About Relationship Between Flux and Concentration Gradient of Particles at Description of Diffusion with the Usage of Random Walk Model

TL;DR

This paper explores the relationship between flux and concentration gradients in diffusion, using a random walk model to compare local and nonlocal diffusion equations and proposing a new modified non-local diffusion equation based on microparameters.

Contribution

It introduces a new modified non-local diffusion equation derived from random walk microparameters, bridging local and nonlocal diffusion models.

Findings

Fundamental solutions of diffusion and telegraph equations are very close over certain periods.

The probability density distribution from the random walk closely matches diffusion solutions.

A new non-local diffusion equation with microparameters is proposed.

Abstract

The fundamental solutions of diffusion equation for the local-equilibrium and nonlocal models are considered as the limiting cases of the solution of a problem related to consideration of the Brownian particles random walks. The differences between fundamental solutions were studied. It was shown that on the period of observation time and distances exceeding the time and space associated with one step of random walk the fundamental solutions of diffusion and telegraph equations are very close to each other. In particular, these fundamental solutions are very close to the values of the probability density distribution of diffusive particles that is obtained from the accurate solution of the random walk problem. The new modified non-local diffusion equation is suggested. It contains only microparameters of the random walk problem.