Entropic Forms and Related Algebras

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Abstract:In this paper, we present a review of recent developments on the κ-deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the κ-formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the κ-deformed version of Kullback-Leibler, "Kerridge" and Brègman divergences. The first statistical manifold derived from the κ-Kullback-Leibler divergence form an invariant geometry with a positive curvature that vanishes in the κ → 0 limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the κ-escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of κ-thermodynamics in the picture of the information geometry.

“…It is worthy to derive the canonical divergence for S κ (2) , that is a Brégman-like divergence obtainable from Equation (36). In fact, by using κ-parentropy it reads…”

Section: Second Statistical Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures

Scarfone

Matsuzoe

Wada

2018

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“…Generalized statistical mechanics, based on κ-entropy [1,3,4], preserves the main features of ordinary Boltzmann-Gibbs statistical mechanics. For this reason, it has attracted the interest of many researchers over the last 16 years, who have studied its foundations and mathematical aspects [5][6][7][8][9][10][11][12], the underlying thermodynamics [13][14][15][16][17], and specific applications of the theory in various scientific and engineering fields. A non-exhaustive list of application areas includes quantum statistics [18][19][20], quantum entanglement [21,22], plasma physics [23][24][25][26][27], nuclear fission [28], astrophysics [29][30][31][32][33][34][35], geomechanics [36], genomics [37], complex networks [38,39], economy [40][41][42][43] and finance [44][45][46]…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle

Kaniadakis

Hristopulos

2018

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker-Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker-Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker-Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in G. Kaniadakis, Physica A 296, 405 (2001), univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker-Planck equation in its most general form.

“…A unifying element in this analysis is the generalized κ-distribution [3][4][5][6], which has proven to be helpful for the description of both kinds of phenomena. We start from the familiar assumption of statistical mechanics according to which the macroscopic distributions (of energy or income) are determined by the microscopic interactions (between molecules or, respectively, individuals).…”

Section: Introductionmentioning

confidence: 99%

Statistics of Binary Exchange of Energy or Money

Bertotti

Modanese

2017

Why does the Maxwell-Boltzmann energy distribution for an ideal classical gas have an exponentially thin tail at high energies, while the Kaniadakis distribution for a relativistic gas has a power-law fat tail? We argue that a crucial role is played by the kinematics of the binary collisions. In the classical case the probability of an energy exchange far from the average (i.e., close to 0% or 100%) is quite large, while in the extreme relativistic case it is small. We compare these properties with the concept of "saving propensity", employed in econophysics to define the fraction of their money that individuals put at stake in economic interactions.