Maximum entropy principle and power-law tailed distributions

Abstract

In ordinary statistical mechanics the Boltzmann-Shannon entropy is related to the Maxwell-Bolzmann distribution $p_i$ by means of a twofold link. The first link is differential and is offered by the Jaynes Maximum Entropy Principle. The second link is algebraic and imposes that both the entropy and the distribution must be expressed in terms of the same function in direct and inverse form. Indeed, the Maxwell-Boltzmann distribution $p_i$ is expressed in terms of the exponential function, while the Boltzmann-Shannon entropy is defined as the mean value of $-\ln (p_i)$. In generalized statistical mechanics the second link is customarily relaxed. Here we consider the question if and how is it possible to select generalized statistical theories in which the above mentioned twofold link between entropy and the distribution function continues to hold, such as in the case of ordinary statistical mechanics. Within this scenario, there emerge new couples of direct-inverse functions, i.e. generalized logarithms $\Lambda (x)$ and generalized exponentials ${\Lambda }^{-1}(x)$, defining coherent and self-consistent generalized statistical theories. Interestingly, all these theories preserve the main features of ordinary statistical mechanics, and predict distribution functions presenting power-law tails. Furthermore, the obtained generalized entropies are both thermodynamically and Lesche stable.

To appear in EPJB. 11 journal pages, 100 references

References in corpus (52)

• Power laws, Pareto distributions and Zipf's law
• How Popular is Your Paper? An Empirical Study of the Citation Distribution
• Scale-free topology of e-mail networks
• Problems with Fitting to the Power-Law Distribution
• Statistical mechanics in the context of special relativity
• Non Linear Kinetics underlying Generalized Statistics
• Statistical mechanics in the context of special relativity II
• Thermodynamic Description of the Relaxation of Two-Dimensional Euler Turbulence Using Tsallis Statistics
• On the Frequency of Severe Terrorist Events
• Economic Fluctuations and Diffusion
• Traffic on complex networks: Towards understanding global statistical properties from microscopic density fluctuations
• Multiplicative processes and power laws
• H-Theorem and Generalized Entropies Within the Framework of Non Linear Kinetics
• Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics
• Equilibrium Distribution of Heavy Quarks in Fokker-Planck Dynamics
• Deformed exponentials and logarithms in generalized thermostatistics
• k-Generalized Statistics in Personal Income Distribution
• The k-generalized distribution: A new descriptive model for the size distribution of incomes
• Deformed logarithms and entropies
• A General Nonlinear Fokker-Planck Equation and its Associated Entropy
• Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski
• Kinetic model for q-deformed bosons and fermions
• Power law correlations in the Southern Oscillation Index fluctuations characterizing El Nino
• Non-gaussian statistics and stellar rotational velocities of main sequence field stars
• Kappa-deformed Statistics and the Formation of a Quark-Gluon Plasma
• Generalized Kinetic Equations for a System of Interacting Atoms and Photons: Theory and Simulations
• Power-law Distribution of Family Names in Japanese Societies
• Two generalizations of the Boltzmann equation
• Stabilities of generalized entropies
• A two-parameter generalization of Shannon-Khinchin Axioms and the uniqueness theorem
• Power-law statistics and stellar rotational velocities in the Pleiades
• Lesche Stability of $\kappa$-Entropy
• $\kappa$-exponential models from the geometrical viewpoint
• Generalized entropy optimized by an arbitrary distribution
• Kinetic Approach to Fractional Exclusion Statistics
• Microscopic dynamics underlying the anomalous diffusion
• $\kappa$-generalization of Gauss' law of error
• The property of kappa-deformed statistics for a relativistic gas in an electromagnetic field: kappa parameter and kappa-distribution
• The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis
• Asymptotic solutions of a nonlinear diffusive equation in the framework of $\kappa$-generalized statistical mechanics
• Canonical partition function for anomalous systems described by the $\kappa$-entropy
• Statistical Mechanical Approach to Human Language
• A k-generalized statistical mechanics approach to income analysis
• Multifractal spectrum of phase space related to generalized thermostatistics
• Nonextensive random-matrix theory based on Kaniadakis entropy
• Conservative Force Fields in Non-Gaussian Statistics
• Statistical descriptions of nonlinear systems at the onset of chaos
• Thermodynamic stability conditions for nonadditive composable entropies
• Entropy production and Pesin-like identity at the onset of chaos
• Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos
• Nonextensive and superstatistical generalizations of random-matrix theory
• Schwinger-Dyson Analysis of Dynamical Symmetry Breaking on a Brane with Bulk Yang-Mills Theory

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