[Paper Review] Miller-Abrahams random resistor network, Mott random walk and 2-scale homogenization

Alessandra Faggionato|arXiv (Cornell University)|Feb 9, 2020

Advanced Mathematical Modeling in Engineering23 references5 citations

TL;DR

This paper establishes the almost sure homogenization of the Miller-Abrahams random resistor network using 2-scale homogenization, proving that its infinite-volume conductivity is governed by an effective diffusion matrix $D$, which matches the limiting diffusion matrix of Mott random walk. The result confirms the physical Mott law for low-temperature conductivity decay and provides a variational characterization of $D$, extending rigorous bounds to the resistor network model.

ABSTRACT

The Miller-Abrahams (MA) random resistor network is given by a complete graph on a marked simple point process with edge conductivities depending on the marks and decaying exponentially in the edge length. As Mott random walk, it is an effective model to study Mott variable range hopping in amorphous solids as doped semiconductors. By using 2-scale homogenization we prove that a.s. the infinite volume conductivity of the MA resistor network is given by an effective homogenized matrix $D$. Moreover $D$ admits a variational characterization and equals the limiting diffusion matrix of Mott random walk. This result clarifies the relation between the two models and it also allows to extend to the MA resistor network the existing bounds on $D$ in agreement with the physical Mott law [12,14]. The latter concerns the low temperature stretched exponential decay of conductivity in amorphous solids. The techniques developed here can be applied to other models, as e.g. the random conductance model [11], without ellipticity assumptions.

Motivation & Objective

• To rigorously establish the homogenization of the Miller-Abrahams random resistor network in the infinite-volume limit.
• To clarify the connection between the Miller-Abrahams resistor network and Mott random walk by showing they share the same limiting diffusion matrix $D$.
• To derive a variational characterization of the effective conductivity matrix $D$ and verify it satisfies the Mott law for low-temperature conductivity decay.
• To extend existing rigorous bounds on $D$ from Mott random walk to the resistor network model, confirming consistency with physical expectations.
• To develop a framework based on 2-scale convergence applicable to general random resistor networks without ellipticity assumptions.

Proposed method

• Uses 2-scale homogenization techniques to analyze the electrical conductivity of the Miller-Abrahams random resistor network on a marked simple point process.
• Applies 2-scale convergence to derive the effective conductivity matrix $D$ in the limit of large system size.
• Relies on the variational characterization of $D$ derived from the energy functional of the system.
• Establishes convergence of the electrical potential and current fields via weak limits in $L^2$ with respect to the random measure $\nu^\varepsilon$.
• Uses antisymmetry of current fields and divergence-free conditions on internal nodes to simplify network current balances.
• Applies the theory of 2-scale convergence to handle the random, long-range conductivities depending on spatial distance and energy marks.

Experimental results

Research questions

• RQ1 Does the infinite-volume conductivity of the Miller-Abrahams random resistor network converge to a deterministic effective matrix $D$ almost surely?
• RQ2 Is the effective matrix $D$ of the resistor network identical to the limiting diffusion matrix of Mott random walk?
• RQ3 Can the variational characterization of $D$ be rigorously derived in the context of the resistor network model?
• RQ4 Do the bounds on $D$ derived for Mott random walk extend to the Miller-Abrahams resistor network, confirming the Mott law for conductivity decay?
• RQ5 Can 2-scale homogenization be applied to random resistor networks without requiring ellipticity conditions on conductivities?

Key findings

• The infinite-volume conductivity of the Miller-Abrahams random resistor network converges almost surely to $D \cdot \rho$, where $D$ is a deterministic effective matrix and $\rho$ is the mean point density.
• The effective matrix $D$ is identical to the limiting diffusion matrix of Mott random walk, establishing a rigorous link between the two models.
• The matrix $D$ admits a variational characterization, which allows for the derivation of upper and lower bounds consistent with the Mott law.
• The bounds on $D$ match the stretched exponential decay predicted by the Mott law: $D(\beta) \sim \exp\bigl(-c\,\beta^{\frac{1+\alpha}{\alpha+d+1}}\bigr)$ for $d \geq 2$.
• The 2-scale homogenization framework developed here applies to general random resistor networks without requiring ellipticity assumptions on the conductivities.
• The results are extended to the electrical potential, showing its homogenization to a harmonic limit under the same scaling.