Hydrodynamic limit for a disordered quantum harmonic chain

TL;DR

This paper rigorously establishes the hydrodynamic limit for a disordered quantum harmonic chain, showing convergence to the Euler equation and revealing effects of Anderson localization and correlation decay.

Contribution

First proof of hydrodynamic limit for a quantum system, demonstrating convergence to Euler equations in a disordered harmonic chain.

Findings

Distribution of elongation, momentum, and energy converges to Euler equation solutions.

Anderson localization decouples mechanical and thermal energy, freezing the temperature profile.

Decay of correlations helps overcome non-product Gibbs state complexities.

Abstract

In this note, 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 Gibbs state, converges to the solution of the Euler equation. There are two main phenomena in this chain which 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 and indicating that the temperature profile will be frozen. 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.