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

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.

💡 Research Summary

The paper addresses a fundamental issue in Monte Carlo (MC) simulations of diffusion when the diffusion coefficient varies spatially, a situation common in many physical and biological systems. For homogeneous media, the standard practice is to draw particle displacements from a Gaussian distribution with zero mean and variance 2 D Δt, where D is the constant diffusion coefficient and Δt the time step. This approach reproduces the exact solution of the diffusion equation because it is mathematically equivalent to a Wiener process. However, the authors demonstrate that the same Gaussian‑step method becomes fundamentally flawed in non‑homogeneous media. When D = D(x) depends on position, the diffusion equation takes the form ∂c/∂t = ∇·