Wigner transport in linear electromagnetic fields

Abstract

Applying a Weyl-Stratonovich transform to the evolution equation of the Wigner function in an electromagnetic field yields a multidimensional gauge-invariant equation which is numerically very challenging to solve. In this work, we apply simplifying assumptions for linear electromagnetic fields and the evolution of an electron in a plane (two-dimensional transport), which reduces the complexity and enables to gain first experiences with a gauge-invariant Wigner equation. We present an equation analysis and show that a finite difference approach for solving the high-order derivatives allows for reformulation into a Fredholm integral equation. The resolvent expansion of the latter contains consecutive integrals, which is favorable for Monte Carlo solution approaches. To that end, we present two stochastic (Monte Carlo) algorithms that evaluate averages of generic physical quantities or directly the Wigner function. The algorithms give rise to a quantum particle model, which interprets quantum transport in heuristic terms.

References in corpus (7)

• From Lorenz to Coulomb and other explicit gauge transformations
• Admissible states in quantum phase space
• Fluid moment hierarchy equations derived from gauge invariant quantum kinetic theory
• Quantum transport and the Wigner distribution function for Bloch electrons in spatially homogeneous electric and magnetic fields
• Interband tunneling effects on materials transport properties using the first principles Wigner distribution
• Dynamics of the effective mass and the anomalous velocity in two-dimensional lattices
• Gauge-Invariant Semi-Discrete Wigner Theory