Entropy minimization for many-body quantum systems

The problem we consider is motivated by a work by B. Nachtergaele and 
H.T. Yau where the Euler equations of fluid dynamics are derived from 
many-body quantum mechanics. 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. 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 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.