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.

##### MSC:

46N50 | Applications of functional analysis in quantum physics |

81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |

##### Keywords:

Kolmogorov-Feller equation; Fock space; WKB asymptotic expansion; Boltzmann equation; Hamilton equations; Hamilton-Jacobi equation; conservation of the total probability; small transfer of momentum
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\textit{V. P. Maslov}, Math. Notes 58, No. 5, 1166--1177 (1995; Zbl 0855.46047); translation from Mat. Zametki 58, No. 5, 694--709 (1995)

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##### References:

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