TY - JOUR

T1 - Einstein relation and linear response in one-dimensional Mott variable-range hopping

AU - Faggionato, Alessandra

AU - Gantert, Nina

AU - Salvi, Michele

N1 - Publisher Copyright:
© 2019 Association des Publications de l'Institut Henri Poincaré. © 2019 Association des Publications de l'Institut Henri Poincaré.

PY - 2019

Y1 - 2019

N2 - We consider one-dimensional Mott variable-range hopping. This random walk is an effective model for the phononinduced 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 (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 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.

AB - We consider one-dimensional Mott variable-range hopping. This random walk is an effective model for the phononinduced 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 (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 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.

KW - Einstein relation

KW - Environment seen from the particle

KW - Linear response

KW - Mott variable-range hopping

KW - Random conductance model

KW - Random walk in random environment

KW - Steady states

UR - http://www.scopus.com/inward/record.url?scp=85074204319&partnerID=8YFLogxK

U2 - 10.1214/18-AIHP925

DO - 10.1214/18-AIHP925

M3 - Article

AN - SCOPUS:85074204319

SN - 0246-0203

VL - 55

SP - 1477

EP - 1508

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 3

ER -