Quantum Hydrodynamics and Quantum Diffusion Models Derived from the Entropy Principle
This work addresses the question of deriving hydrodynamic and diffusion models from a macroscopic limit of quantum kinetic models. This question is of key importance in a certain number of fields such as plasma or semiconductor mesoscopic modeling. The major difficulty to solve when investigating hydrodynamic limits is that of the closure relation (i.e. finding the equation-of-state of the system). This problem is resolved in the classical framework by assuming that the microscopic state is at local thermodynamically equilibrium. Such a state realizes the minimum of the entropy functional subject to local constraints of mass, momentum and energy. We propose an extension of this method to quantum systems. This leads to hydrodynamic models with non-local closure relations. These models preserve the monotony of the entropy functional. The same approach leads to a proposal for quantum extensions of the classical Boltzmann or BGK collision operators. Finally, it allows the investigation of diffusion limits of quantum systems (which are distringuished from hydrodynamic limits by the nature of the scaling) ands lead to quantum extension of the well-established drift-diffusion and energy-transport models.