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2
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5
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6
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ed. A. H. Taub (Pergamon Press, Oxford)
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J. von Neumann: Goettinger Nachr. 1 (1927) 273; Collected Works, ed. A. H. Taub (Pergamon Press, Oxford, 1961) Vol. 1, p. 208.
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10
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2442459454
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(Holt, Reinhart and Winston, New York) 2nd ed., and references therein
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D. ter Haar: Elements of Thermostatistics (Holt, Reinhart and Winston, New York, 1966) 2nd ed., and references therein.
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Ter Haar, D.1
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13
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11744350512
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and references therein
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J. L. Lebowitz: Physica A 194 (1993) 1, and references therein.
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Physica A
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Lebowitz, J.L.1
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14
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2442461819
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J. Bricmont: chao-dyn/9603009
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J. Bricmont: chao-dyn/9603009.
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15
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-
2442459430
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-
note
-
There appears an important difference between H and S if one allows a time dependent Hamiltonian. The density matrix is still transformed by a unitary transformation and thus H stays constant. But S generally changes except for an infinitely slow adiabatic variation of the Hamiltonian.
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-
-
-
16
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33645063164
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J. G. Kirkwood: J. Chem. Phys. 14 (1946) 180; ibid. 15 (1947) 72. N. N. Bogolyubov: Studies in Stat. Mech. 1 (1962) 1. E. D. G. Cohen and T. H. Berlin: Physica 26 (1960) 717.
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Kirkwood, J.G.1
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17
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33645063164
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J. G. Kirkwood: J. Chem. Phys. 14 (1946) 180; ibid. 15 (1947) 72. N. N. Bogolyubov: Studies in Stat. Mech. 1 (1962) 1. E. D. G. Cohen and T. H. Berlin: Physica 26 (1960) 717.
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18
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33645063164
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J. G. Kirkwood: J. Chem. Phys. 14 (1946) 180; ibid. 15 (1947) 72. N. N. Bogolyubov: Studies in Stat. Mech. 1 (1962) 1. E. D. G. Cohen and T. H. Berlin: Physica 26 (1960) 717.
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Studies in Stat. Mech.
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Bogolyubov, N.N.1
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19
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11744348257
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J. G. Kirkwood: J. Chem. Phys. 14 (1946) 180; ibid. 15 (1947) 72. N. N. Bogolyubov: Studies in Stat. Mech. 1 (1962) 1. E. D. G. Cohen and T. H. Berlin: Physica 26 (1960) 717.
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Physica
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, pp. 717
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-
Cohen, E.D.G.1
Berlin, T.H.2
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20
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2442466478
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-
note
-
One can confirm the relation dS/dE = kβ(E) for this solution, and we assume β(E) > 0.
-
-
-
-
21
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-
2442573019
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-
note
-
m,n, which we adopt in the present note.
-
-
-
-
22
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-
2442518658
-
-
note
-
This statistical inference of diagonal elements means that the macroscopic observables 〈E〉 and 〈N〉 are insensitive to whether one uses the original density matrix or the density matrix after one applied the statistical inference.
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-
-
-
23
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0039870649
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-
Sommerfeld Festschrift, ed. P. Debye (Hirzel, Leipzig)
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W. Pauli: Probleme der Modernen Physik, Sommerfeld Festschrift, ed. P. Debye (Hirzel, Leipzig, 1928); R. C. Tolman: The Principles of Statistical Mechanics (Oxford University Press, New York, 1938).
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(1928)
Probleme der Modernen Physik
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Pauli, W.1
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24
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0004026847
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Oxford University Press, New York
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W. Pauli: Probleme der Modernen Physik, Sommerfeld Festschrift, ed. P. Debye (Hirzel, Leipzig, 1928); R. C. Tolman: The Principles of Statistical Mechanics (Oxford University Press, New York, 1938).
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(1938)
The Principles of Statistical Mechanics
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Tolman, R.C.1
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26
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2442464418
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-
note
-
n 〈n|ρ|n〉 ln〈n|ρ|n〉, as required.
