Inhomogeneous Tsallis distributions in the HMF model

TL;DR

This paper investigates inhomogeneous Tsallis distributions within the HMF model, revealing phase transition behaviors and explaining numerical anomalies through polytropic distributions linked to incomplete relaxation and non-ergodicity.

Contribution

It provides the first comprehensive analysis of polytropic distributions in the HMF model and explains numerical anomalies via a novel interpretation of Tsallis distributions.

Findings

Identification of a critical index q_c=3 for phase transition types.

Explanation of caloric curve anomalies using polytropic distributions.

Discovery of simultaneous microcanonical first and second order phase transitions.

Abstract

We study the maximization of the Tsallis functional at fixed mass and energy in the HMF model. We give a thermodynamical and a dynamical interpretation of this variational principle. This leads to q-distributions known as stellar polytropes in astrophysics. We study phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium states. We show that there exists a particular index q_c=3 playing the role of a canonical tricritical point separating first and second order phase transitions in the canonical ensemble and marking the occurence of a negative specific heat region in the microcanonical ensemble. We apply our results to the situation considered by Antoni & Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly displayed on their caloric curve can be explained naturally by assuming that, in this region, the QSSs are polytropes with critical index q_c=3. We qualitatively justify the occurrence of polytropic (Tsallis) distributions with compact support in terms of incomplete relaxation and inefficient mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF model and the first plausible explanation of the surprising result observed numerically by Antoni & Ruffo (1995). In the course of our analysis, we also report an interesting situation where the caloric curve presents both microcanonical first and second order phase transitions.