Non-Gaussian density fluctuations in the Dean-Kawasaki equation

Computing analytically the $n$-point density correlations in systems of interacting particles is a long-standing problem of statistical physics, with a broad range of applications, from the interpretation of scattering experiments in simple liquids, to the understanding of their collective dynamics. For Brownian particles, i.e. with overdamped Langevin dynamics, the microscopic density obeys a stochastic evolution equation, known as the Dean-Kawasaki equation. In spite of the importance of this equation, its complexity makes it very difficult to analyze the statistics of the microscopic density beyond simple Gaussian approximations. In this work, resorting to a path-integral description of the stochastic dynamics and relying on a saddle-point analysis in the limit of high density and weak interactions between the particles, we go beyond the usual linearization of the Dean-Kawasaki equation, and we compute exactly the three- and four-point density correlation functions. This result opens the way to using the Dean-Kawasaki equation beyond the simple Gaussian treatments, and could find applications to understand many fluctuation-related effects in soft and active matter systems.