Monte Carlo simulation with fixed steplength for diffusion processes in nonhomogeneous media

TL;DR

This paper discusses a Monte Carlo simulation method with fixed step lengths for modeling diffusion in nonhomogeneous media, demonstrating its accuracy over traditional Gaussian step methods through a calcium ion diffusion example.

Contribution

It introduces a fixed step length Monte Carlo approach for nonhomogeneous diffusion, improving accuracy over Gaussian methods in such media.

Findings

Fixed step length method produces correct diffusion results in nonhomogeneous media.

Gaussian step length method is erroneous in nonhomogeneous diffusion.

Application to calcium ion diffusion validates the fixed step approach.

Abstract

Monte Carlo simulation is one of the most important tools in the study of diffusion processes. For constant diffusion coefficients, an appropriate Gaussian distribution of particle's steplengths can generate exact results, when compared with integration of the diffusion equation. It is important to notice that the same method is completely erroneous when applied to non-homogeneous diffusion coefficients. A simple alternative, jumping at fixed steplegths with appropriate transition probabilities, produces correct results. Here, a model for diffusion of calcium ions in the neuromuscular junction of the crayfish is used as a test to compare Monte Carlo simulation with fixed and Gaussian steplegth.