A splitting scheme for the quantum Liouville-BGK equation

TL;DR

This paper presents a new efficient numerical splitting scheme for simulating the quantum Liouville-BGK equation, enabling better modeling of quantum particle transport with proven convergence and validation against quantum drift-diffusion models.

Contribution

The paper introduces a novel splitting scheme that simplifies the nonlinear PDE system of the quantum Liouville-BGK equation, with convergence proof and application to quantum drift-diffusion validation.

Findings

The scheme is fully linear in the collision step.

The method converges in the semi-discrete time setting.

Validation against quantum drift-diffusion model confirms accuracy.

Abstract

We introduce in this work an efficient numerical method for the simulation of the quantum Liouville-BGK equation, which models the diffusive transport of quantum particles. The corner stone to the model is the BGK collision operator, obtained by minimizing the quantum free energy under the constraint that the local density of particles is conserved during collisions. This leads to a large system of coupled nonlinear nonlocal PDEs whose resolution is challenging. We then define a splitting scheme that separates the transport and the collision parts, which, exploiting the local conservation of particles, leads to a fully linear collision step. The latter involves the resolution of a constrained optimization problem that is is handled with the nonlinear conjugate gradient algorithm. We prove that the time semi-discrete scheme is convergent, and as an application of our numerical scheme, we validate the quantum drift-diffusion model that is obtained as the diffusive limit of the quantum Liouville-BGK equation.