Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional FokkerPlanck equation is derived directly from a master equation, by considering nonlinearities in the transition rates. Using nonlinear Fokker-Planck equations, the H-theorem is proved; for that, an important relation involving these equations ...

The aim of this paper is to show that the Tsallis-type (q-additive) entropic chain rule allows for a wider class of entropic functionals than previously thought. In particular, we point out that the ensuing entropy solutions (e.g., Tsallis entropy) can be determined uniquely only when one fixes the prescription for handling conditional entropies. By using the concept of Kolmogorov–Nagumo quasi-...

In this article, the effect of the nonextensivity parameter on the energy fluctuations of nonextensive systems is studied in two different versions of the Tsallis statistical mechanics. Once a general expression has been reported for the energy fluctuations in the second version (Tsallis work in 1988), the energy fluctuations of an ideal gas and a harmonic oscillator are studied in the second a...

In statistical physics, Boltzmann-Shannon entropy provides good understanding for the equilibrium states of a number of phenomena. In statistics, the entropy corresponds to the maximum likelihood method, in which Kullback-Leibler divergence connects Boltzmann-Shannon entropy and the expected log-likelihood function. The maximum likelihood estimation has been supported for the optimal performanc...

We prove characterization theorems for relative entropy (also known as Kullback–Leibler divergence), q-logarithmic entropy (also known as Tsallis entropy), and q-logarithmic relative entropy. All three have been characterized axiomatically before, but we show that earlier proofs can be simplified considerably, at the same time relaxing some of the hypotheses.

We revisit the issues on the thermodynamic property of stellar self-gravitating system arising from Tsallis’ non-extensive entropy. Previous papers (Physica A 307 (2002) 185; ibid. (2003) in press (cond-mat/0204315)) have revealed that the extremum-state of Tsallis entropy, so-called stellar polytrope, has consistent thermodynamic structure, which predicts the thermodynamic instability due to t...

Escort distributions are a simple one parameter deformation of an original distribution p. In Tsallis extended thermostatistics, the escort-averages, defined with respect to an escort distribution, have revealed useful in order to obtain analytical results and variational equations, with in particular the equilibrium distributions obtained as maxima of Rényi-Tsallis entropy subject to constrain...

In density estimation task, maximum entropy model (Maxent) can effectively use reliable prior information via certain constraints, i.e., linear constraints without empirical parameters. However, reliable prior information is often insufficient, and the selection of uncertain constraints becomes necessary but poses considerable implementation complexity. Improper setting of uncertain constraints...

There is a conception that Boltzmann-Gibbs statistics cannot yield the long tail distribution. This is the justification for the intensive research of nonextensive entropies (i.e. Tsallis entropy and others). Here the error that caused this misconception is explained and it is shown that a long tail distribution exists in equilibrium thermodynamics for more than a century.

Decision making is still an open issue in the application of Dempster-Shafer evidence theory. A lot of works have been presented for it. In the transferable belief model (TBM), pignistic probabilities based on the basic probability assignments are used for decision making. In this paper, multiscale probability transformation of basic probability assignment based on the belief function and the p...