Four approaches for description of stochastic systems with small and finite inertia

TL;DR

This paper compares four mathematical approaches for simplifying stochastic systems with small or finite inertia, evaluating their accuracy and applicability to passive and active Brownian particles.

Contribution

It introduces and assesses four different methods for eliminating fast variables in stochastic systems, including their accuracy and suitability for passive and active Brownian particles.

Findings

Moment and cumulant formalisms are effective for passive particles.

Hermite basis approaches perform well for active particles.

Accuracy varies with method and system type.

Abstract

We analyse four approaches to elimination of a fast variable, which are applicable to systems like passive Brownian particles: (i) moment formalism, (ii) corresponding cumulant formalism, (iii) Hermite function basis, (iv) formal `cumulants' for the Hermit function basis. The accuracy and its strong order are assessed. The applicability and performance of two first approaches are also demonstrated for active Brownian particles.