Abstract
The classical BBGKY hierarchy is reformulated into a single functional equation. A suitable modification of this (neglecting the anti-irreversible or entropy-destructive process) gives an irreversible dynamics of the many-particle system evolution, which is called the functional random-walk model.
Then, the basic (functional-differential) equation governs formally the time development of the probability density functional of the particle-number density in the one-body phase space. Some topics, mathematical as well as physical, around this equation are presented. For instance, the integral representation of the equation expresses an ensemble of generalized Brownian paths in the function space of the particle-number density; any state of the system approaches irreversibly to the ultimate equilibrium state (if the system is closed), for which the probabilistic mode in the function space is nothing other than of canonical form, etc.
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References
For example, see I. Prigogine, Non-Equilibrium Statistical Mechanics (Interscience Publishers, Inc., New York, 1962);
T. Y. Wu, Kinetic Equations of Gases and Plasmas (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1966).
A. I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover Publ., Inc., New York, 1949).
I. Hosokawa, Prog. Theor. Phys. (Kyoto) 52, 1513 (1974).
, N. N. Bogoliubov, in Studies in Statistical Mechanics, J. de Boer and G. E. Uhlenbeck, Eds. (North-Holland Publ. Co., Amsterdam, 1962), Vol. 1.
I. Hosokawa, J. Hath. Phys. 8, 221 (1967).
I. Hosokawa, J. Math. Phys. 11 657 (1970).
E. Hopf, J. Ratl. Mech. Anal. 1, 87 (1952).
E. A. Novikov, Zh. Exsp. Teor. Fiz. 47, 1919 (1964)
[Soviet-Phys. — JETP 20, 1290 (1965)];I. Hosokawa, J. Phys. Soc. Japan 25, 271 (1968).
Iu. L. Klimontovich, Zh. Exsp. Teor. Fiz. 33, 982 (1957)
; Iu. L. Klimontovich The Statistical Theory of Non-Equilibrium Vnocesses in a ?lasma (Pergamon Press, Inc., New York, 1967).
A. Kolmogorov, Math. Ann. 108, 149 (1933).
R. D. Richtmyer and K. W. Morton, Difference Methods for Ini-tial-Value Problems (Interscience Publishers, Inc., New York, 1967).
I. Hosokawa, J. Hath. Phys. 14, 1374 (1973).
C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1959).
A. H. Gray, Jr., J. Math. Phys. 6, 644 (1965).
A. A. Vlasov, Many-Particle Theory (Moscow-Leningrad, 1950, trans. AEC-tr-3406, Washington, D. C., 1959), Part 2, Chap.1; also the book by Wu in Ref. 1, Chap. 5.
E. T. Jaynes, Phys. Rev. 106, 620 (1957).
I. Hosokawa, J, Stat. Phys. 15, 87 (1976).
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Hosokawa, I. (1978). Functional Equations and Their Treatment in Non-Equilibrium Statistical Mechanics. In: Papadopoulos, G.J., Devreese, J.T. (eds) Path Integrals. NATO Advanced Study Institutes Series, vol 34. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-9140-1_14
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DOI: https://doi.org/10.1007/978-1-4684-9140-1_14
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