Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Obtaining macroscopic laws (e.g., Euler equation/ Heat equation) from interacting particle systems governed by microscopic laws (e.g., Newton/Schrödinger eq.) in proper scaling limits is a very challenging task and has received much attention in recent years. In this talk, I will present the first example (to the best of my knowledge), where one can do this task rigorously for a simple interacting quantum system without any apriori assumptions (CMP, 390, 349–423 (2022)).

The model is a one-dimensional unpinned disordered chain of quantum harmonic oscillators: a hydrodynamic limit in the hyperbolic scaling of time and space is proven; distribution of the elongation, momentum, and energy converges to the solution of the Euler equation in this scaling. Two physical phenomena are behind this proof: Anderson localization decouples the mechanical and thermal energy, providing the closure of the macroscopic equation out of thermal equilibrium and indicating that the temperature profile does not evolve in time. The macroscopic evolution of the mechanical energy results from the divergence of the localization length at the bottom of the spectrum.  

The decay of correlation-type phenomena facilitates dealing with the quantum nature of the system. We also strengthen the above convergence in the sense of ”higher moments” in recent joint work with Francois Huveneers (J. Phys. A: Math. Theor. 55 424005).