The BBGKY Hierarchy and Fokker-Planck Equation for Many-Body Dissipative Randomly Driven Systems

TL;DR

This paper develops a formalism extending the BBGKY hierarchy to dissipative many-body systems driven by non-Gaussian stochastic fields, deriving a Fokker-Planck equation under weak interaction and field assumptions.

Contribution

It generalizes the BBGKY hierarchy for dissipative systems with non-Gaussian stochastic driving and derives a corresponding Fokker-Planck equation.

Findings

Derived a generalized BBGKY hierarchy for dissipative stochastic systems.

Obtained a Fokker-Planck equation for weakly interacting particles in non-Gaussian fields.

Discussed special cases with Gaussian statistics and system homogeneity.

Abstract

By generalizing Bogolyubov's reduced description method, we suggest a formalism to derive kinetic equations for many-body dissipative systems in external stochastic field. As a starting point, we use a stochastic Liouville equation obtained from Hamilton's equations taking dissipation and stochastic perturbations into account. The Liouville equation is then averaged over realizations of the stochastic field by an extension of the Furutsu-Novikov formula to the case of a non-Gaussian field. As the result, a generalization of the classical Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is derived. In order to get a kinetic equation for the one-particle distribution function, we use a regular cut off procedure of the BBGKY hierarchy by assuming weak interaction between the particles and weak intensity of the field. Within this approximation we get the corresponding Fokker-Planck equation for the system in a non-Gaussian stochastic field. Two particular cases by assuming either Gaussian statistics of external perturbation or homogeneity of the system are discussed.