Diffusive Hydrodynamics of Inhomogenous Hamiltonians

TL;DR

This paper develops a hydrodynamic framework for inhomogeneous Hamiltonian systems with conserved quantities, predicting entropy growth and thermalization, and validates it through molecular dynamics simulations of the Toda model.

Contribution

It introduces a general diffusive hydrodynamic equation for inhomogeneous systems with conserved quantities, including a detailed analysis of the Onsager matrix and entropy production.

Findings

Hydrodynamic equation predicts entropy increase and thermal states.

Exact evaluation of the 2-particle-hole contribution to the Onsager matrix.

Simulation confirms thermalization over diffusive timescales in the Toda model.

Abstract

We derive a large-scale hydrodynamic equation, including diffusive and dissipative effects, for systems with generic static position-dependent driving forces coupling to local conserved quantities. We show that this equation predicts entropy increase and thermal states as the only stationary states. The equation applies to any hydrodynamic system with any number of local, PT-symmetric conserved quantities, in arbitrary dimension. It is fully expressed in terms of elements of an extended Onsager matrix. In integrable systems, this matrix admits an expansion in the density of excitations. We evaluate exactly its 2-particle-hole contribution, which dominates at low density, in terms of the scattering phase and dispersion of the quasiparticles, giving a lower bound for the extended Onsager matrix and entropy production. We conclude with a molecular dynamics simulation, demonstrating thermalisation over diffusive time scales in the Toda interacting particle model with an inhomogeneous energy field.