Hydrodynamic limit for a disordered quantum harmonic chain

Obtaining the macroscopic evolution of conserved quantities and their corresponding currents for a "physical" system from its microscopic dynamics also known as hydrodynamic limits is a matter of interest both in Physics and Mathematics communities.
  In this talk, I present a hydrodynamic limit, in the hyperbolic space-time scaling, for a one-dimensional unpinned chain of quantum harmonic oscillators with random masses. This model is among the famous toy-models for studying heat transfer in solid.  To the best of my knowledge, this is among the first examples, where one can prove the hydrodynamic limit for a quantum system rigorously. In fact, I show that the distribution of the elongation, momentum, and energy averaged under the proper Gibbs state converges to the solution of the Euler equation. There are two main phenomena in this chain that let us deduce this result. First is the Anderson localization which decouples the mechanical and thermal energy, providing the closure of the equation for energy and indicating that the temperature profile does not evolve in time. The second phenomena is similar to some sort of decay of correlation phenomena which let us circumvent the difficulties arising from the fact that our Gibbs state is not a product state due to the quantum nature of the system.