Random Time-Scale Invariant Diffusion and Transport Coefficients

TL;DR

This paper studies subdiffusive particle motion in living cells, revealing ergodicity breaking and fluctuations in time-averaged mean squared displacement, and generalizes the Einstein relation for such systems.

Contribution

It introduces a model for ergodicity breaking in subdiffusive systems and derives the distribution of displacement fluctuations and a generalized Einstein relation.

Findings

Distribution for fluctuations of time-averaged MSD derived

Ergodicity breaking confirmed in subdiffusive systems

Generalized Einstein relation established

Abstract

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement $\overline{\delta^2}$ of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that $\overline{\delta^2}$ differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable $\overline{\delta^2}$. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.