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The modeling and simulation of stochastic reaction-diffusion processes is atopic of steady interest that is approached with a wide range of methods.\rev{At the level of particle-resolved descriptions, where chemical reactionsare coupled to the spatial diffusion of individual particles, there existcomprehensive numerical simulation schemes, while the correspondingmathematical formalization is relatively underdeveloped. The aim of this paperis to provide a framework to systematically formulate the probabilisticevolution equation, termed chemical diffusion master equation (CDME), thatgoverns particle-based stochastic reaction-diffusion processes. To account forthe non-conserved and unbounded particle number of this type of open systems,we employ a classical analogue of the quantum mechanical Fock space thatcontains the symmetrized probability densities of the many-particleconfigurations in space. Following field-theoretical ideas of secondquantization, we introduce creation and annihilation operators that act onsingle-particle densities and provide natural representations of symmetrizedprobability densities as well as of reaction and diffusion operators. Theseoperators allow us to consistently and systematically formulate the CDME forarbitrary reaction schemes. The resulting form of the CDME further serves asthe foundation to derive more coarse-grained descriptions of reaction-diffusiondynamics. In this regard, we show that a discretization of the evolutionequation by projection onto a Fock subspace generated by a finite set ofsingle-particle densities leads to a generalized form of the well-knownreaction-diffusion master equation, which supports non-local reactions betweengrid cells and which converges properly in the continuum limit.

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