A no-go theorem for ergodicity and an Einstein relation

We provide a simple no-go theorem for ergodicity and the generalized Einstein relation for anomalous diffusion processes. The theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements (MSD) is broken, and/or the generalized Einstein relation for time averaged diffusivity and mobility is invalid, which is in complete contrast to normal diffusion processes. We also give a general relation for the time averages of drift and MSD for \textit{ergodic} (in the MSD sense) anomalous diffusion processes, showing that the ratio of these quantities depends on the measurement time. The L\'evy walk model is used to exemplify the no-go theorem.