Abstract
We study the Liouville equation in the domain of small deviations from absolute equilibrium. The solution is expressed in terms of amplitudes ofn-body additive functions which are orthogonal with respect to the Gibbs weight factor. In the memory operator approach the memory operators are formally exact continued fractions. We show that with the isolation in the Liouville operator of a one-body additive operatorL o, any memory operator can be written alternatively as an exact infinite series, each term of which can be calculated exactly. This yields improvements of the dressed particle approximation. We discuss the choice ofL o, which is in general time dependent. The theory is developed both for smooth potentials and for hard spheres, where we use pseudo-Liouville operators. The theory can be formulated in an equivalent manner by introducing modified cumulant distributions, which are closely related to the amplitudes. The modified cumulants have the same spatial asymptotic properties as ordinary cumulants, but have superior short-time and small-distance behavior.
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References
E. P. Gross,Ann. Phys. N. Y. 69:42 (1972). Referred to as I.
C. D. Boley,Ann. Phys. N. Y. 86:91 (1974);Phys. Rev. A 11:328 (1975).
G. F. Mazenko,Phys. Rev. 7:209, 222 (1973);9:360 (1974).
E. P. Gross,J. Stat. Phys. 11:503 (1974). Referred to as V.
M. H. Ernst, J. R. Dorfman, W. R. Hoegy, and J. M. J. van Leeuwen,Physica 45:127 (1969); E. G. D. Cohen,Physica 27:163 (1961).
T. Y. Wu,Kinetic Equations of Gases and Plasmas, Addison-Wesley, Reading, Massachusetts (1966); Y. L. Klimontovich,Statistical Theory of Non-Equilibrium Processes in Plasmas, Pergamon.
J. L. Lebowitz, J. K. Percus, and J. Sykes,Phys. Rev. 188, 487 (1969); L. Blum and J. L. Lebowitz,Phys. Rev. 185:273 (1969).
G. F. Mazenko,Phys. Rev. A 3:3121 (1971);6:2545 (1972); G. F. Mazenko, T. Y. C. Wei, and S. Yip,Phys. Rev. A 6:1981 (1972).
J. Sykes,J. Stat. Phys. 8:279 (1973).
M. H. Ernst and J. R. Dorfman,Physica 61:157 (1972).
E. P. Gross,J. Stat. Phys. 9:275, 297 (1973). Referred to as II and III.
P. C. Martin and J. Schwinger,Phys. Rev. 115:1342 (1957); L. P. Kadanoff and G. Baym,Quantum Statistical Mechanics, Benjamin (1962).
D. Forster and P. Martin,Phys. Rev. A 2:1575 (1970); D. Forster,Phys. Rev. A 9:943 (1974).
I. Prigogine,Non-Equilibrium Statistical Mechanics, Interscience (1962); R. Balescu,Statistical Mechanics of Charged Particles, Wiley (1963); P. Résibois, inPhysics of Many Particle Systems, E. Meeron, ed., Gordon and Breach (1966).
A. Z. Ackasu and J. J. Duderstadt,Phys. Rev. 188:479 (1969);Phys. Rev., A 1:905 (1970); R. Zwanzig,Phys. Rev. 144:170 (1966).
Y. Pomeau,Phys. Rev. A 3:11, 74 (1972).
K. Bergeron, E. P. Gross, and R. Varley,J. Stat. Phys. 10:111 (1974). Referred to as IV.
J. V. Sengers, inKinetic Equations, R. L. Liboff and N. Rostoker, eds., Gordon and Breach (1971), p. 137.
H. H. U. Konijnendjck and J. M. J. van Leeuwen,Physica 64:342 (1973).
L. P. Kadanoff,Phys. Rev. 116:89 (1968).
P. Résibois,J. Stat. Phys. 2, 21 (1970); J. W. Dufty,Phys. Rev. A 5:2247 (1972); K. Kawasaki and I. Oppenheim,Phys. Rev. 136:A1519 (1964);139:3A, 649 (1965);139:6A, 1763 (1965); J. R. Dorfman and E. G. D. Cohen,Phys. Rev. A 6:776 (1972).
J. R. Dorfman and E. G. D. Cohen,Phys. Rev. A.
J. Lebowitz and P. Résibois,J. Stat. Phys. 12:483 (1975).
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Work supported by the National Science Foundation.
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Gross, E.P. Formal structure of kinetic theory. J Stat Phys 15, 181–214 (1976). https://doi.org/10.1007/BF01012876
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DOI: https://doi.org/10.1007/BF01012876