A Derivation of the effective Fokker-Planck equation

To study statistical properties of the system one needs to find the probability density function P = P(p,x,t) of the system states distribution in the phase state {x,p}.

By definition, the probability density function is given by averaging over noises of the density function ρ(x,p,t) of the microscopic states distribution in the phase space:

To construct an equation for the macroscopic density function P = P(x,p,t), we exploit the conventional device to proceed from the continuity equation for the microscopic one ρ = ρ(x,p,t):

Inserting the time derivative of the momentum p˙=mx¨ from eq. (1) into eq. (27), we obtain

where the operators Lˆ and Nˆμ are defined as follows:

Within the interaction representation, the microstate density function reads as

to reduce eq. (28) to the form

where

A standard and effective device to solve such a type of stochastic equation is the well–known cumulant expansion method [22]. Neglecting terms of the order O(Rˆμ3), we arrive at the kinetic equation of the form

Within the original representation, the equation for the probability density (26) reads

If the physical time is much more than a correlation scale (t≫τμ), we can replace the upper limit of the integration by t=∞. Then, expanding exponents, we derive to the perturbation expansion

where collision operator

is determined through the commutators

and moments of correlation function

To perform the following calculations, we shall restrict ourselves to considering overdamped systems where the variation scales ts, ℓ, xs, vs, γs, fs, and gs of the time t, the coordinate r, the quantity x, the velocity v≡p/m, the damping coefficient γ(x), the force f(x), and the noise amplitudes gμ(x), respectively, obey the following conditions:

These conditions means a hierarchy of the damping and the deterministic/stochastic forces are characterized by relations

As a result, the dimensionless system of equations (1) takes the form

Respectively, the Fokker–Planck eq. (34) reads

where the operator

has the components

To obtain the usual probability function P(x,t), we consider velocity moments of the initial distribution function P(x,v,t) in the standard form [26]:

We multiply the Fokker–Planck eq. (41) by the factor vn and integrate over velocities. At n = 0, we obtain the equation for the distribution function P≡P0(x,t):

The expression for the first moment P1 (n = 1 and only terms of the first order in ε are kept):

The second moment P2 can be obtained if one puts n = 2 and takes into account only zeroth terms of smallness over the parameter ϵ≪1:

Equations (45)–(47) allow to obtain the Fokker–Planck equation in the Kramers–Moyal form eq. (4).