On the minimization of quantum entropies under local constraints

TL;DR

This paper addresses the challenge of minimizing quantum entropies with local physical constraints, employing a thermodynamically inspired monotonicity approach to overcome mathematical difficulties in the quantum setting.

Contribution

It introduces a novel monotonicity argument to handle the lack of compactness in quantum entropy minimization under local constraints.

Findings

Established a new method for quantum entropy minimization

Overcame mathematical difficulties related to energy constraint

Provided insights applicable to quantum hydrodynamical models

Abstract

This work is concerned with the minimization of quantum entropies under local constraints of density, current, and energy. The problem arises in the work of Degond and Ringhofer about the derivation of quantum hydrodynamical models from first principles, and is an adaptation to the quantum setting of the moment closure strategy by entropy minimization encountered in kinetic equations. The main mathematical difficulty is the lack of compactness needed to recover the energy constraint. We circumvent this issue by a monotonicity argument involving energy, temperature and entropy, that is inspired by some thermodynamical considerations.