Derivation of Euler equations from quantum and classical microscopic dynamics

Abstract

Abstract We derive Euler equations from a Hamiltonian microscopic dynamics. The microscopic system is a one-dimensional disordered harmonic chain, and the dynamics is either quantum or classical. This chain is an Anderson insulator with a symmetry protected mode: thermal fluctuations are frozen while the low modes ensure the transport of elongation, momentum and mechanical energy, that evolve according to Euler equations in an hyperbolic scaling limit. In this paper, we strengthen considerably the results in Bernardin et al (2019 Commun. Math. Phys. 365 215–37); Hannani (2022 Commun. Math. Phys. 390 349–23), where we established a limit in mean starting from a local Gibbs state: we now control the second moment of the fluctuations around the average, yielding a limit in probability, and we enlarge the class of admissible initial states.