A New Approach based on Langevin type Equation for Driven Granular Gas under Gravity

TL;DR

This paper introduces a Langevin equation-based model for the fluctuating motion of a driven granular gas's center of mass, validated by simulations, and explores the limits of fluctuation-dissipation relation validity.

Contribution

The paper develops an analytical Langevin equation framework for granular gases and compares it with simulations, highlighting the relation between inelasticity and fluctuation-dissipation violations.

Findings

Analytical expressions for macroscopic quantities derived from the Langevin model.

Good agreement between theory and simulations for nearly elastic collisions.

Deviations increase with inelasticity, indicating fluctuation-dissipation violation.

Abstract

We propose a novel approach based on a Langevin equation for fluctuating motion of the center of mass of granular media fluidized by energy injection from a bottom plate. In this framework, the analytical solution of the Langevin equation is used to derive analytic expressions for several macroscopic quantities and the power spectrum for the center of mass. In order to test our theory, we performed event-driven molecular dynamics simulations for one- and two-dimensional systems. Energy is injected from a vibrating bottom plate in the one-dimensional case and from a thermal wall at the bottom in the two-dimensional case. We found that the theoretical predictions are in good agreement with the results of those simulations under the assumption that the fluctuation-dissipation relation holds in the case of nearly elastic collisions between particles. However, as the inelasticity of the interparticle collisions increases, the power spectrum for the center of mass obtained by the simulations gradually deviates from the prediction of theoretical curve. Connection between this deviation and violation of the fluctuation-dissipation relation is discussed.