# Multiplicative Langevin Equation to Reproduce Long-time Properties of Nonequilibrium Brownian Motion

## TL;DR

This paper introduces a multiplicative Langevin equation that accurately models long-time properties of nonequilibrium Brownian motion, capturing cumulants and skewness, and highlights its broad applicability beyond the Rayleigh piston model.

## Contribution

The paper proposes a novel multiplicative Langevin equation that reproduces higher-order cumulants and skewness of nonequilibrium Brownian motion, extending modeling capabilities.

## Key findings

- The third cumulant of displacement increases linearly with time.
- The Langevin equation reproduces cumulants, mean, variance, and skewness.
- The method applies broadly to various time sequences in different fields.

## Abstract

We statistically examine long time sequences of Brownian motion for a nonequilibrium version of the Rayleigh piston model and confirm that the third cumulant of a long-time displacement for the nonequilibrium Brownian motion linearly increases with the observation time interval. We identify a multiplicative Langevin equation that can reproduce the cumulants of the long-time displacement up to at least the third order, as well as its mean, variance and skewness. The identified Langevin equation involves a velocity-dependent friction coefficient that breaks the time-reversibility and may act as a generator of the directionality. Our method to find the Langevin equation is not specific to the Rayleigh piston model but may be applied to a general time sequence in various fields.