The transient drift-diffusion model describing the charge transport in semiconductors is investigated in the case that the currents are prescribed. It is shown that the solutions of the drift-diffusion system coupled with Maxwell's equations converge to the solution of the drift-diffusion system coupled with Poisson's equation if the magnetic susceptibility tends to zero. Furthermore it is shown that the densities converge to the thermal equilibrium state for $t\rightarrow\infty$ provided that the boundary conditions are compatible with the thermal equilibrium.
"A quasistationary limit and convergence to equilibrium in the drift diffusion system for semiconductors coupled with Maxwell's equations." Differential Integral Equations 14 (4) 427 - 474, 2001. https://doi.org/10.57262/die/1356123315