Mathematical and Numerical Methods for Complex Quantum Systems

On the moment problem for quantum hydrodynamics

Olivier Pinaud

Colorado State University

We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density n(x)? We give a positive answer in various configurations and characterize the minimizer in dimension one. We give in addition an existence result for a related Quantum Liouville equation with BGK type collision operator. This problem is motivated by a recent theory developed by P. Degond and C. Ringhofer concerning the derivation of quantum hydrodynamics models. This is jointwork with F. Mehats.