Inhomogeneous Tsallis distributions in the HMF model |
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Authors: | P-H Chavanis and A Campa |
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Institution: | 1.Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse,Toulouse,France;2.Complex Systems and Theoretical Physics Unit, Health and Technology Department, Istituto Superiore di Sanità,
and INFN Roma 1, Gruppo Collegato Sanità,Roma,Italy |
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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. |
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Keywords: | |
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