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

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.

Figures (5)

• Figure 1: Velocity autocorrelation functions (VACF) of three-dimensional Brownian particles. Solid line is the analytical hard-sphere result for small volume fractions $\phi$. Dashed line acts as a guide for the eye for the long-time tail corresponding to the exponent $- 5/2$. Circles corresponds to the results of particles that interact via the WCA potential. Squares depict the results of the hard-sphere simulations. Crosses are calculated by directly correlating the displacements per simulation time step of the WCA particles. All other data is calculated using the force autocorrelation function (FACF). See Appendix \ref{['app:WCA']} and Appendix \ref{['app:HS']} for the simulation parameters.
• Figure 2: Comparison between the force-noise correlations and the force autocorrelation functions for Brownian hard-sphere suspensions at different volume fractions. See Appendix \ref{['app:HS']} for the corresponding simulation parameters.
• Figure 3: Velocity autocorrelation function (VACF) for one-dimensional Brownian hard spheres at different densities $\phi$. Circles show the results obtained via the effective force autocorrelation function (FACF). Crosses correspond to data obtained with a standard (noise suppressing) algorithm Frenkel1987. Solid lines are the predicted long-time behaviors [Eq. \ref{['eq:VACFSFD']}]. Inset: Curves for different $\phi$ plotted with respect to the single-file time scale $\tau_{\textrm{SF}}$. See Appendix \ref{['app:HS']} for the corresponding simulation parameters.
• Figure 4: Velocity autocorrelation function (VACF) for driven Brownian hard spheres at different volume fractions $\phi$. The driving force is ${\textbf{F}_0 = (5,0,0)\, k_B T / \sigma}$ and the lines are analytic predictions. VACF results are calculated by evaluating the effective force autocorrelation function of the particle interactions and correcting the results with the analytic result for the cross-correlations (symbols) and the standard Frenkel algorithm Frenkel1987 (crosses). The dashed line corresponds to the asymptotic value after correcting the NC expression with the finite cross-correlation term. See Appendix \ref{['app:HS']} for the corresponding simulation parameters.
• Figure 5: Velocity autocorrelation function (VACF) of an active Brownian particle (ABP) in a two-dimensional harmonic potential with spring constant $k = 1 k_B T / \sigma^2$ for different effective self-propulsion forces $F_{\textrm{eff}}$. Symbols are calculated with the adjusted NC algorithm by evaluating the FACF during the simulations and analytically correcting the results afterwards. Crosses correspond to data obtained with a standard (noise suppressing) algorithm Frenkel1987. Lines are the theoretical predictions of Ref. Caraglio2022. See Appendix \ref{['app:ABPSIM']} for the corresponding simulation parameters.