Stability of inhomogeneous states in mean-field models with a local potential

TL;DR

This paper analytically investigates the linear stability of inhomogeneous states in mean-field models using the Vlasov equation, deriving dispersion relations for specific models and validating results with numerical simulations.

Contribution

It provides an analytical treatment of the linear stability of inhomogeneous states in mean-field models with external fields, including derivation of dispersion relations for specific models.

Findings

Derived dispersion relations for stability analysis

Identified stability thresholds for the models

Validated analytical results with numerical simulations

Abstract

The Vlasov equation is well known to provide a good description of the dynamics of mean-field systems in the $N \to \infty$ limit. This equation has an infinity of stationary states and the case of {\it homogeneous} states, for which the single-particle distribution function is independent of the spatial variable, is well characterized analytically. On the other hand, the inhomogeneous case often requires some approximations for an analytical treatment: the dynamics is then best treated in action-angle variables, and the potential generating inhomogeneity is generally very complex in these new variables. We here treat analytically the linear stability of toy-models where the inhomogeneity is created by an external field. Transforming the Vlasov equation into action-angle variables, we derive a dispersion relation that we accomplish to solve for both the growth rate of the instability and the stability threshold for two specific models: the Hamiltonian Mean-Field model with additional asymmetry and the mean-field $\phi^4$ model. The results are compared with numerical simulations of the $N$-body dynamics. When the {\it inhomogeneous} state is stationary, we expect to observe in the $N$-body dynamics Quasi-Stationary-States (QSS), whose lifetime diverge algebraically with $N$.