BBGKY Hierarchy and Generalised Hydrodynamics

TL;DR

This paper explores the construction of conserved charges in weakly interacting fermionic systems, identifying specific potentials that preserve locality, and demonstrates how Generalized Hydrodynamics naturally arises from the BBGKY hierarchy.

Contribution

It introduces a perturbative method to construct conserved charges in fermionic systems and links GHD to the BBGKY hierarchy, highlighting robustness under perturbations.

Findings

Only Cheon-Shigehara and Calogero-Sutherland potentials satisfy locality conditions.

GHD emerges from the BBGKY hierarchy in weakly interacting fermions.

GHD remains robust under certain non-integrable perturbations.

Abstract

We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally non-local and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our construction to show how Generalized Hydrodynamics (GHD) emerges from the Bogoliubov--Born--Green--Kirkwood--Yvon hierarchy, and argue that GHD in the weak interaction regime is robust under non-integrable perturbations.