in)i([BR2])h(and)e(in)i(the)f(recent)f (lecture)h(notes)g([D2)o(].)415 3744 y(The)h(free)g(Fermi)h(gas)f(is)h (described)e(by)h(the)h(so)f(called)h(CAR)g(\(canonical)e 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b(a)h(comple)o(x)d(conjugation)f Ft(f)33 b Fq(7!)1787 2075 y Fn(\026)1769 2097 y Ft(f)d Fx(on)20 b Fk(h)i Fx(and)f(e)o(xtend)e(it)j(to)f Fn(\000)2598 2109 y Fs(\000)2654 2097 y Fn(\()p Fk(h)p Fn(\))p Fx(.)29 b(Denote)20 b(by)291 2196 y Fn(\012)g Fx(the)g(v)n(acuum)f(v)o(ector)g (and)h Ft(N)29 b Fx(the)20 b(number)f(operator)f(in)i Fn(\000)2070 2208 y Fs(\000)2126 2196 y Fn(\()p Fk(h)p Fn(\))p Fx(.)27 b(Set)937 2377 y Fq(H)1007 2389 y Fo(!)1049 2397 y Fh(T)1122 2377 y Fn(=)c(\000)1262 2389 y Fs(\000)1318 2377 y Fn(\()p Fk(h)p Fn(\))c Fq(\012)f Fn(\000)1579 2389 y Fs(\000)1635 2377 y Fn(\()p Fk(h)p Fn(\))p Ft(;)948 2501 y Fn(\012)1008 2513 y Fo(!)1050 2521 y Fh(T)1122 2501 y Fn(=)23 b(\012)c Fq(\012)f Fn(\012)p Ft(;)738 2642 y(\031)785 2654 y Fo(!)827 2662 y Fh(T)877 2642 y Fn(\()p Ft(a)p Fn(\()p Ft(f)9 b Fn(\)\))23 b(=)g Ft(a)p Fn(\(\()p Ft(I)j Fq(\000)18 b Ft(T)12 b Fn(\))1556 2608 y Fj(1)p Fo(=)p Fj(2)1660 2642 y Ft(f)d Fn(\))18 b Fq(\012)g Ft(I)26 b Fn(+)18 b(\()p Fq(\000)p Ft(I)7 b Fn(\))2160 2608 y Fo(N)2241 2642 y Fq(\012)18 b Ft(a)2368 2608 y Fs(\003)2406 2642 y Fn(\()2455 2621 y(\026)2438 2642 y Ft(T)2499 2608 y Fj(1)p Fo(=)p Fj(2)2621 2620 y Fn(\026)2603 2642 y Ft(f)8 b Fn(\))p Ft(:)291 2824 y Fx(The)26 b(triple)g Fn(\()p Fq(H)751 2836 y Fo(!)793 2844 y Fh(T)844 2824 y Ft(;)14 b(\031)928 2836 y Fo(!)970 2844 y Fh(T)1020 2824 y Ft(;)g Fn(\012)1117 2836 y Fo(!)1159 2844 y Fh(T)1208 2824 y Fn(\))28 b Fx(is)f(the)g(GNS)g(representation)e(of)h(the)g (algebra)g Fn(CAR\()p Fk(h)p Fn(\))i Fx(asso-)291 2924 y(ciated)22 b(to)h Ft(!)652 2936 y Fo(T)704 2924 y Fx(.)33 b(\(This)23 b(representation)d(w)o(as)k(constructed)d(in)i([A)-7 b(W)o(])23 b(and)g(if)g(often)f(called)g(Araki-)291 3023 y(W)-6 b(yss)21 b(representation.\))h(If)e Ft(!)1162 3035 y Fo(T)1235 3023 y Fx(is)h Ft(\034)9 b Fx(-in)m(v)n(ariant,)19 b(the)h(corresponding)d Ft(!)2386 3035 y Fo(T)2438 3023 y Fx(-Liouvillean)h(is)1222 3204 y Ft(L)23 b Fn(=)f(d\000\()p Ft(h)p Fn(\))d Fq(\012)f Ft(I)25 b Fq(\000)19 b 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0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1 0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11 0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0 setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0 0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC /Aquamarine{0.82 0 0.30 0 setcmykcolor}DC/BlueGreen{0.85 0 0.33 0 setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0 0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0 1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC /PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0 setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{ 0.26 0 0.76 0 setcmykcolor}DC/OliveGreen{0.64 0 0.95 0.40 setcmykcolor} DC/RawSienna{0 0.72 1 0.45 setcmykcolor}DC/Sepia{0 0.83 1 0.70 setcmykcolor}DC/Brown{0 0.81 1 0.60 setcmykcolor}DC/Tan{0.14 0.42 0.56 0 setcmykcolor}DC/Gray{0 0 0 0.50 setcmykcolor}DC/Black{0 0 0 1 setcmykcolor}DC/White{0 0 0 0 setcmykcolor}DC end %%EndProcSet TeXDict begin 40258437 52099154 1000 600 600 (f2pspost.dvi) @start %DVIPSBitmapFont: Fa cmr12 14.4 1 /Fa 1 10 df<0207B712C0A491C749C9FCEE3FF8705AAAD8FFE0F01FFC01F8187F6D18FF D80FFE4D13C000071A806C6C4D13006C616E1607A26C616E160FA2017F60AD6D6C4C5AA4 131F6E4C5AA2130F6E4C5AA26D6C5F19FF6D6C94C7FC01015E6E4B5A6D01804A5ADA7FC0 5DDA3FE04A5ADA1FF0EC3FE0DA07FC4A5ADA03FFD9F1FFC8FC020090B512FC033F14F003 0714C09226007FF8C9FCEE1FF0AA4C7EEEFFFE0207B712C0A44E527AD15B>9 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmr10 10 2 /Fb 2 51 df49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmsy10 14.4 1 /Fc 1 83 df<93B7FC031F16FC4AB97E020F18F0023F18FC91BA7E010385490181D98001 8190261FF003DA000780D97F80040080D9FE00171F48481807484892C86C7F484884000F 85121F5B003F4A177F485A4C5F5B00FEC7FC00F8631240C8000F17FF4C5F99C7FC626363 4B481503505A63505A505A4B484B5A50C8FCF101FEF103F84CEC0FF0037FED7FC0953807 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{0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 76 moveto 0 0 lineto 328 0 lineto 328 76 lineto closepath clip newpath -34.4 109.6 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proc char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def 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y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fm(5.2)99 b(Examples)739 679 y Fx(Recall)21 b(that)f(the)g(P)o (auli)g(matrices)g(are)h(de\002ned)e(by)1203 901 y Ft(\033)1250 913 y Fo(x)1315 901 y Fq(\021)1403 784 y Fl(\024)1446 851 y Fn(0)83 b(1)1446 950 y(1)g(0)1613 784 y Fl(\025)1670 901 y Ft(;)180 b(\033)1920 913 y Fo(y)1984 901 y Fq(\021)2071 784 y Fl(\024)2115 851 y Fn(0)83 b Fq(\000)p Fn(i)2124 950 y(i)116 b(0)2327 784 y Fl(\025)2385 901 y Ft(;)180 b(\033)2635 913 y Fo(z)2697 901 y Fq(\021)2784 784 y Fl(\024)2828 851 y Fn(1)115 b(0)2828 950 y(0)83 b Fq(\000)p Fn(1)3059 784 y Fl(\025)3116 901 y Ft(:)739 1132 y Fx(W)-7 b(e)23 b(set)g Ft(\033)1032 1144 y Fs(\006)1115 1132 y Fq(\021)k Fn(\()p Ft(\033)1286 1144 y Fo(x)1348 1132 y Fq(\006)19 b Fn(i)p Ft(\033)1502 1144 y Fo(y)1543 1132 y Fn(\))p Ft(=)p Fn(2)p Fx(.)30 b(Clearly)-5 b(,)22 b Ft(\033)2043 1102 y Fj(2)2040 1152 y Fo(x)2109 1132 y Fn(=)27 b Ft(\033)2251 1102 y Fj(2)2248 1152 y Fo(y)2315 1132 y Fn(=)f Ft(\033)2456 1102 y Fj(2)2453 1152 y Fo(z)2521 1132 y Fn(=)g Ft(I)j Fx(and)22 b Ft(\033)2867 1144 y Fo(x)2909 1132 y Ft(\033)2956 1144 y Fo(y)3023 1132 y Fn(=)27 b Fq(\000)p Ft(\033)3227 1144 y Fo(y)3266 1132 y Ft(\033)3313 1144 y Fo(x)3382 1132 y Fn(=)g(i)p Ft(\033)3544 1144 y Fo(z)3582 1132 y Fx(.)739 1231 y(More)19 b(generally)-5 b(,)19 b(with)f Ft(~)-40 b(\033)26 b Fn(=)d(\()p Ft(\033)1693 1243 y Fo(x)1736 1231 y Ft(;)14 b(\033)1820 1243 y Fo(y)1860 1231 y Ft(;)g(\033)1944 1243 y Fo(z)1982 1231 y Fn(\))21 b Fx(and)e Ft(~)-41 b(u;)11 b(~)-39 b(v)26 b Fq(2)d Fp(R)2465 1201 y Fj(3)2523 1231 y Fx(one)d(has)1534 1407 y Fn(\()o Ft(~)-41 b(u)18 b Fq(\001)f Ft(~)-40 b(\033)s Fn(\)\()m Ft(~)h(v)23 b Fq(\001)16 b Ft(~)-40 b(\033)t Fn(\))23 b(=)e Ft(~)-40 b(u)18 b Fq(\001)d Ft(~)-39 b(v)17 b(I)26 b Fn(+)18 b(i\()o Ft(~)-41 b(u)18 b Fq(\002)d Ft(~)-39 b(v)s Fn(\))19 b Fq(\001)d Ft(~)-40 b(\033)t(:)739 1676 y Fu(Example)28 b(1.)47 b Fx(Assume)28 b(that)g Fn(dim)14 b Fk(h)37 b Fn(=)f(1)p Fx(,)30 b Fr(i.e)o(.,)f Fx(that)e Fk(h)37 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(formulas)f(is)k(called)e(the)g(Jordan-W)m(igner)d(transformation.)291 623 y(It)28 b(is)g(a)h(useful)e(tool)g(in)h(the)g(study)g(of)f(quantum) f(spin)i(systems)g(\(see)g([LMS)o(,)h(AB,)f(Ar3)o(]\).)48 b(F)o(or)291 722 y Ft(\014)t(;)14 b(\026)32 b Fq(2)g Fp(R)p Fx(,)27 b(the)e(quasi-free)e(gauge-in)m(v)n(ariant)e(state)26 b(associated)e(to)h Ft(T)43 b Fn(=)32 b(\()p Ft(I)d Fn(+)22 b(e)2725 692 y Fo(\014)s Fj(\()p Fo(h)p Fs(\000)p Fo(\026)p Fj(\))2952 722 y Fn(\))2984 692 y Fs(\000)p Fj(1)3099 722 y Fx(is)291 822 y(gi)n(v)o(en)c(by)i(the)g(density)g(matrix)1513 922 y Fn(e)1550 892 y Fs(\000)p Fo(\014)s Fj(\()p Fo(H)t Fs(\000)p Fo(\026N)6 b Fj(\))p 1464 959 495 4 v 1464 1037 a Fn(T)-7 b(r)13 b(e)1600 1013 y Fs(\000)p Fo(\014)s Fj(\()p Fo(H)t Fs(\000)p Fo(\026N)6 b Fj(\))1969 978 y Ft(;)291 1150 y Fx(with)514 1378 y Ft(H)30 b Fq(\021)23 b Fn(d\000\()p Ft(h)p Fn(\))g(=)1061 1274 y Fo(n)1022 1299 y Fl(X)1024 1476 y Fo(j)s Fj(=1)1156 1378 y Ft(!)1208 1390 y Fo(j)1256 1378 y Ft(a)1300 1343 y 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2086 y Fh(j)291 2166 y Fx(is)30 b(gi)n(v)o(en)e(by)h Ft(\016)737 2178 y Fo(j)772 2166 y Fn(\()p Fq(\001)p Fn(\))41 b(=)f(i[d\000\()p Ft(h)1229 2178 y Fs(R)1286 2186 y Fh(j)1321 2166 y Fn(\))p Ft(;)14 b Fq(\001)g Fn(])p Fx(.)54 b(Note)29 b(that)h Ft(V)1919 2178 y Fo(j)1994 2166 y Fq(2)41 b Fn(Dom)14 b Ft(\016)2315 2178 y Fo(j)2380 2166 y Fx(if)n(f)29 b Ft(f)2527 2178 y Fo(j)2602 2166 y Fq(2)40 b Fn(Dom)14 b Ft(h)2933 2178 y Fs(R)2990 2186 y Fh(j)3025 2166 y Fx(.)54 b(If)291 2265 y Ft(V)339 2277 y Fo(j)397 2265 y Fq(2)23 b Fn(Dom)14 b Ft(\016)700 2277 y Fo(j)735 2265 y Fx(,)21 b(then)f(the)g(observ)n(able)e(describing)h(the)h(heat)g (\003ux)g(out)g(of)g Fq(R)2526 2277 y Fo(j)2582 2265 y Fx(is)675 2442 y Fn(\010)735 2454 y Fo(j)793 2442 y Fn(=)i Ft(\025\016)965 2454 y Fo(j)1001 2442 y Fn(\()p Ft(V)1081 2454 y Fo(j)1117 2442 y Fn(\))h(=)f Ft(\025)p Fn(\()p Ft(a)p Fn(\(1\))e Fq(\012)e Ft(a)1636 2408 y Fs(\003)1674 2442 y Fn(\(i)p Ft(h)1777 2454 y Fs(R)1834 2462 y Fh(j)1869 2442 y Ft(f)1910 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b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx(still)j(well-de\002ned)e(and)g(the)o(y)h(are)g(equal)f(to)h (the)g(limit)h(of)f(the)g(\003ux)g(observ)n(ables)e(corresponding)739 623 y(to)g(\002nite-dimensional)e(approximations.)863 723 y(The)27 b(\002rst)g(la)o(w)f(of)h(thermodynamics)c(\(ener)o(gy)h (conserv)n(ation\))g(has)j(been)e(v)o(eri\002ed)h(in)g(Sub-)739 822 y(section)20 b(4.3\227for)e(an)o(y)i Ft(\034)1475 834 y Fo(\025)1518 822 y Fx(-in)m(v)n(ariant)f(state)h Ft(\021)k Fx(one)c(has)1940 980 y Fo(M)1915 1005 y Fl(X)1917 1182 y Fo(j)s Fj(=1)2048 1084 y Ft(\021)s Fn(\(\010)2184 1096 y Fo(j)2220 1084 y Fn(\))j(=)g(0)p Ft(:)739 1350 y Fx(The)d(analogous)e(statement)i(for)g(char)o(ge)e(currents)h(is)i (pro)o(v)o(ed)d(in)j(a)f(similar)h(w)o(ay)-5 b(.)24 b(By)d(\(6.38\),) 1764 1508 y Fo(M)1738 1533 y Fl(X)1741 1710 y Fo(j)s Fj(=1)1872 1612 y Fq(J)1928 1624 y Fo(j)1986 1612 y Fn(=)2099 1556 y(d)p 2084 1593 77 4 v 2084 1669 a(d)p Ft(t)2184 1612 y(\034)2229 1578 y Fo(t)2220 1632 y(\025)2264 1612 y Fn(\()p Ft(N)2363 1624 y Fs(S)2412 1612 y Fn(\))p Fq(j)2467 1624 y Fo(t)p Fj(=0)2581 1612 y Ft(;)739 1878 y Fx(and)f(so)g(for)g(an) o(y)f Ft(\034)1267 1890 y Fo(\025)1311 1878 y Fx(-in)m(v)n(ariant)f (state)j Ft(\021)j Fx(one)c(has)1942 2036 y Fo(M)1916 2061 y Fl(X)1919 2238 y Fo(j)s Fj(=1)2050 2140 y Ft(\021)s Fn(\()p Fq(J)2182 2152 y Fo(j)2218 2140 y Fn(\))j(=)g(0)p Ft(:)977 b Fx(\(6.39\))739 2458 y Fm(6.3)99 b(The)26 b(equi)o(v)o(alent)g(fr)n(ee)g(F)n(ermi)e(gas)739 2614 y Fx(In)c(this)h(subsection)e(we)i(shall)g(sho)n(w)f(ho)n(w)g(to)g(use) h(the)f(e)o(xponential)e(la)o(w)j(for)f(fermionic)e(systems)739 2714 y(to)i(map)g(the)g(SEBB)i(model)d(to)h(a)h(free)f(Fermi)g(gas.)k (Let)917 2994 y Fk(h)g Fq(\021)e Fp(C)d Fq(\010)f Fk(h)1276 3006 y Fs(R)1360 2994 y Fn(=)23 b Fp(C)18 b Fq(\010)1609 2827 y Fl(0)1609 2977 y(@)1710 2890 y Fo(M)1682 2915 y Fl(M)1687 3092 y Fo(j)s Fj(=1)1821 2994 y Fk(h)1864 3006 y Fs(R)1921 3014 y Fh(j)1957 2827 y Fl(1)1957 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Fr(-isomorphism.)327 699 y(\(ii\))41 b(F)-9 b(or)21 b(any)f Ft(\025;)14 b(t)23 b Fq(2)g Fp(R)e Fr(one)f(has)g Ft(\036)f Fq(\016)f Ft(\034)1495 669 y Fo(t)1486 723 y(\025)1553 699 y Fn(=)27 b(~)-47 b Ft(\034)1685 669 y Fo(t)1676 723 y Fs(\000)p Fo(\025)1790 699 y Fq(\016)18 b Ft(\036)p Fr(.)304 875 y(\(iii\))41 b Ft(!)26 b Fn(=)j(~)-48 b Ft(!)21 b Fq(\016)d Ft(\036)p Fr(.)314 1051 y(\(iv\))40 b(F)-9 b(or)21 b Ft(j)28 b Fn(=)23 b(1)p Ft(;)14 b Fq(\001)g(\001)g(\001)27 b Ft(;)14 b(M)9 b Fr(,)20 b(one)f(has)916 1218 y Fn(~)907 1238 y(\010)967 1250 y Fo(j)1025 1238 y Fq(\021)k Ft(\036)p Fn(\(\010)1254 1250 y Fo(j)1289 1238 y Fn(\))h(=)e Fq(\000)p Ft(\025)14 b Fn(\()p Ft(a)1635 1204 y Fs(\003)1673 1238 y Fn(\(i)p Ft(h)1776 1250 y Fo(j)1812 1238 y Ft(f)1853 1250 y Fo(j)1887 1238 y Fn(\))p Ft(a)p Fn(\(1\))19 b(+)f Ft(a)2215 1204 y Fs(\003)2253 1238 y Fn(\(1\))p Ft(a)p Fn(\(i)p Ft(h)2506 1250 y Fo(j)2541 1238 y Ft(f)2582 1250 y Fo(j)2617 1238 y Fn(\)\))c Ft(;)470 1426 y Fr(and)1036 1512 y Fn(~)1007 1533 y Fq(J)1063 1545 y Fo(j)1122 1533 y Fq(\021)22 b Ft(\036)p Fn(\()p Fq(J)1346 1545 y Fo(j)1382 1533 y Fn(\))i(=)e Fq(\000)p Ft(\025)p Fn(\()p Ft(a)1714 1499 y Fs(\003)1753 1533 y Fn(\(i)p Ft(f)1849 1545 y Fo(j)1884 1533 y Fn(\))p Ft(a)p Fn(\(1\))c(+)g Ft(a)2211 1499 y Fs(\003)2249 1533 y Fn(\(1\))p Ft(a)p Fn(\(i)p Ft(f)2495 1545 y Fo(j)2530 1533 y Fn(\)\))p Ft(:)291 1728 y Fu(Pr)o(oof)o(.)k Fx(Clearly)-5 b(,)26 b Ft(\036)g Fx(is)f(a)h Fq(\003)p Fx(-isomorphism)c(from)i Fq(B)s Fn(\(\000)1894 1740 y Fs(\000)1949 1728 y Fn(\()p Fp(C)e Fq(\010)g Fk(h)p Fn(\)\))k Fx(onto)e Fq(B)s Fn(\(\000)2598 1740 y Fs(\000)2653 1728 y Fn(\()p Fp(C)p Fn(\))f Fq(\012)e Fn(\000)2938 1740 y Fs(\000)2994 1728 y Fn(\()p Fk(h)p Fn(\)\))p Fx(.)291 1828 y(Using)31 b(the)g(canonical)f(injections)g Fp(C)44 b Fq(!)f Fk(h)32 b Fx(and)e Fk(h)1857 1840 y Fs(R)1962 1828 y Fq(!)44 b Fk(h)32 b Fx(we)f(can)g(identify)f Fq(O)2795 1840 y Fs(S)2876 1828 y Fx(and)h Fq(O)3094 1840 y Fs(R)291 1927 y Fx(with)22 b(the)g(subalgebras)f(of)1115 1906 y Fn(~)1092 1927 y Fq(O)k Fx(generated)c(by)g Ft(a)p Fn(\(1)f Fq(\010)g Fn(0\))i Fx(and)g Fq(f)p Ft(a)p Fn(\(0)d Fq(\010)h Ft(f)9 b Fn(\))14 b Fq(j)g Ft(f)35 b Fq(2)27 b Fk(h)2694 1939 y Fs(R)2755 1927 y Fq(g)p Fx(.)k(W)m(ith)23 b(this)291 2027 y(identi\002cation,)18 b(\(5.35\))h(gi)n(v)o(es)918 2215 y Ft(\036)p Fn(\()p Ft(a)p Fn(\()p Ft(\013)p Fn(\))i Fq(\012)d Ft(I)25 b Fn(+)18 b(\()p Fq(\000)p Ft(I)7 b Fn(\))1580 2180 y Fo(N)1633 2188 y Fd(S)1698 2215 y Fq(\012)18 b Ft(a)p Fn(\()p Ft(f)9 b Fn(\)\))24 b(=)e Ft(a)p Fn(\()p Ft(\013)p Fn(\))e(+)e Ft(a)p Fn(\()p Ft(f)9 b Fn(\))p Ft(;)291 2402 y Fx(for)19 b Ft(\013)24 b Fq(2)f Fp(C)e Fx(and)e Ft(f)32 b Fq(2)23 b Fk(h)978 2414 y Fs(R)1040 2402 y Fx(.)i(W)-7 b(e)22 b(conclude)c(that)1464 2590 y Ft(\036)p Fn(\()p Ft(A)i Fq(\012)e Ft(I)7 b Fn(\))23 b(=)g Ft(A;)973 b Fx(\(6.41\))291 2777 y(for)26 b(an)o(y)g Ft(A)35 b Fq(2)g(O)814 2789 y Fs(S)864 2777 y Fx(.)45 b(In)26 b(particular)m(,)g(since)h Ft(b)35 b Fq(\021)g Fn(\()p Fq(\000)p Ft(I)7 b Fn(\))1931 2747 y Fo(N)1984 2755 y Fd(S)2066 2777 y Fn(=)34 b([)p Ft(a)p Fn(\(1\))p Ft(;)14 b(a)2419 2747 y Fs(\003)2457 2777 y Fn(\(1\)])36 b Fq(2)f(O)2778 2789 y Fs(S)2827 2777 y Fx(,)29 b(we)e(ha)n(v)o(e)291 2877 y Ft(\036)p Fn(\()p Ft(b)20 b Fq(\012)f Ft(I)7 b Fn(\))27 b(=)f Ft(b)p Fx(.)31 b(Relation)22 b Ft(b)1133 2847 y Fj(2)1197 2877 y Fn(=)k Ft(I)j Fx(yields)22 b Ft(\036)p Fn(\()p Ft(I)28 b Fq(\012)19 b Ft(a)p Fn(\()p Ft(f)9 b Fn(\)\))27 b(=)f Ft(b)14 b(a)p Fn(\()p Ft(f)9 b Fn(\))p Fx(.)31 b(Since)22 b Fn([)p Ft(b;)14 b(a)p Fn(\()p Ft(f)9 b Fn(\)])26 b(=)h(0)p Fx(,)22 b(we)291 2977 y(conclude)c(that)i(for)g Ft(A)j Fq(2)h(O)1104 2989 y Fs(R)1140 3208 y Ft(\036)p Fn(\()p Ft(I)j Fq(\012)18 b Ft(A)p Fn(\))23 b(=)1572 3091 y Fl(\032)1676 3157 y Ft(A)132 b Fx(if)21 b Ft(A)i Fq(2)h(O)2174 3121 y Fj(+)2172 3181 y Fs(R)2233 3157 y Fx(,)1676 3258 y Ft(b)14 b(A)82 b Fx(if)21 b Ft(A)i Fq(2)h(O)2174 3223 y Fs(\000)2172 3283 y(R)2233 3258 y Fx(,)2954 3208 y(\(6.42\))291 3456 y(where)j Fq(O)590 3420 y Fs(\006)588 3480 y(R)677 3456 y Fx(denote)f(the)i(e)n(v)o(en)f(and)g(odd)f(parts)i(of)f Fq(O)1891 3468 y Fs(R)1952 3456 y Fx(.)48 b(Equ.)e(\(6.41\))26 b(and)h(\(6.42\))f(sho)n(w)h(that)291 3565 y Ft(\036)p Fn(\()p Fq(O)r Fn(\))36 b Fq(\032)630 3544 y Fn(~)607 3565 y Fq(O)r Fx(.)44 b(Since)975 3544 y Fn(~)952 3565 y Fq(O)37 b Fn(=)1154 3498 y Fl(\012)1193 3565 y Fq(O)1259 3577 y Fs(S)1309 3565 y Ft(;)14 b Fq(O)1414 3530 y Fj(+)1412 3589 y