Time averaged Einstein relation and fluctuating diffusivities for the L\'evy walk

TL;DR

This paper analyzes the fluctuations and ergodic properties of the time-averaged mean squared displacement in Lévy walk models, revealing non-ergodic behavior and deriving generalized Einstein relations for biased motion.

Contribution

It provides analytical expressions for fluctuations of the time-averaged MSD in Lévy walks and explores ergodicity breaking and bias response, extending understanding of anomalous diffusion.

Findings

Ballistic phase exhibits non-ergodicity and fluctuation analysis.

Numerical evidence of apparent ergodicity breaking in sub-ballistic diffusion.

Generalized Einstein relations derived for biased Lévy walk motion.

Abstract

The L\'evy walk model is a stochastic framework of enhanced diffusion with many applications in physics and biology. Here we investigate the time averaged mean squared displacement $\bar{\delta^2}$ often used to analyze single particle tracking experiments. The ballistic phase of the motion is non-ergodic and we obtain analytical expressions for the fluctuations of $\bar{\delta^2}$. For enhanced sub-ballistic diffusion we observe numerically apparent ergodicity breaking on long time scales. As observed by Akimoto \textit{Phys. Rev. Lett.} \textbf{108}, 164101 (2012) deviations of temporal averages $\bar{\delta^2}$ from the ensemble average $< x^2 >$ depend on the initial preparation of the system, and here we quantify this discrepancy from normal diffusive behavior. Time averaged response to a bias is considered and the resultant generalized Einstein relations are discussed.