Einstein relation and linear response in one-dimensional Mott variable-range hopping

We consider one-dimensional Mott variable-range hopping. This random walk is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization at low carrier density. We introduce a bias and prove the linear response as well as the Einstein relation, under an assumption on the exponential moments of the distances between neighboring points. In a previous paper (Ann. Inst. Henri Poincare Probab. Stat. 54 (2018) 1165-1203) we gave conditions on ballisticity, and proved that in the ballistic case the environment viewed from the particle approaches, for almost any initial environment, a given steady state which is absolutely continuous with respect to the original law of the environment. Here, we show that this bias-dependent steady state has a derivative at zero in terms of the bias (linear response), and use this result to get the Einstein relation. Our approach is new: instead of using e.g. perturbation theory or regeneration times, we show that the Radon-Nikodym derivative of the bias-dependent steady state with respect to the equilibrium state in the unbiased case satisfies an L-P -bound, p > 2, uniformly for small bias. This L-P-bound yields, by a general argument not involving our specific model, the statement about the linear response.