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Einstein relation and linear response in one-dimensional Mott variable-range hopping. (English. French summary) Zbl 1467.60087

Summary: 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 Poincaré, Probab. Stat. 54, No. 3, 1165–1203 (2018; Zbl 1401.60182)] 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.

MSC:

60K37 Processes in random environments
60J25 Continuous-time Markov processes on general state spaces
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Citations:

Zbl 1401.60182
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References:

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