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This work is devoted to the analysis of the quantum drift-diffusion modelderived by Degond et al. The model is obtained as the diffusive limit of thequantum Liouville-BGK equation, where the collision term is defined after alocal quantum statistical equilibrium. The corner stone of the model is theclosure relation between the density and the current, which is nonlinear andnonlocal, and is the main source of the mathematical difficulties. The questionof the existence of solutions has been open since the derivation of the model,and we provide here a first result in a one-dimensional periodic setting. Theproof is based on an approximation argument, and exploits some properties ofthe minimizers of an appropriate quantum free energy. We investigate as wellthe long time behavior, and show that the solutions converge exponentially fastto the equilibrium. This is done by deriving a non-commutative logarithmicSobolev inequality for the local quantum statistical equilibrium.

The quantum drift-diffusion model: existence and exponential convergence to the equilibrium