  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 (2020)] 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.