-
-
-
-
29
-
-
2442549360
-
-
note
-
Intuitively, one can see a possible origin of the violation of time reversal symmetry in this time averaging. One cannot reproduce the original density matrix form the time averaged one, and in this sense the use of macroscopic observables induces the violation of microscopic time reversal invariance.
-
-
-
-
30
-
-
2442523858
-
-
note
-
ρ{τ(ρ )}. In practice, however, the time dependence of an observable always appears in the combination Tr ρ(t)Ô in our treatment. One may thus use the time scale given by the maximum of τ which satisfies (3.10) for a given physical system specified by ρ(t).
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-
-
-
32
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0011093691
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P. Bocchieri and A. Loinger: Phys. Rev. 107 (1957) 337; I. C. Percival: J. Math. Phys. 2 (1961) 235.
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Percival, I.C.1
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33
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2442572859
-
-
note
-
One may avoid the recurrence theorem by appealing to the thermodynamic limit with V → large and the particle density N/V kept finite, but at the same time one looses the dynamical information of the system. We know very little about the dynamical properties of a many particle system, and the reccurence theorem provides precious information.
-
-
-
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34
-
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2442513874
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-
note
-
-13 eV.
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-
-
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35
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2442496926
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-
note
-
A similar consideration is applicable to the analysis of a time dependent Hamiltonian if one analyzes the behavior of the system after switching off the time dependent perturbation.
-
-
-
-
36
-
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2442553565
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-
note
-
This picture suggests that the bulk of entropy generation takes place within the time scale of relaxation measured by macroscopic observables.
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37
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36149028105
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H. B. Callen and T. A. Welton: Phys. Rev. 83 (1951) 34; R. Kubo: J. Phys. Soc. Jpn. 12 (1957) 570; H. Nakano: Int. J. Mod. Phys. B 7 (1993) 2397 and references therein.
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Callen, H.B.1
Welton, T.A.2
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38
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12344271673
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H. B. Callen and T. A. Welton: Phys. Rev. 83 (1951) 34; R. Kubo: J. Phys. Soc. Jpn. 12 (1957) 570; H. Nakano: Int. J. Mod. Phys. B 7 (1993) 2397 and references therein.
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Kubo, R.1
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39
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36149028105
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and references therein
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H. B. Callen and T. A. Welton: Phys. Rev. 83 (1951) 34; R. Kubo: J. Phys. Soc. Jpn. 12 (1957) 570; H. Nakano: Int. J. Mod. Phys. B 7 (1993) 2397 and references therein.
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0042829027
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L. Van Hove: Physica 25 (1959) 268; ibid. 21 (1955) 517.
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43
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0000937315
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Lieb, E.H.2
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Bombelli, L.1
Koul, R.K.2
Lee, J.3
Sorkin, R.D.4
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45
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0039768613
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H. Araki and E. H. Lieb: Commun. Math. Phys. 18 (1970) 160; L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin: Phys. Rev. D 34 (1986) 373; M. Srednicki: Phys. Rev. Lett. 71 (1993) 666; H. Terashima: Phys. Rev. D 61 (2000) 104016, and references therein.
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Srednicki, M.1
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46
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17144371503
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and references therein
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H. Araki and E. H. Lieb: Commun. Math. Phys. 18 (1970) 160; L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin: Phys. Rev. D 34 (1986) 373; M. Srednicki: Phys. Rev. Lett. 71 (1993) 666; H. Terashima: Phys. Rev. D 61 (2000) 104016, and references therein.
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Terashima, H.1
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48
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C. Jarzynski: Phys. Rev. Lett. 78 (1997) 2690; Phys. Rev. E 56 (1997) 5018; J. Stat. Phys. 96 (1999) 415.
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C. Jarzynski: Phys. Rev. Lett. 78 (1997) 2690; Phys. Rev. E 56 (1997) 5018; J. Stat. Phys. 96 (1999) 415.
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H. Tasaki: cond-mat/0011321, and references therein
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H. Tasaki: cond-mat/0011321, and references therein.
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55
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2442524240
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note
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0 for an arbitrary small unitary transformation U, which includes time development generated by a time-dependent Hamiltonian.39) Our analysis in this paper is complementary to that of ref. 39 where no analysis of relaxation after an extemal perturbation is given.
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