Information Geometry and its Applications III

Abstract Jan Naudts

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Jan Naudts  (Universiteit Antwerpen, Belgium)
Friday, August 06, 2010, room Hörsaal 1
The generalized exponential family in statistical physics: a framework for microcanonical phase transitions

Statistical models belonging to a generalized exponential family automatically satisfy the variational principle, which is a stronger statement than the maximum entropy principle. It implies a dual structure which in thermodynamics is known since long in the form of Legendre transforms and their inverses. The entropy function in the variational principle is essentially unique. For convenience, let us call it the stable entropy function. For the special case of the q-exponential family this is Tsallis' entropy function with parameter 2-q.

Statistical models belonging to the q-exponential family occur frequently in statistical physics. In particular, the configurational probability distribution of any model of classical mechanics, when considered as a function of the total energy, belongs to the q-exponential family, with a parameter q which tends to 1 when the number of degrees of freedom tends to infinity. It is well-known from the canonical case of the Boltzmann-Gibbs distribution that the variational principle implies a property of stability, which roughly means that phase transitions cannot occur in systems with a finite number of degrees of freedom. This stability holds also for models belonging to a generalized exponential family.

If the stable entropy function is replaced by an increasing function of itself then the maximum entropy principle is still satisfied, but the variational principle is violated. In particular, if Tsallis' entropy is replaced by that of Rényi, then the stability property gets lost. In physical models the lack of stability means that phase transitions may occur. We show that, when Rényi's entropy function is used, the simple model of the pendulum exhibits a first order phase transition between small angle librational motion at low values of the energy and full rotational motion at high energies.

 

Date and Location

August 02 - 06, 2010
University of Leipzig
Augustusplatz
04103 Leipzig
Germany

Scientific Organizers

Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Information Theory of Cognitive Systems Group
Germany
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Paolo Gibilisco
Università degli Studi di Roma "Tor Vergata"
Facoltà di Economia
Italy
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František Matúš
Academy of Sciences of the Czech Republic
Institute of Information Theory and Automation
Czech Republic
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Scientific Committee

Shun-ichi Amari
RIKEN
Brain Science Institute, Mathematical Neuroscience Laboratory
Japan
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Imre Csiszár
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
Hungary
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Dénes Petz
Budapest University of Technology and Economics
Department for Mathematical Analysis
Hungary
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Giovanni Pistone
Collegio Carlo Alberto, Moncalieri
Italy
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Administrative Contact

Antje Vandenberg
Max Planck Institute for Mathematics in the Sciences
Contact by Email
Phone: (++49)-(0)341-9959-552
Fax: (++49)-(0)341-9959-555

05.04.2017, 12:42