A route to the hydrodynamic limit of a reaction-diffusion master equation using gradient structures

Abstract

The reaction-diffusion master equation (RDME) is a lattice-based stochastic model for spatially resolved cellular processes. It is often interpreted as an approximation to spatially continuous reaction-diffusion models, which, in the limit of an infinitely large population, may be described by means of reaction-diffusion partial differential equations (RDPDEs). Analyzing and understanding the relation between different mathematical models for reaction-diffusion dynamics is a research topic of steady interest. In this work, we explore a route to the hydrodynamic limit of the RDME which uses gradient structures. Specifically, we elaborate on a method introduced in [J. Maas, A. Mielke.: Modeling of chemical reactions systems with detailed balance using gradient structures. J. Stat. Phys. (181), 2257-2303] in the context of well-mixed reaction networks by showing that, once it is complemented with an appropriate limit procedure, it can be applied to spatially extended systems with diffusion. Under the assumption of detailed balance, we write down a gradient structure for the RDME and use the method to produce a gradient structure for its hydrodynamic limit, namely, for the corresponding RDPDE.

Generalized gradient structures for measure-valued population dynamics and their large-population limit

“…Meanwhile, powerful variational tools have been developed in the last decade for studying meanfield interacting jump processes and their limits under the assumption of detailed balance. To highlight only a few: [EFLS16] studied mean-field limits for measure-dependent jump processes; [Erb16] proved the convergence of the spatially-homogeneous Kac-process to the Boltzmann equation; [Sch19] investigated the macroscopic limit of Becker-Döring models; [KJZ19] showed hydrodynamic limits for zero-range and exclusion processes; [MM20] discussed convergence and higher-order approximations for chemical reaction networks, an approach that was subsequently used in the setting of discretized reaction-diffusion equations in [MSW22].…”