A quantum drift-diffusion model derived from an entropy minimization
Abstract
In this work in collaboration with P. Degond, S. Gallego (Toulouse) and\nC. Ringhofer (Arizona State University), we give an overview of a\nrecently derived quantum drift-diffusion model. A quantum local\nequilibrium is defined as a minimizer of the quantum free energy subject\nto local moment constraints (the local mass density is given). This\nequilibrium relates a thermodynamic parameter (the chemical potential) to\nthe density in a non-local way. We derive collision operators for quantum\nkinetic models which decrease the quantum entropy and relax towards those\nquantum equilibria. Then, through a diffusion limit of the quantum\nkinetic equation, we establish a transport model at a diffusive scale,\nwhich is a quantum extension of the classical drift-diffusion model. We\nremark that a perturbative analysis of this model in a semiclassical\nasymptotics enables to recover the well-known density-gradient model. We\nshall present 1D numerical results which show that this quantum\ndrift-diffusion model is able to reproduce the major features of quantum\ntransport in resonant tunneling structures.\n\n
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