# Energy-transport systems for optical lattices: derivation, analysis, simulation

## TL;DR

This paper derives and analyzes energy-transport equations for fermions in optical lattices, exploring their mathematical properties and numerical behavior, including existence of solutions and steady-state convergence.

## Contribution

It provides a formal derivation of energy-transport equations from Boltzmann equations and proves existence of weak solutions with numerical validation.

## Key findings

- Particle density converges to a constant steady state with high initial energy density.
- Nonconstant steady states can occur depending on initial conditions.
- The first-order expansion improves the physical realism of the model.

## Abstract

Energy-transport equations for the transport of fermions in optical lattices are formally derived from a Boltzmann transport equation with a periodic lattice potential in the diffusive limit. The limit model possesses a formal gradient-flow structure like in the case of the energy-transport equations for semiconductors. At the zeroth-order high temperature limit, the energy-transport equations reduce to the whole-space logarithmic diffusion equation which has some unphysical properties. Therefore, the first-order expansion is derived and analyzed. The existence of weak solutions to the time-discretized system for the particle and energy densities with periodic boundary conditions is proved. The difficulties are the nonstandard degeneracy and the quadratic gradient term. The main tool of the proof is a result on the strong convergence of the gradients of the approximate solutions. Numerical simulations in one space dimension show that the particle density converges to a constant steady state if the initial energy density is sufficiently large, otherwise the particle density converges to a nonconstant steady state.