Eur. Phys. J. B 76, 581-611 (2010)
https://doi.org/10.1140/epjb/e2010-00243-x

Inhomogeneous Tsallis distributions in the HMF model

¹ Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse, 31062 Toulouse, France
² Complex Systems and Theoretical Physics Unit, Health and Technology Department, Istituto Superiore di Sanità, and INFN Roma 1, Gruppo Collegato Sanità, 00161 Roma, Italy

Corresponding author: ^a chavanis@irsamc.ups-tlse.fr

Received: 12 January 2010
Revised: 20 May 2010
Published online: 6 August 2010

Abstract

We study the maximization of the Tsallis functional at fixed mass and energy in the Hamiltonian Mean Field (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 and 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 and 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.