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

TL;DR

This paper proves that the noise-cancellation algorithm for analyzing Brownian particle dynamics is exact at equilibrium due to the proportionality between velocity autocorrelation and force autocorrelation, and it identifies nonequilibrium signatures.

Contribution

The work provides a rigorous theoretical foundation for the noise-cancellation method, showing its exactness at equilibrium and its potential to detect nonequilibrium behavior.

Findings

VACF is proportional to negative FACF at equilibrium

Cross-correlations vanish in equilibrium, validating the NC algorithm

Finite cross-correlations indicate nonequilibrium states

Abstract

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 et al. Phys. Rev. Lett. 123, 168001 (2019)] addresses 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.