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 (delta(2)) over bar 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 (delta(2)) over bar differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable (delta(2)) over bar. 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.