On the entropy method and exponential convergence to equilibrium for a recombination-drift-diffusion system with self-consistent potential

We consider a Shockley-Read-Hall recombination-drift-diffusion model coupled to Poisson's equation and subject to boundary conditions, which imply conservation of the total charge. As main result, we derive an explicit functional inequality between relative entropy and entropy production rate, which implies exponential convergence to equilibrium with explicit constant and rate. We report that the key entropy-entropy production inequality ought rather not to be formulated on the states space of the parabolic-elliptic system, but on the reduced states space of the associated nonlocal drift-diffusion problem, where the Poisson equation is replaced by the corresponding Green function.