Space-time velocity correlation function for random walks

TL;DR

This paper introduces space-time velocity correlation functions for single-particle random walks, revealing long-lasting correlations and velocity echo effects, with implications for cold atom and nanoscale charge transport studies.

Contribution

It extends the concept of correlation functions to single-particle random walks, uncovering long-range temporal correlations and velocity echo phenomena.

Findings

Velocity echo effect identified in random walks.

Correlations extend much longer than traditional temporal correlations.

Relevance demonstrated for cold atom and nanoscale charge transport experiments.

Abstract

Space-time correlation functions constitute a useful instrument from the research toolkit of continuous-media and many-body physics. We adopt here this concept for single-particle random walks and demonstrate that the corresponding space-time velocity auto-correlation functions reveal correlations which extend in time much longer than estimated with the commonly employed temporal correlation functions. A generic feature of considered random-walk processes is an effect of velocity echo identified by the existence of time-dependent regions where most of the walkers are moving in the direction opposite to their initial motion. We discuss the relevance of the space-time velocity correlation functions for the experimental studies of cold atom dynamics in an optical potential and charge transport on micro- and nano-scales.