Fs(R)1473 3565 y Ft(;)g Fq(O)1578 3530 y Fs(\000)1576 3589 y(R)1637 3498 y Fl(\013)1676 3565 y Fx(,)28 b(it)g(follo)n(ws)e (from)f Ft(\036)p Fn(\()p Fq(O)2408 3577 y Fs(S)2481 3565 y Fq(\012)e Ft(I)7 b Fn(\))35 b(=)f Fq(O)2844 3577 y Fs(S)2893 3565 y Fx(,)28 b Ft(\036)p Fn(\()p Ft(I)j Fq(\012)291 3679 y(O)359 3643 y Fj(+)357 3703 y Fs(R)418 3679 y Fn(\))23 b(=)g Fq(O)629 3643 y Fj(+)627 3703 y Fs(R)709 3679 y Fx(and)c Ft(\036)p Fn(\()p Ft(b)g Fq(\012)f(O)1136 3643 y Fs(\000)1134 3703 y(R)1196 3679 y Fn(\))23 b(=)g Fq(O)1407 3643 y Fs(\000)1405 3703 y(R)1487 3679 y Fx(that)d Ft(\036)p Fn(\()p Fq(O)r Fn(\))k Fq(\033)1948 3658 y Fn(~)1925 3679 y Fq(O)r Fx(.)i(This)20 b(pro)o(v)o(es)f(P)o(art)h (\(i\).)415 3781 y(From)g(\(5.35\))e(we)j(can)f(see)g(that)h Ft(U)1429 3751 y Fs(\000)p Fj(1)1518 3781 y Ft(H)1587 3793 y Fj(0)1624 3781 y Ft(U)32 b Fn(=)22 b(d\000\()p Ft(h)1978 3793 y Fj(0)2016 3781 y Fn(\))f Fx(and)e(from)g(\(6.41\))g (and)g(\(6.42\))g(that)863 3969 y Ft(U)929 3934 y Fs(\000)p Fj(1)1018 3969 y Ft(V)1066 3981 y Fo(j)1102 3969 y Ft(U)32 b Fn(=)22 b Ft(\036)p Fn(\()p Ft(V)1407 3981 y Fo(j)1443 3969 y Fn(\))i(=)e Ft(a)p Fn(\(1\))14 b Ft(b)g(a)1844 3934 y Fs(\003)1882 3969 y Fn(\()p Ft(f)1955 3981 y Fo(j)1990 3969 y Fn(\))k(+)g Ft(a)2167 3934 y Fs(\003)2205 3969 y Fn(\(1\))c Ft(b)g(a)p Fn(\()p Ft(f)2492 3981 y Fo(j)2527 3969 y Fn(\))p Ft(:)291 4156 y Fx(Since)20 b(it)h(also)f(follo)n(ws)g (from)f(CAR)j(that)1079 4344 y Ft(a)p Fn(\(1\))14 b Ft(b)23 b Fn(=)f Fq(\000)p Ft(a)p Fn(\(1\))p Ft(;)180 b(a)1851 4310 y Fs(\003)1889 4344 y Fn(\(1\))14 b Ft(b)22 b Fn(=)h Ft(a)2199 4310 y Fs(\003)2237 4344 y Fn(\(1\))p Ft(;)588 b Fx(\(6.43\))291 4531 y(we)20 b(get)304 4719 y Ft(U)370 4685 y Fs(\000)p Fj(1)459 4719 y Ft(V)507 4731 y Fo(j)542 4719 y Ft(U)32 b Fn(=)22 b Fq(\000)p Ft(a)p Fn(\(1\))14 b Ft(a)991 4685 y Fs(\003)1029 4719 y Fn(\()p Ft(f)1102 4731 y Fo(j)1137 4719 y Fn(\))19 b(+)f Ft(a)1315 4685 y Fs(\003)1353 4719 y Fn(\(1\))c Ft(a)p Fn(\()p Ft(f)1590 4731 y Fo(j)1624 4719 y Fn(\))24 b(=)e Fq(\000)p Ft(a)p Fn(\(1\))14 b Ft(a)2040 4685 y Fs(\003)2078 4719 y Fn(\()p Ft(f)2151 4731 y Fo(j)2186 4719 y Fn(\))k Fq(\000)g Ft(a)p Fn(\()p Ft(f)2436 4731 y Fo(j)2471 4719 y Fn(\))c Ft(a)2561 4685 y Fs(\003)2599 4719 y Fn(\(1\))24 b(=)e Fq(\000)p Fn(d\000\()p Ft(v)3051 4731 y Fo(j)3086 4719 y Fn(\))p Ft(:)291 4907 y Fx(Therefore)d Ft(U)706 4876 y Fs(\000)p Fj(1)795 4907 y Ft(H)864 4919 y Fo(\025)908 4907 y Ft(U)34 b Fn(=)26 b(d\000\()p Ft(h)1268 4919 y Fs(\000)p Fo(\025)1364 4907 y Fn(\))c Fx(from)f(which)g(P)o(art)h(\(ii\))g(follo)n(ws.)29 b(A)23 b(similar)f(computation)291 5006 y(yields)e(P)o(art)g(\(i)n (v\).)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 739 232 a Fx(38)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)863 523 y Fx(It)i(remains)e(to)h(pro)o(v)o(e)e(P)o(art)i(\(iii\).)27 b(Using)20 b(the)h(morphism)e Ft(\022)24 b Fx(\(recall)d(Equ.)26 b(\(5.32\)\))18 b(to)j(e)o(xpress)739 623 y(the)f(e)n(v)o(en)f(and)h (odd)f(parts)h(of)g Ft(B)27 b Fq(2)d(O)1828 635 y Fs(R)1889 623 y Fx(,)d(we)f(can)g(re)n(write)g(\(6.41\))e(and)i(\(6.42\))e(as) 1298 812 y Ft(\036)p Fn(\()p Ft(A)h Fq(\012)f Ft(B)t Fn(\))24 b(=)e Ft(A)p Fn(\()p Ft(B)i Fn(+)18 b Ft(\022)r Fn(\()p Ft(B)t Fn(\)\))p Ft(=)p Fn(2)g(+)g Ft(A)c(b)g Fn(\()p Ft(B)23 b Fq(\000)18 b Ft(\022)r Fn(\()p Ft(B)t Fn(\)\))p Ft(=)p Fn(2)p Ft(;)739 1002 y Fx(from)h(which)h(we)g(easily)h (get)1361 1191 y Ft(\036)p Fn(\()p Ft(A)e Fq(\012)f Ft(B)t Fn(\))23 b(=)g Ft(Aa)p Fn(\(1\))p Ft(a)2072 1157 y Fs(\003)2110 1191 y Fn(\(1\))p Ft(B)g Fn(+)18 b Ft(Aa)2491 1157 y Fs(\003)2529 1191 y Fn(\(1\))p Ft(a)p Fn(\(1\))p Ft(\022)r Fn(\()p Ft(B)t Fn(\))p Ft(:)739 1381 y Fx(It)24 b(follo)n(ws)f(from)g (the)g(f)o(actorization)f(property)g(\(5.30\))f(and)j(the)f(in)m(v)n (ariance)f(property)f(\(5.33\))h(of)739 1480 y(quasi-free)d(states)i (that)1067 1670 y Fn(~)-49 b Ft(!)21 b Fq(\016)d Ft(\036)p Fn(\()p Ft(A)i Fq(\012)e Ft(B)t Fn(\))83 b(=)89 b(~)-48 b Ft(!)r Fn(\()p Ft(Aa)p Fn(\(1\))p Ft(a)2111 1635 y Fs(\003)2150 1670 y Fn(\(1\)\))7 b(~)-49 b Ft(!)s Fn(\()p Ft(B)t Fn(\))19 b(+)25 b(~)-49 b Ft(!)s Fn(\()p Ft(Aa)2769 1635 y Fs(\003)2807 1670 y Fn(\(1\))p Ft(a)p Fn(\(1\)\))7 b(~)-49 b Ft(!)s Fn(\()p Ft(B)t Fn(\))1621 1794 y(=)89 b(~)-48 b Ft(!)r Fn(\()p Ft(Aa)p Fn(\(1\))p Ft(a)2111 1760 y Fs(\003)2150 1794 y Fn(\(1\))p Ft(B)23 b Fn(+)18 b Ft(Aa)2531 1760 y Fs(\003)2569 1794 y Fn(\(1\))p Ft(a)p Fn(\(1\))p Ft(B)t Fn(\))1621 1919 y(=)89 b(~)-48 b Ft(!)r Fn(\()p Ft(AB)t Fn(\))24 b(=)30 b(~)-49 b Ft(!)s Fn(\()p Ft(A)p Fn(\))7 b(~)-49 b Ft(!)s Fn(\()p Ft(B)t Fn(\))1621 2043 y(=)83 b Ft(!)1821 2055 y Fs(S)1869 2043 y Fn(\()p Ft(A)p Fn(\))p Ft(!)2047 2055 y Fs(R)2109 2043 y Fn(\()p Ft(B)t Fn(\))24 b(=)f Ft(!)s Fn(\()p Ft(A)18 b Fq(\012)g Ft(B)t Fn(\))p Ft(:)739 2233 y Fe(\003)863 2393 y Fx(By)24 b(Theorem)c(6.1,)j(the)g(SEBB)g(model)f(can)h(be)f(equi)n(v)n(alently)f (described)g(by)h(the)h Ft(C)3348 2362 y Fs(\003)3386 2393 y Fx(-dyna-)739 2492 y(mical)15 b(system)h Fn(\()1237 2471 y(~)1218 2492 y Ft(O)s(;)i Fn(~)-46 b Ft(\034)1357 2504 y Fs(\000)p Fo(\025)1453 2492 y Fn(\))16 b Fx(and)f(the)h (reference)d(state)k Ft(!)2313 2502 y Fj(~)2300 2517 y Fo(T)2352 2492 y Fx(.)24 b(The)15 b(heat)g(and)g(char)o(ge)f(\003ux)h (observ)n(ables)739 2602 y(are)872 2581 y Fn(~)863 2602 y(\010)923 2614 y Fo(j)980 2602 y Fx(and)1151 2581 y Fn(~)1122 2602 y Fq(J)1178 2614 y Fo(j)1214 2602 y Fx(.)30 b(Since)22 b(the)g(change)f Ft(\025)26 b Fq(!)g(\000)p Ft(\025)d Fx(af)n(fects)e(neither)g(the)h(model)f(nor)g(the)h(results,) h Fr(in)739 2711 y(the)j(sequel)f(we)i(will)g(work)g(with)f(the)g (system)h Fn(\()2168 2690 y(~)2145 2711 y Fq(O)s Ft(;)18 b Fn(~)-46 b Ft(\034)2287 2723 y Fo(\025)2330 2711 y Fn(\))27 b Fr(and)e(we)i(will)g(dr)l(op)f(the)f Fq(\030)p Fr(.)43 b Fx(Hence,)26 b(we)739 2810 y(will)21 b(use)f(the)g Ft(C)1207 2780 y Fs(\003)1246 2810 y Fx(-algebra)f Fq(O)25 b Fn(=)e(CAR\()p Fp(C)c Fq(\010)f Fk(h)2138 2822 y Fs(R)2199 2810 y Fn(\))j Fx(and)f Ft(C)2458 2780 y Fs(\003)2496 2810 y Fx(-dynamics)1651 3000 y Ft(\034)1696 2965 y Fo(t)1687 3020 y(\025)1731 3000 y Fn(\()p Ft(A)p Fn(\))k(=)f(e)2006 2965 y Fj(i)p Fo(t)p Fj(d\000\()p Fo(h)2193 2974 y Fh(\025)2231 2965 y Fj(\))2261 3000 y Ft(A)p Fn(e)2360 2965 y Fs(\000)p Fj(i)p Fo(t)p Fj(d\000\()p Fo(h)2599 2974 y Fh(\025)2638 2965 y Fj(\))2668 3000 y Ft(;)739 3189 y Fx(with)d(the)f(reference)f (state)j Ft(!)s Fx(,)f(the)f(quasi-free)f(gauge-in)m(v)n(ariant)e 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5018 y Fs(S)1843 5006 y Fn(\))p Ft(a)p Fn(\()p Ft(f)9 b Fn(\)\))14 b Fq(\000)f Ft(\025)h Fn(\()q Ft(a)2296 4972 y Fs(\003)2334 5006 y Fn(\(i)p Ft(s)2428 5018 y Fs(R)2489 5006 y Ft(f)9 b Fn(\))p Ft(a)p Fn(\(1\))19 b(+)f Ft(a)2867 4972 y Fs(\003)2905 5006 y Fn(\(1\))p Ft(a)p Fn(\(i)p Ft(s)3149 5018 y Fs(R)3210 5006 y Ft(f)9 b Fn(\)\))14 b Ft(:)42 b Fx(\(6.44\))p eop end %%Page: 39 39 TeXDict begin 39 38 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(39)291 523 y(The)19 b(entrop)o(y)g(balance)g(equation)1103 756 y Fn(En)n(t)o(\()p Ft(!)j Fq(\016)c Ft(\034)1446 721 y Fo(t)1437 776 y(\025)1480 756 y Fq(j)p Ft(!)s Fn(\))23 b(=)g Fq(\000)1780 643 y Fl(Z)1862 663 y Fo(t)1826 831 y Fj(0)1906 756 y Ft(!)s Fn(\()p Ft(\034)2038 721 y Fo(s)2029 776 y(\025)2074 756 y Fn(\()p Ft(\033)s Fn(\)\))14 b(d)p Ft(s;)291 980 y Fx(holds)g(and)h(so,)h(as)g(in)g(Subsection)e(3.2,)h (the)g(entrop)o(y)f(production)e(of)j(an)o(y)g(NESS)g Ft(!)2671 992 y Fj(+)2749 980 y Fq(2)24 b Fn(\006)2888 992 y Fj(+)2943 980 y Fn(\()p Ft(!)s(;)14 b(\034)3103 992 y Fo(\025)3146 980 y Fn(\))291 1079 y Fx(is)23 b(non-ne)o(gati)n(v) o(e.)j(In)c(f)o(act,)g(it)h(is)g(not)e(dif)n(\002cult)h(to)g(sho)n(w)g (that)g(the)g(entrop)o(y)e(production)f(of)j Ft(!)3022 1091 y Fj(+)3099 1079 y Fx(is)291 1179 y(independent)h(of)j Ft(\015)31 b Fx(as)c(long)e(as)h Ft(\015)39 b Fq(2)34 b Fn(\(0)p Ft(;)14 b Fn(1\))26 b Fx(\(see)g(Proposition)f(5.3)g(in)h ([JP4]\).)42 b Fr(In)26 b(the)g(sequel,)291 1279 y(whene)o(ver)f(we)i (speak)f(about)f(the)i(entr)l(opy)e(pr)l(oduction,)h(we)h(will)h (assume)e(that)g Ft(\015)39 b Fn(=)34 b(1)p Ft(=)p Fn(2)25 b Fr(and)291 1378 y(hence)19 b(that)1008 1478 y Ft(\033)26 b Fn(=)d Fq(\000)p Ft(\025)14 b Fn(\()p Ft(a)1372 1444 y Fs(\003)1410 1478 y Fn(\(i)p Ft(s)1504 1490 y Fs(R)1566 1478 y Ft(f)9 b Fn(\))p Ft(a)p Fn(\(1\))18 b(+)g Ft(a)1943 1444 y Fs(\003)1981 1478 y Fn(\(1\))p Ft(a)p Fn(\(i)p Ft(s)2225 1490 y Fs(R)2286 1478 y Ft(f)9 b Fn(\)\))14 b Ft(:)517 b Fx(\(6.45\))291 1624 y(In)19 b(particular)m(,)g(if)1277 1733 y Ft(T)1326 1745 y Fs(R)1383 1753 y Fh(j)1440 1733 y Fn(=)k(\()p Ft(I)j Fn(+)18 b(e)1742 1694 y Fo(\014)1780 1702 y Fh(j)1810 1694 y Fj(\()p Fo(h)1875 1702 y Fd(R)1925 1715 y Fh(j)1960 1694 y Fs(\000)p Fo(\026)2052 1702 y Fh(j)2083 1694 y Fj(\))2113 1733 y Fn(\))p Ft(;)291 1879 y Fx(then)h Ft(s)493 1891 y Fo(j)551 1879 y Fn(=)k Fq(\000)p Ft(\014)751 1891 y Fo(j)785 1879 y Fn(\()p Ft(h)865 1891 y Fs(R)922 1899 y Fh(j)976 1879 y Fq(\000)18 b Ft(\026)1109 1891 y Fo(j)1144 1879 y Fn(\))p Fx(,)j(and)1265 2146 y Ft(\033)26 b Fn(=)d Fq(\000)1530 2043 y Fo(M)1504 2067 y Fl(X)1507 2244 y Fo(j)s Fj(=1)1638 2146 y Ft(\014)1685 2158 y Fo(j)1720 2146 y Fn(\(\010)1812 2158 y Fo(j)1866 2146 y Fq(\000)18 b Ft(\026)1999 2158 y Fo(j)2034 2146 y Fq(J)2090 2158 y Fo(j)2125 2146 y Fn(\))p Ft(:)774 b Fx(\(6.46\))415 2406 y(W)-7 b(e)31 b(\002nish)g(with)f(the)g(follo)n (wing)e(remark.)54 b(In)30 b(the)g(physics)f(literature,)j(the)e (Hamiltonian)291 2506 y(\(6.40\))17 b(is)j(sometimes)f(called)g(the)g Fr(W)-5 b(igner)n(-W)d(eissk)o(opf)19 b(atom)g Fx([WW])h(\(see)f([JKP]) g(for)g(references)291 2606 y(and)i(additional)h(information\).)28 b(The)22 b(operators)f(of)i(this)g(type)e(are)i(also)g(often)e(called)h Fr(F)-5 b(riedric)o(h)291 2705 y(Hamiltonians)30 b Fx([Fr)o(].)59 b(The)32 b(point)e(we)i(wish)g(to)g(emphasize)f(is)h(that)g(such)f (Hamiltonians)f(are)291 2805 y(often)23 b(used)h(as)h(to)o(y)f(models)g (which)g(allo)n(w)g(for)g(simple)g(mathematical)f(analysis)i(of)f (physically)291 2905 y(important)18 b(phenomena.)291 3141 y Fm(6.4)99 b(Assumptions)291 3297 y Fx(In)28 b(this)i(subsection) e(we)i(describe)e(a)i(set)f(of)g(assumptions)g(under)e(which)i(we)g (shall)h(study)e(the)291 3397 y(thermodynamics)17 b(of)j(the)g(SEBB)h (model.)291 3592 y Fu(Assumption)i(\(SEBB1\))g Fk(h)1113 3604 y Fs(R)1170 3612 y Fh(j)1233 3592 y Fn(=)j Ft(L)1381 3562 y Fj(2)1418 3592 y Fn(\(\()p Ft(e)1521 3604 y Fs(\000)1577 3592 y Ft(;)14 b(e)1653 3604 y Fj(+)1708 3592 y Fn(\))p Ft(;)g Fn(d)p Ft(r)r Fn(\))24 b Fx(for)e(some)g Fq(\0001)27 b Ft(<)g(e)2542 3604 y Fs(\000)2625 3592 y Ft(<)f(e)2755 3604 y Fj(+)2837 3592 y Fq(\024)h(1)c Fx(and)291 3692 y Ft(h)339 3704 y Fs(R)396 3712 y Fh(j)451 3692 y Fx(is)e(the)g (operator)d(of)i(multiplication)f(by)g Ft(r)r Fx(.)415 3887 y(The)31 b(assumption)f(\(SEBB1\))h(yields)g(that)h Fk(h)1752 3899 y Fs(R)1856 3887 y Fn(=)44 b Ft(L)2022 3857 y Fj(2)2058 3887 y Fn(\(\()p Ft(e)2161 3899 y Fs(\000)2218 3887 y Ft(;)14 b(e)2294 3899 y Fj(+)2348 3887 y Fn(\))p Ft(;)g Fn(d)p Ft(r)r Fn(;)g Fp(C)2599 3857 y Fo(M)2674 3887 y Fn(\))32 b Fx(and)f(that)g Ft(h)3094 3899 y Fs(R)291 3986 y Fx(is)h(the)g(operator)e(of)h(multiplication)f(by)i Ft(r)r Fx(.)60 b(W)m(ith)32 b(a)h(slight)e(ab)n(use)h(of)f(the)h (notation)f(we)h(will)291 4086 y(sometimes)18 b(denote)g Ft(h)954 4098 y Fs(R)1011 4106 y Fh(j)1065 4086 y Fx(and)h Ft(h)1253 4098 y Fs(R)1333 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Ft(g)1277 4969 y Fo(j)1311 4957 y Fn(\()p Ft(t)p Fn(\))24 b Fq(\021)1517 4844 y Fl(Z)1600 4864 y Fo(e)1631 4872 y Fi(+)1563 5033 y Fo(e)1594 5041 y Fd(\000)1696 4957 y Fn(e)1733 4923 y Fj(i)p Fo(tr)1827 4957 y Fq(j)p Ft(f)1891 4969 y Fo(j)1926 4957 y Fn(\()p Ft(r)r Fn(\))p Fq(j)2052 4923 y Fj(2)2104 4957 y Fn(d)p Ft(r)n(;)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 739 232 a Fx(40)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx(belong)f(to)h Ft(L)1128 493 y Fj(1)1165 523 y Fn(\()p Fp(R)p Ft(;)14 b Fn(d)p Ft(t)p Fn(\))p Fx(.)863 733 y(Assumption)20 b(\(SEBB2\))g(implies)g(that)g(the)g(function)1354 971 y Ft(G)p Fn(\()p Ft(z)t Fn(\))k Fq(\021)1637 858 y Fl(Z)1720 878 y Fo(e)1751 886 y Fi(+)1683 1046 y Fo(e)1714 1054 y Fd(\000)1826 914 y Fq(j)p Ft(f)9 b Fn(\()p Ft(r)r Fn(\))p Fq(j)2025 884 y Fj(2)p 1826 951 238 4 v 1853 1027 a Ft(r)22 b Fq(\000)c Ft(z)2088 971 y Fn(d)p Ft(r)25 b Fn(=)e Fq(\000)p Fn(i)2386 858 y Fl(Z)2469 878 y 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Fj(+)1949 3217 y Fn(\))21 b Fx(and)f Fq(j)p Ft(f)9 b Fn(\()p Ft(")2287 3229 y Fj(0)2324 3217 y Fn(\))p Fq(j)23 b(6)p Fn(=)g(0)p Fx(.)863 3427 y(W)-7 b(e)22 b(set)940 3647 y Ft(F)12 b Fn(\()p Ft(r)r Fn(\))24 b Fq(\021)f Ft(")1259 3659 y Fj(0)1314 3647 y Fq(\000)18 b Ft(r)j Fq(\000)d Ft(\025)1586 3613 y Fj(2)1624 3647 y Ft(G)p Fn(\()p Ft(r)k Fq(\000)c Fn(i)p Ft(o)p Fn(\))24 b(=)e Ft(")2108 3659 y Fj(0)2164 3647 y Fq(\000)c Ft(r)j Fq(\000)d Ft(\025)2436 3613 y Fj(2)2487 3534 y Fl(Z)2570 3555 y Fo(e)2601 3563 y Fi(+)2533 3723 y Fo(e)2564 3731 y Fd(\000)2731 3591 y Fq(j)p Ft(f)9 b Fn(\()p Ft(r)2875 3561 y Fs(0)2899 3591 y Fn(\))p Fq(j)2954 3561 y Fj(2)p 2677 3628 370 4 v 2677 3704 a Ft(r)2716 3680 y Fs(0)2758 3704 y Fq(\000)18 b Ft(r)j Fn(+)d(i)p Ft(o)3069 3647 y Fn(d)p Ft(r)3154 3613 y Fs(0)3179 3647 y Ft(:)201 b Fx(\(6.48\))739 3890 y(By)24 b(a)g(well-kno)n(wn)e(result)i(in)g (harmonic)e(analysis)h(\(see,)i(e.g.,)f([Ja)o(])g(or)g(an)o(y)f (harmonic)f(analysis)739 3990 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2088 4549 a Ft(r)2127 4525 y Fs(0)2169 4549 y Fq(\000)f Ft(r)j Fn(+)d(i)p Ft(o)2500 4492 y Fn(d)p Ft(r)2585 4458 y Fs(0)2609 4492 y Ft(;)739 4705 y Fx(is)j(also)g(in)f Fk(h)1098 4717 y Fs(R)1159 4705 y Fx(.)863 4807 y(The)i(main)g (spectral)f(and)h(scattering)f(theoretic)g(results)h(on)g Ft(h)2678 4819 y Fo(\025)2744 4807 y Fx(are)g(gi)n(v)o(en)e(in)i(the)g (follo)n(wing)739 4907 y(Theorem)h(which)h(is)h(an)g(easy)f (consequence)e(of)j(the)f(techniques)f(described)h(in)g([Ja].)38 b(Its)25 b(proof)739 5006 y(can)20 b(be)g(found)e(in)j([JKP].)