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Nonextensive Entropy, Prior PDFs and Spontaneous Symmetry Breaking

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      Abstract

      We show that using nonextensive entropy can lead to spontaneous symmetry breaking when a parameter changes its value from that applicable for a symmetric domain, as in field theory. We give the physical reasons and also show that even for symmetric Dirichlet priors, such a defnition of the entropy and the parameter value can lead to asymmetry when entropy is maximized.

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      Most cited references 6

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      Possible generalization of Boltzmann-Gibbs statistics

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        Statistical mechanics in the context of special relativity

         G. Kaniadakis (2002)
        In the present effort we show that \(S_{\kappa}=-k_B \int d^3p (n^{1+\kappa}-n^{1-\kappa})/(2\kappa)\) is the unique existing entropy obtained by a continuous deformation of the Shannon-Boltzmann entropy \(S_0=-k_B \int d^3p n \ln n\) and preserving unaltered its fundamental properties of concavity, additivity and extensivity. Subsequently, we explain the origin of the deformation mechanism introduced by \(\kappa\) and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter \(\kappa\) which results to depend on the light speed \(c\) and reduces to zero as \(c \to \infty\) recovering in this way the ordinary statistical mechanics and thermodynamics. The novel statistical mechanics constructed starting from the above entropy, preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural, economic and social sciences. Finally, in order to test the correctness and predictability of the theory, as working example we consider the cosmic rays spectrum, which spans 13 decades in energy and 33 decades in flux, finding a high quality agreement between our predictions and observed data. PACS number(s): 05.20.-y, 51.10.+y, 03.30.+p, 02.20.-a
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          Non Linear Kinetics underlying Generalized Statistics

            (2001)
          The purpose of the present effort is threefold. Firstly, it is shown that there exists a principle, that we call Kinetical Interaction Principle (KIP), underlying the non linear kinetics in particle systems, independently on the picture (Kramers, Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand, to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and, on the other hand, suggests how we can introduce naturally new distributions. Finally, as a working example of the approach to the non linear kinetics here presented, a new non extensive statistics is constructed and studied starting from a one-parameter deformation of the exponential function holding the relation \(f(-x)f(x)=1\).
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            Author and article information

            Journal
            06 October 2008
            2008-10-26
            0810.1072

            http://creativecommons.org/licenses/by/3.0/

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            Some typos and confusing lines have been fixed
            cond-mat.stat-mech nlin.AO physics.data-an

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