Volume 111, Number 2, July 2015
|Number of page(s)||6|
|Published online||05 August 2015|
A geometric entropy detecting the Erdös-Rényi phase transition
1 QSTAR & INO-CNR - Largo E. Fermi 2, 50125 Firenze, Italy
2 School of Science and Technology, University of Camerino - 62032 Camerino, Italy
3 INFN-Sezione di Perugia - Via A. Pascoli, 06123 Perugia, Italy
4 Aix-Marseille University - Marseille, France
5 CNRS, Centre de Physique Théorique, UMR7332 - 13288 Marseille, France
Received: 9 June 2015
Accepted: 10 July 2015
We propose a method to associate a differentiable Riemannian manifold to a generic many-degrees-of-freedom discrete system which is not described by a Hamiltonian function. Then, in analogy with classical statistical mechanics, we introduce an entropy as the logarithm of the volume of the manifold. The geometric entropy so defined is able to detect a paradigmatic phase transition occurring in random graphs theory: the appearance of the “giant component” according to the Erdös-Rényi theorem.
PACS: 02.50.Cw – Probability theory / 02.40.Ky – Riemannian geometries / 89.75.-k – Complex systems
© EPLA, 2015
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