Time-averaged Einstein relation and fluctuating diffusivities for the Levy walk

The Levy walk model is a stochastic framework of enhanced diffusion with many applications in physics and biology. Here we investigate the time-averaged mean squared displacement (delta(2)) over bar often used to analyze single particle tracking experiments. The ballistic phase of the motion is nonergodic and we obtain analytical expressions for the fluctuations of (delta(2)) over bar. For enhanced subballistic diffusion we observe numerically apparent ergodicity breaking on long time scales. As observed by Akimoto [Phys. Rev. Lett. 108, 164101 (2012)], deviations of temporal averages delta(2) from the ensemble average < x(2)> depend on the initial preparation of the system, and here we quantify this discrepancy from normal diffusive behavior. Time-averaged response to a bias is considered and the resultant generalized Einstein relations are discussed. DOI: 10.1103/PhysRevE.87.030104