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(41)291 523 y Fu(Theor)o(em)20 b(6.2)40 b Fr(Suppose)17 b(that)h(Assumptions)f Fx(\(SEBB1\),)i(\(SEBB2\))f Fr(and)g Fx(\(SEBB4\))g Fr(hold.)24 b(Then)291 623 y(ther)m(e)c(e)n(xists)h(a)g (constant)e Fn(\003)k Ft(>)f Fn(0)e Fr(suc)o(h)g(that,)g(for)g(any)g Fn(0)j Ft(<)f Fq(j)p Ft(\025)p Fq(j)i Ft(<)f 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Fn(\()p Ft(W)685 1581 y Fs(\000)742 1569 y Ft( )s Fn(\)\()p Ft(r)r Fn(\))j(=)f Ft(g)s Fn(\()p Ft(r)r Fn(\))c Fq(\000)g Ft(\025F)12 b Fn(\()p Ft(r)r Fn(\))1511 1535 y Fs(\000)p Fj(1)1615 1427 y Fl(")1664 1569 y Ft(\013)18 b Fq(\000)g Ft(\025)1880 1456 y Fl(Z)1964 1477 y Fo(e)1995 1485 y Fi(+)1927 1645 y Fo(e)1958 1653 y Fd(\000)2088 1491 y Fn(\026)2070 1513 y Ft(f)9 b Fn(\()p Ft(r)2191 1483 y Fs(0)2215 1513 y Fn(\))19 b Fq(\001)f Ft(g)s Fn(\()p Ft(r)2421 1483 y Fs(0)2445 1513 y Fn(\))p 2070 1550 408 4 v 2089 1626 a Ft(r)2128 1602 y Fs(0)2171 1626 y Fq(\000)g Ft(r)j Fn(+)d(i)p Ft(o)2501 1569 y Fn(d)p Ft(r)2586 1535 y Fs(0)2610 1427 y Fl(#)2673 1569 y Ft(f)9 b Fn(\()p Ft(r)r Fn(\))p Ft(:)105 b Fx(\(6.50\))415 1843 y(Needless)32 b(to)f(say)-5 b(,)34 b(the)e(thermodynamics)c(of)j(the)g(SEBB)i(model)d (can)i(be)f(studied)g(under)291 1943 y(much)17 b(more)g(general)g (assumptions)g(than)h(\(SEBB1\)-\(SEBB4\).)24 b(Ho)n(we)n(v)o(er)m(,)16 b(these)i(assumptions)291 2042 y(allo)n(w)i(us)g(to)h(describe)e(the)h (results)h(of)f([AJPP])g(with)h(the)f(least)h(number)d(of)i (technicalities.)415 2142 y(P)o(arenthetically)-5 b(,)20 b(we)h(note)g(that)h(the)f(SEBB)i(model)d(is)i(ob)o(viously)e(time-re)n (v)o(ersal)f(in)m(v)n(ariant.)291 2241 y(Write)h Ft(f)541 2253 y Fo(j)576 2241 y Fn(\()p Ft(r)r Fn(\))k(=)f(e)828 2211 y Fj(i)p Fo(\022)879 2219 y Fh(j)909 2211 y Fj(\()p Fo(r)r Fj(\))998 2241 y Fq(j)p Ft(f)1062 2253 y Fo(j)1097 2241 y Fn(\()p Ft(r)r Fn(\))p Fq(j)p Fx(,)f(and)d(let)832 2415 y Fk(j)p Fn(\()p Ft(\013)h Fq(\010)e Fn(\()p Ft(g)1115 2427 y Fj(1)1152 2415 y Ft(;)c Fq(\001)g(\001)g(\001)27 b Ft(;)14 b(g)1390 2427 y Fo(M)1463 2415 y Fn(\)\))24 b(=)31 b(\026)-50 b Ft(\013)19 b Fq(\010)f Fn(\(e)1863 2381 y Fj(2i)p Fo(\022)1947 2389 y Fi(1)1986 2415 y Fn(\026)-45 b Ft(g)2023 2427 y Fj(1)2060 2415 y Ft(;)14 b Fq(\001)g(\001)g(\001)27 b Ft(;)14 b Fn(e)2295 2381 y Fj(2i)p Fo(\022)2379 2389 y Fh(M)2447 2415 y Fn(\026)-44 b Ft(g)2485 2427 y Fo(M)2558 2415 y Fn(\))p Ft(;)291 2588 y Fx(where)519 2587 y Fn(\026)528 2588 y Fq(\001)35 b Fx(denotes)19 b(the)h(usual)g(comple)o(x)f (conjugation.)j(Then)d(the)h(map)1361 2762 y Fk(r)p Fn(\()p Ft(A)p Fn(\))k(=)f(\000\()p Fk(j)p Fn(\))p Ft(A)p Fn(\000\()p Fk(j)1939 2728 y Fs(\000)p Fj(1)2029 2762 y Fn(\))p Ft(:)291 2935 y Fx(is)e(a)f(time)h(re)n(v)o(ersal)e(and)h Ft(!)j Fx(is)e(time)f(re)n(v)o(ersal)g(in)m(v)n(ariant.)415 3035 y(Finally)-5 b(,)21 b(as)h(an)f(e)o(xample,)f(consider)h(a)g (concrete)g(SEBB)h(model)f(where)f(each)h(reserv)n(oir)g(is)h(a)291 3135 y(semi-in\002nite)h(wire)h(in)f(the)h(tight-binding)d (approximation)g(described)h(in)i(Example)f(5)h(of)f(Sub-)291 3234 y(section)e(5.2.)28 b(Thus,)21 b(for)g(each)g Ft(j)5 b Fx(,)22 b Fk(h)1329 3246 y Fs(R)1386 3254 y Fh(j)1446 3234 y Fn(=)j Ft(`)1571 3204 y Fj(2)1608 3234 y Fn(\()p Fp(Z)1695 3246 y Fj(+)1751 3234 y Fn(\))d Fx(and)f Ft(h)1995 3246 y Fs(R)2052 3254 y Fh(j)2109 3234 y Fx(is)h(the)g(discrete)f (Laplacian)f(on)h Fp(Z)3099 3246 y Fj(+)291 3334 y Fx(with)k(Dirichlet) g(boundary)d(condition)i(at)h Fn(0)p Fx(.)40 b(Choosing)24 b Ft(f)2037 3346 y Fo(j)2104 3334 y Fn(=)32 b Ft(\016)2238 3346 y Fj(1)2301 3334 y Fx(we)26 b(obtain,)f(in)g(the)h(spectral)291 3433 y(representation)18 b(of)i Ft(h)920 3445 y Fs(R)977 3453 y Fh(j)1012 3433 y Fx(,)1295 3616 y Fk(h)1338 3628 y Fs(R)1395 3636 y Fh(j)1513 3616 y Fn(=)83 b Ft(L)1718 3582 y Fj(2)1755 3616 y Fn(\(\()p Fq(\000)p Fn(1)p Ft(;)14 b Fn(1\))p Ft(;)g Fn(d)p Ft(r)r Fn(\))p Ft(;)1290 3741 y(h)1338 3753 y Fs(R)1395 3761 y Fh(j)1513 3741 y Fn(=)83 b Ft(r)n(;)1218 3940 y(f)1268 3900 y Fj(#)1259 3963 y Fo(j)1326 3940 y Fn(\()p Ft(r)r Fn(\))h(=)1661 3809 y Fl(r)p 1744 3809 71 4 v 1758 3884 a Fn(2)p 1754 3921 51 4 v 1754 3997 a Ft(\031)1814 3940 y Fn(\(1)18 b Fq(\000)g Ft(r)2028 3906 y Fj(2)2066 3940 y Fn(\))2098 3906 y Fj(1)p Fo(=)p Fj(4)2203 3940 y Ft(:)291 4144 y Fx(Thus,)h(Assumptions)h (\(SEBB1\))g(and)f(\(SEBB4\))i(hold.)j(Since,)c(as)h Ft(t)i Fq(!)g(1)p 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4901 y Fj(0)1946 4889 y Fq(j)p 1704 4926 266 4 v 1771 5002 a Fn(2)p Ft(M)1979 4945 y(:)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 739 232 a Fx(42)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fv(7)119 b(Thermodynamics)30 b(of)f(the)h(SEBB)g(model)739 712 y Fx(Throughout)i(this)j(and)g(the)g(ne)o(xt)g(section)g(we)g(will) h(assume)f(that)h(Assumptions)e(\(SEBB1\)-)739 812 y(\(SEBB4\))20 b(hold.)739 1059 y Fm(7.1)99 b(Non-equilibrium)26 b(steady)f(states)739 1218 y Fx(In)c(this)i(subsection)e(we)h(sho)n(w)f(that)h(the)g(SEBB)h (model)e(has)h(a)g(unique)f(NESS)h Ft(!)3109 1230 y Fo(\025)p Fj(+)3226 1218 y Fx(which)f(does)739 1317 y(not)h(depend)f(on)h(the)h (choice)e(of)i(the)f(initial)h(state)g Ft(\021)31 b Fq(2)c(N)2432 1329 y Fo(!)2481 1317 y Fx(.)32 b(Recall)23 b(that)g(the)f(reference)f (state)i Ft(!)739 1417 y Fx(of)c(the)h(SEBB)g(model)f(is)i(the)e (quasi-free)f(gauge-in)m(v)n(ariant)d(state)21 b(generated)d(by)h Ft(T)34 b Fn(=)22 b Ft(T)3327 1429 y Fs(S)3392 1417 y Fq(\010)16 b Ft(T)3522 1429 y Fs(R)3582 1417 y Fx(,)739 1517 y(where)j Ft(T)1011 1529 y Fs(S)1083 1517 y Fn(=)k Ft(\015)k Fq(2)d Fn(\(0)p Ft(;)14 b Fn(1\))20 b Fx(and)g Ft(T)1715 1529 y Fs(R)1799 1517 y Fn(=)i Fq(\010)1951 1529 y Fo(j)1986 1517 y Ft(\032)2029 1529 y Fo(j)2064 1517 y Fn(\()p Ft(r)r Fn(\))p Fx(.)739 1708 y Fu(Theor)o(em)e(7.1)40 b Fr(Let)19 b Fn(\003)j Ft(>)h Fn(0)18 b Fr(be)g(the)g(constant)f(intr) l(oduced)f(in)i(Theor)m(em)g(6.2.)23 b(Then,)18 b(for)g(any)f(r)m(eal) 739 1808 y Ft(\025)k Fr(suc)o(h)f(that)g Fn(0)i Ft(<)h Fq(j)p Ft(\025)p Fq(j)h Ft(<)e Fn(\003)f Fr(the)f(following)f(hold:)799 1979 y(\(i\))41 b(The)20 b(limit)1785 2084 y Ft(\013)1838 2048 y Fj(+)1838 2109 y Fo(\025)1893 2084 y Fn(\()p Ft(A)p Fn(\))k Fq(\021)44 b Fn(lim)2131 2133 y Fo(t)p Fs(!1)2302 2084 y Ft(\034)2347 2048 y Fs(\000)p Fo(t)2338 2106 y Fj(0)2447 2084 y Fq(\016)18 b Ft(\034)2552 2049 y Fo(t)2543 2104 y(\025)2587 2084 y Fn(\()p Ft(A)p Fn(\))p Ft(;)667 b Fx(\(7.51\))919 2277 y Fr(e)n(xists)32 b(for)f(all)g Ft(A)43 b Fq(2)g(O)r Fr(.)57 b(Mor)m(eo)o(ver)-9 b(,)33 b Fn(Ran)14 b Ft(\013)2314 2241 y Fj(+)2314 2302 y Fo(\025)2412 2277 y Fn(=)42 b Fq(O)2585 2289 y Fs(R)2678 2277 y Fr(and)29 b Ft(\013)2886 2241 y Fj(+)2886 2302 y Fo(\025)2973 2277 y Fr(is)j(an)e(isomorphism)919 2376 y(between)20 b(the)g Ft(C)1399 2346 y Fs(\003)1437 2376 y Fr(-dynamical)e(systems)k Fn(\()p Fq(O)r Ft(;)14 b(\034)2279 2388 y Fo(\025)2323 2376 y Fn(\))21 b Fr(and)f Fn(\()p Fq(O)2620 2388 y Fs(R)2681 2376 y Ft(;)14 b(\034)2754 2388 y Fs(R)2815 2376 y Fn(\))p Fr(.)776 2549 y(\(ii\))41 b(Let)21 b Ft(!)1098 2561 y Fo(\025)p Fj(+)1215 2549 y Fq(\021)h Ft(!)1354 2561 y Fs(R)1434 2549 y Fq(\016)c Ft(\013)1547 2514 y Fj(+)1547 2574 y Fo(\025)1602 2549 y Fr(.)26 b(Then)1955 2735 y Fn(lim)1934 2785 y Fo(t)p Fs(!1)2105 2735 y Ft(\021)c Fq(\016)c Ft(\034)2273 2701 y Fo(t)2264 2756 y(\025)2331 2735 y Fn(=)23 b Ft(!)2471 2747 y Fo(\025)p Fj(+)2565 2735 y Ft(;)919 2945 y Fr(for)d(all)h Ft(\021)26 b Fq(2)d(N)1358 2957 y Fo(!)1407 2945 y Fr(.)753 3118 y(\(iii\))41 b Ft(!)971 3130 y Fo(\025)p Fj(+)1086 3118 y Fr(is)21 b(the)f(gaug)o (e-in)m(variant)d(quasi-fr)m(ee)i(state)h(on)g Fq(O)j Fr(g)o(ener)o(ated)c(by)1953 3304 y Ft(T)2002 3316 y Fj(+)2080 3304 y Fq(\021)j Ft(W)2257 3270 y Fs(\003)2245 3325 y(\000)2302 3304 y Ft(T)2351 3316 y Fs(R)2411 3304 y Ft(W)2489 3316 y Fs(\000)2546 3304 y Ft(;)919 3490 y Fr(wher)m(e)e Ft(W)1217 3502 y Fs(\000)1295 3490 y Fr(is)h(the)f(wave)g(oper)o(ator)f(of)i(Theor)m(em)e(6.2.)739 3680 y Fu(Pr)o(oof)o(.)30 b Fx(Recall)i(that)g Ft(\034)1434 3650 y Fo(t)1425 3703 y(\025)1501 3680 y Fx(is)h(a)f(group)e(of)i (Bogoliubo)o(v)d(automorphisms,)j Fr(i.e)o(.,)i Ft(\034)3178 3650 y Fo(t)3169 3703 y(\025)3213 3680 y Fn(\()p Ft(a)3289 3650 y Fj(#)3348 3680 y Fn(\()p Ft(f)9 b Fn(\)\))45 b(=)739 3779 y Ft(a)783 3749 y Fj(#)841 3779 y Fn(\(e)910 3749 y Fj(i)p Fo(th)993 3758 y Fh(\025)1037 3779 y Ft(f)9 b Fn(\))p Fx(.)25 b(Hence,)19 b(for)h(an)o(y)f(observ)n(able)g(of)h (the)g(form)1748 3966 y Ft(A)k Fn(=)e Ft(a)1965 3931 y Fj(#)2024 3966 y Fn(\()p Ft( )2110 3978 y Fj(1)2147 3966 y Fn(\))14 b Fq(\001)g(\001)g(\001)g Ft(a)2348 3931 y Fj(#)2407 3966 y Fn(\()p Ft( )2493 3978 y Fo(n)2538 3966 y Fn(\))p Ft(;)810 b Fx(\(7.52\))1203 4157 y Ft(\034)1248 4121 y Fs(\000)p Fo(t)1239 4179 y Fj(0)1348 4157 y Fq(\016)18 b Ft(\034)1453 4123 y Fo(t)1444 4177 y(\025)1488 4157 y Fn(\()p Ft(A)p Fn(\))24 b(=)e Ft(a)1769 4123 y Fj(#)1828 4157 y Fn(\(e)1897 4123 y Fs(\000)p Fj(i)p Fo(th)2032 4131 y Fi(0)2068 4157 y Fn(e)2105 4123 y Fj(i)p Fo(th)2188 4132 y Fh(\025)2231 4157 y Ft( )2285 4169 y Fj(1)2322 4157 y Fn(\))14 b Fq(\001)g(\001)g(\001)g Ft(a)2523 4123 y Fj(#)2582 4157 y Fn(\(e)2651 4123 y Fs(\000)p Fj(i)p Fo(th)2786 4131 y Fi(0)2822 4157 y Fn(e)2859 4123 y Fj(i)p Fo(th)2942 4132 y Fh(\025)2985 4157 y Ft( )3039 4169 y Fo(n)3084 4157 y Fn(\))p Ft(:)739 4312 y Fx(It)20 b(follo)n(ws)g (from)f(Theorem)g(6.2)g(that)1374 4498 y Fn(lim)1353 4548 y Fo(t)p Fs(!1)1525 4498 y Ft(\034)1570 4462 y Fs(\000)p Fo(t)1561 4520 y Fj(0)1670 4498 y Fq(\016)f Ft(\034)1775 4463 y Fo(t)1766 4518 y(\025)1810 4498 y Fn(\()p Ft(A)p Fn(\))24 b(=)e Ft(a)2091 4463 y Fj(#)2150 4498 y Fn(\()p Ft(W)2260 4510 y Fs(\000)2317 4498 y Ft( )2371 4510 y Fj(1)2408 4498 y Fn(\))14 b Fq(\001)g(\001)g(\001)g Ft(a)2609 4463 y Fj(#)2667 4498 y Fn(\()p Ft(W)2777 4510 y Fs(\000)2834 4498 y Ft( )2888 4510 y Fo(n)2933 4498 y Fn(\))p Ft(:)739 4707 y Fx(Since)j(the)h(linear)e(span)h(of)g(set)h(of)f(elements)g(of)g (the)h(form)e(\(7.52\))f(is)k(dense)e(in)g Fq(O)r Fx(,)i(the)e(limit)h (\(7.51\))739 4807 y(e)o(xists)j(and)f(is)h(gi)n(v)o(en)e(by)h(the)h (Bogoliubo)o(v)d(morphism)g Ft(\013)2416 4771 y Fj(+)2416 4832 y Fo(\025)2472 4807 y Fn(\()p Ft(a)2548 4777 y Fj(#)2607 4807 y Fn(\()p Ft(f)9 b Fn(\)\))24 b(=)f Ft(a)2909 4777 y Fj(#)2967 4807 y Fn(\()p Ft(W)3077 4819 y Fs(\000)3134 4807 y Ft(f)9 b Fn(\))p Fx(.)26 b(Since)21 b Ft(W)3547 4819 y Fs(\000)739 4907 y Fx(is)k(a)g(unitary)e(operator)f(between)i Fk(h)h Fx(and)f Fk(h)2000 4919 y Fs(R)2061 4907 y Fx(,)i Fn(Ran)13 b Ft(\013)2323 4871 y Fj(+)2323 4932 y Fo(\025)2409 4907 y Fn(=)31 b(CAR\()p Fk(h)2763 4919 y Fs(R)2825 4907 y Fn(\))g(=)f Fq(O)3049 4919 y Fs(R)3110 4907 y Fx(,)c(which)d(pro)o(v) o(es)739 5006 y(P)o(art)d(\(i\).)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(43)415 523 y(Since)16 b Ft(h)664 535 y Fs(R)741 523 y Fx(has)h(purely)d(absolutely)h(continuous)f(spectrum,)h(it)i (follo)n(ws)e(from)g(our)g(discussion)291 623 y(of)32 b(quasi-free)f(states)j(in)e(Subsection)g(5.1)g(that)h Ft(!)1818 635 y Fs(R)1912 623 y Fx(is)g(mixing)f(for)g Ft(\034)2444 593 y Fo(t)2435 643 y Fj(0)2474 623 y Fx(.)62 b(P)o(art)33 b(\(ii\))g(is)g(thus)g(a)291 722 y(restatement)19 b(of)h(Proposition)f(3.9.)415 834 y(If)h Ft(A)j Fn(=)g Ft(a)708 804 y Fs(\003)746 834 y Fn(\()p Ft( )832 846 y Fo(n)878 834 y Fn(\))14 b Fq(\001)g(\001)g(\001)g Ft(a)1079 804 y Fs(\003)1117 834 y Fn(\()p Ft( )1203 846 y Fj(1)1240 834 y Fn(\))p Ft(a)p Fn(\()p Ft(\036)1397 846 y Fj(1)1435 834 y Fn(\))g Fq(\001)g(\001)g(\001)g Ft(a)p Fn(\()p Ft(\036)1717 846 y Fo(m)1781 834 y Fn(\))21 b Fx(is)g(an)f(element)g (of)g Fq(O)r Fx(,)h(then)557 1040 y Ft(!)612 1006 y Fj(+)667 1040 y Fn(\()p Ft(A)p Fn(\))84 b(=)e Ft(!)1076 1052 y Fs(R)1137 1040 y Fn(\()p Ft(a)1213 1006 y Fs(\003)1251 1040 y Fn(\()p Ft(W)1361 1052 y Fs(\000)1418 1040 y Ft( )1472 1052 y Fo(n)1517 1040 y Fn(\))14 b Fq(\001)g(\001)g(\001)g Ft(a)1718 1006 y Fs(\003)1756 1040 y Fn(\()p Ft(W)1866 1052 y Fs(\000)1923 1040 y Ft( )1977 1052 y Fj(1)2014 1040 y Fn(\))p Ft(a)p Fn(\()p Ft(W)2200 1052 y Fs(\000)2257 1040 y Ft(\036)2306 1052 y Fj(1)2344 1040 y Fn(\))g Fq(\001)g(\001)g (\001)g Ft(a)p Fn(\()p Ft(W)2655 1052 y Fs(\000)2711 1040 y Ft(\036)2760 1052 y Fo(m)2824 1040 y Fn(\)\))877 1165 y(=)82 b Ft(\016)1061 1177 y Fo(n;m)1199 1165 y Fn(det)14 b Fq(f)p Fn(\()p Ft(W)1480 1177 y Fs(\000)1536 1165 y Ft(\036)1585 1177 y Fo(i)1613 1165 y Ft(;)g(T)1699 1177 y Fs(R)1760 1165 y Ft(W)1838 1177 y Fs(\000)1894 1165 y Ft( )1948 1177 y Fo(j)1983 1165 y Fn(\))p Fq(g)877 1289 y Fn(=)82 b Ft(\016)1061 1301 y Fo(n;m)1199 1289 y Fn(det)14 b Fq(f)p Fn(\()p Ft(\036)1451 1301 y Fo(i)1479 1289 y Ft(;)g(T)1565 1301 y Fj(+)1619 1289 y Ft( )1673 1301 y Fo(j)1708 1289 y Fn(\))p Fq(g)p Ft(:)291 1496 y Fx(and)19 b(P)o(art)h(\(iii\))h(follo)n(ws.)j Fe(\003)291 1872 y Fm(7.2)99 b(The)26 b(Hilbert-Schmidt)g(condition)291 2050 y Fx(Since)32 b Ft(!)j Fx(and)c Ft(!)799 2062 y Fo(\025)p Fj(+)926 2050 y Fx(are)h(f)o(actor)f(states,)36 b(the)o(y)31 b(are)h(either)g(quasi-equi)n(v)n(alent)d(\()p Fq(N)2729 2062 y Fo(!)2822 2050 y Fn(=)44 b Fq(N)2999 2062 y Fo(!)3041 2071 y Fh(\025)p Fi(+)3127 2050 y Fx(\))291 2150 y(or)28 b(disjoint)h(\()p Fq(N)764 2162 y Fo(!)837 2150 y Fq(\\)c(N)985 2162 y Fo(!)1027 2171 y Fh(\025)p Fi(+)1152 2150 y Fn(=)38 b Fq(;)p Fx(\).)51 b(Since)29 b Fn(Ker)13 b Ft(T)50 b Fn(=)39 b(Ker)13 b(\()p Ft(I)32 b Fq(\000)24 b Ft(T)12 b Fn(\))39 b(=)g Fq(f)p Fn(0)p Fq(g)p Fx(,)30 b(we)f(also)g(ha)n(v)o(e)291 2250 y Fn(Ker)12 b Ft(T)487 2262 y Fj(+)575 2250 y Fn(=)32 b(Ker)13 b(\()p Ft(I)29 b Fq(\000)22 b Ft(T)1053 2262 y Fj(+)1108 2250 y Fn(\))32 b(=)h Fq(f)p Fn(0)p Fq(g)p Fx(,)25 b(and)g(so)h Ft(!)i Fx(and)d Ft(!)1966 2262 y Fo(\025)p Fj(+)2086 2250 y Fx(are)g(quasi-equi)n(v)n(alent)e(if)n(f)i(the)o(y)f(are)291 2349 y(unitarily)19 b(equi)n(v)n(alent.)415 2461 y(Let)k Ft(\013)29 b(>)e Fn(0)p Fx(.)33 b(A)23 b(function)f Ft(h)27 b Fn(:)h(\()p Ft(e)1400 2473 y Fs(\000)1456 2461 y Ft(;)14 b(e)1532 2473 y Fj(+)1587 2461 y Fn(\))28 b Fq(!)g Fp(C)c Fx(is)f Ft(\013)p Fx(-H\366lder)f(continuous)f(if)i(there)g(e)o(xists)g (a)291 2560 y(constant)c Ft(C)27 b Fx(such)20 b(that)g(for)g(all)h Ft(r)n(;)14 b(r)1325 2530 y Fs(0)1372 2560 y Fq(2)23 b Fn(\()p Ft(e)1521 2572 y Fs(\000)1577 2560 y Ft(;)14 b(e)1653 2572 y Fj(+)1708 2560 y Fn(\))p Fx(,)21 b Fq(j)p Ft(h)p Fn(\()p Ft(r)r Fn(\))e Fq(\000)f Ft(h)p Fn(\()p Ft(r)2177 2530 y Fs(0)2202 2560 y Fn(\))p Fq(j)23 b(\024)g Ft(C)6 b Fq(j)p Ft(r)21 b Fq(\000)d Ft(r)2636 2530 y Fs(0)2660 2560 y Fq(j)2683 2530 y Fo(\013)2731 2560 y Fx(.)291 2774 y Fu(Theor)o(em)i(7.2)40 b Fr(Assume)22 b(that)g(all)g(the)g(densities)g Ft(\032)1796 2786 y Fo(j)1831 2774 y Fn(\()p Ft(r)r Fn(\))h Fr(ar)m(e)f(the)g(same)g(and)f (equal)g(to)h Ft(\032)p Fn(\()p Ft(r)r Fn(\))p Fr(.)32 b(As-)291 2874 y(sume)19 b(further)g(that)g(the)g(functions)f Ft(\032)p Fn(\()p Ft(r)r Fn(\))1471 2843 y Fj(1)p Fo(=)p Fj(2)1596 2874 y Fr(and)g Fn(\(1)c Fq(\000)g Ft(\032)p Fn(\()p Ft(r)r Fn(\)\))2085 2843 y Fj(1)p Fo(=)p Fj(2)2211 2874 y Fr(ar)m(e)19 b Ft(\013)p Fr(-H\366lder)g(continuous)e(for)291 2973 y(some)j Ft(\013)j(>)g Fn(1)p Ft(=)p Fn(2)p Fr(.)h(Then)c(the)g (oper)o(ator)o(s)779 3180 y Fn(\()p Ft(T)860 3192 y Fj(+)915 3180 y Fn(\))947 3145 y Fj(1)p Fo(=)p Fj(2)1070 3180 y Fq(\000)e Ft(T)1214 3145 y Fj(1)p Fo(=)p Fj(2)1484 3180 y Fr(and)164 b Fn(\()p Ft(I)26 b Fq(\000)18 b Ft(T)2000 3192 y Fj(+)2055 3180 y Fn(\))2087 3145 y Fj(1)p Fo(=)p Fj(2)2210 3180 y Fq(\000)g Fn(\()p Ft(I)26 b Fq(\000)18 b Ft(T)12 b Fn(\))2563 3145 y Fj(1)p Fo(=)p Fj(2)291 3386 y Fr(ar)m(e)19 b(Hilbert-Sc)o(hmidt.)k(In)c(particular)-9 b(,)19 b(the)h(r)m(efer)m(ence)f(state)h Ft(!)j Fr(and)18 b(the)i(NESS)f Ft(!)2684 3398 y Fo(\025)p Fj(+)2798 3386 y Fr(ar)m(e)h(unitar)n(-)291 3486 y(ily)g(equivalent)f(and)g Fn(Ep\()p Ft(!)1093 3498 y Fo(\025)p Fj(+)1187 3486 y Fn(\))24 b(=)e(0)p Fr(.)291 3687 y Fu(Remark.)i Fx(W)-7 b(e)19 b(will)g(pro)o(v)o(e)d(this)j(theorem)e(in)h(Appendix)f(9.2.)23 b(Although)17 b(the)h(H\366lder)f(continuity)291 3787 y(assumption)25 b(is)i(certainly)f(not)g(optimal,)h(it)h(co)o(v)o(ers)d (most)h(cases)i(of)e(interest)g(and)g(allo)n(ws)h(for)f(a)291 3887 y(technically)19 b(simple)h(proof.)415 3998 y(Theorem)h(7.2)h (requires)f(a)i(comment.)31 b(By)23 b(the)f(general)g(principles)f(of)h (statistical)i(mechan-)291 4098 y(ics,)34 b(one)d(e)o(xpects)g(that)g Fn(Ep\()p Ft(!)1213 4110 y Fo(\025)p Fj(+)1307 4098 y Fn(\))44 b(=)f(0)32 b Fx(if)f(and)g(only)f(if)i(all)g(the)f(reserv)n (oirs)g(are)g(in)h Fr(thermal)291 4197 y(equilibrium)25 b Fx(at)h(the)g(same)g(in)m(v)o(erse)f(temperature)f Ft(\014)31 b Fx(and)26 b(chemical)f(potential)g Ft(\026)i Fx(\(see)f(Section)291 4297 y(4.3)h(in)h([JP4]\).)49 b(This)28 b(is)h(not)f(the)g(case)h(in)f(the)g(SEBB)i(model)d(because)g (the)i(perturbations)c Ft(V)3119 4309 y Fo(j)291 4397 y Fx(are)d(chosen)g(in)g(such)g(a)h(special)g(w)o(ay)g(that)f(the)h (coupled)e(dynamics)g(is)i(still)h(gi)n(v)o(en)d(by)h(a)h(Bogoli-)291 4496 y(ubo)o(v)16 b(automorphism.)22 b(F)o(ollo)n(wing)17 b(the)i(strate)o(gy)e(of)h([JP4],)h(one)f(can)g(sho)n(w)g(that)h(the)f (Planck)g(la)o(w)291 4596 y Ft(\032)p Fn(\()p Ft(r)r Fn(\))26 b(=)e(\(1)19 b(+)g(e)766 4566 y Fo(\014)s Fj(\()p Fo(r)r Fs(\000)p Fo(\026)p Fj(\))988 4596 y Fn(\))1020 4566 y Fs(\000)p Fj(1)1131 4596 y Fx(can)i(be)g(deduced)f(from)g(the)h (stability)g(requirement)e Fn(Ep\()p Ft(!)2872 4608 y Fo(\025)p Fj(+)2966 4596 y Fn(\))26 b(=)e(0)291 4695 y Fx(for)g(a)i(more)f(general)f(class)i(of)g(interactions)e Ft(V)1687 4707 y Fo(j)1722 4695 y Fx(.)41 b(F)o(or)26 b(reasons)e(of)i(space)f(we)h(will)g(not)f(discuss)291 4795 y(this)20 b(subject)g(in)h(detail)f(in)g(these)h(lecture)e(notes)h (\(the)g(interested)g(reader)f(may)g(consult)h([AJPP]\).)415 4907 y(W)-7 b(e)19 b(will)g(see)g(belo)n(w)e(that)i(the)f(entrop)o(y)e (production)f(of)j(the)g(SEBB)i(model)d(is)i(non-v)n(anishing)291 5006 y(whene)n(v)o(er)f(the)i(density)g(operators)e(of)i(the)g(reserv)n (oirs)g(are)g(not)g(identical.)