An Iterative Energy Estimate for Degenerate Einstein model of Brownian motion

TL;DR

This paper analyzes a degenerate Einstein Brownian motion model where particle collision intervals depend inversely on local particle density, leading to localization effects, and proposes conditions ensuring finite propagation speed of the particle distribution.

Contribution

It introduces a structural condition linking free jump frequency and time interval to particle density, guaranteeing finite speed of propagation in the degenerate model.

Findings

Established a structural condition for finite speed of propagation.

Demonstrated localization of particle distribution due to degeneracy.

Analyzed the impact of density-dependent parameters on particle dynamics.

Abstract

We consider the degenerate Einsteins Brownian motion model when the time interval of the moving particles before the collisions, is reciprocal to the number of particles per unit volume u(x,t), at the point of observation x at time t. The parameter 0 < tau < C, which controls the characteristics of the fluid, almost increases unboundedly, as u approaches 0. This degeneration leads to the localization of the particle distribution in the media. In the paper, we present a structural condition of the time interval and the frequency of these free jumps, as functions of u which guarantees the finite speed of propagation of u.