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

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 [12] 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.

1D Mott Variable-Range Hopping with External Field

“…Hence, due to Proposition 2, the family of Radon-Nykodim derivatives dQ λ dQ0 , λ ∈ [0, λ 0 ], is relatively compact for the L 2 (Q 0 )-weak topology. In [6] we then prove that any limit point of this family is given by 1. As a byproduct of the representation Q λ (f ) = Q 0 ( dQ λ dQ0 f ) and of the weak convergence dQ λ dQ0 1, we get the continuity of (8) at λ = 0.…”

“…Theorem 3 [6] Suppose that E[e pZ0 ] < ∞ for some p ≥ 2 and let q be the conjugate exponent of p, i.e. q satisfies 1 p + 1 q = 1.…”

“…Without entering into the details of the proof (which can be found in [6]) we give some comments on the derivation of Theorem 3 from Proposition 2. To this aim, we take for simplicity p = 2.…”

“…As discussed with more details in [6] as ε goes to zero the family of potential forms ∇g f ε converges in L 2 (M ) to a potential form h f :…”

“…Theorem 4 [6] Suppose E(e pZ0 ) < ∞ for some p > 2. Then, for any f ∈ H −1 ∩ L 2 (Q 0 ), ∂ λ=0 Q λ (f ) exists.…”