Brownian Simulations and Uni-Directional Flux in Diffusion

TL;DR

This paper investigates the concept of unidirectional flux in Brownian diffusion simulations, revealing that classical approaches can produce infinite flux artifacts and proposing a physically consistent simulation method.

Contribution

It introduces a new Brownian dynamics simulation approach that accurately maintains boundary concentrations without spurious boundary layers, addressing artifacts from classical diffusion approximations.

Findings

Unidirectional fluxes are infinite in classical diffusion models due to approximation artifacts.

The unidirectional flux needed for boundary maintenance is proportional to concentration and inversely proportional to the square root of time step.

A new simulation method aligns with physical Brownian particle behavior and avoids spurious boundary effects.

Abstract

Brownian dynamics simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages. The continuum baths are connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the unidirectional flux in the opposite direction. The classical diffusion equation defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener path integral, which can split the net flux into unidirectional fluxes. These unidirectional fluxes are infinite, though the net flux is finite and agrees with classical theory. We find that the infinite unidirectional flux is an artifact caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails on time scales shorter than the relaxation time $1/\gamma$ of the Langevin equation. We find the unidirectional flux (source strength) needed to maintain average boundary concentrations in a manner consistent with the physics of Brownian particles. This unidirectional flux is proportional to the concentration and inversely proportional to $\sqrt{\Delta t}$ to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers.