Nonlinear Fokker-Planck-Kolmogorov Equation in the Semiclassical Coherent Trajectory Approximation

Abstract

A semiclassical asymptotic of the Cauchy problem is constructed for the two-dimensional Fokker-Planck equation with nonlocal nonlinearity in the class of trajectory concentrated functions based on the complex WKB-Maslov method. The system of Einstein-Ehrenfest equations describing the dynamics of average values of the coordinate operator and centered moments is derived. The results obtained are illustrated by a number of examples.

Stochastic models of heat and nuclear particle transfer based on generalized equation of Fokker-Planck-Kolmogorov

“…for which the fol low ing equa tions are sat is fied Since PB and its de riv a tives are zero at in fin ity, we ob tain that the right hand side of the last equa tion is zero ¶ ¶ ¶ ¶ = 0 (13) and get a prob lem for a par a bolic equa tion with uni tary unique ness con di tions. From the unique ness of the so lution for such a prob lem, we con clude that P t x ( , , )…”

Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity

“…The group properties of the semiclassical approximation were considered in quantum mechanics and in some models of the quantum field theory [21]. The semiclassical method for solving the Cauchy problem in the class of TCF's has been developed for the Hartree type equation [17,18,19,20] and for the one-dimensional FPKE [11,22,23]. For the Hatree type equation, symmetry operators have been found in the TCF class.…”