Entropy minimization for many-body quantum systems

TL;DR

This paper proves the existence of local quantum Gibbs states with prescribed densities by entropy minimization, facilitating the derivation of fluid dynamics equations from many-body quantum mechanics.

Contribution

It establishes the construction of local quantum Gibbs states from prescribed densities in both fermionic and bosonic systems, addressing a key mathematical challenge.

Findings

Existence of local quantum Gibbs states under mild conditions.

Development of auxiliary optimization problems to handle lack of compactness.

Application of monotonicity properties of equilibrium entropies.

Abstract

The problem considered here is motivated by a work by B. Nachtergaele and H.T. Yau where the Euler equations of fluid dynamics are derived from manybody quantum mechanics, see [10]. A crucial concept in their work is that of local quantum Gibbs states, which are quantum statistical equilibria with prescribed particle, current, and energy densities at each point of space (here R d , d $\ge$ 1). They assume that such local Gibbs states exist, and show that if the quantum system is initially in a local Gibbs state, then the system stays, in an appropriate asymptotic limit, in a Gibbs state with particle, current, and energy densities now solutions to the Euler equations. Our main contribution in this work is to prove that such local quantum Gibbs states can be constructed from prescribed densities under mild hypotheses, in both the fermionic and bosonic cases. The problem consists in minimizing the von Neumann entropy in the quantum grand canonical picture under constraints of local particle, current, and energy densities. The main mathematical difficulty is the lack of compactness of the minimizing sequences to pass to the limit in the constraints. The issue is solved by defining auxiliary constrained optimization problems, and by using some monotonicity properties of equilibrium entropies.