2017/04/10 by Marcel Braukhoff, Ansgar Jüngel, Braukhoff, Marcel +1
Computer Science · Engineering · Materials Science · Mathematics · #35K59 #35K65 #35Q20 #82B40 #Advanced Materials Characterization Techniques #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Thermal properties of materials #math.AP #msc:35K59 #msc:35K65 #msc:35Q20 #msc:82B40
paper · pdf · doi:10.48550/arxiv.1704.02845
arxiv created 2017/04/10 · openalex publication_date 2017/04/10 · arxiv updated 2017/04/11 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
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.