Solution of a generalised Boltzmann's equation for non-equilibrium charged particle transport via localised and delocalised states

TL;DR

This paper develops a comprehensive phase-space kinetic model for charged particle transport involving localized and delocalized states, providing analytical solutions and revealing complex behaviors like transient negative transport coefficients and fractional diffusion regimes.

Contribution

It introduces a generalized Boltzmann equation framework with analytical solutions for non-equilibrium charged particle transport, including novel insights into dispersive diffusion and trapping effects.

Findings

Analytical expressions for center of mass velocity and diffusivity.

Observation of transient negative transport coefficients.

Derivation of a generalized diffusion equation encompassing normal and fractional diffusion.

Abstract

We present a general phase-space kinetic model for charged particle transport through combined localised and delocalised states, capable of describing scattering collisions, trapping, detrapping and losses. The model is described by a generalised Boltzmann equation, for which an analytical solution is found in Fourier-Laplace space. The velocity of the centre of mass (CM) and the diffusivity about it are determined analytically, together with the flux transport coefficients. Transient negative values of the free particle CM transport coefficients can be observed due to the trapping to, and detrapping from, localised states. A Chapman-Enskog type perturbative solution technique is applied, confirming the analytical results and highlighting the emergence of a density gradient representation in the weak-gradient hydrodynamic regime. A generalised diffusion equation with a unique global time operator is shown to arise, reducing to the standard diffusion equation and a Caputo fractional diffusion equation in the normal and dispersive limits. A subordination transformation is used to solve the generalised diffusion equation by mapping from the solution of a corresponding standard diffusion equation.