Geodesic
Nonlinear generalized master equations and accounting for initial correlations
Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 2, pp. 304-330 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: nonlinear master equation, initial correlation, irreversibility. 
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     author = {V. F. Los'},
     title = {Nonlinear generalized master equations and accounting for initial correlations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {304--330},
     year = {2009},
     volume = {160},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_160_2_a5/}
}
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V. F. Los'. Nonlinear generalized master equations and accounting for initial correlations. Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 2, pp. 304-330. http://geodesic.mathdoc.fr/item/TMF_2009_160_2_a5/

[1] N. N. Bogolyubov, Problemy dinamicheskoi teorii v statisticheskoi fizike, Gostekhizdat, M.-L., 1946 | MR | MR | Zbl

[2] J. R. Dorfman, E. G. D. Cohen, J. Math. Phys., 8:2 (1967), 282–297 | DOI

[3] Dzh. Fertsiger, G. Kaper, Matematicheskaya teoriya protsessov perenosa v gazakh, Mir, M., 1976

[4] Yu. L. Klimontovich, Statisticheskaya fizika, Nauka, M., 1982 | MR

[5] O. E. Lanford III, “Time evolution of large classical systems”, Dynamical Systems, Theory and Applications, Lecture Notes in Phys., 38, ed. E. J. Moser, Springer, 1975, 1–111 | DOI | MR | Zbl

[6] D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti, J. Stat. Phys., 116:1–4 (2004), 381–410 | DOI | MR | Zbl

[7] N. G. Van Kampen, J. Stat. Phys., 115:3–4 (2004), 1057–1072 | DOI | MR | Zbl

[8] S. Nakajima, Progr. Theoret. Phys., 20:6 (1958), 948–959 | DOI | MR | Zbl

[9] R. Zwanzig, J. Chem. Phys., 33:5 (1960), 1338–1341 | DOI | MR

[10] I. Prigozhin, Neravnovesnaya statisticheskaya mekhanika, Sinergetika: ot proshlogo k buduschemu, Editorial URSS, M., 2009 | MR | Zbl

[11] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems, Oxford Univ. Press, Oxford, 2002 | MR | Zbl

[12] B. Bellomo, G. Compagno, F. Petruccione, J. Phys. A, 38:47 (2005), 10203–10216 | DOI | MR | Zbl

[13] F. Shibata, Y. Takahashi, N. Hashitsume, J. Stat. Phys., 17:4 (1977), 171–187 | DOI | MR

[14] F. Shibata, T. Arimitsu, J. Phys. Soc. Japan, 49:3 (1980), 891–897 | DOI

[15] V. F. Los, J. Phys. A, 34:33 (2001), 6389–6403 | DOI | Zbl

[16] V. F. Los, J. Stat. Phys., 119:1–2 (2005), 241–271 | DOI | MR | Zbl

[17] Dzh. Ulenbek, Dzh. Ford, Lektsii po statisticheskoi mekhanike, Mir, M., 1965 | MR | Zbl

[18] R. Balesku, Ravnovesnaya i neravnovesnaya statisticheskaya mekhanika, T. 2, Mir, M., 1978 | MR | MR | Zbl

[19] Yu. L. Klimontovich, Kineticheskaya teoriya neidealnogo gaza i neidealnoi plazmy, Nauka, M., 1975 | MR | MR

[20] S. Tasaki, K. Yuasa, P. Facchi, G. Kimura, H. Nakasato, I. Ohba, S. Pascazio, Ann. Phys., 322:3 (2007), 631–656 | DOI | MR | Zbl

[21] K. Yuasa, S. Tasaki, P. Facchi, G. Kimura, H. Nakasato, I. Ohba, S. Pascazio, Ann. Phys., 322:3 (2007), 657–676 | DOI | MR | Zbl