Random Time-Scale Invariant Diffusion and Transport Coefficients

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.