Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales

TL;DR

This paper develops a multiscale framework connecting probabilistic particle-level models of open reaction-diffusion systems with macroscopic concentration descriptions, enabling consistent theory and simulation across scales.

Contribution

It introduces a unified mathematical approach to model open reaction-diffusion systems at both microscopic and macroscopic levels, including interactions with reservoirs.

Findings

Established probabilistic descriptions for particle-level reaction-diffusion.

Derived macroscopic concentration equations from particle models.

Developed numerical schemes validated with illustrative examples.

Abstract

Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual particle/agent level, and in realistic applications, they often display interaction with their environment through energy or material exchange with a reservoir. This requires intricate mathematical considerations, especially in the case of material exchange since the varying number of particles/agents results in ``on-the-fly'' modification of the system dimension. In this work, we first overview the probabilistic description of reaction-diffusion processes at the particle level, which readily handles varying number of particles. We then extend this model to consistently incorporate interactions with macroscopic material reservoirs. Based on the resulting expressions, we bridge the probabilistic description with macroscopic concentration-based descriptions for linear and nonlinear reaction-diffusion systems, as well as for an archetypal open reaction-diffusion system. Using these mathematical bridges across scales, we finally develop numerical schemes for open reaction-diffusion systems, which we implement in two illustrative examples. This work establishes a methodological workflow to bridge particle-based probabilistic descriptions with macroscopic concentration-based descriptions of reaction-diffusion in open settings, laying the foundations for a multiscale theoretical framework upon which to construct theory and simulation schemes that are consistent across scales.