When velocity autocorrelations mirror force autocorrelations: Exact noise-cancellation in interacting Brownian systems

Resolving the mean-squared displacement (MSD) and velocity autocorrelation function (VACF) of interacting Brownian particles remains a central challenge in simulations of soft-matter systems, especially at low densities where particle-particle interactions are sparse and signals are dominated by thermal noise. A recently proposed noise-cancellation (NC) algorithm [Mandal ] addressed this by decomposing particle trajectories into two components: free Brownian motion and interaction-induced displacements. The NC approximation enhances signal clarity by neglecting cross-correlations between the total particle displacements and the extracted interaction-induced contributions of the trajectories—an assumption that has so far lacked rigorous theoretical justification. In this work, we establish an exact theoretical relation between the VACF, the force autocorrelation function (FACF) characterizing the interaction-induced contributions, and these cross-correlations, which is valid for Brownian systems. We show that in thermal equilibrium, the cross-correlations vanish for Brownian systems because the VACF is strictly proportional to the negative FACF, which establishes the NC algorithm as an exact method. In contrast, for Brownian nonequilibrium systems, the cross-correlations remain finite, providing a direct fingerprint of nonequilibrium physics in such systems and a criterion to distinguish equilibrium from nonequilibrium states. Here, suitable corrections must be applied for the NC method to remain accurate. Our results expand the scope of the NC algorithm to a broad range of soft-matter systems in and out of equilibrium, where it has the potential to strongly enhance the resolution of VACF data obtained through simulations in future studies.