Field theories and quantum methods for stochastic reaction-diffusion systems

TL;DR

This work presents a unified, basis-independent field-theoretic framework for stochastic reaction-diffusion systems with varying particle numbers and spatial structure. It starts from well-established second-quantized formulations (CME/RDME) and develops unifying representations that encompass Doi, Doi–Peliti path integrals, and Galerkin discretizations, connecting microscopic CRNs to macroscopic PDEs. The approach provides tools for generating functions, large-deviation statistics, and perturbative RG analysis, enabling rigorous links between particle-level parameters and emergent coarse-grained models. By outlining a consistent discretization and path-integral methodology, the paper offers a versatile toolkit for multi-scale modeling across physics, chemistry, ecology, epidemiology, and socio-economic systems, with potential impacts on numerical methods and stochastic PDEs.

Abstract

Complex systems are composed of many particles or agents that move and interact with one another. The underlying mathematical framework to model many of these systems must incorporate the spatial transport of particles and their interactions, as well as changes to their copy numbers, all of which can be formulated in terms of stochastic reaction-diffusion processes. The probabilistic representation of these processes is complex because of combinatorial aspects arising due to nonlinear interactions and varying particle numbers. This review presents the main field theory representations of stochastic reaction-diffusion systems, which handle these issues `under-the-hood'. First, we focus on bringing techniques familiar to theoretical physicists -- such as second quantization, Fock space, and path integrals -- back into the classical domain of reaction-diffusion systems. We demonstrate how various field theory representations can all be unified under a single basis-independent representation. We then extend existing quantum-based methods and notation to work directly on the level of the unifying representation, and we illustrate how they can be used to consistently obtain previous known results, such as numerical discretizations and relations between model parameters at multiple scales. Throughout the work, we contextualize how these representations mirror well-known models of chemical physics depending on their spatial resolution, as well as the corresponding macroscopic limits. The framework presented here may find applications in a diverse set of scientific fields, including physical chemistry, theoretical ecology, epidemiology, game theory and socio-economical models of complex systems. The presentation is done in a self-contained educational and unifying manner such that it can be followed by researchers across several fields.

Figures (5)

• Figure 1: Diagram showing the relations between the main theories and models discussed in this review, along with the section(s) where they are discussed. a. All the field theory representations emerge as special cases of the basis-independent representation of \ref{['sec:stochMecRD']}. These are categorized according to their spatial resolution determined by the basis of the space/subspace used. b. The stochastic chemical physics models corresponding to the field theory representations are shown depending on their spatial resolution. As the number of particles becomes larger, deterministic macroscopic models emerge. The basis-independent formulation can be used to formulate the CDME, and thus other chemical physics models like the discretized reaction-diffusion partial differential equation (RD-PDE) for arbitrary choices of basis functions (\ref{['sec:pelitipathint']}).
• Figure 2: Diagram illustrating how to construct the generator for a system with a general one species reaction with rate $\lambda$ in the well-mixed case.
• Figure 3: Structure of the phase space of the chemical diffusion master equation. a. Phase space for a system with one chemical species $A$, consisting of discrete sets of continuous diffusion domains $\mathbb{X}$ (e.g. a bounded region within $\mathbb{R}^3$), one for each particle in the current configuration. b. Analogous phase space but for a system with two chemical species $A$ and $B$. The arrows denote possible transitions between first neighbors. Reactions can in general transition between any two subsets in the phase space (not depicted). Figure adapted from del2024open.
• Figure 4: Diagram illustrating how to construct the generator for a system with a general one species reaction with reaction rate function $\lambda (\mathbf{y};\mathbf{x})$ for the basis-independent case. This case takes into account the spatial dynamics in continuous space, so it generalizes the diagram presented for the well-mixed case, (\ref{['fig:genwellmixdiag']}).
• Figure 5: Diagram illustrating the construction of the generator for a system with a general one species reaction with reaction rate function $\lambda (\mathbf{y};\mathbf{x})$ in the Doi formulation. This case takes into account the spatial dynamics in continuous space, and it is a special case of the basis-independent case presented in \ref{['fig:genbasiindependiag']}.