[Paper Review] Einstein relation and linear response in one-dimensional Mott variable-range hopping

Alessandra Faggionato, Nina Gantert+1 more|arXiv (Cornell University)|Aug 31, 2017

Random Matrices and Applications34 references3 citations

TL;DR

This paper establishes the linear response and Einstein relation in one-dimensional Mott variable-range hopping with a bias, proving that the steady-state distribution of the environment viewed from the particle is differentiable at zero bias. Using an $L^p$-bound ($p>2$) on the Radon-Nikodym derivative of the biased steady state with respect to the equilibrium measure, the authors derive the linear response and Einstein relation without relying on perturbation theory or regeneration times, offering a novel analytical framework for disordered transport systems.

ABSTRACT

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.

Motivation & Objective

• To establish the linear response and Einstein relation in one-dimensional Mott variable-range hopping under a bias.
• To analyze the bias-dependent steady state of the environment viewed from the particle, showing its differentiability at zero bias.
• To develop a new analytical method based on $L^p$-bounds ($p>2$) on the Radon-Nikodym derivative of the biased steady state with respect to the equilibrium measure.
• To avoid traditional tools like perturbation theory or regeneration times, offering a general and robust framework for disordered systems.

Proposed method

• Derive an $L^p$-bound ($p>2$) for the Radon-Nikodym derivative of the bias-dependent steady state with respect to the equilibrium state in the unbiased case.
• Use the $L^p$-bound to infer linear response via a general argument independent of the specific model.
• Employ a coupling argument involving the environment measure and the particle's position, leveraging moment conditions on the distances between impurities.
• Apply Hölder’s inequality and exponential moment assumptions to control the tail behavior of the environment’s jump rates.
• Use weak convergence in $L^2$-spaces and Kakutani’s theorem to prove convergence of the steady-state density to the equilibrium measure.
• Verify the necessary moment conditions on the environment (e.g., ${\mathbb{E}}[e^{pZ_0}] < \infty$) to ensure the $L^p$-bound holds uniformly for small biases.

Experimental results

Research questions

• RQ1 Does the steady-state distribution of the environment viewed from the particle exhibit differentiability at zero bias in one-dimensional Mott variable-range hopping?
• RQ2 Can the linear response of the system to a small bias be rigorously derived without using perturbation theory or regeneration times?
• RQ3 Is the Einstein relation valid in this disordered, long-range hopping model under the given moment assumptions?
• RQ4 What is the role of the $L^p$-bound on the Radon-Nikodym derivative in establishing linear response?
• RQ5 How do the exponential moment conditions on the distances between impurities affect the validity of the linear response and Einstein relation?

Key findings

• The bias-dependent steady state of the environment viewed from the particle is differentiable at zero bias, confirming the linear response property.
• The Radon-Nikodym derivative of the biased steady state with respect to the equilibrium measure satisfies an $L^p$-bound for $p>2$, uniformly for small biases.
• This $L^p$-bound implies the linear response via a general argument, independent of the specific dynamics.
• The Einstein relation holds in the one-dimensional Mott variable-range hopping model under the assumption that $\mathbb{E}[e^{(\lambda_0+4\delta)Z_{-1}+3\delta Z_0}] < \infty$ and $\mathbb{E}[e^{pZ_0}] < \infty$.
• The method avoids traditional tools such as regeneration times or perturbation theory, offering a new route to linear response in disordered systems.
• The result is robust under the given moment conditions, which ensure the convergence of the environment measure under the particle's motion.