Hydrodynamic Limit for a Disordered Quantum Harmonic Chain

Abstract

We study the hydrodynamic limit, in the hyperbolic space-time scaling, for a one-dimensional unpinned chain of quantum harmonic oscillators with random masses. To the best of our knowledge, this is among the first examples where one can prove the hydrodynamic limit for a quantum system rigorously. In fact, we prove that after hyperbolic rescaling of time and space, the distribution of the elongation, momentum, and energy averaged under the proper locally Gibbs state converges to the solution of the Euler equation. Moreover, our result indicates that the temperature profile is frozen in any space-time scale; in particular, the thermal diffusion coefficient vanishes. There are two main phenomena in this chain that enable us to deduce this result. First is the Anderson localization, which decouples the mechanical and thermal energy, providing the closure of the equation for energy. The second phenomenon 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.