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

Faggionato, Alessandra; Gantert, Nina; Salvi, Michele (2017), Einstein relation and linear response in one-dimensional Mott variable-range hopping. https://basepub.dauphine.fr/handle/123456789/17526

Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-01646209
Date
2017
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Pages
35
Metadata
Show full item record
Author(s)
Faggionato, Alessandra

Gantert, Nina
Lehrstuhl fur Warscheinlichkeitstheorie
Salvi, Michele
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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 \cite{FGS} 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.
Subjects / Keywords
Mott variable-range hopping; random walk in random environment; randomconductance model; environment seen from the particle; steady states; linear response; Einstein relation

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