Exact Large-Scale Correlations in Diffusive Systems with General Interactions

Characterizing the statistical properties of classical interacting particle systems is a long-standing question. For Brownian particles, the microscopic density obeys a stochastic evolution equation, known as the Dean-Kawasaki equation. This equation remains mostly formal and linearization (or higher-order expansions) is required to obtain explicit expressions for physical observables, with a range of validity not easily defined. Here, by combining macroscopic fluctuation theory with equilibrium statistical mechanics, we provide a systematic alternative to the Dean-Kawasaki framework to characterize large-scale correlations. This approach enables us to obtain explicit and exact results for the large-scale behavior of dynamical observables such as tracer cumulants and bath-tracer correlations in one dimension, both in and out of equilibrium. In particular, we reveal a generic nonmonotonic spatial structure in the response of the bath following a temperature quench. Our approach applies to a broad class of interaction potentials and extends naturally to higher dimensions.