Dynamics and Thermodynamics of the mean field d-HMF model out-of-equilibrium

TL;DR

This paper investigates the out-of-equilibrium dynamics and thermodynamics of the d-HMF model, revealing ensemble equivalence, Boltzmann-Gibbs equilibrium distributions, and characterizing quasi-stationary states using molecular dynamics and Vlasov equation solutions.

Contribution

It provides analytical and numerical descriptions of the d-HMF model's equilibrium and out-of-equilibrium states, including the characterization of QSS via stationary Vlasov solutions.

Findings

Ensemble equivalence established between canonical and microcanonical.

Boltzmann-Gibbs distribution matches simulations.

QSS described by q-exponential stationary solutions.

Abstract

We study the d-HMF model proposed by Atenas and Curilef, a mean field model with long-range interactions inspired by the dipole-dipole interaction. Among the challenges of this thesis is: the resolution of the d-HMF model in the canonical and microcanonical ensembles and the analytical and numerical description of the distribution function of the system. This model has been studied both in equilibrium and out of equilibrium. In the equilibrium, analytical solutions have been found for internal energy per particle and temperature using standard procedures through statistical mechanics such as the calculation of the partition function in the canonical ensemble and the calculation of the number of microstates accessible in the microcanonical ensemble. The results indicate that there is an equivalence of ensembles. Additionally, we found the Boltzmann-Gibbs (BG) equilibrium distribution, which coincides with the analytical results of the canonical and microcanonical ensembles and molecular dynamics simulations. Then we focused on describing the Quasi-Stationary-States (QSS) present in this system, by means of molecular dynamics simulations. Two types of QSS states are found out of equilibrium. For its description, a combination of two techniques is used; the methods of molecular dynamics and stationary solutions of the Vlasov equation associated with the equations of motion of the system. From molecular we describe the marginal distributions in moments and orientations, which were describe anaytically by means of stationary solutions of Vlasov equation. The stationary solutions tested in this thesis correspond to distribution functions of the q-exponential type. In particular, it is found that one of these solutions describes precisely the marginal distributions of the moments and the orientations of one of the QSS states found with the molecular dynamics.