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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Possible generalization of Boltzmann-Gibbs statistics

Constantino Tsallis (1988)

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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