Projection operator approach to master equations for coarse grained occupation numbers in non-ideal quantum gases

TL;DR

This paper develops a projection operator method to derive simplified, finite-dimensional equations of motion for coarse-grained occupation numbers in non-ideal quantum gases, enabling easier calculation of transport properties.

Contribution

It introduces a novel projection operator approach to obtain finite-dimensional rate matrices for coarse-grained occupation numbers in quantum gases, simplifying analysis and computation.

Findings

Derived a linear collision term as a finite-dimensional rate matrix.

Applied the method to a 3D Anderson model with weak disorder.

Facilitated calculation of transport coefficients.

Abstract

We aim at deriving an equation of motion for specific sums of momentum mode occupation numbers from models for electrons in periodic lattices experiencing elastic scattering, electron-phonon scattering or electron-electron scattering. These sums correspond to "grains" in momentum space. This equation of motion is supposed to involve only a moderate number of dynamical variables and/or exhibit a sufficiently simple structure such that neither its construction nor its analyzation/solution requires substantial numerical effort. To this we end compute, by means of a projection operator technique, a linear(ized) collision term which determines the dynamics of the above grain-sums. This collision term results as non-singular, finite dimensional rate matrix and may thus be inverted regardless of any symmetry of the underlying model. This facilitates calculations of, e.g., transport coefficients, as we demonstrate for a 3-dim. Anderson model featuring weak disorder.