Einstein relation and linear response in one-dimensional Mott variable-range hopping
| Autor: | Michele Salvi, Alessandra Faggionato, Nina Gantert |
|---|---|
| Přispěvatelé: | Dipartimento di Matematica Guido Castelnuovo, Università degli Studi di Roma 'La Sapienza' = Sapienza University [Rome], Institut für Mathematische Statistik, Universitaet Muenster, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Università degli Studi di Roma 'La Sapienza' [Rome], Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin] (WIAS), Forschungsverbund Berlin e.V. (FVB) (FVB), Lehrstuhl fur Warscheinlichkeitstheorie, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
Steady states [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] 60K37 60J25 60G55 82D30 FOS: Physical sciences Mott variable-range hopping random walk in random environment random conductance model environment seen from the particle steady states linear response Einstein relation 01 natural sciences Variable-range hopping Environment seen from the particle 010104 statistics & probability 60J25 [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Random conductance model FOS: Mathematics [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech] 0101 mathematics Linear response Condensed Matter - Statistical Mechanics Mathematical Physics ComputingMilieux_MISCELLANEOUS Mathematics Random walk in random environment Statistical Mechanics (cond-mat.stat-mech) 82D30 010102 general mathematics Probability (math.PR) Mathematical Physics (math-ph) ddc [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] 60K37 60G55 Statistics Probability and Uncertainty Humanities Mathematics - Probability |
| Zdroj: | Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2019, 55 (3), pp.1477-1508. ⟨10.1214/18-AIHP925⟩ Ann. Inst. H. Poincaré Probab. Statist. 55, no. 3 (2019), 1477-1508 Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2019, 55 (3), pp.1477-1508. ⟨10.1214/18-AIHP925⟩ |
| ISSN: | 0246-0203 1778-7017 |
| DOI: | 10.1214/18-AIHP925⟩ |
| Popis: | 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 $L^p$-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. 35 pages, 1 figure |
| Databáze: | OpenAIRE |
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