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 739 232 a Fx(44)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fm(7.3)99 b(The)26 b(heat)g(and)f(char)o(ge)h(\003uxes)739 679 y Fx(Recall)19 b(that)f(the)g(observ)n(ables)f(describing)f(heat)j (and)e(char)o(ge)g(currents)g(out)h(of)g(the)g Ft(j)5 b Fx(-th)18 b(reserv)n(oir)739 778 y(are)1448 961 y Fn(\010)1508 973 y Fo(j)1625 961 y Fn(=)83 b Ft(\025)p Fn(\()p Ft(a)1897 927 y Fs(\003)1936 961 y Fn(\(i)p Ft(r)r(f)2071 973 y Fo(j)2107 961 y Fn(\))p Ft(a)p Fn(\(1\))18 b(+)g Ft(a)2434 927 y Fs(\003)2472 961 y Fn(\(1\))p Ft(a)p Fn(\(i)p Ft(r)r(f)2757 973 y Fo(j)2793 961 y Fn(\)\))p Ft(;)1451 1086 y Fq(J)1507 1098 y Fo(j)1625 1086 y Fn(=)83 b Ft(\025)p Fn(\()p Ft(a)1897 1051 y Fs(\003)1936 1086 y Fn(\(i)p Ft(f)2032 1098 y Fo(j)2067 1086 y Fn(\))p Ft(a)p Fn(\(1\))19 b(+)f Ft(a)2395 1051 y Fs(\003)2433 1086 y Fn(\(1\))p Ft(a)p Fn(\(i)p Ft(f)2679 1098 y Fo(j)2714 1086 y Fn(\)\))p Ft(:)739 1269 y Fx(The)i(e)o(xpectation)e(of)i(the)g(currents)f(in)i(the)f (state)h Ft(!)2224 1281 y Fo(\025)p Fj(+)2339 1269 y Fx(are)f(thus)1293 1451 y Ft(!)1345 1463 y Fo(\025)p Fj(+)1439 1451 y Fn(\(\010)1531 1463 y Fo(j)1567 1451 y Fn(\))83 b(=)g(i)p Ft(\025!)1953 1463 y Fo(\025)p Fj(+)2047 1384 y Fl(\000)2085 1451 y Ft(a)2129 1417 y Fs(\003)2167 1451 y Fn(\()p Ft(r)r(f)2279 1463 y Fo(j)2315 1451 y Fn(\))p Ft(a)p Fn(\(1\))19 b Fq(\000)f Ft(a)2643 1417 y Fs(\003)2681 1451 y Fn(\(1\))p Ft(a)p Fn(\()p Ft(r)r(f)2943 1463 y Fo(j)2978 1451 y Fn(\))3010 1384 y Fl(\001)1682 1576 y Fn(=)83 b(2)p Ft(\025)p Fn(Im)14 b(\()p Ft(r)r(f)2145 1588 y Fo(j)2180 1576 y Ft(;)g(T)2266 1588 y Fj(+)2321 1576 y Fn(1\))1682 1700 y(=)83 b(2)p Ft(\025)p Fn(Im)14 b(\()p Ft(W)2143 1712 y Fs(\000)2199 1700 y Ft(r)r(f)2279 1712 y Fo(j)2315 1700 y Ft(;)g(T)2401 1712 y Fs(R)2461 1700 y Ft(W)2539 1712 y Fs(\000)2596 1700 y Fn(1\))p Ft(;)739 1883 y Fx(and)1335 2066 y Ft(!)1387 2078 y Fo(\025)p Fj(+)1481 2066 y Fn(\()p Fq(J)1569 2078 y Fo(j)1605 2066 y Fn(\))83 b(=)f(i)p Ft(\025!)1990 2078 y Fo(\025)p Fj(+)2085 1999 y Fl(\000)2123 2066 y Ft(a)2167 2032 y Fs(\003)2205 2066 y Fn(\()p Ft(f)2278 2078 y Fo(j)2313 2066 y Fn(\))p Ft(a)p Fn(\(1\))19 b Fq(\000)f Ft(a)2641 2032 y Fs(\003)2679 2066 y Fn(\(1\))p Ft(a)p Fn(\()p 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Fj(2)2382 4940 y Fn(\()p Ft(\032)2457 4952 y Fo(j)2492 4940 y Fn(\()p Ft(r)r Fn(\))d Fq(\000)e Ft(\032)2741 4952 y Fo(k)2782 4940 y Fn(\()p Ft(r)r Fn(\)\))2992 4883 y Ft(r)r Fn(d)p Ft(r)p 2927 4920 253 4 v 2927 4997 a Fq(j)p Ft(F)12 b Fn(\()p Ft(r)r Fn(\))p Fq(j)3141 4973 y Fj(2)3191 4940 y Ft(:)189 b Fx(\(7.53\))p eop end %%Page: 45 45 TeXDict begin 45 44 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(45)291 523 y(In)19 b(a)i(completely)e(similar)h(w)o(ay)g(one)g (obtains)481 753 y Ft(!)533 765 y Fo(\025)p Fj(+)627 753 y Fn(\()p Fq(J)715 765 y Fo(j)751 753 y Fn(\))j(=)g(2)p Ft(\031)s(\025)1034 719 y Fj(4)1111 650 y Fo(M)1086 674 y Fl(X)1085 853 y Fo(k)q Fj(=1)1220 640 y Fl(Z)1303 661 y Fo(e)1334 669 y Fi(+)1266 829 y Fo(e)1297 837 y Fd(\000)1399 753 y Fq(j)p Ft(f)1463 765 y Fo(j)1498 753 y Fn(\()p Ft(r)r Fn(\))p Fq(j)1624 719 y Fj(2)1662 753 y Fq(j)p Ft(f)1726 765 y Fo(k)1767 753 y Fn(\()p Ft(r)r Fn(\))p Fq(j)1893 719 y Fj(2)1932 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Fr(is)393 4664 y Ft(!)445 4676 y Fo(\025)p Fj(+)539 4664 y Fn(\()p Ft(\033)s Fn(\))j(=)f Ft(\031)s(\025)863 4630 y Fj(4)964 4561 y Fo(M)939 4585 y Fl(X)915 4764 y Fo(j;k)q Fj(=1)1096 4551 y Fl(Z)1179 4572 y Fo(e)1210 4580 y Fi(+)1142 4740 y Fo(e)1173 4748 y Fd(\000)1286 4608 y Fq(j)p Ft(f)1350 4620 y Fo(j)1384 4608 y Fn(\()p Ft(r)r Fn(\))p Fq(j)1510 4578 y Fj(2)1549 4608 y Fq(j)p Ft(f)1613 4620 y Fo(k)1653 4608 y Fn(\()p Ft(r)r Fn(\))p Fq(j)1779 4578 y Fj(2)p 1286 4645 V 1425 4721 a Fq(j)p Ft(F)12 b Fn(\()p Ft(r)r Fn(\))p Fq(j)1639 4697 y Fj(2)1842 4664 y Fn(\()p Ft(s)1913 4676 y Fo(j)1948 4664 y Fn(\()p Ft(r)r Fn(\))20 b Fq(\000)e Ft(s)2193 4676 y Fo(k)2234 4664 y Fn(\()p Ft(r)r Fn(\)\))d(\()p Ft(\032)2459 4676 y Fo(k)2500 4664 y Fn(\()p Ft(r)r Fn(\))20 b Fq(\000)e Ft(\032)2749 4676 y Fo(j)2784 4664 y Fn(\()p Ft(r)r Fn(\)\))29 b(d)p Ft(r)n(:)291 4907 y Fr(In)24 b(particular)-9 b(,)24 b Fn(Ep\()p Ft(!)946 4919 y Fj(+)1001 4907 y Fn(\))31 b Fq(\025)g Fn(0)24 b Fr(\(something)g(we)h(alr)m(eady) e(know)i(fr)l(om)g(the)f(g)o(ener)o(al)g(principles\))291 5006 y(and)19 b Fn(Ep)o(\()p Ft(!)622 5018 y Fj(+)678 5006 y Fn(\))k(=)g(0)d Fr(if)h(and)e(only)h(if)h Ft(\032)1369 5018 y Fj(1)1429 5006 y Fn(=)i Fq(\001)14 b(\001)g(\001)22 b Fn(=)h Ft(\032)1767 5018 y Fo(M)1841 5006 y Fr(.)p eop end %%Page: 46 46 TeXDict begin 46 45 bop 739 232 a Fx(46)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)863 523 y Fx(Since)k Ft(!)i Fx(and)d Ft(!)1346 535 y Fo(\025)p Fj(+)1464 523 y Fx(are)g(f)o(actor)g(states,)i(the)o(y)e(are)g(either)g (quasi-equi)n(v)n(alent)d(or)j(disjoint.)34 b(By)739 623 y(Theorem)16 b(3.4,)i(if)h Fn(Ep)o(\()p Ft(!)1457 635 y Fo(\025)p Fj(+)1552 623 y Fn(\))k Ft(>)g Fn(0)p Fx(,)18 b(then)g Ft(!)1990 635 y Fo(\025)p Fj(+)2103 623 y Fx(is)h(not)f Ft(!)s Fx(-normal.)23 b(Hence,)18 b(Theorem)e(7.3)i(implies)739 722 y(that)23 b(if)g(the)g(densities)g Ft(\032)1443 734 y Fo(j)1501 722 y Fx(are)g(not)f(all)h(equal,)g(then)f (the)h(reference)e(state)j Ft(!)i Fx(and)c(the)h(NESS)g Ft(!)3509 734 y Fo(\025)p Fj(+)739 822 y Fx(are)d(disjoint)g(states.) 863 923 y(Until)31 b(the)f(end)g(of)g(this)g(section)g(we)h(will)g (assume)f(that)g(the)g(ener)o(gy)e(density)i(of)g(the)g Ft(j)5 b Fx(-th)739 1022 y(reserv)n(oir)19 b(is)1718 1145 y Ft(\032)1761 1157 y Fo(\014)1799 1165 y Fh(j)1830 1157 y Fo(\026)1870 1165 y Fh(j)1905 1145 y Fn(\()p Ft(r)r Fn(\))25 b Fq(\021)2340 1089 y Fn(1)p 2130 1126 461 4 v 2130 1205 a(1)18 b(+)g(e)2310 1180 y Fo(\014)2348 1188 y Fh(j)2379 1180 y Fj(\()p Fo(r)r Fs(\000)p Fo(\026)2530 1188 y Fh(j)2561 1180 y Fj(\))2601 1145 y Ft(;)739 1331 y Fx(where)24 b Ft(\014)1014 1343 y Fo(j)1075 1331 y Fx(is)i(the)f(in)m(v)o(erse)f(temperature)f(and)h Ft(\026)2161 1343 y Fo(j)2228 1331 y Fq(2)32 b Fp(R)26 b Fx(is)g(the)f(chemical)f (potential)g(of)h(the)g Ft(j)5 b Fx(-th)739 1431 y(reserv)n(oir)-5 b(.)24 b(Then,)19 b(by)h(\(6.46\),)e Fn(Ep\()p Ft(!)1819 1443 y Fo(\025)p Fj(+)1913 1431 y Fn(\))j Fx(can)f(be)g(written)g(as) 1424 1616 y Fn(Ep)o(\()p Ft(!)1610 1628 y Fo(\025)p Fj(+)1705 1616 y Fn(\))j(=)g(Ep)1950 1636 y Fj(heat)2080 1616 y Fn(\()p Ft(!)2164 1628 y Fo(\025)p Fj(+)2258 1616 y Fn(\))c(+)f(Ep)2494 1636 y Fj(c)n(harge)2685 1616 y Fn(\()p Ft(!)2769 1628 y Fo(\025)p Fj(+)2863 1616 y Fn(\))p Ft(;)739 1801 y Fx(where)1583 1966 y Fn(Ep)1686 1986 y Fj(heat)1815 1966 y Fn(\()p Ft(!)1899 1978 y Fo(\025)p Fj(+)1993 1966 y Fn(\))24 b(=)e Fq(\000)2240 1862 y Fo(M)2215 1887 y Fl(X)2217 2064 y Fo(j)s Fj(=1)2348 1966 y Ft(\014)2395 1978 y Fo(j)2430 1966 y Ft(!)2482 1978 y Fo(\025)p Fj(+)2577 1966 y Fn(\(\010)2669 1978 y Fo(j)2704 1966 y Fn(\))p Ft(;)739 2202 y Fx(is)f(interpreted)d (as)j(the)g(entrop)o(y)d(production)g(due)h(to)i(the)f(heat)g(\003ux)o (es)g(and)1551 2465 y Fn(Ep)1654 2486 y Fj(c)n(harge)1844 2465 y Fn(\()p Ft(!)1928 2477 y Fo(\025)p Fj(+)2022 2465 y Fn(\))k(=)2190 2362 y Fo(M)2165 2387 y Fl(X)2168 2563 y Fo(j)s Fj(=1)2299 2465 y Ft(\014)2346 2477 y Fo(j)2381 2465 y Ft(\026)2431 2477 y Fo(j)2466 2465 y Ft(!)2518 2477 y Fo(\025)p Fj(+)2612 2465 y Fn(\()p Fq(J)2700 2477 y Fo(j)2736 2465 y Fn(\))p Ft(:)739 2733 y Fx(as)d(the)f(entrop)o(y)f (production)e(due)j(to)g(the)g(electric)g(currents.)739 2978 y Fm(7.5)99 b(Equilibrium)26 b(corr)n(elation)f(functions)739 3136 y Fx(In)20 b(this)h(subsection)e(we)h(compute)f(the)h(inte)o (grated)f(current-current)d(correlation)j(functions)1439 3380 y Ft(L)1496 3392 y Fo(\032)1535 3380 y Fn(\()p Ft(A;)14 b(B)t Fn(\))24 b Fq(\021)55 b Fn(lim)1876 3433 y Fo(T)9 b Fs(!1)2080 3324 y Fn(1)p 2080 3361 42 4 v 2080 3437 a(2)2146 3267 y Fl(Z)2229 3287 y Fo(T)2192 3456 y Fs(\000)p Fo(T)2310 3380 y Ft(!)2362 3392 y Fo(\032)p Fj(+)2451 3380 y Fn(\()p Ft(\034)2528 3346 y Fo(t)2519 3400 y(\025)2563 3380 y Fn(\()p Ft(A)p Fn(\))p Ft(B)t Fn(\))14 b(d)p Ft(t;)739 3613 y Fx(where)25 b Ft(A)i Fx(and)f Ft(B)k Fx(are)c(heat)g(or)g(char)o (ge)e(\003ux)i(observ)n(ables)f(and)g Ft(!)2703 3625 y Fo(\032)p Fj(+)2819 3613 y Fx(denotes)g(the)h(NESS)h Ft(!)3509 3625 y Fo(\025)p Fj(+)739 3713 y Fx(in)d(the)g(equilibrium)e (case)j Ft(\032)1577 3725 y Fj(1)1644 3713 y Fn(=)30 b Fq(\001)14 b(\001)g(\001)31 b Fn(=)e Ft(\032)2004 3725 y Fo(M)2108 3713 y Fn(=)h Ft(\032)p Fx(.)37 b(T)-7 b(o)25 b(do)e(this,)j(note)d(that)i Fn(\010)3066 3725 y Fo(l)3121 3713 y Fn(=)30 b(d\000\()p Ft(')3400 3725 y Fo(l)3426 3713 y Fn(\))25 b Fx(and)739 3812 y Fq(J)795 3824 y Fo(l)844 3812 y Fn(=)d(d\000\()p Ft(j)1095 3824 y Fo(l)1121 3812 y Fn(\))f Fx(where)1576 3997 y Ft(')1630 4009 y Fo(l)1738 3997 y Fn(=)83 b(i[)p Ft(h)1980 4009 y Fs(R)2037 4018 y Fh(l)2065 3997 y Ft(;)14 b(\025v)s Fn(])24 b(=)f Fq(\000)p Fn(i[)p Ft(h)2487 4009 y Fo(\025)2530 3997 y Ft(;)14 b(h)2615 4009 y Fs(R)2672 4017 y Fh(j)2706 3997 y Fn(])p Ft(;)1596 4122 y(j)1630 4134 y Fo(l)1738 4122 y Fn(=)83 b(i[)p Ft(p)1974 4134 y Fo(j)2009 4122 y Ft(;)14 b(\025v)s Fn(])24 b(=)e Fq(\000)p Fn(i[)p Ft(h)2430 4134 y Fo(\025)2473 4122 y Ft(;)14 b(p)2552 4134 y Fo(j)2587 4122 y Fn(])p Ft(;)739 4307 y Fx(are)23 b(\002nite)g(rank)f(operators.)31 b(W)-7 b(e)24 b(will)g(only)e(consider)g Ft(L)2410 4319 y Fo(\032)2448 4307 y Fn(\(\010)2540 4319 y Fo(j)2575 4307 y Ft(;)14 b Fn(\010)2672 4319 y Fo(k)2713 4307 y Fn(\))p Fx(,)24 b(the)f(other)f(cases)i(are)f(com-)739 4407 y(pletely)d(similar)-5 b(.)863 4508 y(Using)25 b(the)f(CAR,)h(F)o (ormula)e(\(5.29\))g(and)h(the)g(f)o(act)g(that)h Ft(!)2581 4520 y Fo(\032)p Fj(+)2670 4508 y Fn(\(\010)2762 4520 y Fo(l)2788 4508 y Fn(\))31 b(=)f(0)p Fx(,)25 b(one)f(easily)h(sho)n (ws)739 4607 y(that)1260 4711 y Ft(!)1312 4723 y Fo(\032)p Fj(+)1401 4711 y Fn(\()p Ft(\034)1478 4677 y Fo(t)1469 4731 y(\025)1513 4711 y Fn(\(\010)1605 4723 y Fo(j)1641 4711 y Fn(\)\010)1733 4723 y Fo(k)1774 4711 y Fn(\))e(=)g(T)-7 b(r)13 b(\()p Ft(T)2097 4723 y Fj(+)2152 4711 y Fn(e)2189 4677 y Fj(i)p Fo(th)2272 4686 y Fh(\025)2315 4711 y Ft(')2369 4723 y Fo(j)2404 4711 y Fn(e)2441 4677 y Fs(\000)p Fj(i)p Fo(th)2576 4686 y Fh(\025)2619 4711 y Fn(\()p Ft(I)26 b Fq(\000)18 b Ft(T)2845 4723 y Fj(+)2899 4711 y Fn(\))p Ft(')2985 4723 y Fo(k)3027 4711 y Fn(\))p Ft(:)739 4864 y Fx(Since)1532 4991 y Fn(e)1569 4957 y Fj(i)p Fo(th)1652 4966 y Fh(\025)1694 4991 y Ft(')1748 5003 y Fo(j)1784 4991 y Fn(e)1821 4957 y Fs(\000)p Fj(i)p Fo(th)1956 4966 y Fh(\025)2022 4991 y Fn(=)k Fq(\000)2199 4935 y Fn(d)p 2184 4972 77 4 v 2184 5048 a(d)p Ft(t)2270 4991 y Fn(e)2307 4957 y Fj(i)p Fo(th)2390 4966 y Fh(\025)2433 4991 y Ft(h)2481 5003 y Fs(R)2538 5011 y Fh(j)2573 4991 y Fn(e)2610 4957 y Fs(\000)p Fj(i)p Fo(th)2745 4966 y Fh(\025)2787 4991 y Ft(;)p eop end %%Page: 47 47 TeXDict begin 47 46 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(47)291 523 y(the)20 b(inte)o(gration)e(can)i(be)g(e)o(xplicitly)f (performed)f(and)h(we)i(ha)n(v)o(e)618 736 y Ft(L)675 748 y Fo(\032)713 736 y Fn(\(\010)805 748 y Fo(j)841 736 y Ft(;)14 b Fn(\010)938 748 y Fo(k)978 736 y Fn(\))24 b(=)e 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Fo(j)1712 4907 y Ft(;)14 b Fq(J)1805 4919 y Fo(k)1846 4907 y Fn(\))27 b Fq(\024)e Fn(0)d Fx(for)g Ft(j)31 b Fq(6)p Fn(=)26 b Ft(k)f Fx(while)d Ft(L)2667 4919 y Fo(\032)2705 4907 y Fn(\(\010)2797 4919 y Fo(j)2832 4907 y Ft(;)14 b Fn(\010)2929 4919 y Fo(j)2964 4907 y Fn(\))27 b Fq(\025)e Fn(0)291 5006 y Fx(and)19 b Ft(L)488 5018 y Fo(\032)526 5006 y Fn(\()p Fq(J)614 5018 y Fo(j)650 5006 y Ft(;)14 b Fq(J)743 5018 y Fo(j)778 5006 y Fn(\))23 b Fq(\025)g Fn(0)p Fx(.)p eop end %%Page: 48 48 TeXDict begin 48 47 bop 739 232 a Fx(48)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fm(7.6)99 b(Onsager)25 b(r)n(elations.)31 b(K)o(ubo)24 b(f)n(ormulas.)739 679 y Fx(Let)c Ft(\014)917 691 y Fj(eq)1006 679 y Fx(and)g Ft(\026)1197 691 y Fj(eq)1286 679 y Fx(be)g(gi)n(v)o(en) f(equilibrium)f(v)n(alues)h(of)h(the)g(in)m(v)o(erse)f(temperature)f (and)i(the)g(chemi-)739 778 y(cal)i(potential.)27 b(The)21 b(af)n(\002nities)g(\(thermodynamic)d(forces\))i(conjugated)f(to)i(the) g(currents)g Fn(\010)3427 790 y Fo(j)3483 778 y Fx(and)739 878 y Fq(J)795 890 y Fo(j)851 878 y Fx(are)1453 978 y Ft(X)1522 990 y Fo(j)1580 978 y Fn(=)i Ft(\014)1715 990 y Fj(eq)1802 978 y Fq(\000)18 b Ft(\014)1932 990 y Fo(j)1967 978 y Ft(;)179 b(Y)2217 990 y Fo(j)2276 978 y Fn(=)22 b Ft(\014)2410 990 y Fo(j)2445 978 y Ft(\026)2495 990 y Fo(j)2549 978 y Fq(\000)c Ft(\014)2679 990 y Fj(eq)2747 978 y Ft(\026)2797 990 y Fj(eq)2866 978 y Ft(:)739 1106 y Fx(Indeed,)g(it)j(follo)n(ws)f(from)f(the)h(conserv)n(ations)f(la)o (ws)h(\(4.12\))f(and)g(\(6.39\))g(that)1379 1329 y Fn(Ep\()p Ft(!)1566 1341 y Fo(\025)p Fj(+)1661 1329 y Fn(\))k(=)1829 1225 y Fo(M)1804 1250 y Fl(X)1806 1427 y Fo(j)s Fj(=1)1937 1329 y Fn(\()q Ft(X)2039 1341 y Fo(j)2087 1329 y Ft(!)2139 1341 y Fo(\025)p Fj(+)2233 1329 y Fn(\(\010)2325 1341 y Fo(j)2361 1329 y Fn(\))18 b(+)g Ft(Y)2542 1341 y Fo(j)2591 1329 y Ft(!)2643 1341 y Fo(\025)p Fj(+)2738 1329 y 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Fs(S)1168 3039 y Fn(=)1256 2922 y Fl(\024)1300 2989 y Fn(0)100 b(0)1300 3088 y(0)82 b Ft(")1463 3100 y Fj(0)1500 2922 y Fl(\025)1558 3039 y Ft(;)180 b(!)1813 3051 y Fs(S)1885 3039 y Fn(=)1972 2922 y Fl(\024)2016 2989 y Fn(1)18 b Fq(\000)g Ft(\015)91 b Fn(0)2091 3088 y(0)157 b Ft(\015)2337 2922 y Fl(\025)2395 3039 y Ft(:)415 3273 y Fx(Let)24 b Ft(A)29 b Fq(2)g(O)791 3285 y Fs(S)864 3273 y Fx(be)23 b(an)h(observ)n(able)d(of)i(the)g (small)h(system.)35 b(W)-7 b(e)24 b(will)g(study)f(the)g(e)o (xpectation)291 3373 y(v)n(alues)1498 3479 y Ft(!)s Fn(\()p Ft(\034)1630 3436 y Fo(t=\025)1728 3411 y Fi(2)1621 3504 y Fo(\025)1765 3479 y Fn(\()p Ft(A)p Fn(\)\))p Ft(;)1008 b Fx(\(8.58\))291 3635 y(as)18 b Ft(\025)24 b Fq(!)f Fn(0)p Fx(.)h(If)17 b Ft(A)24 b Fn(=)e Ft(a)933 3605 y Fj(#)992 3635 y Fn(\(1\))p Fx(,)d(then)e(\(8.58\))f(v)n(anishes,)h (so)h(we)h(need)e(only)g(to)h(consider)e(the)i(Abelian)291 3734 y(2-dimensional)j(e)n(v)o(en)h(subalgebra)g Fq(O)1419 3699 y Fj(+)1417 3759 y Fs(S)1504 3734 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4561 y Fl(X)1817 4738 y Fo(j)s Fj(=1)1948 4640 y Ft(!)2000 4652 y Fs(S)t Fj(+)2100 4640 y Fn(\()p Fq(J)2188 4652 y Fj(fgr)p Fo(;j)2323 4640 y Fn(\))g(=)g(0)p Ft(;)291 4907 y Fx(follo)n(w)18 b(from)f(the)i(de\002nition)f(of)h(the) f(\003ux)o(es)h(and)f(the)h(relation)f Ft(K)2194 4919 y Fj(S)2234 4907 y Fn(\()p Ft(!)2318 4919 y Fs(S)t Fj(+)2418 4907 y Fn(\))24 b(=)e(0)p Fx(.)j(Of)19 b(course,)f(the)o(y)291 5006 y(also)i(follo)n(w)g(easily)g(from)f(the)h(abo)o(v)o(e)f(e)o (xplicit)g(formulas.)p eop end %%Page: 54 54 TeXDict begin 54 53 bop 739 232 a Fx(54)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)863 523 y Fx(Until)h(the)f(end)g(of)g(this)g(subsection)g(we)g(will)h (assume)f(that)1773 737 y Ft(\032)1816 749 y Fo(j)1851 737 y Fn(\()p Ft(r)r Fn(\))k(=)2285 681 y(1)p 2076 718 461 4 v 2076 797 a(1)18 b(+)g(e)2256 772 y Fo(\014)2294 780 y Fh(j)2324 772 y Fj(\()p Fo(r)r Fs(\000)p Fo(\026)2475 780 y Fh(j)2506 772 y Fj(\))2546 737 y Ft(:)863 970 y 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3725 y Fj(\()p Fo(H)1305 3733 y Fd(S)1348 3725 y Fs(\000)p Fo(\026)1440 3733 y Fi(eq)1500 3725 y Fj(\))1530 3760 y Ft(=)p Fn(T)-7 b(r)o(\(e)1726 3725 y Fs(\000)p Fo(\014)1816 3733 y Fi(eq)1875 3725 y Fj(\()p Fo(H)1955 3733 y Fd(S)1998 3725 y Fs(\000)p Fo(\026)2090 3733 y Fi(eq)2150 3725 y Fj(\))2180 3760 y Fn(\))23 b(=)2323 3642 y Fl(\024)2366 3709 y Fn(\(1)c(+)f(e)2579 3679 y Fs(\000)p Fo(\014)2669 3687 y Fi(eq)2727 3679 y Fo(")2758 3687 y Fi(0)2795 3709 y Fn(\))2827 3679 y Fs(\000)p Fj(1)3228 3709 y Fn(0)2621 3809 y(0)337 b(\(1)18 b(+)g(e)3212 3779 y Fo(\014)3250 3787 y Fi(eq)3309 3779 y Fo(")3340 3787 y Fi(0)3377 3809 y Fn(\))3409 3779 y Fs(\000)p Fj(1)3498 3642 y Fl(\025)3556 3760 y Ft(;)739 3999 y Fx(the)24 b(corresponding)d(NESS.)k(As)g(in)f (Subsection)f(7.6,)i(the)f(af)n(\002nities)g(\(thermodynamic)d (forces\))739 4099 y(are)i Ft(X)933 4111 y Fo(j)996 4099 y Fn(=)28 b Ft(\014)1136 4111 y Fj(eq)1225 4099 y Fq(\000)21 b Ft(\014)1358 4111 y Fo(j)1416 4099 y Fx(and)i Ft(Y)1608 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Fx(At)k(the)f(end)g(of)g(Subsection)g(4.5)g (we)h(ha)n(v)o(e)f(brie\003y)f(discussed)i(the)f(passage)g(from)g(the)h (micro-)291 1649 y(scopic)19 b(to)g(the)g(FGR)i(thermodynamics.)g(W)-7 b(e)21 b(no)n(w)d(return)g(to)i(this)g(subject)f(in)g(the)g(conte)o(xt) f(of)h(the)291 1748 y(SEBB)25 b(model.)35 b(The)23 b(ne)o(xt)g(theorem) g(is)h(a)h(mathematically)d(rigorous)g(v)o(ersion)h(of)g(the)h (heuristic)291 1848 y(statement)j(that)g(the)g(FGR)h(thermodynamics)d (is)j(the)f(\002rst)h(non-tri)n(vial)d(contrib)n(ution)g(\(in)i Ft(\025)p Fx(\))h(to)291 1948 y(the)20 b(microscopic)e(thermodynamics.) 291 2122 y Fu(Theor)o(em)i(8.2)100 b Fr(\(i\))41 b(F)-9 b(or)20 b(any)g(dia)o(gonal)e(observable)h Ft(A)k Fq(2)h(O)2165 2134 y Fs(S)2214 2122 y Fr(,)1405 2296 y Fn(lim)1394 2350 y Fo(\025)p Fs(!)p Fj(0)1546 2296 y Ft(!)1598 2308 y Fo(\025)p Fj(+)1692 2296 y Fn(\()p Ft(A)p Fn(\))g(=)f Ft(!)1982 2308 y Fs(S)t Fj(+)2082 2296 y Fn(\()p Ft(A)p Fn(\))p Ft(:)327 2529 y Fr(\(ii\))41 b(F)-9 b(or)21 b Ft(j)28 b Fn(=)23 b(1)p Ft(;)14 b Fq(\001)g(\001)g(\001)27 b Ft(;)14 b(M)9 b Fr(,)598 2703 y Fn(lim)586 2757 y Fo(\025)p Fs(!)p Fj(0)739 2703 y Ft(\025)787 2669 y Fs(\000)p Fj(2)876 2703 y Ft(!)928 2715 y Fo(\025)p Fj(+)1022 2703 y Fn(\(\010)1114 2715 y Fo(j)1150 2703 y Fn(\))23 b(=)g Ft(!)1345 2715 y Fs(S)t Fj(+)1444 2703 y Fn(\(\010)1536 2715 y Fj(fgr)p Fo(;j)1671 2703 y Fn(\))p Ft(;)192 b Fn(lim)1906 2757 y Fo(\025)p Fs(!)p Fj(0)2059 2703 y Ft(\025)2107 2669 y Fs(\000)p Fj(2)2196 2703 y Ft(!)2248 2715 y Fo(\025)p Fj(+)2342 2703 y Fn(\()p Fq(J)2430 2715 y Fo(j)2466 2703 y Fn(\))23 b(=)g Ft(!)2661 2715 y Fs(S)t Fj(+)2761 2703 y Fn(\()p Fq(J)2849 2715 y Fj(fgr)p Fo(;j)2984 2703 y Fn(\))p Ft(:)304 2942 y Fr(\(iii\))41 b(Let)21 b Ft(s)636 2954 y Fo(j)694 2942 y Fq(\021)i Fn(log)14 b Ft(\032)946 2954 y Fo(j)981 2942 y Fn(\()p Ft(")1052 2954 y Fj(0)1089 2942 y Fn(\))p Ft(=)p Fn(\(1)k Fq(\000)g Ft(\032)1381 2954 y Fo(j)1416 2942 y Fn(\()p Ft(")1487 2954 y Fj(0)1524 2942 y Fn(\)\))k Fr(and)d(de\002ne)g(the)h(FGR)g(entr)l(opy)g(pr)l (oduction)e(by)1135 3200 y Ft(\033)1182 3212 y Fj(fgr)1290 3200 y Fq(\021)k Fn(2)p Ft(\031)1508 3096 y Fo(M)1483 3121 y Fl(X)1485 3298 y Fo(j)s Fj(=1)1617 3200 y Fq(j)p Ft(f)1681 3212 y Fo(j)1715 3200 y Fn(\()p Ft(")1786 3212 y Fj(0)1824 3200 y Fn(\))p Fq(j)1879 3166 y Fj(2)1916 3200 y Ft(s)1955 3212 y Fo(j)2004 3083 y Fl(\024)2087 3150 y Fq(\000)p Ft(\032)2195 3162 y Fo(j)2229 3150 y Fn(\()p Ft(")2300 3162 y Fj(0)2338 3150 y Fn(\))2048 3249 y(1)c Fq(\000)g Ft(\032)2234 3261 y Fo(j)2269 3249 y Fn(\()p Ft(")2340 3261 y Fj(0)2377 3249 y Fn(\))2409 3083 y Fl(\025)2467 3200 y Ft(:)470 3457 y Fr(Then)1275 3556 y Fn(lim)1263 3611 y Fo(\025)p Fs(!)p Fj(0)1416 3556 y Ft(\025)1464 3522 y Fs(\000)p Fj(2)1567 3556 y Fn(Ep\()p Ft(!)1754 3568 y Fo(\025)p Fj(+)1848 3556 y Fn(\))24 b(=)e Ft(!)2043 3568 y Fs(S)t Fj(+)2143 3556 y Fn(\()p Ft(\033)2222 3568 y Fj(fgr)2307 3556 y Fn(\))p Ft(:)415 3758 y Fx(The)i(proof)f(of)i(this)g(theorem)e(is)j(an)f(inte)o (gration)d(e)o(x)o(ercise.)38 b(W)-7 b(e)26 b(will)f(restrict)g (ourselv)o(es)e(to)291 3857 y(an)h(outline)f(of)h(the)h(proof)d(of)i(P) o(art)h(\(i\))f(and)g(se)n(v)o(eral)f(comments.)36 b(Let)25 b Ft(A)31 b Fn(=)f Ft(N)2621 3869 y Fs(S)2701 3857 y Fn(=)g Ft(a)2840 3827 y Fs(\003)2878 3857 y Fn(\(1\))p Ft(a)p Fn(\(1\))p Fx(.)291 3957 y(Then)696 4118 y Ft(!)748 4130 y Fo(\025)p Fj(+)842 4118 y Fn(\()p Ft(A)p Fn(\))24 b(=)f(\()p Ft(W)1190 4130 y Fs(\000)1246 4118 y Fn(1)p Ft(;)14 b(T)1374 4130 y Fs(R)1434 4118 y Ft(W)1512 4130 y Fs(\000)1569 4118 y Fn(1\))23 b(=)1779 4014 y Fo(M)1753 4039 y Fl(X)1756 4216 y Fo(j)s Fj(=1)1887 4118 y Ft(\025)1935 4084 y Fj(2)1987 4005 y Fl(Z)2070 4025 y Fo(e)2101 4033 y Fi(+)2033 4194 y Fo(e)2064 4202 y Fd(\000)2176 4062 y Fq(j)p Ft(f)2240 4074 y Fo(j)2275 4062 y Fn(\()p Ft(r)r Fn(\))p Fq(j)2401 4032 y Fj(2)p 2176 4099 264 4 v 2181 4175 a Fq(j)p Ft(F)12 b Fn(\()p Ft(r)r Fn(\))p Fq(j)2395 4151 y Fj(2)2449 4118 y Ft(\032)2492 4130 y Fo(j)2527 4118 y Fn(\()p Ft(r)r Fn(\))i(d)p Ft(r)n(;)291 4345 y Fx(and)1181 4506 y Ft(!)1233 4518 y Fs(S)t Fj(+)1332 4506 y Fn(\()p Ft(A)p Fn(\))24 b(=)1595 4402 y Fo(M)1570 4427 y Fl(X)1573 4604 y Fo(j)s Fj(=1)1714 4450 y Fq(j)p Ft(f)1778 4462 y Fo(j)1812 4450 y Fn(\()p Ft(")1883 4462 y Fj(0)1921 4450 y Fn(\))p Fq(j)1976 4419 y Fj(2)p 1714 4487 300 4 v 1727 4563 a Fq(j)p Ft(f)9 b Fn(\()p Ft(")1871 4575 y Fj(0)1908 4563 y Fn(\))p Fq(j)1963 4539 y Fj(2)2023 4506 y Ft(\032)2066 4518 y Fo(j)2101 4506 y Fn(\()p Ft(")2172 4518 y Fj(0)2209 4506 y Fn(\))p Ft(:)291 4732 y Fx(Hence,)19 b(to)h(pro)o(v)o(e)f(P)o(art)h(\(i\))g(we)h(need)e(to)i(sho)n(w)f(that) 900 4957 y Fn(lim)889 5011 y Fo(\025)p Fs(!)p Fj(0)1041 4957 y Ft(\025)1089 4923 y Fj(2)1141 4844 y Fl(Z)1224 4864 y Fo(e)1255 4872 y Fi(+)1187 5033 y Fo(e)1218 5041 y Fd(\000)1330 4901 y Fq(j)p Ft(f)1394 4913 y Fo(j)1429 4901 y Fn(\()p Ft(r)r Fn(\))p Fq(j)1555 4871 y Fj(2)p 1330 4938 264 4 v 1335 5014 a Fq(j)p Ft(F)12 b Fn(\()p Ft(r)r Fn(\))p Fq(j)1549 4990 y Fj(2)1603 4957 y Ft(\032)1646 4969 y Fo(j)1681 4957 y Fn(\()p Ft(r)r Fn(\))i(d)p Ft(r)27 b Fn(=)2006 4901 y Fq(j)p Ft(f)2070 4913 y Fo(j)2105 4901 y Fn(\()p Ft(")2176 4913 y Fj(0)2213 4901 y Fn(\))p Fq(j)2268 4871 y Fj(2)p 2006 4938 300 4 v 2019 5014 a Fq(j)p Ft(f)9 b Fn(\()p Ft(")2163 5026 y Fj(0)2200 5014 y Fn(\))p Fq(j)2255 4990 y Fj(2)2315 4957 y Ft(\032)2358 4969 y Fo(j)2393 4957 y Fn(\()p Ft(")2464 4969 y Fj(0)2501 4957 y Fn(\))p Ft(:)p eop end %%Page: 56 56 TeXDict begin 56 55 bop 739 232 a Fx(56)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx(By)28 b(Assumption)f(\(SEBB2\),)i Ft(R)q Fn(\()p Ft(r)r Fn(\))38 b Fq(\021)e Fn(Re)14 b Ft(G)p Fn(\()p Ft(r)28 b Fq(\000)23 b Fn(i)p Ft(o)p Fn(\))29 b Fx(and)e Ft(\031)s Fq(j)p Ft(f)9 b Fn(\()p Ft(r)r Fn(\))p Fq(j)2838 493 y Fj(2)2914 523 y Fn(=)36 b(Im)14 b Ft(G)p Fn(\()p Ft(r)27 b Fq(\000)d Fn(i)p Ft(o)p Fn(\))29 b Fx(are)739 623 y(bounded)24 b(continuous)g(functions.)41 b(The)26 b(same)h(is)g(true)f(for)f Ft(\032)2574 635 y Fo(j)2609 623 y Fn(\()p Ft(r)r Fn(\))j Fx(by)e(Assumption)f(\(SEBB3\).)739 722 y(Since)1484 822 y Ft(F)12 b Fn(\()p Ft(r)r Fn(\))25 b(=)d Ft(")1803 834 y Fj(0)1859 822 y Fq(\000)c Ft(r)j Fq(\000)d Ft(\025)2131 788 y Fj(2)2169 822 y Ft(R)q Fn(\()p Ft(r)r Fn(\))h(+)f(i)p Ft(\025)2509 788 y Fj(2)2547 822 y Ft(\031)s Fq(j)p Ft(f)9 b Fn(\()p Ft(r)r Fn(\))p Fq(j)2796 788 y Fj(2)2835 822 y Ft(;)739 971 y Fx(we)20 b(ha)n(v)o(e)992 1074 y Fl(Z)1075 1094 y Fo(e)1106 1102 y Fi(+)1038 1262 y Fo(e)1069 1270 y Fd(\000)1181 1130 y Fq(j)p Ft(f)1245 1142 y Fo(j)1280 1130 y Fn(\()p Ft(r)r Fn(\))p Fq(j)1406 1100 y Fj(2)p 1181 1167 264 4 v 1186 1243 a Fq(j)p Ft(F)12 b Fn(\()p Ft(r)r Fn(\))p Fq(j)1400 1219 y Fj(2)1454 1187 y Ft(\032)1497 1199 y Fo(j)1532 1187 y Fn(\()p Ft(r)r Fn(\))i(d)p Ft(r)27 b Fn(=)1847 1074 y Fl(Z)1930 1094 y Fo(e)1961 1102 y Fi(+)1893 1262 y Fo(e)1924 1270 y Fd(\000)2406 1130 y Fq(j)p Ft(f)2470 1142 y Fo(j)2505 1130 y Fn(\()p Ft(r)r Fn(\))p Fq(j)2631 1100 y Fj(2)2670 1130 y Ft(\032)2713 1142 y Fo(j)2748 1130 y Fn(\()p Ft(r)r Fn(\))p 2036 1167 1187 4 v 2036 1243 a(\()p Ft(r)22 b Fq(\000)c Ft(")2249 1255 y Fj(0)2304 1243 y Fn(+)g Ft(\025)2435 1219 y Fj(2)2473 1243 y Ft(R)q Fn(\()p Ft(r)r Fn(\)\))2672 1219 y Fj(2)2729 1243 y Fn(+)g Ft(\031)2862 1219 y Fj(2)2899 1243 y Ft(\025)2947 1219 y Fj(4)2985 1243 y Fq(j)p Ft(f)9 b Fn(\()p Ft(r)r Fn(\))p Fq(j)3184 1219 y Fj(4)3246 1187 y Fn(d)p Ft(r)n(:)739 1425 y Fx(Using)31 b(the)h(abo)o(v)o(e-mentioned) 27 b(continuity)j(and)h(boundedness)e(properties)h(it)i(is)h(not)e (hard)g(to)739 1524 y(sho)n(w)20 b(that)1050 1740 y Fn(lim)1038 1794 y Fo(\025)p Fs(!)p Fj(0)1190 1740 y Ft(\025)1238 1705 y Fj(2)1290 1627 y Fl(Z)1373 1647 y Fo(e)1404 1655 y Fi(+)1336 1815 y Fo(e)1367 1823 y Fd(\000)1479 1683 y Fq(j)p Ft(f)1543 1695 y Fo(j)1578 1683 y Fn(\()p Ft(r)r Fn(\))p Fq(j)1704 1653 y Fj(2)p 1479 1721 264 4 v 1484 1797 a Fq(j)p Ft(F)12 b Fn(\()p Ft(r)r Fn(\))p Fq(j)1698 1773 y Fj(2)1753 1740 y Ft(\032)1796 1752 y Fo(j)1830 1740 y Fn(\()p Ft(r)r Fn(\))i(d)p Ft(r)1047 2052 y Fn(=)23 b Ft(\032)1178 2064 y Fo(j)1213 2052 y Fn(\()p Ft(")1284 2064 y Fj(0)1321 2052 y Fn(\))p Fq(j)p Ft(f)1417 2064 y Fo(j)1452 2052 y Fn(\()p Ft(")1523 2064 y Fj(0)1560 2052 y Fn(\))p Fq(j)1615 2018 y Fj(2)1692 2052 y Fn(lim)1680 2106 y Fo(\025)p Fs(!)p Fj(0)1833 2052 y Ft(\025)1881 2018 y Fj(2)1932 1939 y Fl(Z)2016 1960 y Fo(e)2047 1968 y Fi(+)1979 2128 y Fo(e)2010 2136 y Fd(\000)2672 1996 y Fn(d)p Ft(r)p 2122 2033 1187 4 v 2122 2109 a Fn(\()p Ft(r)e Fq(\000)d Ft(")2334 2121 y Fj(0)2390 2109 y Fn(+)g Ft(\025)2521 2085 y Fj(2)2558 2109 y Ft(R)q Fn(\()p Ft(r)r Fn(\)\))2757 2085 y Fj(2)2814 2109 y Fn(+)g Ft(\031)2947 2085 y Fj(2)2985 2109 y Ft(\025)3033 2085 y Fj(4)3071 2109 y Fq(j)p Ft(f)9 b Fn(\()p Ft(r)r Fn(\))p Fq(j)3270 2085 y Fj(4)1047 2364 y Fn(=)23 b Ft(\032)1178 2376 y Fo(j)1213 2364 y Fn(\()p Ft(")1284 2376 y Fj(0)1321 2364 y Fn(\))p Fq(j)p Ft(f)1417 2376 y Fo(j)1452 2364 y Fn(\()p Ft(")1523 2376 y Fj(0)1560 2364 y Fn(\))p Fq(j)1615 2330 y Fj(2)1678 2364 y Fn(lim)1667 2419 y Fo(\025)p Fs(!)p Fj(0)1819 2364 y Ft(\025)1867 2330 y Fj(2)1919 2251 y Fl(Z)2002 2272 y Fs(1)1965 2440 y(\0001)2380 2308 y Fn(d)p Ft(r)p 2111 2345 626 4 v 2111 2421 a(r)2150 2397 y Fj(2)2206 2421 y Fn(+)18 b Ft(\031)2339 2397 y Fj(2)2377 2421 y Ft(\025)2425 2397 y Fj(4)2463 2421 y Fq(j)p Ft(f)9 b Fn(\()p Ft(")2607 2433 y Fj(0)2643 2421 y Fn(\))p Fq(j)2698 2397 y Fj(4)1047 2604 y Fn(=)1145 2548 y Fq(j)p Ft(f)1209 2560 y Fo(j)1244 2548 y Fn(\()p Ft(")1315 2560 y Fj(0)1352 2548 y Fn(\))p Fq(j)1407 2518 y Fj(2)p 1145 2585 300 4 v 1158 2661 a Fq(j)p Ft(f)g Fn(\()p Ft(")1302 2673 y Fj(0)1339 2661 y Fn(\))p Fq(j)1394 2637 y Fj(2)1454 2604 y Ft(\032)1497 2616 y Fo(j)1532 2604 y Fn(\()p Ft(")1603 2616 y Fj(0)1640 2604 y Fn(\))p Ft(:)739 2829 y Fx(The)28 b(proofs)e(of)i(P)o(arts)h(\(ii\))f(and)g(\(iii\))g(are)g(similar)-5 b(.)49 b(Clearly)-5 b(,)30 b(under)c(additional)h(re)o(gularity)f(as-) 739 2928 y(sumptions)19 b(one)g(can)h(get)g(information)d(on)j(the)g (rate)g(of)f(con)m(v)o(er)o(gence)d(in)k(P)o(arts)h(\(i\)-\(iii\).)j (Finally)-5 b(,)739 3028 y(it)19 b(is)h(not)e(dif)n(\002cult)g(to)h (sho)n(w)-5 b(,)18 b(using)g(the)g(K)o(ubo)g(formulas)f(described)g(in) i(Subsection)f(7.6)g(and)g(8.3,)739 3128 y(that)1541 3227 y Fn(lim)1529 3282 y Fo(\025)p Fs(!)p Fj(0)1681 3227 y Ft(\025)1729 3193 y Fs(\000)p Fj(2)1819 3227 y Ft(L)1876 3239 y Fo(\032)1914 3227 y Fn(\()p Ft(A;)c(B)t Fn(\))24 b(=)f Ft(L)2313 3239 y Fj(fgr)2396 3227 y Fn(\()p Ft(A)2490 3239 y Fj(fgr)2574 3227 y Ft(;)14 b(B)2674 3239 y Fj(fgr)2758 3227 y Fn(\))p Ft(;)739 3404 y Fx(where)j Ft(A)p Fx(,)h Ft(B)k Fx(are)17 b(the)h(microscopic)d(heat)j(or)f(char)o (ge)e(\003ux)i(observ)n(ables)f(and)h Ft(A)3044 3416 y Fj(fgr)3128 3404 y Fx(,)h Ft(B)3230 3416 y Fj(fgr)3332 3404 y Fx(are)f(their)739 3504 y(FGR)k(counterparts.)739 3784 y Fv(9)119 b(A)m(ppendix)739 3986 y Fm(9.1)99 b(Structural)27 b(theor)n(ems)739 4142 y Fu(Pr)o(oof)15 b(of)i(Theor)o(em)h(3.3)81 b Fx(Recall)18 b(that)g Ft(\031)1986 4154 y Fo(!)2034 4142 y Fn(\()p Fq(O)r Fn(\))2166 4112 y Fs(00)2227 4142 y Fx(is)g(the)g(Banach)e(space)i(dual)e(of)h Fq(N)3204 4154 y Fo(!)3252 4142 y Fx(.)25 b(If)17 b Ft(A)23 b Fq(2)h(O)739 4242 y Fx(and)901 4221 y Fn(~)880 4242 y Ft(A)f Fq(2)g Ft(\031)1090 4254 y Fo(!)1139 4242 y Fn(\()p Fq(O)r Fn(\))1271 4211 y Fs(00)1335 4242 y Fx(is)e(a)g(weak-)p Fq(\003)e Fx(accumulation)f(point)h(of)h(the)g(net)1829 4425 y Fn(1)p 1829 4462 42 4 v 1835 4538 a Ft(t)1894 4368 y Fl(Z)1977 4389 y Fo(t)1940 4557 y Fj(0)2020 4481 y Ft(\031)2067 4493 y Fo(!)2116 4481 y Fn(\()p Ft(\034)2193 4447 y Fo(s)2184 4502 y(V)2242 4481 y Fn(\()p Ft(A)p Fn(\)\))14 b(d)p Ft(s;)739 4724 y(t)23 b Fq(\025)g Fn(0)p Fx(,)c(it)g(follo)n(ws)g(from) e(the)i(asymptotic)f(abelianness)g(in)h(mean)f(that)2830 4703 y Fn(~)2808 4724 y Ft(A)24 b Fq(2)f Ft(\031)3019 4736 y Fo(!)3067 4724 y Fn(\()p Fq(O)r Fn(\))3199 4694 y Fs(0)3224 4724 y Fx(.)i(Since)19 b Ft(!)j Fx(is)739 4824 y(a)e(f)o(actor)f(state)h(we)f(ha)n(v)o(e)g Ft(\031)1518 4836 y Fo(!)1567 4824 y Fn(\()p Fq(O)r Fn(\))1699 4794 y Fs(0)1739 4824 y Fq(\\)d Ft(\031)1857 4836 y Fo(!)1905 4824 y Fn(\()p Fq(O)r Fn(\))2037 4794 y Fs(00)2103 4824 y Fn(=)23 b Fp(C)p Ft(I)k Fx(and)19 b(therefore,)e(for)i(an)o(y)g Ft(\021)26 b Fq(2)e(N)3264 4836 y Fo(!)3312 4824 y Fx(,)c(one)e(has) 1928 5006 y Ft(\021)s Fn(\()2026 4985 y(~)2004 5006 y Ft(A)q Fn(\))23 b(=)g Ft(!)s Fn(\()2318 4985 y(~)2297 5006 y Ft(A)p Fn(\))p Ft(:)989 b Fx(\(9.67\))p eop end %%Page: 57 57 TeXDict begin 57 56 