Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation

TL;DR

This paper presents an efficient numerical algorithm for solving a matrix-valued quantum Boltzmann equation with spin considerations, using spectral methods and Fourier transforms, and investigates its convergence to equilibrium.

Contribution

The paper introduces a novel spectral Fourier-based numerical scheme for the matrix-valued quantum Boltzmann equation, accommodating spin degrees of freedom.

Findings

Efficient spectral accuracy collision kernel evaluation.

Numerical simulations demonstrate convergence to thermal equilibrium.

The method handles various boundary conditions effectively.

Abstract

We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are $2 \times 2$ matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.