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 δ2̄ 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 δ2̄ differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable δ2̄. 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.