# Linearized regime of the generalized hydrodynamics with diffusion

## TL;DR

This paper analyzes the linearized generalized hydrodynamics with diffusion in the Lieb-Liniger model, highlighting differences between ballistic and diffusive dynamics for various initial states.

## Contribution

It constructs a general solution for the linearized hydrodynamics including diffusion and explores the evolution of different initial states beyond the linear regime.

## Key findings

- Ballistic evolution causes spatial scrambling.
- Diffusive evolution scrambles quasi-particle content.
- Zero momentum mode exhibits non-linear evolution.

## Abstract

We consider the generalized hydrodynamics including the recently introduced diffusion term for an initially inhomogeneous state in the Lieb-Liniger model. We construct a general solution to the linearized hydrodynamics equation in terms of the eigenstates of the evolution operator and study two prototypical classes of initial states: delocalized and localized spatially. We exhibit some general features of the resulting dynamics, among them, we highlight the difference between the ballistic and diffusive evolution. The first one governs a spatial scrambling, the second, a scrambling of the quasi-particles content. We also go one step beyond the linear regime and discuss the evolution of the zero momentum mode that does not evolve in the linear regime.