Transport in quantum multi-barrier systems as random walks on a lattice

TL;DR

This paper establishes a precise connection between quantum multi-barrier systems and stochastic random walks on a lattice, demonstrating their equivalence in both ordered and disordered configurations through analytical and numerical methods.

Contribution

It introduces a novel mapping between quantum wave amplitudes and lattice random walk probabilities, extending the equivalence to disordered barrier setups.

Findings

Stationary density profiles match between quantum and stochastic models.

The quantum-stochastic equivalence holds even with random barrier parameters.

Analytical and numerical results confirm the model's robustness.

Abstract

A quantum finite multi-barrier system, with a periodic potential, is considered and exact expressions for its plane wave amplitudes are obtained using the Transfer Matrix method [10]. This quantum model is then associated with a stochastic process of independent random walks on a lattice, by properly relating the wave amplitudes with the hopping probabilities of the particles moving on the lattice and with the injection rates from external particle reservoirs. Analytical and numerical results prove that the stationary density profile of the particle system overlaps with the quantum mass density profile of the stationary Schrodinger equation, when the parameters of the two models are suitably matched. The equivalence between the quantum model and a stochastic particle system would mainly be fruitful in a disordered setup. Indeed, we also show, here, that this connection, analytically proven to hold for periodic barriers, holds even when the width of the barriers and the distance between barriers are randomly chosen.