Random time-scale invariant diffusion and transport coefficients.

TLDR: This work investigates this type of ergodicity breaking within the continuous time random walk model and derives the distribution for the fluctuations of the random variable delta2[over], revealing a generalization of the Einstein relation.

Abstract: Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement $\overline{{\ensuremath{\delta}}^{2}}$ 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 $\overline{{\ensuremath{\delta}}^{2}}$ differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable $\overline{{\ensuremath{\delta}}^{2}}$. 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.