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(57)415 523 y(Let)21 b Ft(\026;)14 b(\027)28 b Fq(2)23 b(N)849 535 y Fo(!)918 523 y Fx(and)d Ft(\026)1109 535 y Fj(+)1187 523 y Fq(2)j Fn(\006)1325 535 y Fj(+)1380 523 y Fn(\()p Ft(\026;)14 b(\034)1535 535 y Fo(V)1594 523 y Fn(\))p Fx(.)25 b(Let)c Ft(t)1834 535 y Fo(\013)1904 523 y Fq(!)i(1)e Fx(be)f(a)h(net)f(such)g(that)1132 775 y Fn(lim)1168 825 y Fo(\013)1289 719 y Fn(1)p 1271 756 78 4 v 1271 832 a Ft(t)1301 844 y Fo(\013)1373 662 y Fl(Z)1456 683 y Fo(t)1481 691 y Fh(\013)1419 851 y Fj(0)1499 775 y Ft(\026)f Fq(\016)f Ft(\034)1673 741 y Fo(s)1664 796 y(V)1722 775 y Fn(\()p Ft(A)p Fn(\))c(d)p Ft(s)23 b Fn(=)g Ft(\026)2108 787 y Fj(+)2163 775 y Fn(\()p Ft(A)p Fn(\))p Ft(;)291 1011 y Fx(for)e(all)i Ft(A)k Fq(2)h(O)r Fx(.)k(P)o(assing)22 b(to)g(a)h(subnet,)f(we)h(may)e(also)i(assume)f (that)h(for)e(all)i Ft(A)k Fq(2)h(O)d Fx(and)d(some)291 1111 y Ft(\027)332 1123 y Fj(+)410 1111 y Fq(2)h Fn(\006)548 1123 y Fj(+)603 1111 y Fn(\()p Ft(\027)q(;)14 b(\034)750 1123 y Fo(V)808 1111 y Fn(\))p Fx(,)1139 1286 y Fn(lim)1175 1336 y Fo(\013)1296 1230 y Fn(1)p 1278 1267 V 1278 1343 a Ft(t)1308 1355 y Fo(\013)1379 1173 y Fl(Z)1462 1194 y Fo(t)1487 1202 y Fh(\013)1425 1362 y Fj(0)1506 1286 y Ft(\027)23 b Fq(\016)18 b Ft(\034)1675 1252 y Fo(s)1666 1307 y(V)1724 1286 y Fn(\()p Ft(A)p Fn(\))c(d)p Ft(s)24 b Fn(=)f Ft(\027)2102 1298 y Fj(+)2157 1286 y Fn(\()p Ft(A)p Fn(\))p Ft(:)291 1500 y Fx(By)e(the)g(Banach-Alaoglu)d(theorem,) i(for)g(an)o(y)g Ft(A)25 b Fq(2)f(O)g Fx(there)c(e)o(xists)i(a)f (subnet)f Ft(t)2649 1512 y Fo(\015)2692 1500 y Fn(\()p Ft(A)p Fn(\))i Fx(of)f(the)f(net)291 1600 y Ft(t)321 1612 y Fo(\013)389 1600 y Fx(and)f Ft(A)591 1570 y Fj(#)673 1600 y Fq(2)24 b Ft(\031)799 1612 y Fo(!)847 1600 y Fn(\()p Fq(O)r Fn(\))979 1570 y Fs(00)1043 1600 y Fx(such)c(that,)g(for)g(all)h Ft(\021)26 b Fq(2)d(N)1817 1612 y Fo(!)953 1860 y Fn(lim)992 1910 y Fo(\015)1171 1804 y Fn(1)p 1092 1841 200 4 v 1092 1917 a Ft(t)1122 1929 y Fo(\015)1165 1917 y Fn(\()p Ft(A)p Fn(\))1316 1747 y Fl(Z)1399 1768 y Fo(t)1424 1776 y Fh(\015)1462 1768 y Fj(\()p Fo(A)p Fj(\))1362 1936 y(0)1540 1860 y Ft(\021)s Fn(\()p Ft(\031)1663 1872 y Fo(!)1712 1860 y Fn(\()p Ft(\034)1789 1826 y Fo(s)1780 1881 y(V)1838 1860 y Fn(\()p Ft(A)p Fn(\)\)\))14 b(d)p Ft(s)25 b Fn(=)d Ft(\021)s Fn(\()p Ft(A)2377 1826 y Fj(#)2437 1860 y Fn(\))p Ft(:)291 2115 y Fx(Hence,)28 b Ft(\026)603 2127 y Fj(+)658 2115 y Fn(\()p Ft(A)p Fn(\))38 b(=)e Ft(\026)p Fn(\()p Ft(A)1067 2085 y Fj(#)1126 2115 y Fn(\))29 b Fx(and)e Ft(\027)1376 2127 y Fj(+)1431 2115 y Fn(\()p Ft(A)p Fn(\))37 b(=)g Ft(\027)5 b Fn(\()p Ft(A)1836 2085 y Fj(#)1895 2115 y Fn(\))p Fx(.)48 b(By)28 b(\(9.67\))d(we)j(also)g(ha)n(v)o(e)f Ft(\026)p Fn(\()p Ft(A)2962 2085 y Fj(#)3021 2115 y Fn(\))37 b(=)291 2214 y Ft(!)s Fn(\()p Ft(A)440 2184 y Fj(#)498 2214 y Fn(\))24 b(=)e Ft(\027)5 b Fn(\()p Ft(A)781 2184 y Fj(#)841 2214 y Fn(\))21 b Fx(and)f(so)g Ft(\026)1179 2226 y Fj(+)1234 2214 y Fn(\()p Ft(A)p Fn(\))k(=)f Ft(\027)1513 2226 y Fj(+)1568 2214 y Fn(\()p Ft(A)p Fn(\))p Fx(.)j(W)-7 b(e)22 b(conclude)c(that)i Ft(\026)2386 2226 y Fj(+)2465 2214 y Fn(=)i Ft(\027)2593 2226 y Fj(+)2669 2214 y Fx(and)e(that)1299 2409 y Fn(\006)1359 2421 y Fj(+)1415 2409 y Fn(\()p Ft(\026;)14 b(\034)1570 2421 y Fo(V)1628 2409 y Fn(\))23 b Fq(\032)g Fn(\006)1831 2421 y Fj(+)1886 2409 y Fn(\()p Ft(\027)q(;)14 b(\034)2033 2421 y Fo(V)2091 2409 y Fn(\))p Ft(:)291 2604 y Fx(By)20 b(symmetry)-5 b(,)18 b(the)j(re)n(v)o(erse)e(inclusion) g(also)h(holds)g(and)1305 2799 y Fn(\006)1365 2811 y Fj(+)1420 2799 y Fn(\()p Ft(\026;)14 b(\034)1575 2811 y Fo(V)1633 2799 y Fn(\))23 b(=)g(\006)1836 2811 y Fj(+)1891 2799 y Fn(\()p Ft(!)s(;)14 b(\034)2051 2811 y Fo(V)2109 2799 y Fn(\))291 2993 y Fx(for)19 b(all)i Ft(\026)i Fq(2)g(N)731 3005 y Fo(!)780 2993 y Fx(.)i Fe(\003)291 3310 y Fu(Pr)o(oof)h(of)j (Theor)o(em)f(3.6)82 b Fx(T)-7 b(o)28 b(pro)o(v)o(e)f(this)i(theorem)e (we)i(use)g(the)f(correspondence)d(between)291 3409 y Ft(!)s Fx(-normal)17 b(states)j(and)e(elements)h(of)g(the)g(standard)f (cone)g Fq(P)27 b Fx(obtained)17 b(from)h Ft(!)23 b Fx(\(see)c (Proposition)291 3509 y(37)g(in)i([Pi]\);)f(this)g(is)i(possible)d (since)i Ft(!)i Fx(is)e(modular)e(by)h(assumption.)415 3614 y(Note)27 b(that)g(if)h Fn(Ker)12 b Ft(L)1039 3626 y Fo(V)1132 3614 y Fq(6)p Fn(=)36 b Fq(f)p Fn(0)p Fq(g)p Fx(,)27 b(then)g(there)f(is)i(an)f Ft(!)s Fx(-normal,)g Ft(\034)2364 3626 y Fo(V)2422 3614 y Fx(-in)m(v)n(ariant)e(state)j Ft(\021)s Fx(.)46 b(By)291 3714 y(Theorem)24 b(3.3,)j Fn(\006)830 3726 y Fj(+)885 3714 y Fn(\()p Ft(!)s(;)14 b(\034)1045 3726 y Fo(V)1103 3714 y Fn(\))35 b(=)f(\006)1329 3726 y Fj(+)1384 3714 y Fn(\()p Ft(\021)s(;)14 b(\034)1533 3726 y Fo(V)1591 3714 y Fn(\))28 b Fx(and)d(ob)o(viously)g Fn(\006)2211 3726 y Fj(+)2266 3714 y Fn(\()p Ft(\021)s(;)14 b(\034)2415 3726 y Fo(V)2473 3714 y Fn(\))35 b(=)f Fq(f)p Ft(\021)s Fq(g)p Fx(.)43 b(T)-7 b(w)o(o)27 b(non-)291 3814 y(zero)c(elements)g(in)h Fn(Ker)13 b Ft(L)1071 3826 y Fo(V)1153 3814 y Fx(therefore)22 b(yield)h(the)h(same)g(v)o(ector)e (state)j(and)e(are)h(represented)e(by)291 3913 y(the)g(same)g(v)o (ector)e(in)j(the)e(standard)g(cone,)h Fr(i.e)o(.,)g Fn(Ker)12 b Ft(L)1900 3925 y Fo(V)1977 3913 y Fq(\\)21 b(P)29 b Fx(is)22 b(a)h(one-dimensional)c(half-line.)291 4013 y(Recall)h(that)h(an)o(y)e Ft(\020)30 b Fq(2)23 b Fk(h)995 4025 y Fo(!)1064 4013 y Fx(can)d(be)g(uniquely)f(decomposed) e(as)1312 4208 y Ft(\020)30 b Fn(=)22 b Ft(\020)1501 4220 y Fj(1)1557 4208 y Fq(\000)c Ft(\020)1676 4220 y Fj(2)1732 4208 y Fn(+)g(i)p Ft(\020)1874 4220 y Fj(3)1930 4208 y Fq(\000)h Fn(i)p Ft(\020)2073 4220 y Fj(4)2110 4208 y Ft(;)291 4402 y Fx(with)i Ft(\020)496 4414 y Fo(i)545 4402 y Fx(in)g Fq(P)7 b Fx(.)28 b(Since)21 b Ft(e)990 4372 y Fj(i)p Fo(tL)1080 4380 y Fh(V)1154 4402 y Fx(preserv)o(es)f(the) h(standard)f(cone,)g Ft(e)2152 4372 y Fj(i)p Fo(tL)2242 4380 y Fh(V)2295 4402 y Ft(\020)31 b Fn(=)24 b Ft(\020)k Fx(if)n(f)21 b Ft(e)2652 4372 y Fj(i)p Fo(tL)2742 4380 y Fh(V)2795 4402 y Ft(\020)2831 4414 y Fo(i)2884 4402 y Fn(=)j Ft(\020)3009 4414 y Fo(i)3058 4402 y Fx(for)291 4502 y(all)f Ft(i)g Fx(\()p Fr(i.e)o(.,)g Ft(\020)658 4514 y Fo(i)713 4502 y Fq(2)29 b Fn(Ker)12 b Ft(L)1001 4514 y Fo(V)1079 4502 y Fq(\\)21 b(P)30 b Fx(for)22 b(all)h Ft(i)p Fx(\).)33 b(Hence,)23 b Fn(Ker)12 b Ft(L)2041 4514 y Fo(V)2122 4502 y Fx(is)24 b(one-dimensional)19 b(and)k(P)o(art)f(\(i\))291 4602 y(follo)n(ws.)415 4707 y(The)k(proof)e(of)i(P)o(art)g(\(ii\))g(is)h(simple.)43 b(An)o(y)25 b(NESS)i Ft(\021)37 b Fq(2)d Fn(\006)2153 4719 y Fj(+)2208 4707 y Fn(\()p Ft(!)s(;)14 b(\034)2368 4719 y Fo(V)2426 4707 y Fn(\))27 b Fx(can)f(be)g(uniquely)e(de-)291 4807 y(composed)h(as)j Ft(\021)793 4819 y Fo(n)862 4807 y Fn(+)23 b Ft(\021)991 4819 y Fo(s)1055 4807 y Fx(where)j Ft(\021)1326 4819 y Fo(n)1408 4807 y Fq(\034)36 b Ft(!)30 b Fx(and)d Ft(\021)1798 4819 y Fo(s)1869 4807 y Fq(?)36 b Ft(!)s Fx(.)46 b(Since)27 b Ft(\021)k Fx(is)d Ft(\034)2495 4819 y Fo(V)2553 4807 y Fx(-in)m(v)n(ariant,)f Ft(\021)2962 4819 y Fo(n)3035 4807 y Fx(and)291 4907 y Ft(\021)332 4919 y Fo(s)390 4907 y Fx(are)22 b(also)h Ft(\034)707 4919 y Fo(V)765 4907 y Fx(-in)m(v)n(ariant.)29 b(Therefore)20 b Ft(\021)1526 4919 y Fo(n)1594 4907 y Fx(is)j(represented)e(by)h(a)g (v)o(ector)f Ft(\020)30 b Fx(in)22 b Fn(Ker)13 b Ft(L)2830 4919 y Fo(V)2907 4907 y Fq(\\)20 b(P)7 b Fx(.)32 b(If)291 5006 y Fn(Ker)12 b Ft(L)495 5018 y Fo(V)576 5006 y Fn(=)22 b Fq(f)p Fn(0)p Fq(g)p Fx(,)d(then)h Ft(\021)1034 5018 y Fo(n)1102 5006 y Fn(=)j(0)d Fx(and)g Ft(\021)26 b Fq(?)d Ft(!)s Fx(.)p eop end %%Page: 58 58 TeXDict begin 58 57 bop 739 232 a Fx(58)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)863 523 y Fx(It)26 b(remains)f(to)h(pro)o(v)o(e)e(P)o(art)h(\(iii\))h (\(see)g(Theorem)e(44)h(in)h(the)f(lecture)g(notes)h([Pi]\).)41 b(Let)25 b Ft(')34 b Fq(2)739 623 y Fn(Ker)13 b Ft(L)944 635 y Fo(V)1024 623 y Fx(be)22 b(a)h(separating)e(v)o(ector)g(for)h Fk(M)1985 635 y Fo(!)2033 623 y Fx(.)32 b(Let)23 b Ft(B)31 b Fq(2)d Ft(\031)2444 635 y Fo(!)2492 623 y Fn(\()p Fq(O)r Fn(\))2624 593 y Fs(0)2671 623 y Fx(be)23 b(such)f(that)g Fq(k)p Ft(B)t(')p Fq(k)27 b Fn(=)g(1)22 b Fx(and)739 722 y(let)f Ft(\027)884 734 y Fo(B)962 722 y Fx(be)f(the)g(v)o(ector)f (state)i(associated)f(to)g Ft(B)t(')p Fx(,)h Ft(\027)2234 734 y Fo(B)2291 722 y Fn(\()p Fq(\001)p Fn(\))j(=)f(\()p Ft(B)t(';)14 b Fq(\001)p Ft(B)t(')p Fn(\))p Fx(.)27 b(F)o(or)19 b(an)o(y)h Ft(A)j Fq(2)g Ft(\031)3388 734 y Fo(!)3437 722 y Fn(\()p Fq(O)r Fn(\))p Fx(,)1117 893 y Fn(1)p 1117 930 42 4 v 1123 1006 a Ft(t)1183 836 y Fl(Z)1266 856 y Fo(t)1229 1025 y Fj(0)1309 949 y Ft(\027)1350 961 y Fo(B)1407 882 y Fl(\000)1445 949 y Ft(\034)1490 915 y Fo(s)1481 969 y(V)1539 949 y Fn(\()p Ft(A)p Fn(\))1665 882 y Fl(\001)1718 949 y Fn(d)p Ft(s)g Fn(=)1923 893 y(1)p 1923 930 V 1929 1006 a Ft(t)1989 836 y Fl(Z)2072 856 y Fo(t)2035 1025 y Fj(0)2115 882 y Fl(\000)2153 949 y Ft(B)t(';)28 b(e)2364 915 y Fj(i)p Fo(sL)2460 923 y Fh(V)2514 949 y Ft(\031)2561 961 y Fo(!)2609 949 y Fn(\()p Ft(A)p Fn(\))p Ft(e)2774 915 y Fs(\000)p Fj(i)p Fo(sL)2922 923 y Fh(V)2976 949 y Ft(B)t(')3097 882 y Fl(\001)3150 949 y Fn(d)p Ft(s)1826 1280 y Fn(=)1913 1163 y Fl(\022)1985 1224 y Fn(1)p 1985 1261 V 1991 1337 a Ft(t)2050 1167 y Fl(Z)2133 1188 y Fo(t)2096 1356 y Fj(0)2176 1280 y Ft(e)2215 1246 y Fs(\000)p Fj(i)p Fo(sL)2363 1254 y Fh(V)2416 1280 y Ft(B)2483 1246 y Fs(\003)2508 1280 y Ft(B)18 b(')c Fn(d)p Ft(s;)28 b(\031)2840 1292 y Fo(!)2888 1280 y Fn(\()p Ft(A)p Fn(\))p Ft(')3068 1163 y Fl(\023)3144 1280 y Ft(:)739 1493 y Fx(Hence,)19 b(by)h(the)g(v)n(on)g(Neumann)f(er)o(godic)f (theorem,)1034 1717 y Ft(\027)1075 1729 y Fo(B)s Fj(+)1183 1717 y Fn(\()p Ft(A)p Fn(\))24 b Fq(\021)44 b Fn(lim)1421 1767 y Fo(t)p Fs(!1)1602 1661 y Fn(1)p 1602 1698 V 1608 1774 a Ft(t)1667 1604 y Fl(Z)1750 1625 y Fo(t)1713 1793 y Fj(0)1793 1717 y Ft(\027)1834 1729 y Fo(B)1892 1650 y Fl(\000)1930 1717 y Ft(\034)1975 1683 y Fo(s)1966 1738 y(V)2024 1717 y Fn(\()p Ft(A)p Fn(\))2150 1650 y Fl(\001)2202 1717 y Fn(d)p Ft(s)23 b Fn(=)2398 1650 y Fl(\000)2436 1717 y Ft(P)2489 1729 y Fj(Ker)11 b Fo(L)2653 1737 y Fh(V)2707 1717 y Ft(B)2774 1683 y Fs(\003)2798 1717 y Ft(B)18 b(';)c(\031)3017 1729 y Fo(!)3066 1717 y Fn(\()p Ft(A)p Fn(\))p Ft(')3246 1650 y Fl(\001)3285 1717 y Ft(;)739 1933 y Fx(where)25 b Ft(P)1021 1945 y Fj(Ker)11 b Fo(L)1185 1953 y Fh(V)1266 1933 y Fx(is)27 b(the)f(projection)f(on)g Fn(Ker)13 b Ft(L)2154 1945 y Fo(V)2211 1933 y Fx(.)44 b(Since)26 b Ft(')h Fx(is)g(c)o(yclic)f(for)g Ft(\031)3045 1945 y Fo(!)3093 1933 y Fn(\()p Fq(O)r Fn(\))3225 1903 y Fs(0)3249 1933 y Fx(,)j(for)c(e)n(v)o(ery)739 2032 y Ft(n)f Fq(2)g Fp(N)d Fx(we)g(can)g(\002nd)f(a)h Ft(B)1499 2044 y Fo(n)1566 2032 y Fx(such)f(that)h Fq(k)p Ft(!)g Fq(\000)d Ft(\027)2124 2044 y Fo(B)2174 2052 y Fh(n)2219 2032 y Fq(k)24 b Ft(<)f Fn(1)p Ft(=n)p Fx(.)j(The)21 b(sequence)e Ft(\027)3071 2044 y Fo(B)3121 2052 y Fh(n)3187 2032 y Fx(is)j(Cauchy)e(in)739 2132 y(norm)f(and)h(for)f(all)i Ft(!)1349 2144 y Fj(+)1427 2132 y Fq(2)i Fn(\006)1565 2144 y Fj(+)1620 2132 y Fn(\()p Ft(!)s(;)14 b(\034)1780 2144 y Fo(V)1838 2132 y Fn(\))p Fx(,)1556 2301 y Fq(k)p Ft(!)1650 2313 y Fj(+)1722 2301 y Fq(\000)k Ft(\027)1846 2313 y Fo(B)1896 2321 y Fh(n)1937 2313 y Fj(+)1992 2301 y Fq(k)23 b(\024)g(k)p Ft(!)d Fq(\000)e Ft(\027)2383 2313 y Fo(B)2433 2321 y Fh(n)2478 2301 y Fq(k)23 b Ft(<)g Fn(1)p Ft(=n:)739 2470 y Fx(This)32 b(implies)h(that)f(the)g(norm)f (limit)i(of)f Ft(\027)2029 2482 y Fo(B)2079 2490 y Fh(n)2157 2470 y Fx(is)h(the)f(unique)f(NESS)i(in)f Fn(\006)3031 2482 y Fj(+)3086 2470 y Fn(\()p Ft(!)s(;)14 b(\034)3246 2482 y Fo(V)3304 2470 y Fn(\))p Fx(.)62 b(Since)739 2570 y Ft(\027)780 2582 y Fo(B)830 2590 y Fh(n)871 2582 y Fj(+)949 2570 y Fq(2)23 b(N)1095 2582 y Fo(!)1164 2570 y Fx(and)d Fq(N)1373 2582 y Fo(!)1442 2570 y Fx(is)h(a)g(norm)e(closed) h(subset)g(of)g Fq(O)2391 2539 y Fs(\003)2429 2570 y Fx(,)h(this)g(NESS)f(is)h Ft(!)s Fx(-normal.)i Fe(\003)739 2849 y Fm(9.2)99 b(The)26 b(Hilbert-Schmidt)g(condition)739 3005 y Fu(Pr)o(oof)16 b(of)h(Theor)o(em)g(7.2)82 b Fx(W)-7 b(e)19 b(will)f(pro)o(v)o(e)e(that)i Ft(T)2250 2962 y Fj(1)p Fo(=)p Fj(2)2238 3025 y(+)2362 3005 y Fq(\000)9 b Ft(T)2497 2975 y Fj(1)p Fo(=)p Fj(2)2617 3005 y Fx(is)19 b(Hilbert-Schmidt.)j(The)17 b(proof)739 3114 y(that)g Fn(\()p Ft(I)12 b Fq(\000)6 b Ft(T)1081 3126 y Fj(+)1136 3114 y Fn(\))1168 3084 y Fj(1)p Fo(=)p Fj(2)1278 3114 y Fq(\000)g Fn(\()p Ft(I)12 b Fq(\000)6 b Ft(T)12 b Fn(\))1593 3084 y Fj(1)p Fo(=)p Fj(2)1713 3114 y Fx(is)18 b(also)f (Hilbert-Schmidt)e(is)i(identical.)24 b(F)o(or)16 b(an)h(elementary)e (in-)739 3214 y(troduction)h(to)j(Hilbert-Schmidt)e(operators)h (\(which)f(suf)n(\002ces)i(for)f(the)h(proof)e(belo)n(w\))h(the)h (reader)739 3314 y(may)h(consult)f(Section)h(VI.6)g(in)g([RS].)863 3413 y(By)i(our)e(general)g(assumptions,)g(the)h(functions)f Ft(f)9 b Fn(\()p Ft(r)r Fn(\))22 b Fx(and)f Ft(F)12 b Fn(\()p Ft(r)r Fn(\))2772 3383 y Fs(\000)p Fj(1)2884 3413 y Fx(are)21 b(bounded)d(and)j(con-)739 3513 y(tinuous.)39 b(By)26 b(the)f(assumption)f(of)h(Theorem)f(7.2,)i(all)f(the)h (densities)f Ft(\032)2867 3525 y Fo(j)2902 3513 y Fn(\()p Ft(r)r Fn(\))i Fx(are)f(the)f(same)g(and)739 3613 y(equal)19 b(to)i Ft(\032)p Fn(\()p Ft(r)r Fn(\))p Fx(.)27 b(Hence,)1725 3793 y Ft(T)1774 3805 y Fs(R)1858 3793 y Fn(=)1973 3690 y Fo(M)1945 3715 y Fl(M)1951 3891 y Fo(j)s Fj(=1)2085 3793 y Ft(\032)2128 3805 y Fo(j)2163 3793 y Fn(\()p Ft(r)r Fn(\))d(=)f Ft(\032)p Fn(\()p Ft(h)2501 3805 y Fs(R)2562 3793 y Fn(\))p Ft(:)739 4035 y Fx(Let)g Ft(p)915 4047 y Fs(R)1000 4035 y Fx(be)g(the)g(orthogonal)d(projection)h(on)i(the)g (reserv)n(oir)f(Hilbert)h(space)g Fk(h)3029 4047 y Fs(R)3090 4035 y Fx(.)35 b(Since)23 b Ft(T)3415 4004 y Fj(1)p Fo(=)p Fj(2)3539 4035 y Fq(\000)739 4148 y Ft(T)800 4104 y Fj(1)p Fo(=)p Fj(2)788 4172 y Fs(R)926 4148 y Fn(=)g Ft(T)1075 4104 y Fj(1)p Fo(=)p Fj(2)1063 4172 y Fs(S)1178 4148 y Fx(,)e Ft(T)1281 4104 y Fj(1)p Fo(=)p Fj(2)1269 4168 y(+)1384 4148 y Fn(\()p Ft(I)26 b Fq(\000)18 b Ft(p)1603 4160 y Fs(R)1664 4148 y Fn(\))p Fx(,)j Fn(\()p Ft(I)k Fq(\000)18 b Ft(p)1956 4160 y Fs(R)2017 4148 y Fn(\))p Ft(T)2110 4104 y Fj(1)p Fo(=)p Fj(2)2098 4168 y(+)2234 4148 y Fx(are)j(ob)o(viously)d(Hilbert-Schmidt,)g(it)j(suf)n(\002ces) 739 4270 y(to)e(sho)n(w)g(that)g Ft(p)1202 4282 y Fs(R)1263 4270 y Ft(T)1324 4227 y Fj(1)p Fo(=)p Fj(2)1312 4291 y(+)1428 4270 y Ft(p)1470 4282 y Fs(R)1545 4270 y Fq(\000)14 b Ft(T)1685 4227 y Fj(1)p Fo(=)p Fj(2)1673 4295 y Fs(R)1808 4270 y Fx(is)20 b(a)g(Hilbert-Schmidt)d(operator)g(on)i(the)g(Hilbert)g (space)g Fk(h)3521 4282 y Fs(R)3582 4270 y Fx(.)739 4370 y(Since)1393 4469 y Ft(p)1435 4481 y Fs(R)1496 4469 y Ft(T)1557 4426 y Fj(1)p Fo(=)p Fj(2)1545 4490 y(+)1660 4469 y Ft(p)1702 4481 y Fs(R)1781 4469 y Fq(\000)f Ft(T)1925 4426 y Fj(1)p Fo(=)p Fj(2)1913 4494 y Fs(R)2052 4469 y Fn(=)23 b Fq(\000)p Ft(p)2247 4481 y Fs(R)2307 4469 y Ft(W)2397 4435 y Fs(\003)2385 4490 y(\000)2441 4469 y Fn([)p Ft(W)2542 4481 y Fs(\000)2599 4469 y Ft(p)2641 4481 y Fs(R)2702 4469 y Ft(;)14 b(T)2800 4426 y Fj(1)p Fo(=)p Fj(2)2788 4494 y Fs(R)2903 4469 y Fn(])p Ft(;)739 4634 y Fx(it)24 b(suf)n(\002ces)g(to)g(sho)n(w)g(that)g Ft(K)35 b Fq(\021)30 