A probabilistic framework for particle-based reaction-diffusion dynamics using classical Fock space representations

TL;DR

This paper introduces a probabilistic framework for particle-based reaction-diffusion systems using classical Fock space, enabling systematic formulation of evolution equations and connections to existing master equations.

Contribution

It develops a Fock space-based formalism for stochastic reaction-diffusion processes, providing a systematic way to derive evolution equations and generalized master equations.

Findings

Formulation of the chemical diffusion master equation (CDME) using Fock space.

Derivation of a generalized reaction-diffusion master equation supporting non-local reactions.

Establishment of a continuum limit where the generalized master equation converges.

Abstract

The modeling and simulation of stochastic reaction-diffusion processes is a topic of steady interest that is approached with a wide range of methods. \rev{At the level of particle-resolved descriptions, where chemical reactions are coupled to the spatial diffusion of individual particles, there exist comprehensive numerical simulation schemes, while the corresponding mathematical formalization is relatively underdeveloped. The aim of this paper is to provide a framework to systematically formulate the probabilistic evolution equation, termed chemical diffusion master equation (CDME), that governs particle-based stochastic reaction-diffusion processes. To account for the non-conserved and unbounded particle number of this type of open systems, we employ a classical analogue of the quantum mechanical Fock space that contains the symmetrized probability densities of the many-particle configurations in space. Following field-theoretical ideas of second quantization, we introduce creation and annihilation operators that act on single-particle densities and provide natural representations of symmetrized probability densities as well as of reaction and diffusion operators. These operators allow us to consistently and systematically formulate the CDME for arbitrary reaction schemes. The resulting form of the CDME further serves as the foundation to derive more coarse-grained descriptions of reaction-diffusion dynamics. In this regard, we show that a discretization of the evolution equation by projection onto a Fock subspace generated by a finite set of single-particle densities leads to a generalized form of the well-known reaction-diffusion master equation, which supports non-local reactions between grid cells and which converges properly in the continuum limit.