Brownian motion: non-equilibrium states from equilibrium trajectories -- recovering hydrodynamic regimes from prepared displacement measurements

TL;DR

This paper demonstrates how equilibrium trajectories of Brownian particles can reveal non-equilibrium fluid-particle interactions and hydrodynamic regimes by analyzing displacement moments, confirming and extending known scaling laws.

Contribution

It introduces a method to extract non-equilibrium fluid dynamics from single Brownian trajectories using moments analysis, connecting microscopic behavior to hydrodynamics.

Findings

Confirmed the $t^{5/2}$-scaling law for short-time displacement due to fluid inertia.

Indicated a possible $t^4$-scaling at very short times considering correlated stochastic forces.

Isolated thermal-hydrodynamic acceleration from other factors in particle motion.

Abstract

Owing to the Chapman-Kolmogorov equation for Markovian dynamics,any equilibrium trajectory of a Brownian particle in a solvent fluid can be viewed as the superposition of an uncountable number of non-equilibrium states. This property permits the unraveling of fine details of fluid-particle interactions at microscales defined by its non-equilibrium properties from the analysis of a single Brownian trajectory and to connect them to the hydrodynamics of the solvent fluid, simply considering the lower-order (second) moments of particle position in trapped conditions. In this way, the acceleration due to thermal-hydrodynamic fluctuational forces is isolated from the other factors and the short-time displacement statistics is completely determined by the correlation properties of the fluctuational thermal-hydrodynamic force. This approach not only confirms the $t^{5/2}$-law obtained by Boynewicz et al. (2026), related to fluid inertial effects, but indicates that this scaling may be superseded by a $t^4$-scaling at very short times once the correlated nature of the stochastic forcings is taken into account. The latter result is related to the regularity properties of particle velocity realizations.