Multinomial Diffusion Equation

TL;DR

This paper introduces a microscopic diffusion model that accurately captures fluctuations at small scales, demonstrating its equivalence to classical diffusion equations in the large particle limit and outperforming traditional models in simulations.

Contribution

The paper presents a new microscopic diffusion model that incorporates fluctuations and proves its equivalence to the stochastic diffusion equation as particle number grows large.

Findings

The microscopic model reproduces correct ensemble statistics.

Classical stochastic diffusion equation fails to capture fluctuations at small scales.

Numerical simulations confirm the model's accuracy in representing diffusion-induced fluctuations.

Abstract

We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles $N \to \infty$, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.