The velocity of 1d Mott variable-range hopping with external field

Abstract

Mott variable range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps.We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias.Keywords: random walk in random environment, disordered media, ballisticity, environment viewed from the walker, electron transport in disordered solids.

1D Mott Variable-Range Hopping with External Field

“…We point out that the above 1d results hold also for a larger class of jump rates and energy distributions ν [2]. Moreover, we stress that the above bounds (5) and (6) refer to the diffusion matrix D(β) and not to the conductivity matrix σ(β). On the other hand, believing in the Einstein relation (which states that σ(β) = βD(β)), the above bounds would extend to σ(β), hence one would recover lower and upper bounds on the conductivity matrix in agreement with the physical Mott law for d ≥ 2 and with the Arrenhius-type decay for d = 1.…”

“…We also point out that one can easily prove that the random walk X ω,λ t is well defined since λ ∈ [0, 1). The following result, obtained in [5], concerns the ballistic/sub-ballistic regime: As discussed in [5], the condition E e (1−λ)Z0 = ∞ does not imply that v X (λ) = 0. On the other hand, if (Z k ) k∈Z are i.i.d.…”

Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls

“…This process is the simplified version of the Variable-Range Hopping model used in Physics (cfr. [FGS18]), and essentially corresponds to a random walk among random conductances with a different type of bias. In [BS19] we identify the correct scaling exponent for the walk as a specific function of the field intensity λ (in contrast with the BiRC where the scaling α does not depend on λ).…”

Regularity of biased 1D random walks in random environment

“…Results concerning the continuity of the speed have been obtained e.g. for a random walk in a one-dimensional percolation model [16] and for the 1d Mott random walk [10] (in [16] also the differentiability has been studied). The behavior of one dimensional RWRE's that are transient but with zerospeed has been studied in [8,9,20] for i.i.d.…”

“…The solid line represents the function σ 2 (λ) in the case of i.i.d. conductances sampled from a uniform random variable in the interval[1,10]. This is plotted against the function σ 2 (λ) (dotted line) in the deterministic case of constant conductances, cfr.…”