Constrained minimizers of the von Neumann entropy and their characterization

TL;DR

This paper characterizes the unique minimizer of the von Neumann entropy under local constraints on particle density, current, and energy, showing it solves a nonlinear eigenvalue problem with implications for quantum hydrodynamics.

Contribution

It introduces a novel characterization of the local constrained minimizer of von Neumann entropy as a solution to a self-consistent nonlinear eigenvalue problem.

Findings

The minimizer is a self-adjoint positive trace class operator.

The minimizer satisfies a self-consistent nonlinear eigenvalue problem.

The approach involves parametrizing the feasible set and deriving the Euler-Lagrange equation.

Abstract

We consider in this work the problem of minimizing the von Neumann entropy under the constraints that the density of particles, the current, and the kinetic energy of the system is fixed at each point of space. The unique minimizer is a self-adjoint positive trace class operator, and our objective is to characterize its form. We will show that this minimizer is solution to a self-consistent nonlinear eigenvalue problem. One of the main difficulties in the proof is to parametrize the feasible set in order to derive the Euler-Lagrange equation, and we will proceed by constructing an appropriate form of perturbations of the minimizer. The question of deriving quantum statistical equilibria is at the heart of the quantum hydrody-namical models introduced by Degond and Ringhofer in [5]. An original feature of the problem is the local nature of constraints, i.e. they depend on position, while more classical models consider the total number of particles, the total current and the total energy in the system to be fixed.