An Iterative Energy Estimate for Degenerate Einstein model of Brownian motion

TL;DR

This paper develops an iterative energy estimate for a degenerate Einstein model of Brownian motion, analyzing how particle jump intervals depend on particle density and ensuring finite propagation speed in the medium.

Contribution

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

Findings

Derived conditions for jump interval and frequency based on particle density.

Proved finite speed of propagation under these conditions.

Analyzed the degeneracy effects on particle spread.

Abstract

We consider the degenerate Einstein's Brownian motion model for the case when the time interval ($\tau$) of particle Jumps before collision (free jumps) reciprocal to the number of particles per unit volume $u(x,t) > 0$ at the point of observation $x$ at time $t$. The parameter $0 < \tau \leq C < \infty$, controls characteristic of the fluid "almost decreases" to $ 0 $ when $u \rightarrow \infty$. This degeneration leads to the localisation of the spread of particle propagation in the media. In our report we will present a structural condition of the time interval of free jumps - $\tau$ and the frequency of these free jumps $\phi$ as functions of $u$ which guarantees the finite speed of propagation of $u$.