Synchronous vs. asynchronous dynamics of diffusion-controlled reactions

TL;DR

This paper introduces an analytical approach to study how synchronous and asynchronous random walks affect reaction times on lattices, revealing parity effects and optimal diffusion strategies for different lattice sizes.

Contribution

It develops a new analytical method based on the ruin problem to analyze the impact of synchronicity on reaction efficiency in lattice-based diffusion models.

Findings

Purely synchronous reactions are most efficient for odd lattice sizes.

A mix of synchronous and asynchronous events minimizes encounter time for even sizes.

Monte Carlo simulations confirm the parity effects across 1D, 2D, and 3D lattices.

Abstract

An analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice. At each time step, either both walkers diffuse simultaneously with probability $p$ (synchronous event) or one of them diffuses while the other remains immobile with complementary probability (asynchronous event). Reaction takes place through same site occupation or position exchange. We study the influence of the degree of synchronicity $p$ of the walkers and the lattice size $N$ on the global reaction's efficiency. For odd $N$, the purely synchronous case ($p=1$) is always the most effective one, while for even $N$, the encounter time is minimized by a combination of synchronous and asynchronous events. This new parity effect is fully confirmed by Monte Carlo simulations on 1d lattices as well as for 2d and 3d lattices. In contrast, the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of $p$. The relevance of the model for several research areas is briefly discussed.