Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated with the Mittag–Leffler function. This family gives an extension to the generaliz...

The relationships among reversible Carnot cycles, the absence of perpetual motion machines, and the existence of a nondecreasing globally unique entropy function form the starting point of many textbook presentations of the foundations of thermodynamics. However, the thermal fluctuation phenomena associated with statistical mechanics has been argued to restrict the domain of validity of this ba...

The formalism of statistical mechanics can be generalized by starting from more general measures of information than the Shannon entropy and maximizing those subject to suitable constraints. We discuss some of the most important examples of information measures that are useful for the description of complex systems. Examples treated are the Rényi entropy, Tsallis entropy, Abe entropy, Kaniadaki...

The formalism of statistical mechanics can be generalised by starting from more general measures of information than the Shannon entropy and maximising those subject to suitable constraints. We discuss some of the most important examples of information measures that are useful for the description of complex systems. Examples treated are the Rényi entropy, Tsallis entropy, Abe entropy, Kaniadaki...

Thermodynamic semirings are deformed additive structures on characteristic one semirings, defined using a binary information measure. The algebraic properties of the semiring encode thermodynamical and information theoretic properties of the entropy function. Besides the case of the Shannon entropy, which arises in the context of geometry over the field with one element and the Witt constructio...

We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature Tc and unstable below it. Then, we ...

The Boltzmann-Gibbs statistical mechanics and the standard thermodynamics are very useful in the discussion of extensive systems. However, they are not good to study situations where nonextensive e ects occur, for instance, systems involving long-range interactions (e. g., d = 3 gravitation)[1, 2], long-range microscopic memory[3], and systems with a relevant (multi)fractal-like structure. On t...

This paper introduces a new hybrid approach for learning systems that builds on the theory of nonextensive statistical mechanics. The proposed learning scheme uses only the sign of the gradient, and combines adaptive stepsize local searches with global search steps that make use of an annealing schedule inspired from nonextensive statistics, as proposed by Tsallis. The performance of the hybrid...

Our experiments on turbulent flow in a rotating annulus yield probability distribution functions (PDFs) for velocity increments δv( ), where is the separation between points. We fit these PDFs to a form derived for turbulent flows by Beck, who used the Tsallis nonextensive statistical mechanics formalism. For slow rotation rates, we find that the fit parameter q is 1.25 for small . At large , q...

1. The Physics of Information 2 2. Thermodynamics 2 2.1. The laws 3 2.2. Free energy 6 3. Statistical mechanics 7 3.1. Definitions and postulates 8 3.2. A simple model system of magnetic spins 9 3.3. The Maxwell-Boltzmann distribution 10 3.4. Free energy revisited 11 3.5. Gibbs entropy 12 4. Nonlinear dynamics 12 4.1. The ergodic hypothesis 13 4.2. Chaos and limits to prediction 14 4.3. Quantif...