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Time averaged Einstein relation and fluctuating diffusivities for the Lévy walk

The Lévy 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 $\bar{δ^2}$ often used to analyze single particle tracking experiments. The ballistic phase of the motion is non-ergodic and we obtain analytical expressions for the fluctuations of $\bar{δ^2}$. For enhanced sub-ballistic diffusion we observe numerically apparent ergodicity breaking on long time scales. As observed by Akimoto \textit{Phys. Rev. Lett.} \textbf{108}, 164101 (2012) deviations of temporal averages $\bar{δ^2}$ 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.