Multiplicative Langevin equation to reproduce long-time properties of nonequilibrium Brownian motion

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.

Effective Langevin equations leading to large deviation function of time-averaged velocity for a nonequilibrium Rayleigh piston

“…where the symbol denotes the anti-Itô product. This equation was numerically found in [48]. In this form, a fluctuationdissipation-like relation holds even out of equilibrium.…”

“…It will be shown later that T is the kinetic temperature of the piston to lowest order in . We now introduce the rescaled dimensionless variables used in [48]…”

“…As a necessity condition, it should precisely reproduce the large deviation function for the microscopic model. Such an effective model was discovered numerically in [48]. Fortunately, the microscopic model is so simple that we can calculate the large deviation function in a perturbative expansion.…”