Abstract
We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.
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References
N. N. Bogoliubov, Problems of a Dynamical Theory in Statistical Physics [in Russian], Gostekhizdat, Moscow (1946); English transl. (Stud. Statist. Mech., Vol. 1), North-Holland, Amsterdam (1962).
J. R. Dorfman and E. G. D. Cohen, J. Math. Phys., 8, 282–297 (1967).
J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland, Amsterdam (1972).
Yu. L. Klimontovich, Statistical Physics [in Russian], Nauka, Moscow (1982); English transl., Harwood Academic, Chur (1986).
O. E. Lanford III, “Time evolution of large classical systems,” in: Dynamical Systems, Theory and Applications (Lect. Notes Phys., Vol. 38, E. J. Moser, ed.), Vol. 38, Springer, Berlin (1975), pp. 1–111.
D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti, J. Stat. Phys., 116, 381–410 (2004).
N. G. Van Kampen, J. Stat. Phys., 115, 1057–1072 (2004).
S. Nakajima, Progr. Theoret. Phys., 20, 948–959 (1958).
R. Zwanzig, J. Chem. Phys., 33, 1338–1341 (1960).
I. Prigogine, Non-Equilibrium Statistical Mechanics (Monogr. Statist. Phys. and Thermodynamics, Vol. 1), Interscience, New York (1962).
H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford Univ. Press, Oxford (2002).
B. Bellomo, G. Compagno, and F. Petruccione, J. Phys. A, 38, 10203–10216 (2005).
F. Shibata, Y. Takahashi, and N. Hashitsume, J. Stat. Phys., 17, 171–187 (1977).
F. Shibata and T. Arimitsu, J. Phys. Soc. Japan, 49, 891–897 (1980).
V. F. Los, J. Phys. A, 34, 6389–6403 (2001).
V. F. Los, J. Stat. Phys., 119, 241–271 (2005).
G. E. Uhlenbeck and G. W. Ford, Lectures in Statistical Mechanics, Vol. 1, Amer.Math. Soc., Providence, R. I. (1963).
R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York (1975).
Yu. L. Klimontovich, Kinetic Theory of Nonideal Gases and Nonideal Plasmas [in Russian], Nauka, Moscow (1975); English transl. (Internat. Ser. Natural Philosophy, Vol. 105), Pergamon, Oxford (1982).
S. Tasaki, K. Yuasa, P. Facchi, G. Kimura, H. Nakasato, I. Ohba, and S. Pascazio, Ann. Phys., 322, 631–656 (2007).
K. Yuasa, S. Tasaki, P. Facchi, G. Kimura, H. Nakasato, I. Ohba, and S. Pascazio, Ann. Phys., 322, 657–676 (2007).
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Dedicated to the memory of Academician N. N. Bogoliubov
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 2, pp. 304–330, August, 2009.
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Los, V.F. Nonlinear generalized master equations and accounting for initial correlations. Theor Math Phys 160, 1124–1143 (2009). https://doi.org/10.1007/s11232-009-0105-4
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DOI: https://doi.org/10.1007/s11232-009-0105-4