Einstein model of the movement of small particles in a stationary liquid revisited: Finite Propagation Speed

TL;DR

This paper revisits Einstein's particle diffusion model, addressing its paradox of infinite propagation speed by introducing concentration-dependent diffusion, resulting in a finite propagation speed under specific conditions.

Contribution

It modifies Einstein's model by making the diffusion matrix dependent on concentration, proving finite propagation speed with nonlinear PDE techniques.

Findings

Finite propagation speed achieved with concentration-dependent diffusion.

Demonstrates the model's consistency with thermodynamics.

Uses nonlinear PDE methods to establish results.

Abstract

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix dependent of the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentration. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality and the Ladyzhenskaya-Uraltseva iteration lemma.