On local quantum Gibbs states

We address in this work the problem of minimizing quantum entropies under local constraints. suppose that macroscopic quantities, such as particle density, current, and kinetic energy, are fixed at each point [Formula: see text] look for a density operator over text], an entropy functional. Such minimizers referred to Gibbs states. This setting is contrast with classical prescribing global constraints, where total number particles, energy system fixed. The question arises, instance, derivation fluid models from dynamics. prove, fairly general conditions, admits unique constrained minimizer. Due lack compactness, main difficulty proof show limits sequences satisfy constraint. tackle issue by introducing simpler auxiliary minimization using monotonicity argument involving entropy.