8788 2023 eng 1 64 article 0 2023-01-20 -- -- The chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction--diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation for reaction--diffusion processes. The CDME consists of an infinite ordered family of Fokker--Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators --linear operators representing chemical reactions. These operators change the number of particles in the system, and thus transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally we discuss applications to multiscale simulations of biochemical systems among other future prospects. Journal of Mathematical Physics 10.1063/5.0129620 2210.02268 2023-01-02 publish yes Mauricio del Razo Stefanie Winkelmann Stefanie Winkelmann Rupert Klein Felix Höfling Numerical Mathematics Computational Systems Biology SFB1114-C3 Winkelmann, Stefanie Höfling, Felix Modeling and Simulation of Complex Processes MathPlusAA1-5 del Razo Sarmina, Mauricio DFG-OpenMultiscaleBiochem