Inhomogeneous Tsallis distributions in the HMF model

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.

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Cited by in corpus (20)

• Influence of the interaction range on the thermostatistics of a classical many-body system
• The Brownian Mean Field model
• Out-of-equilibrium phase transitions in the HMF model: a closer look
• Controlling the Range of Interactions in the Classical Inertial Ferromagnetic Heisenberg Model: Analysis of Metastable States
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Generalization of the possible algebraic basis of $q$-triplets
• Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics
• Kinetic theory of two-dimensional point vortices with collective effects
• Caloric curves fitted by polytropic distributions in the HMF model
• Thermodynamics of the HMF model with a magnetic field
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• Analysis of Tsallis' classical partition function's poles
• On Linear Damping Around Inhomogeneous Stationary States of the Vlasov-HMF Model
• Rescuing the MaxEnt treatment for $q$-generalized entropies
• Verlinde's emergent gravity in an $\mathbf{n}-$dimensional, non-additive Tsallis' scenario
• Hidden correlations entailed by q-non additivity render the q-monoatomic gas highly non trivial
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• The quasilinear theory in the approach of long-range systems to quasi-stationary states
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