b Fn([)p Ft(W)1823 4646 y Fs(\000)1879 4634 y Ft(p)1921 4646 y Fs(R)1982 4634 y Ft(;)14 b(T)2080 4590 y Fj(1)p Fo(=)p Fj(2)2068 4658 y Fs(R)2183 4634 y Fn(])25 b Fx(is)g(a)f(Hilbert-Schmidt)e(operator)g(on)h Fk(h)3387 4646 y Fs(R)3449 4634 y Fx(.)36 b(By)739 4733 y(Theorem)18 b(6.2,)i(for)f Ft(g)26 b Fq(2)d Fk(h)1509 4745 y Fs(R)1571 4733 y Fx(,)1137 4957 y Fn(\()p Ft(K)6 b(g)s Fn(\)\()p Ft(r)r Fn(\))24 b(=)f Ft(\025)1584 4923 y Fj(2)1639 4901 y Ft(f)9 b Fn(\()p Ft(r)r Fn(\))p 1631 4938 170 4 v 1631 5014 a Ft(F)j Fn(\()p Ft(r)r Fn(\))1824 4844 y Fl(Z)1907 4864 y Fo(e)1938 4872 y Fi(+)1870 5033 y Fo(e)1901 5041 y Fd(\000)2014 4901 y Ft(\032)p Fn(\()p Ft(r)2128 4871 y Fs(0)2152 4901 y Fn(\))2184 4871 y Fj(1)p Fo(=)p Fj(2)2307 4901 y Fq(\000)18 b Ft(\032)p Fn(\()p Ft(r)r Fn(\))2536 4871 y Fj(1)p Fo(=)p Fj(2)p 2014 4938 628 4 v 2143 5014 a Ft(r)2182 4990 y Fs(0)2225 5014 y Fq(\000)g Ft(r)j Fn(+)d(i)p Ft(o)2669 4935 y Fn(\026)2652 4957 y Ft(f)8 b Fn(\()p Ft(r)2772 4923 y Fs(0)2797 4957 y Fn(\))18 b Fq(\001)h Ft(g)s Fn(\()p Ft(r)3003 4923 y Fs(0)3027 4957 y Fn(\))14 b(d)p Ft(r)3158 4923 y Fs(0)3182 4957 y Ft(:)p eop end %%Page: 59 59 TeXDict begin 59 58 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(59)291 523 y(Let)20 b Ft(K)493 535 y Fo(ij)572 523 y Fx(be)g(an)g(operator)e(on)i Ft(L)1228 493 y Fj(2)1265 523 y Fn(\(\()p Ft(e)1368 535 y Fs(\000)1424 523 y Ft(;)14 b(e)1500 535 y Fj(+)1555 523 y Fn(\))p Ft(;)g Fn(d)p Ft(r)r Fn(\))22 b Fx(de\002ned)d(by)672 758 y Fn(\()p Ft(K)775 770 y Fo(ij)834 758 y Ft(h)p Fn(\)\()p Ft(r)r Fn(\))24 b(=)f Ft(\025)1177 724 y Fj(2)1225 702 y Ft(f)1266 714 y Fo(i)1293 702 y Fn(\()p Ft(r)r Fn(\))p 1225 739 173 4 v 1227 815 a Ft(F)12 b Fn(\()p Ft(r)r Fn(\))1421 645 y Fl(Z)1504 666 y Fo(e)1535 674 y Fi(+)1467 834 y Fo(e)1498 842 y Fd(\000)1610 702 y Ft(\032)p Fn(\()p Ft(r)1724 672 y Fs(0)1748 702 y Fn(\))1780 672 y Fj(1)p Fo(=)p Fj(2)1904 702 y Fq(\000)18 b Ft(\032)p Fn(\()p 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2618 y Fj(1)p Fo(=)p Fj(2)2397 2652 y Fq(\000)18 b Ft(\032)p Fn(\()p Ft(r)r Fn(\))2626 2618 y Fj(1)p Fo(=)p Fj(2)2732 2652 y Fn(\))2782 2631 y(\026)2764 2652 y Ft(f)2805 2664 y Fo(j)2839 2652 y Fn(\()p Ft(r)r Fn(\))p Ft(h)2990 2664 y Fj(2)3029 2652 y Fn(\()p Ft(r)r Fn(\))1897 2830 y(=)82 b(0)p Ft(;)291 3003 y Fx(\(see)20 b(the)g(Lecture)g([Ja)o(]\),)g(we)h (obtain)651 3229 y Ft(g)691 3241 y Fj(2)728 3229 y Fn(\()p Ft(r)r Fn(\))j(=)f(lim)953 3283 y Fo(\017)p Fs(#)p Fj(0)1072 3116 y Fl(Z)1155 3136 y Fo(e)1186 3144 y Fi(+)1118 3305 y Fo(e)1149 3313 y Fd(\000)1261 3173 y Fn(\()p Ft(r)1332 3143 y Fs(0)1375 3173 y Fq(\000)18 b Ft(r)r Fn(\)\()p Ft(\032)p Fn(\()p Ft(r)1675 3143 y Fs(0)1701 3173 y Fn(\))1733 3143 y Fj(1)p Fo(=)p Fj(2)1856 3173 y Fq(\000)g Ft(\032)p Fn(\()p Ft(r)r Fn(\))2085 3143 y Fj(1)p Fo(=)p Fj(2)2190 3173 y Fn(\))p 1261 3210 962 4 v 1503 3286 a(\()p Ft(r)1574 3262 y Fs(0)1616 3286 y Fq(\000)g Ft(r)r Fn(\))1770 3262 y Fj(2)1827 3286 y Fn(+)g Ft(\017)1944 3262 y Fj(2)2250 3207 y Fn(\026)2233 3229 y Ft(f)2274 3241 y Fo(j)2308 3229 y Fn(\()p Ft(r)2379 3195 y Fs(0)2404 3229 y Fn(\))p Ft(h)2484 3241 y Fj(2)2521 3229 y Fn(\()p Ft(r)2592 3195 y Fs(0)2616 3229 y Fn(\))c(d)p Ft(r)2747 3195 y Fs(0)2772 3229 y Ft(:)291 3475 y Fx(Since)20 b Ft(f)537 3487 y Fo(j)592 3475 y Fx(and)g Ft(h)781 3487 y Fj(2)839 3475 y Fx(are)g(bounded)d(and)j Ft(\032)p Fn(\()p Ft(r)r Fn(\))1554 3445 y Fj(1)p Fo(=)p Fj(2)1680 3475 y Fx(is)1767 3442 y Fj(1)p 1767 3456 34 4 v 1767 3504 a(2)1810 3475 y Fx(-H\366lder)f (continuous,)e(we)k(ha)n(v)o(e)716 3730 y Fn(sup)558 3804 y Fo(\017>)p Fj(0)p Fo(;r)r Fs(2)p Fj(\()p Fo(e)826 3812 y Fd(\000)874 3804 y Fo(;e)925 3812 y Fi(+)973 3804 y Fj(\))1118 3585 y Fl(\014)1118 3635 y(\014)1118 3685 y(\014)1118 3734 y(\014)1118 3784 y(\014)1146 3617 y(Z)1229 3638 y Fo(e)1260 3646 y Fi(+)1192 3806 y Fo(e)1223 3814 y Fd(\000)1335 3674 y Fn(\()p Ft(r)1406 3644 y Fs(0)1449 3674 y Fq(\000)d Ft(r)r Fn(\)\()p Ft(\032)p Fn(\()p Ft(r)1749 3644 y Fs(0)1775 3674 y Fn(\))1807 3644 y Fj(1)p Fo(=)p Fj(2)1930 3674 y Fq(\000)g Ft(\032)p Fn(\()p Ft(r)r Fn(\))2159 3644 y Fj(1)p Fo(=)p Fj(2)2264 3674 y Fn(\))p 1335 3711 962 4 v 1577 3787 a(\()p Ft(r)1648 3763 y Fs(0)1691 3787 y Fq(\000)g Ft(r)r Fn(\))1845 3763 y Fj(2)1901 3787 y Fn(+)g Ft(\017)2018 3763 y Fj(2)2325 3708 y Fn(\026)2307 3730 y Ft(f)2348 3742 y Fo(j)2382 3730 y Fn(\()p Ft(r)2453 3696 y Fs(0)2478 3730 y Fn(\))p Ft(h)2558 3742 y Fj(2)2595 3730 y Fn(\()p Ft(r)2666 3696 y Fs(0)2690 3730 y Fn(\))c(d)p Ft(r)2821 3696 y Fs(0)2846 3585 y Fl(\014)2846 3635 y(\014)2846 3685 y(\014)2846 3734 y(\014)2846 3784 y(\014)1026 3992 y Fq(\024)83 b Ft(C)111 b Fn(sup)1253 4066 y Fo(r)r Fs(2)p Fj(\()p Fo(e)1388 4074 y Fd(\000)1437 4066 y Fo(;e)1488 4074 y Fi(+)1535 4066 y Fj(\))1575 3879 y Fl(Z)1658 3900 y Fo(e)1689 3908 y Fi(+)1621 4068 y Fo(e)1652 4076 y Fd(\000)1782 3914 y Fn(\026)1764 3936 y Ft(f)1805 3948 y Fo(j)1839 3936 y Fn(\()p Ft(r)1910 3906 y Fs(0)1935 3936 y Fn(\))p Ft(h)2015 3948 y Fj(2)2052 3936 y 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Fj(2)2258 4914 y Fq(\000)g Ft(\032)p Fn(\()p Ft(r)r Fn(\))2487 4884 y Fj(1)p Fo(=)p Fj(2)2593 4914 y Fn(\))p 1664 4951 962 4 v 1905 5027 a(\()p Ft(r)1976 5003 y Fs(0)2019 5027 y Fq(\000)g Ft(r)r Fn(\))2173 5003 y Fj(2)2230 5027 y Fn(+)g Ft(\017)2347 5003 y Fj(2)2635 4970 y Ft(:)p eop end %%Page: 60 60 TeXDict begin 60 59 bop 739 232 a Fx(60)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx(W)-7 b(e)21 b(denote)e(by)h Fq(k)e(\001)h(k)1358 535 y Fj(HS)1468 523 y Fx(the)h(Hilbert-Schmidt)e(norm.)24 b(Then)1592 748 y Fq(k)p Ft(K)1705 760 y Fo(ij;\017)1807 748 y Fq(k)1849 714 y Fj(2)1849 769 y(HS)1961 748 y Fn(=)2049 635 y Fl(Z)2146 748 y Fq(j)p Ft(k)2212 760 y Fo(\017)2244 748 y Fn(\()p Ft(r)n(;)14 b(r)2387 714 y Fs(0)2411 748 y Fn(\))p Fq(j)2466 714 y Fj(2)2518 748 y Fn(d)p Ft(r)i Fn(d)p Ft(r)2702 714 y Fs(0)2727 748 y Ft(:)739 985 y Fx(Since)j Ft(\032)p Fn(\()p Ft(r)r Fn(\))1089 955 y Fj(1)p Fo(=)p Fj(2)1215 985 y Fx(is)h Ft(\013)p 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Fj(\))2715 1320 y Ft(:)739 1553 y Fx(Therefore,)g(since)i Fn(2\(1)e Fq(\000)g Ft(\013)p Fn(\))24 b Ft(<)e Fn(1)p Fx(,)e(we)h(conclude)d(that)1357 1778 y Fn(sup)1363 1845 y Fo(\017>)p Fj(0)1496 1778 y Fq(k)p Ft(K)1609 1790 y Fo(ij;\017)1710 1778 y Fq(k)1752 1744 y Fj(2)1752 1798 y(HS)1864 1778 y Fn(=)23 b(sup)1958 1845 y Fo(\017>)p Fj(0)2091 1665 y Fl(Z)2188 1778 y Fq(j)p Ft(k)2254 1790 y Fo(\017)2286 1778 y Fn(\()p Ft(r)n(;)14 b(r)2429 1744 y Fs(0)2453 1778 y Fn(\))p Fq(j)2508 1744 y Fj(2)2560 1778 y Fn(d)p Ft(r)i Fn(d)p Ft(r)2744 1744 y Fs(0)2792 1778 y Ft(<)22 b Fq(1)p Ft(:)739 2017 y Fx(The)d(Hilbert-Schmidt)f (class)i(of)g(operators)e(on)h Ft(L)2208 1987 y Fj(2)2244 2017 y Fn(\(\()p Ft(e)2347 2029 y Fs(\000)2404 2017 y Ft(;)14 b(e)2480 2029 y Fj(+)2534 2017 y Fn(\))p Ft(;)g Fn(d)p Ft(r)r Fn(\))22 b Fx(is)e(a)g(Hilbert)g(space)f(with)h(the)739 2117 y(inner)j(product)f Fn(\()p Ft(X)r(;)14 b(Y)k Fn(\))30 b(=)f(T)-7 b(r\()p Ft(X)1768 2087 y Fs(\003)1806 2117 y Ft(Y)18 b Fn(\))p Fx(.)37 b(Since)23 b Fq(f)p Ft(K)2283 2129 y Fo(ij;\017)2385 2117 y Fq(g)2427 2129 y Fo(\017>)p Fj(0)2567 2117 y Fx(is)i(a)f(bounded)d(set)k(in)f(this)g(Hilbert)739 2225 y(space,)j(there)f(is)h(a)g(sequence)e Ft(\017)1676 2237 y Fo(n)1755 2225 y Fq(!)34 b Fn(0)26 b Fx(and)g(a)h (Hilbert-Schmidt)d(operator)3042 2204 y Fn(~)3020 2225 y Ft(K)3091 2237 y Fo(ij)3176 2225 y Fx(such)i(that)g(for)739 2325 y(an)o(y)19 b(Hilbert-Schmidt)g(operator)f Ft(X)27 b Fx(on)20 b Ft(L)1993 2295 y Fj(2)2030 2325 y Fn(\(\()p Ft(e)2133 2337 y Fs(\000)2189 2325 y Ft(;)14 b(e)2265 2337 y Fj(+)2320 2325 y Fn(\))p Ft(;)g Fn(d)p Ft(r)r Fn(\))p Fx(,)1605 2507 y Fn(lim)1576 2557 y Fo(n)p Fs(!1)1763 2507 y Fn(T)-7 b(r)o(\()p Ft(X)1956 2473 y Fs(\003)1994 2507 y Ft(K)2065 2519 y Fo(ij;\017)2163 2527 y Fh(n)2208 2507 y Fn(\))23 b(=)g(T)-7 b(r)o(\()p Ft(X)2544 2473 y Fs(\003)2604 2486 y Fn(~)2582 2507 y Ft(K)2653 2519 y Fo(ij)2711 2507 y Fn(\))p Ft(:)739 2724 y Fx(T)g(aking)21 b Ft(X)33 b Fn(=)26 b(\()p Ft(h)1263 2736 y Fj(1)1300 2724 y Ft(;)14 b Fq(\001)p Fn(\))p Ft(h)1440 2736 y Fj(2)1477 2724 y Fx(,)23 b(where)e Ft(h)1794 2736 y Fo(i)1848 2724 y Fq(2)27 b Ft(L)1987 2694 y Fj(2)2024 2724 y Fn(\(\()p Ft(e)2127 2736 y Fs(\000)2183 2724 y Ft(;)14 b(e)2259 2736 y Fj(+)2314 2724 y Fn(\))p Ft(;)g Fn(d)p Ft(r)r Fn(\))23 b Fx(are)f(bounded)e(and)h(continuous,)f(we)739 2824 y(deri)n(v)o(e)h(from)g(\(9.69\))g(that)i Fn(\()p Ft(h)1602 2836 y Fj(1)1639 2824 y Ft(;)1698 2803 y Fn(~)1676 2824 y Ft(K)1747 2836 y Fo(ij)1805 2824 y Ft(h)1853 2836 y Fj(2)1890 2824 y Fn(\))k(=)g(\()p Ft(h)2121 2836 y Fj(1)2159 2824 y Ft(;)14 b(K)2267 2836 y Fo(ij)2324 2824 y Ft(h)2372 2836 y Fj(2)2410 2824 y Fn(\))p Fx(.)32 b(Since)22 b(the)h(set)g(of)f(such)h Ft(h)p Fx(')-5 b(s)23 b(is)g(dense)739 2933 y(in)d Ft(L)881 2902 y Fj(2)918 2933 y Fn(\(\()p Ft(e)1021 2945 y Fs(\000)1077 2933 y Ft(;)14 b(e)1153 2945 y Fj(+)1208 2933 y Fn(\))p Ft(;)g Fn(d)p Ft(r)r Fn(\))p Fx(,)1459 2912 y Fn(~)1437 2933 y Ft(K)1508 2945 y Fo(ij)1589 2933 y Fn(=)22 b Ft(K)1747 2945 y Fo(ij)1826 2933 y Fx(and)e(so)g Ft(K)2132 2945 y Fo(ij)2211 2933 y Fx(is)h(Hilbert-Schmidt.)i Fe(\003)p eop end %%Page: 61 61 TeXDict begin 61 60 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(61)291 523 y Fv(Refer)n(ences)291 712 y Fx([AB])147 b(Araki,)21 b(H.,)i(Barouch,)e(E.:)28 b(On)22 b(the)g(dynamics)f(and)g (er)o(godic)f(properties)h(of)g(the)h Ft(X)7 b(Y)18 b Fx(-)609 812 y(model.)h(J.)i(Stat.)f(Phys.)g Fu(31)p Fx(,)f(327)h(\(1983\).)291 985 y([AJPP])78 b(Aschbacher)m(,)21 b(W)-8 b(.,)23 b(Jak\232i)1348 984 y(\264)1343 985 y(c,)g(V)-11 b(.,)24 b(P)o(autrat,)e(Y)-11 b(.,)23 b(Pillet,)g(C.-A.:)30 b(T)m(ransport)21 b(properties)g(of)609 1085 y(ideal)f(Fermi)g(gases)g (\(in)g(preparation\).)291 1258 y([AM])128 b(Aizenstadt,)18 b(V)-11 b(.V)g(.,)19 b(Malyshe)n(v)-5 b(,)18 b(V)-11 b(.A.:)24 b(Spin)19 b(interaction)e(with)i(an)f(ideal)h(Fermi)g(gas.)f (J.)609 1358 y(Stat.)i(Phys.)g Fu(48)p Fx(,)g(51)f(\(1987\).)291 1531 y([Ar1])132 b(Araki,)28 b(H.:)40 b(Relati)n(v)o(e)28 b(entrop)o(y)e(of)h(states)h(of)f(v)n(on)g(Neumann)f(algebras.)h(Publ.) g(Res.)609 1631 y(Inst.)20 b(Math.)g(Sci.)g(K)n(yoto)f(Uni)n(v)-5 b(.)20 b Fu(11)p Fx(,)f(809)g(\(1975/76\).)291 1804 y([Ar2])132 b(Araki,)21 b(H.:)27 b(Relati)n(v)o(e)21 b(entrop)o(y)f(of)h(states)h (of)f(v)n(on)f(Neumann)g(algebras)g(II.)h(Publ.)g(Res.)609 1904 y(Inst.)f(Math.)g(Sci.)g(K)n(yoto)f(Uni)n(v)-5 b(.)20 b Fu(13)p Fx(,)f(173)g(\(1977/78\).)291 2077 y([Ar3])132 b(Araki,)31 b(H.:)43 b(On)29 b(the)g Ft(X)7 b(Y)18 b Fx(-model)28 b(on)h(tw)o(o-sided)f(in\002nite)h(chain.)g(Publ.)g(Res.)h (Inst.)609 2177 y(Math.)20 b(Sci.)g(K)n(yoto)f(Uni)n(v)-5 b(.)19 b Fu(20)p Fx(,)h(277)f(\(1984\).)291 2350 y([ArM])100 b(Araki,)20 b(H.,)h(Masuda,)f(T)-6 b(.:)27 b(Positi)n(v)o(e)21 b(cones)f(and)g Ft(L)2078 2320 y Fo(p)2116 2350 y Fx(-spaces)h(for)f(v) n(on)h(Neumann)e(alge-)609 2450 y(bras.)h(Publ.)f(RIMS)i(K)n(yoto)e (Uni)n(v)-5 b(.)19 b Fu(18)p Fx(,)h(339)f(\(1982\).)291 2623 y([At])179 b(Attal,)19 b(S.:)25 b(Elements)19 b(of)g(operator)e (algebras)h(and)g(modular)g(theory)-5 b(.)17 b(V)-11 b(olume)18 b(I)h(of)g(this)609 2723 y(series.)291 2896 y([A)-7 b(W])131 b(Araki,)31 b(H.,)h(W)-6 b(yss,)32 b(W)-8 b(.:)45 b(Representations)28 b(of)h(canonical)f(anti-commutation)f (rela-)609 2996 y(tions.)20 b(Helv)-5 b(.)19 b(Phys.)h(Acta)g Fu(37)p Fx(,)g(136)f(\(1964\).)291 3169 y([BFS])115 b(Bach,)32 b(V)-11 b(.,)33 b(Fr\366hlich,)e(J.,)h(Sigal,)g(I.:)45 b(Return)29 b(to)h(equilibrium.)e(J.)i(Math.)g(Phys.)f Fu(41)p Fx(,)609 3269 y(3985)19 b(\(2000\).)291 3442 y([BLR])101 b(Bonetto,)34 b(F)-7 b(.,)36 b(Lebo)n(witz,)e(J.L.,)g(Re)o (y-Bellet,)h(L.:)49 b(F)o(ourier)31 b(La)o(w:)49 b(A)32 b(challenge)f(to)609 3542 y(theorists.)20 b(In)g Fr(Mathematical)f (Physics)h(2000.)e Fx(Imp.)i(Coll.)g(Press,)h(London)d(\(2000\).)291 3715 y([BM])133 b(Botvich,)23 b(D.D.,)h(Malyshe)n(v)-5 b(,)22 b(V)-11 b(.A.:)31 b(Unitary)23 b(equi)n(v)n(alence)e(of)i (temperature)e(dynam-)609 3815 y(ics)g(for)e(ideal)h(and)g(locally)f (perturbed)f(Fermi)i(Gas.)g(Commun.)f(Math.)g(Phys.)h Fu(61)p Fx(,)f(209)609 3914 y(\(1978\).)291 4088 y([BR1])110 b(Bratteli,)35 b(O,)d(Robinson)f(D.)h(W)-8 b(.:)49 b Fr(Oper)o(ator)31 b(Alg)o(ebr)o(as)g(and)g(Quantum)g(Statistical)609 4187 y(Mec)o(hanics)19 b(1.)h Fx(Springer)m(,)e(Berlin)i(\(1987\).)291 4361 y([BR2])110 b(Bratteli,)35 b(O,)d(Robinson)f(D.)h(W)-8 b(.:)49 b Fr(Oper)o(ator)31 b(Alg)o(ebr)o(as)g(and)g(Quantum)g (Statistical)609 4460 y(Mec)o(hanics)19 b(2.)h Fx(Springer)m(,)e (Berlin)i(\(1996\).)291 4634 y([BSZ])110 b(Baez,)24 b(J.C.,)h(Se)o (gal,)e(I.E.,)h(Zhou,)f(Z.:)31 b Fr(Intr)l(oduction)22 b(to)h(alg)o(ebr)o(aic)g(and)f(constructive)609 4733 y(quantum)c(\002eld)i(theory)-5 b(.)20 b Fx(Princeton)f(Uni)n(v)o (ersity)g(Press,)i(Princeton)e(NJ,)h(\(1991\).)291 4907 y([CG1])105 b(Cohen,)25 b(E.G.D.,)g(Galla)n(v)n(otti,)h(G.:)35 b(Dynamical)23 b(ensembles)h(in)h(stationary)f(states.)i(J.)609 5006 y(Stat.)20 b(Phys.)g Fu(80)p Fx(,)g(931)f(\(1995\).)p eop end %%Page: 62 62 TeXDict begin 62 61 bop 739 232 a Fx(62)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx([CG2])105 b(Cohen,)22 b(E.G.D.,)h(Galla)n(v)n(otti,)g(G.:)31 b(Dynamical)21 b(ensembles)h(in)h(nonequilibrium)c(sta-)1057 623 y(tistical)i(mechanics.)e(Phys.)h(Re)n(v)-5 b(.)20 b(Lett.)g Fu(74)p Fx(,)f(2694)g(\(1995\).)739 797 y([Da1])123 b(Da)n(vies,)34 b(E.B.:)48 b(Mark)o(o)o(vian)29 b(master)i(equations.)f (Commun.)g(Math.)h(Phys.)g Fu(39)p Fx(,)i(91)1057 897 y(\(1974\).)739 1071 y([Da2])123 b(Da)n(vies,)21 b(E.B.:)k(Mark)o(o)o (vian)18 b(master)i(equations)f(II.)h(Math.)g(Ann.)f Fu(219)p Fx(,)g(147)g(\(1976\).)739 1245 y([De])165 b(Dell'Antonio,)18 b(G.F)-7 b(.:)26 b(Structure)19 b(of)h(the)g(algebra)f(of)h(some)g (free)g(systems.)g(Commun.)1057 1345 y(Math.)g(Phys.)f Fu(9)p Fx(,)h(81)g(\(1968\).)739 1519 y([DGM])68 b(De)46 b(Groot,)52 b(S.R.,)g(Mazur)m(,)f(P)-9 b(.:)77 b Fr(Non-Equilibrium)44 b(Thermodynamics.)g Fx(North-)1057 1619 y(Holland,)19 b(Amsterdam)g(\(1969\).)739 1793 y([D1])160 b(Derezi)1286 1792 y(\264)1279 1793 y(nski,)19 b(J.:)26 b(Fermi)20 b(Golden)f(Rule)i(and)f(open)f(quantum)f(systems.)j(This)f(v)n(olume.) 739 1968 y([D2])160 b(Derezi)1286 1967 y(\264)1279 1968 y(nski,)22 b(J.:)30 b(Inroduction)19 b(to)j(representations)f(of)g (canonical)g(commutation)f(and)1057 2067 y(anticommutation)13 b(relations.)i(Lecture)g(notes)h(of)g(the)g(Nordfjordeid)d(Summer)i (School)1057 2167 y("Lar)o(ge)k(Coulomb)g(Systems\227Quantum)f (Electrodynamics",)g(August)h(2003.)739 2341 y([Di])179 b(Dirac)24 b(P)-9 b(.A.M.:)31 b(The)24 b(quantum)e(theory)g(of)h(the)h (emission)f(and)g(absorption)f(of)i(radia-)1057 2441 y(tion.)c(Proc.)f(Ro)o(y)-5 b(.)20 b(Soc.)g(London,)e(Ser)-5 b(.)21 b(A)f Fu(114)p Fx(,)f(243)h(\(1927\).)739 2615 y([DJ])170 b(Derezi)1286 2614 y(\264)1279 2615 y(nski,)17 b(J.,)h(Jak\232i)1719 2614 y(\264)1714 2615 y(c,)g(V)-11 b(.:)24 b(Return)17 b(to)g(equilibrium)e(for)i(P)o(auli-Fierz)f (systems.)h(Ann.)1057 2715 y(Henri)j(Poincar\351)f Fu(4)p Fx(,)h(739)f(\(2003\).)739 2889 y([DJP])124 b(Derezi)1286 2888 y(\264)1279 2889 y(nski,)19 b(J.,)h(Jak\232i)1723 2888 y(\264)1718 2889 y(c,)g(V)-11 b(.,)20 b(Pillet,)g(C.