How Electronic Dynamics with Pauli Exclusion Produces Fermi-Dirac Statistics

TL;DR

This paper derives a new electron dynamics equation that ensures populations relax to Fermi-Dirac distribution, incorporating Pauli exclusion effects and extending Redfield theory to many-electron systems.

Contribution

It introduces a novel kinetic equation for electron populations that accounts for Pauli blocking and generalizes Redfield theory for many-electron systems.

Findings

Relaxation rates vary due to blocking effects.

Electronic populations relax to Fermi-Dirac distribution.

Bath localization influences electron density matrix.

Abstract

It is important that any dynamics method approaches the correct population distribution at long times. In this paper, we derive a one-body reduced density matrix dynamics for electrons in energetic contact with a bath. We obtain a remarkable equation of motion which shows that in order to reach equilibrium properly, rates of electron transitions depend on the density matrix. Even though the bath drives the electrons towards a Boltzmann distribution, hole blocking factors in our equation of motion cause the electronic populations to relax to a Fermi-Dirac distribution. These factors are an old concept, but we show how they can be derived with a combination of time-dependent perturbation theory, and the extended normal ordering of Mukherjee and Kutzelnigg. The resulting non-equilibrium kinetic equations generalize the usual Redfield theory to many-electron systems, while ensuring that the orbital occupations remain between zero and one. In numerical applications of our equations, we show that relaxation rates of molecules are not constant because of the blocking effect. Other applications to model atomic chains are also presented which highlight the importance of treating both dephasing and relaxation. Finally we show how the bath localizes the electron density matrix.