A Dean-Kawasaki equation for reaction diffusion systems driven by Poisson noise

TL;DR

The paper develops a Poisson-noise generalization of the Dean–Kawasaki equation to reaction–diffusion systems with unary switching, providing a stochastic PDE driven by spatio-temporal Poisson fields that is exact in law for the underlying particle system. By employing weak coupling and a McKean–Vlasov mean-field framework, it derives a deterministic mean-field limit as $N\to\infty$ while preserving Poisson fluctuations at finite $N$ and presents an alternative noise representation that yields a closed SPDE for the empirical field. The work also discusses the relation between weak and strong interactions, offering a Gaussian-limit reduction and two illustrative examples (homogeneous run-and-tumble and field-dependent non-reciprocal ballistic particles) to demonstrate the framework’s applicability and consistency with known results. It closes by outlining extensions to non-conservative and higher-order reactions and highlighting potential numerical and analytical directions for future research, with implications for fluctuating hydrodynamics in low-particle-number regimes.

Abstract

We derive a stochastic partial differential equation that describes the fluctuating behaviour of reaction-diffusion systems of N particles, undergoing Markovian, unary reactions. This generalises the work of Dean [J. Phys. A: Math. and Gen., 29 (24), L613, (1996)] through the inclusion of random Poisson fields. Our approach is based on weak interactions, which has the dual benefit that the resulting equations asymptotically converge (in the N to infinity limit) on a variation of a McKean- Vlasov diffusion, whilst still being related to the case of Dean-like strong interactions via a trivial rescaling. Various examples are presented, alongside a discussion of possible extensions to more complicated reaction schemes.