-A.:)25 b(Perturbation)18 b(theory)g(of)h Ft(W)3198 2859 y Fs(\003)3237 2889 y Fx(-dynamics,)1057 2989 y(KMS-states)i(and)f(Liouvillean.)e(Re)n (v)-5 b(.)20 b(Math.)f(Phys.)h Fu(15)p Fx(,)g(447)f(\(2003\).)739 3163 y([Do])160 b(Dorfman,)24 b(J.R.:)35 b Fr(An)24 b(Intr)l(oduction)e (to)j(Chaos)f(in)h(Nonequilibrium)e(Statistical)h(Me-)1057 3263 y(c)o(hanics.)19 b Fx(Cambridge)g(Uni)n(v)o(ersity)g(Press,)h (Cambridge)f(\(1999\))739 3437 y([Ei])188 b(Einstein,)33 b(A.:)47 b(Zur)31 b(Quantentheorie)d(der)j(Strahlung.)e(Physik.)h (Zeitschr)-5 b(.)31 b Fu(18)p Fx(,)i(121)1057 3537 y(\(1917\).)20 b(This)j(paper)f(is)h(reprinted)e(in:)30 b(v)n(an)22 b(der)h(W)-7 b(aerden,)22 b(B.L.,)h Fr(Sour)m(ces)f(of)h(Quan-)1057 3636 y(tum)d(Mec)o(hanics.)f Fx(Do)o(v)o(er)m(,)f(Ne)n(w)j(Y)-9 b(ork)19 b(\(1967\).)739 3811 y([EM])137 b(Ev)n(ans,)16 b(D.J.,)g(Morriss,)g(G.P)-9 b(.:)22 b Fr(Statistical)15 b(Mec)o(hanics)g(of)g(Non-Equilibrium)e(Liquids.)1057 3910 y Fx(Academic)19 b(Press,)i(Ne)n(w)g(Y)-9 b(ork)19 b(\(1990\).)739 4085 y([Fer])151 b(Fermi,)19 b(E.:)25 b Fr(Nuclear)19 b(Physics.)h Fx(Notes)f(compiled)f(by)h(Orear)g(J.,)h (Rosenfeld)e(A.H.)h(and)1057 4184 y(Schluter)h(R.A.)g(The)g(Uni)n(v)o (ersity)f(of)h(Chicago)f(Press,)i(Chicago,)e(1950.)739 4359 y([FM1])100 b(Fr\366hlich,)16 b(J.,)i(Merkli,)e(M.:)24 b(Thermal)15 b(Ionization.)f(Mathematical)i(Physics,)h(Analysis)1057 4458 y(and)j(Geometry)f Fu(7)p Fx(,)h(239)f(\(2004\).)739 4633 y([FM2])100 b(Fr\366hlich,)18 b(J.,)h(Merkli,)f(M.:)25 b(Another)17 b(return)g(of)h("return)f(to)i(equilibrium".)d(Commun.) 1057 4732 y(Math.)k(Phys.,)f Fu(251)p Fx(,)g(235)h(\(2004\).)739 4907 y([FMS])96 b(Fr\366hlich,)19 b(J.,)h(Merkli,)f(M.,)h(Sigal,)f (I.M.:)25 b(Ionization)18 b(of)h(atoms)h(in)g(a)g(thermal)f(\002eld.)h (J.)1057 5006 y(Stat.)h(Phys.)e Fu(116)p Fx(,)g(311)h(\(2004\).)p eop end %%Page: 63 63 TeXDict begin 63 62 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(63)291 523 y([FMSU])40 b(Fr\366hlich,)34 b(J.,)h(Merkli,)g(M.,)g (Schw)o(arz,)f(S.,)h(Ueltschi,)g(D.:)49 b(Statistical)33 b(mechanics)609 623 y(of)24 b(thermodynamic)c(processes.)k(In)g Fr(A)g(Gar)m(den)f(of)i(Quanta)p Fx(,)e(345.)g(W)-7 b(orld)24 b(Scienti\002c)609 722 y(Publishing,)19 b(Ri)n(v)o(er)g(Edge)h(NJ)h (\(2003\).)291 895 y([FMU])82 b(Fr\366hlich,)18 b(J.,)i(Merkli,)f(M.,)h (Ueltschi,)g(D.:)25 b(Dissipati)n(v)o(e)19 b(transport:)24 b(thermal)19 b(contacts)609 995 y(and)g(tunneling)g(junctions.)g(Ann.)g (Henri)h(Poincar\351)f Fu(4)p Fx(,)h(897)g(\(2004\).)291 1167 y([Fr])188 b(Friedrichs,)15 b(K.)h(O.:)23 b Fr(P)-7 b(erturbation)14 b(of)i(Spectr)o(a)e(in)h(Hilbert)h(Space)p Fx(,)f(AMS,)h(Pro)o(vidence)609 1267 y(\(1965\).)291 1440 y([Ga1])123 b(Galla)n(v)n(otti,)44 b(G.:)63 b(Nonequilibrium)36 b(thermodynamics.)g(Preprint,)42 b(mp-arc)c(03-11)609 1539 y(\(2003\).)291 1712 y([Ga2])123 b(Galla)n(v)n(otti,)21 b(G.:)27 b(Entrop)o(y)19 b(production)g(in)i(nonequilibrium)c (thermodynamics:)24 b(a)e(re-)609 1812 y(vie)n(w)-5 b(.)19 b(Preprint,)g(arXi)n(v)h(cond-mat/0312657)15 b(\(2003\).)291 1984 y([GVV1])40 b(Goderis,)24 b(D.,)h(V)-9 b(erbeure,)22 b(A.,)j(V)-9 b(ets,)25 b(P)-9 b(.:)33 b(Noncommutati)n(v)o(e)21 b(central)i(limits.)h(Probab)m(.)609 2084 y(Theory)18 b(Related)j(Fields)f Fu(82)g Fx(527)g(\(1989\).)291 2257 y([GVV2])40 b(Goderis,)24 b(V)-11 b(.,)25 b(V)-9 b(erbeure,)23 b(A.,)i(V)-9 b(ets,)25 b(P)-9 b(.:)33 b(Quantum)23 b(central)g(limit)i (and)e(coarse)h(grain-)609 2356 y(ing.)31 b(In)h Fr(Quantum)f(pr)l (obability)g(and)g(applications,)i(V)-11 b(.)33 b Fx(Lecture)e(Notes)i (in)f(Math.,)609 2456 y Fu(1442)p Fx(,)18 b(178)i(\(1988\).)291 2628 y([GVV3])40 b(Goderis,)22 b(D.,)i(V)-9 b(erbeure,)21 b(A.,)j(V)-9 b(ets,)24 b(P)-9 b(.:)30 b(About)22 b(the)h(mathematical)f (theory)f(of)i(quan-)609 2728 y(tum)d(\003uctuations.)f(In)i Fr(Mathematical)e(Methods)h(in)h(Statistical)f(Mec)o(hanics.)f Fx(Leuv)o(en)609 2828 y(Notes)26 b(Math.)g(Theoret.)e(Phys.)h(Ser)-5 b(.)27 b(A)f(Math.)g(Phys.,)g Fu(1)p Fx(,)h(31.)f(Leuv)o(en)e(Uni)n(v) -5 b(.)25 b(Press,)609 2927 y(Leuv)o(en)18 b(\(1989\).)291 3100 y([GVV4])40 b(Goderis,)24 b(D.,)h(V)-9 b(erbeure,)24 b(A.,)h(V)-9 b(ets,)25 b(P)-9 b(.:)34 b(Theory)22 b(of)i(quantum)f (\003uctuations)g(and)g(the)609 3200 y(Onsager)c(relations.)h(J.)g (Stat.)h(Phys.)f Fu(56)p Fx(,)f(721)h(\(1989\).)291 3372 y([GVV5])40 b(Goderis,)30 b(D.,)g(V)-9 b(erbeure,)29 b(A.,)h(V)-9 b(ets,)31 b(P)-9 b(.:)42 b(Dynamics)28 b(of)g (\003uctuations)f(for)h(quantum)609 3472 y(lattice)20 b(systems.)h(Commun.)d(Math.)i(Phys.)g Fu(128)p Fx(,)f(533)g(\(1990\).) 291 3645 y([GVV6])40 b(Goderis,)16 b(D.,)i(V)-9 b(erbeure,)15 b(A.,)j(V)-9 b(ets,)18 b(P)-9 b(.:)24 b(About)15 b(the)i(e)o(xactness)g (of)f(the)h(linear)f(response)609 3744 y(theory)-5 b(.)18 b(Commun.)h(Math.)g(Phys.)h Fu(136)p Fx(,)f(265)g(\(1991\).)291 3917 y([Haa])128 b(Haak)o(e,)38 b(F)-7 b(.:)54 b Fr(Statistical)35 b(T)-5 b(r)m(eatment)35 b(of)f(Open)g(Systems)h(by)g(Gener)o(alized)f (Master)609 4017 y(Equation.)18 b Fx(Springer)g(T)m(racts)j(in)f (Modern)f(Physics)h Fu(66)p Fx(,)f(Springer)m(,)f(Berlin)j(\(1973\).) 291 4189 y([Ha])165 b(Haag,)19 b(R.:)26 b Fr(Local)20 b(Quantum)f(Physics.)h Fx(Springer)m(,)e(Ne)n(w)j(Y)-9 b(ork)19 b(\(1993\).)291 4362 y([Ja])193 b(Jak\232i)780 4361 y(\264)775 4362 y(c,)20 b(V)-11 b(.:)26 b(T)-7 b(opics)20 b(in)g(spectral)g(theory)-5 b(.)19 b(V)-11 b(olume)19 b(I)i(of)e(this)i(series.)291 4535 y([JKP])124 b(Jak\232i)780 4534 y(\264)775 4535 y(c,)21 b(V)-11 b(.,)21 b(Kritche)n(vski,)e(E.,)i (Pillet,)h(C.-A.:)k(Mathematical)20 b(theory)f(of)h(the)h(W)m(igner)n (-)609 4634 y(W)-7 b(eissk)o(opf)26 b(atom.)g(Lecture)f(notes)h(of)g (the)g(Nordfjordeid)d(Summer)i(School)h("Lar)o(ge)609 4734 y(Coulomb)19 b(Systems\227Quantum)f(Electrodynamics",)g(August)h (2003.)291 4907 y([Jo])188 b(Jo)o(ye,)35 b(A.:)49 b(Introduction)29 b(to)j(quantum)e(statistical)k(mechanics.)c(V)-11 b(olume)32 b(I)g(of)g(this)609 5006 y(series.)p eop end %%Page: 64 64 TeXDict begin 64 63 bop 739 232 a Fx(64)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx([JP1])142 b(Jak\232i)1228 522 y(\264)1223 523 y(c,)24 b(V)-11 b(.,)24 b(Pillet,)g(C.-A.:)31 b(On)23 b(a)h(model)e(for)g(quantum)f(friction)h(II:)h(Fermi')-5 b(s)23 b(golden)1057 623 y(rule)j(and)f(dynamics)f(at)j(positi)n(v)o(e) d(temperature.)g(Commun.)g(Math.)h(Phys.)h Fu(176)p Fx(,)g(619)1057 722 y(\(1996\).)739 886 y([JP2])142 b(Jak\232i)1228 885 y(\264)1223 886 y(c,)20 b(V)-11 b(.,)19 b(Pillet,)h(C.-A.:)25 b(On)18 b(a)i(model)e(for)g(quantum)f(friction)h(III:)h(Er)o(godic)d (proper)n(-)1057 986 y(ties)21 b(of)f(the)g(spin-boson)e(system.)j (Commun.)d(Math.)i(Phys.)g Fu(178)p Fx(,)f(627)g(\(1996\).)739 1150 y([JP3])142 b(Jak\232i)1228 1149 y(\264)1223 1150 y(c,)21 b(V)-11 b(.,)21 b(Pillet,)g(C.-A.:)26 b(Spectral)20 b(theory)f(of)i(thermal)e(relaxation.)g(J.)i(Math.)f(Phys.)1057 1250 y Fu(38)p Fx(,)g(1757)e(\(1997\).)739 1414 y([JP4])142 b(Jak\232i)1228 1413 y(\264)1223 1414 y(c,)32 b(V)-11 b(.,)32 b(Pillet,)g(C.-A.:)44 b(Mathematical)28 b(theory)g(of)h (non-equilibrium)d(quantum)1057 1513 y(statistical)c(mechanics.)c(J.)j (Stat.)g(Phys.)f Fu(108)p Fx(,)f(787)g(\(2002\).)739 1678 y([JP5])142 b(Jak\232i)1228 1677 y(\264)1223 1678 y(c,)18 b(V)-11 b(.,)17 b(Pillet,)i(C.-A.:)k(Non-equilibrium)13 b(steady)k(states)h(for)e(\002nite)h(quantum)d(sys-)1057 1777 y(tems)21 b(coupled)d(to)j(thermal)e(reserv)n(oirs.)g(Commun.)g (Math.)h(Phys.)f Fu(226)p Fx(,)g(131)h(\(2002\).)739 1941 y([JP6])142 b(Jak\232i)1228 1940 y(\264)1223 1941 y(c,)18 b(V)-11 b(.,)17 b(Pillet,)h(C-A.:)24 b(On)16 b(entrop)o(y)f(production)f(in)j(quantum)e(statistical)i(mechan-)1057 2041 y(ics.)k(Commun.)d(Math.)i(Phys.)g Fu(217)p Fx(,)f(285)g (\(2001\).)739 2205 y([JP7])142 b(Jak\232i)1228 2204 y(\264)1223 2205 y(c,)19 b(V)-11 b(.,)20 b(Pillet,)f(C.-A.:)25 b(A)19 b(note)g(on)f(the)h(entrop)o(y)e(production)f(formula.)h (Contemp.)1057 2305 y(Math.)j Fu(327)p Fx(,)f(175)g(\(2003\).)739 2469 y([JPR1])87 b(Jak\232i)1228 2468 y(\264)1223 2469 y(c,)23 b(V)-11 b(.,)22 b(Pillet,)h(C.-A.,)f(Re)o(y-Bellet,)g(L.:)28 b(Fluctuation)21 b(of)g(entrop)o(y)f(production)g(in)1057 2568 y(classical)h(statistical)h(mechanics.)c(In)i(preparation.)739 2732 y([JPR2])87 b(Jak\232i)1228 2731 y(\264)1223 2732 y(c,)21 b(V)-11 b(.,)20 b(Pillet,)h(C.-A.,)f(Re)o(y-Bellet,)g(L.:)25 b(In)20 b(preparation.)739 2896 y([KR])147 b(Kadison)20 b(R.V)-11 b(.,)22 b(Ringrose)f(J.R.:)28 b Fr(Fundamentals)18 b(of)j(the)h(Theory)e(of)i(Oper)o(ator)e(Alg)o(e-)1057 2996 y(br)o(as)g(II:)f(Advanced)f(Theory)p Fx(,)h(Graduate)g(Studies)h (in)g(Mathematics)f Fu(16)p Fx(,)g(AMS,)h(Pro)o(v-)1057 3096 y(idence)g(\(1997\).)739 3260 y([LeSp])86 b(Lebo)n(witz,)25 b(J.,)i(Spohn,)e(H.:)36 b(Irre)n(v)o(ersible)23 b(thermodynamics)f(for) j(quantum)e(systems)1057 3359 y(weakly)d(coupled)e(to)j(thermal)e (reserv)n(oirs.)g(Adv)-5 b(.)19 b(Chem.)h(Phys.)g Fu(39)p Fx(,)f(109)g(\(1978\).)739 3524 y([Li])188 b(Lindblad,)21 b(G.:)31 b(Completely)21 b(positi)n(v)o(e)h(maps)h(and)f(entrop)o(y)f (inequalities.)g(Commun.)1057 3623 y(Math.)f(Phys.)f Fu(40)p Fx(,)h(147)f(\(1975\).)739 3787 y([LMS])91 b(Lieb,)53 b(E.H.,)g(Schulz,)g(T)-6 b(.,)54 b(Mathis,)f(D.:)79 b(T)-7 b(w)o(o)48 b(soluble)e(models)g(of)h(an)g(anti-)1057 3887 y(ferromagnetic)18 b(chain.)h(Ann.)g(Phys.)h Fu(28)p Fx(,)f(407,)g(\(1961\).)739 4051 y([Ma])151 b(Matsui,)20 b(T)-6 b(.:)26 b(On)20 b(the)g(algebra)g(of)g(\003uctuation)f(in)h (quantum)f(spin)h(chains.)g(Ann.)f(Henri)1057 4151 y(Poincar\351)g Fu(4)p Fx(,)h(63)g(\(2003\).)739 4315 y([Me1])109 b(Merkli,)17 b(M.:)24 b(Positi)n(v)o(e)16 b(commutators)f(in)i(non-equilibrium)c (quantum)i(statistical)j(me-)1057 4414 y(chanics.)i(Commun.)e(Math.)i (Phys.)g Fu(223)p Fx(,)f(327)g(\(2001\).)739 4578 y([Me2])109 b(Merkli,)19 b(M.:)26 b(Stability)20 b(of)f(equilibria)g(with)h(a)h (condensate.)d(Commun.)g(Math.)i(Phys.,)1057 4678 y(in)g(press.)739 4842 y([Me3])109 b(Merkli,)20 b(M.:)25 b(The)20 b(ideal)g(quantum)e (gas.)i(V)-11 b(olume)20 b(I)g(of)g(this)h(series.)739 5006 y([Mes])119 b(Messiah,)20 b(A.:)26 b Fr(Quantum)19 b(Mec)o(hanics.)g(V)-9 b(olume)19 b(II.)g Fx(W)m(ile)o(y)-5 b(,)20 b(Ne)n(w)g(Y)-9 b(ork.)p eop end %%Page: 65 65 TeXDict begin 65 64 bop 291 232 a Fw(T)-7 b(opics)20 b(in)g(non-equilibrium)c(quantum)j(statistical)i(mechanics)898 b Fx(65)291 523 y([Og])160 b(Ogata,)17 b(Y)-11 b(.:)24 b(The)17 b(stability)g(of)g(the)g(non-equilibrium)c(steady)k(states.)h (Commun.)e(Math.)609 623 y(Phys.)j Fu(245)p Fx(,)g(577)h(\(2004\).)291 797 y([OP])156 b(Ohya,)26 b(M.,)h(Petz,)h(D.:)37 b Fr(Quantum)24 b(Entr)l(opy)i(and)f(its)i(Use)o(.)f Fx(Springer)n(-V)-9 b(erlag,)24 b(Berlin)609 897 y(\(1993\).)291 1071 y([P)o(a])180 b(P)o(ais,)35 b(A.:)49 b Fr("Subtle)31 b(is)i(the)f(Lor)m(d...",)h(The) f(Science)f(and)g(Life)i(of)f(Albert)g(Einstein.)609 1171 y Fx(Oxford)18 b(Uni)n(v)o(ersity)h(Press,)i(Oxford)e(\(1982\).) 291 1345 y([Pi])193 b(Pillet,)28 b(C.-A.:)38 b(Quantum)25 b(dynamical)g(systems)i(and)f(their)g(KMS-states.)g(V)-11 b(olume)26 b(I)609 1445 y(of)20 b(this)g(series.)291 1619 y([PoSt])105 b(Po)n(wers,)35 b(R.)e(T)-6 b(.,)36 b(Stormer)m(,)e(E.:)50 b(Free)32 b(states)i(of)e(the)h(canonical)e (anticommutation)609 1719 y(relations.)19 b(Commun.)g(Math.)h(Phys.)f Fu(16)p Fx(,)h(1)g(\(1969\).)291 1893 y([RC])152 b(Rondoni,)24 b(L.,)j(Cohen,)e(E.G.D.:)35 b(Gibbs)25 b(entrop)o(y)e(and)i(irre)n(v)o (ersible)e(thermodynam-)609 1993 y(ics.)e(Nonlinearity)d Fu(13)p Fx(,)i(1905)e(\(2000\).)291 2167 y([Re])170 b(Re)o(y-Bellet,)20 b(L.:)25 b(Open)20 b(classical)h(systems.)f(V)-11 b(olume)20 b(II)g(of)g(this)g(series.)291 2341 y([Ri])184 b(Rideau,)18 b(G.:)25 b(On)18 b(some)h(representations)d(of)j(the)f(anticommutation) e(relations.)i(Com-)609 2441 y(mun.)h(Math.)h(Phys.)f Fu(9)p Fx(,)h(229)g(\(1968\).)291 2615 y([Ro1])123 b(Robinson,)41 b(D.W)-8 b(.:)61 b(Return)37 b(to)h(equilibrium.)e(Commun.)g(Math.)i (Phys.)f Fu(31)p Fx(,)k(171)609 2715 y(\(1973\).)291 2889 y([Ro2])123 b(Robinson,)45 b(D.W)-8 b(.:)67 b Ft(C)1322 2859 y Fs(\003)1361 2889 y Fx(-algebras)40 b(in)h(quantum)e (statistical)j(mechanics.)e(In)h Ft(C)3089 2859 y Fs(\003)3127 2889 y Fr(-)609 2989 y(alg)o(ebr)o(as)20 b(and)g(their)i(Applications)e (to)h(Statistical)g(Mec)o(hanics)f(and)h(Quantum)e(F)l(ield)609 3088 y(Theory)p Fx(,)h(\(D.)g(Kastler)g(editor\).)f(North-Holand,)e (Amsterdam)i(\(1976\).)291 3263 y([Ru1])123 b(Ruelle,)27 b(D.:)36 b(Natural)25 b(nonequilibrium)c(states)27 b(in)e(quantum)f (statistical)j(mechanics.)609 3362 y(J.)21 b(Stat.)f(Phys.)g Fu(98)p Fx(,)f(57)h(\(2000\).)291 3537 y([Ru2])123 b(Ruelle,)30 b(D.:)42 b(Entrop)o(y)27 b(production)e(in)k(quantum)d(spin)j(systems.) f(Commun.)f(Math.)609 3636 y(Phys.)19 b Fu(224)p Fx(,)g(3)i(\(2001\).) 291 3811 y([Ru3])123 b(Ruelle,)30 b(D.:)41 b(T)-7 b(opics)28 b(in)g(quantum)e(statistical)k(mechanics)d(and)g(operator)f(algebras.) 609 3910 y(Preprint,)19 b(mp-arc)g(01-257)f(\(2001\).)291 4085 y([Ru4])123 b(Ruelle,)30 b(D.:)42 b(Smooth)27 b(dynamics)g(and)h (ne)n(w)g(theoretical)f(ideas)h(in)h(nonequilibrium)609 4184 y(statistical)21 b(mechanics.)e(J.)i(Stat.)g(Phys.)e Fu(95)p Fx(,)h(393)f(\(1999\).)291 4359 y([Ru5])123 b(Ruelle,)37 b(D.:)53 b(Extending)32 b(the)i(de\002nition)f(of)h(entrop)o(y)e(to)i (nonequilibrium)d(steady)609 4458 y(states.)21 b(Proc.)e(Nat.)i(Acad.)f (Sci.)g(USA)h Fu(100)p Fx(,)e(3054)g(\(2003\).)291 4633 y([Ru6])123 b(Ruelle,)23 b(D.:)31 b(A)24 b(remark)d(on)i(the)g(equi)n (v)n(alence)d(of)j(isokinetic)f(and)h(isoener)o(getic)e(ther)n(-)609 4732 y(mostats)f(in)h(the)f(thermodynamic)d(limit.)j(J.)h(Stat.)g (Phys.)e Fu(100)p Fx(,)g(757)h(\(2000\).)291 4907 y([Ru7])123 b(Ruelle,)19 b(D.:)25 b(Con)m(v)o(ersations)17 b(on)h(nonequilibrium)d (physics)j(with)h(an)g(e)o(xtraterrestrial.)609 5006 y(Physics)h(T)-7 b(oday)19 b Fu(57)p Fx(,)h(48)f(\(2004\).)p eop end %%Page: 66 66 TeXDict begin 66 65 bop 739 232 a Fx(66)1655 b Fw(Aschbacher)m(,)18 b(Jak\232i)3079 231 y(\264)3074 232 y(c,)i(P)o(autrat,)g(Pillet)739 523 y Fx([RS])161 b(Reed,)30 b(M.,)g(Simon,)f(B.:)42 b Fr(Methods)27 b(of)i(Modern)e(Mathematical)g(Physics,)j(I.)e(Func-) 1057 623 y(tional)20 b(Analysis)p Fx(,)g(London,)d(Academic)j(Press)h (\(1980\).)739 789 y([Sp])174 b(Spohn,)17 b(H.:)24 b(An)18 b(algebraic)f(condition)e(for)j(the)f(approach)f(to)i(equilibrium)e(of) h(an)h(open)1057 888 y Ft(N)9 b Fx(-le)n(v)o(el)19 b(system,)h(Lett.)h (Math.)f(Phys.)f Fu(2)p 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b(approach)d(to)j(equilibrium)d(in)j (quantum)e(statistics.)i(Physica)f Fu(23)p Fx(,)1057 2051 y(441.)739 2217 y([VH3])100 b(v)n(an)19 b(Ho)o(v)o(e,)f(L.:)24 b(Master)19 b(equation)f(and)h(approach)d(to)k(equilibrium)d(for)h (quantum)f(sys-)1057 2316 y(tems.)h(In)g Fr(Fundamental)e(pr)l(oblems)i (in)g(statistical)h(mec)o(hanics)p Fx(,)e(compiled)g(by)g(E.G.D.)1057 2416 y(Cohen,)i(North-Holand,)f(Amsterdam)h(1962.)739 2582 y([WW])106 b(W)-7 b(eissk)o(opf,)41 b(V)-11 b(.,)42 b(W)m(igner)m(,)e(E.:)60 b(Berechnung)35 b(der)h(nat\374rlichen)g (Linienbreite)f(auf)1057 2682 y(Grund)19 b(der)h(Diracschen)f (Lichttheorie.)f(Zeitschrift)i(f\374r)f(Physik)h Fu(63)p Fx(,)f(54)h(\(1930\).)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0506111027508--