Asymptotic property of current for a conduction model of Fermi particles on finite lattice

TL;DR

This paper analyzes the asymptotic behavior of stationary current in a finite lattice conduction model of Fermi particles, deriving formulas and relations for various potentials and noise conditions, including extensions to higher dimensions.

Contribution

It introduces a simple current formula applicable to various potentials and noise scenarios, linking current decay to transfer matrix properties and extending results to higher-dimensional systems.

Findings

Current decays exponentially in the Anderson model without noise.

Explicit 1/N scaling of current when noise exists but potential does not.

In 3D, current scales with cross section and inversely with length.

Abstract

In this paper, we introduce a conduction model of Fermi particles on a finite sample, and investigate the asymptotic behavior of stationary current for large sample size. In our model a sample is described by a one-dimensional finite lattice on which Fermi particles injected at both ends move under various potentials and noise from the environment. We obtain a simple current formula. The formula has broad applicability and is used to study various potentials. When the noise is absent, it provides the asymptotic behavior of the current in terms of a transfer matrix. In particular, for dynamically defined potential cases, a relation between exponential decay of the current and the Lyapunov exponent of a relevant transfer matrix is obtained. For example, it is shown that the current decays exponentially for the Anderson model. On the other hand, when the noise exists but the potential does not, an explicit form of the current is obtained, which scales as 1/N for large sample size N. Moreover, we provide an extension to higher dimensional systems. For a three-dimensional case, it is shown that the current increases in proportion to cross section and decreases in inverse proportion to the length of the sample.