Time- and ensemble-averages in evolving systems: the case of Brownian particles in random potentials

Abstract

Anomalous diffusion is a ubiquitous phenomenon in complex systems. It is often quantified using time- and ensemble-averages to improve statistics, although time averages represent a non-local measure in time and hence can be difficult to interpret. We present a detailed analysis of the influence of time- and ensemble-averages on dynamical quantities by investigating Brownian particles in a rough potential energy landscape (PEL). Initially, the particle ensemble is randomly distributed, but the occupancy of energy values evolves towards the equilibrium distribution. This relaxation manifests itself in the time evolution of time- and ensemble-averaged dynamical measures. We use Monte Carlo simulations to study particle dynamics in a potential with a Gaussian distribution of energy values, where the long-time limit of the diffusion coefficient is known from theory. In our experiments, individual colloidal particles are exposed to a laser speckle pattern inducing a non-Gaussian roughness and are followed by optical microscopy. The relaxation depends on the kind and degree of roughness of the PEL. It can be followed and quantified by the time- and ensemble-averaged mean squared displacement. Moreover, the heterogeneity of the dynamics is characterized using single-trajectory analysis. The results of this work are relevant for the correct interpretation of single-particle tracking experiments in general.