Dynamical and thermodynamical stability of isothermal distributions in the HMF model

TL;DR

This paper presents a new, simple variational method to derive the stability conditions of isothermal distributions in the HMF model, extending previous thermodynamical approaches and applicable to more general cases.

Contribution

It introduces a straightforward variational formalism for stability analysis in the HMF model, applicable to both homogeneous and inhomogeneous distributions, and can be extended to other distribution types.

Findings

Analytical stability conditions for isothermal distributions in the HMF model.

The formalism simplifies stability analysis and can be applied to more general distribution functions.

Provides a clear illustration of the method through explicit calculations.

Abstract

We provide a new derivation of the conditions of dynamical and thermodynamical stability of homogeneous and inhomogeneous isothermal distributions in the Hamiltonian Mean Field (HMF) model. This proof completes the original thermodynamical approach of Inagaki [Prog. Theor. Phys. 90, 557 (1993)]. Our formalism, based on variational principles, is simple and the method can be applied to more general situations. For example, it can be used to settle the dynamical stability of polytropic distributions with respect to the Vlasov equation [Chavanis & Campa, arXiv:1001.2109]. For isothermal distributions, the calculations can be performed fully analytically, providing therefore a clear illustration of the method.