Small and Finite Inertia in Stochastic Systems: Moment and Cumulant Formalisms

TL;DR

This paper compares moment and cumulant formalisms for eliminating fast velocity variables in stochastic systems, deriving simplified equations that are effective for small inertia and applicable to active Brownian particles.

Contribution

It demonstrates that cumulant formalism efficiently captures small inertia effects with fewer elements than moments, avoiding complex eigenfunction expansions.

Findings

Cumulant formalism requires only three elements for small inertia.

Moment formalism needs five elements for the same case.

Both approaches are comparable in efficiency to Hermite function expansion.

Abstract

We analyze two approaches to elimination of a fast variable (velocity) in stochastic systems: moment and cumulant formalisms. With these approaches, we obtain the corresponding Smoluchovski-type equations, which contain only the coordinate/phase variable. The adiabatic elimination of velocity in terms of cumulants and moments requires the first three elements. However, for the case of small inertia, the corrected Smoluchowski equation in terms of moments requires five elements, while in terms of cumulants the same first three elements are sufficient. Compared to the method based on the expansion of the velocity distribution in Hermite functions, the considered approaches have comparable efficiency, but do not require individual mathematical preparation for the case of active Brownian particles, where one has to construct a new basis of eigenfunctions instead of the Hermite ones.