Exact propagation without analytical solutions

TL;DR

This paper introduces a simulation algorithm that efficiently propagates molecule pairs over large time steps with high accuracy, avoiding complex analytical solutions and enabling faster stochastic simulations across multiple dimensions.

Contribution

The proposed method provides a fast, accurate, and flexible simulation algorithm that does not rely on simplifying assumptions and can be applied in various dimensions.

Findings

Significantly faster than traditional schemes based on full solutions

Maintains accuracy comparable to exact analytical solutions

Applicable in 1D, 2D, and 3D simulations

Abstract

We present a simulation algorithm that accurately propagates a molecule pair using large time steps without the need to invoke the full exact analytical solutions of the Smoluchowski diffusion equation. Because the proposed method only uses uniform and Gaussian random numbers, it allows for position updates that are two to three orders of magnitude faster than those of a corresponding scheme based on full solutions, while mantaining the same degree of accuracy. Neither simplifying nor ad hoc assumptions that are foreign to the underlying Smoluchowski theory are employed, instead, the algorithm faithfully incorporates the individual elements of the theoretical model. The method is flexible and applicable in 1, 2 and 3 dimensions, suggesting that it may find broad usage in various stochastic simulation algorithms. We demonstrate the algorithm for the case of a non-reactive, irreversible and reversible reacting molecule pair.