An inverse problem in quantum statistical physics

TL;DR

This paper proves the uniqueness of the quantum free energy minimizer with a given local density in one dimension, establishing a rigorous foundation for local quantum equilibrium concepts used in quantum hydrodynamics.

Contribution

It provides the first proof of the unique minimizer of quantum free energy under local density constraints in one dimension, linking it to quantum thermodynamic equilibrium.

Findings

Unique minimizer exists in 1D for given local density

The minimizer is a quantum Maxwellian with a chemical potential

Supports the rigorous foundation of quantum hydrodynamic models

Abstract

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 to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential.