2017/08/31 by Alessandra Faggionato, Faggionato, A., Nina Gantert +3
Mathematics · Physics and Astronomy · #60G55 #60J25 #60K37 #82D30 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1708.09610
openalex publication_date 2017/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider one-dimensional Mott variable-range hopping with 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 \citeFGS 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 Lp-bound, p>2, uniformly for small bias. This Lp-bound yields, by a general argument not involving our specific model, the statement about the linear response.