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Asymptotics of the solutions to the \(N\)-particle Kolmogorov-Feller equations and the asymptotics of the solution to the Boltzmann equation in the region of large deviations. (English. Russian original) Zbl 0855.46047
Math. Notes 58, No. 5, 1166-1177 (1995); translation from Mat. Zametki 58, No. 5, 694-709 (1995).
Summary: We construct a representation in which the asymptotics of the solution to the Kolmogorov-Feller equation in the Fock space \(\Gamma (L_1 (\mathbb{R}^n))\) is of a form similar to the WKB asymptotic expansion; namely, the Boltzmann equation in \(L_1 (\mathbb{R}^n)\) plays the role of the Hamilton equations, the linearized Boltzmann equation extended to \(\Gamma (L_1 (\mathbb{R}^n))\) plays the role of the transport equation, and the Hamilton-Jacobi equation follows from the conservation of the total probability for the solutions of the Boltzmann equation. We also construct the asymptotics of the solution to the Boltzmann equation with small transfer of momentum; this asymptotics is given by the tunnel canonical operator corresponding to the self-consistent characteristic equation.

46N50 Applications of functional analysis in quantum physics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
Full Text: DOI
[1] V. P. Maslov and S. É. Tariverdiev, ”Asymptotics of the Kolmogorov-Feller equation for a system of a large number of particles,” in:Probability Theory, Mathematical Statistics, Theoretical Cybernetics [in Russian], Vol. 19, VINITI, Moscow (1982), pp. 85–126. · Zbl 0517.60100
[2] V. P. Maslov and A. M. Chebotarev, ”Random fields corresponding to the Bogolyubov, Vlasov, and Boltzmann chains,”Teor. Mat. Fiz. [Theoret. and Math. Phys.],54, 78–88 (1983).
[3] V. P. Maslov and O. Yu. Shvedov, ”An asymptotic formula for theN-particle density function asN and violation of the chaos hypothesis,”Russ. J. Math. Phys.,2, No. 2, 217–234 (1994). · Zbl 0916.35097
[4] V. P. Maslov and O. Yu. Shvedov, ”The spectrum of theN-particle Hamiltonian for largeN and superfluidity,”Dokl. Ross. Akad. Nauk [Russian Math. Dokl.],335, No. 1, 42–46 (1994).
[5] V. P. Maslov and O. Yu. Shvedov, ”Quantization in the vicinity of classical solutions in theN-particle problem and superfluidity,”Teor. Mat. Fiz. [Theoret. and Math. Phys.],98, No. 2, 266–288 (1994).
[6] V. P. Maslov and O. Yu. Shvedov, ”Steady-state asymptotic solutions in the many-body problem and the derivation of integral equations with jumping nonlinearity,”Differentsial’nye Uravneniya [Differential Equations],31, No. 2, 312–325 (1995). · Zbl 0855.35024
[7] V. P. Maslov and O. Yu. Shvedov, ”The complex WKB method in Fock space,”Dokl. Ross. Akad. Nauk [Russian Math. Dokl.],340, No. 1, 42–47 (1995). · Zbl 0884.34065
[8] J.-P. Serre,Lie Algebras and Lie Groups, Benjamin, New York-Amsterdam (1965). · Zbl 0132.27803
[9] A. A. Arsen’ev,Lectures on Kinetic Equations [in Russian], Nauka, Moscow (1992).
[10] H. Tanaka, ”Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,”Z. Wahrshr.,46, 67–105 (1978). · Zbl 0389.60079
[11] V. P. Maslov and A. M. Chebotarev, ”Cluster expansions and secondary quantization,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],203, 168–185 (1994). · Zbl 0901.47054
[12] V. P. Maslov, ”The equation of self-consistent field,” in:Current Problems of Mathematics, Itogi Nauki i Tekhniki [in Russian], Vol. 11, VINITI, Moscow (1978), pp. 153–234.
[13] A. M. Chebotarev, ”The logarithmic asymptotics of the solution to the Cauchy problem for the Boltzmann equation,”Mat. Model.,7, No. 10 (1995). · Zbl 0974.76585
[14] V. P. Maslov,Perturbation Theory and Asymptotic Methods [in Russian], Nauka, Moscow (1988). · Zbl 0653.35002
[15] M. V. Fedoryuk,The Saddle-Point Method [in Russian], Nauka, Moscow (1977). · Zbl 0463.41020
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