L. Nivanen

Generalized algebra within a nonextensive statistics

Abstract By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators: addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g. associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element are no more universal within the generalized algebra.

Temperature and pressure in nonextensive thermostatistics

The definitions of temperature in the nonextensive statistical thermodynamics based on Tsallis entropy are analyzed. A definition of pressure is proposed for nonadditive systems by using a nonadditive effective volume. The thermodynamics of nonadditive photon gas is discussed on this basis. We show that the Stefan-Boltzmann law can be preserved within nonextensive thermodynamics.

On the generalized entropy pseudoadditivity for complex systems

We show that Abes general pseudoadditivity for entropy prescribed by thermal equilibrium in nonextensive systems holds not only for entropy, but also for energy. The application of this general pseudoadditivity to Tsallis entropy tells us that the factorization of the probability of a composite system into a product of the probabilities of the subsystems is just a consequence of the existence of thermal equilibrium and not due to the independence of the subsystems.

How to fit the degree distribution of the air network

We investigate three different approaches for fitting the degree distributions of China-, US- and the composite China+US air network, in order to reveal the nature of such distributions and the potential theoretical background on which they are based. Our first approach is the fitting with q-statistics probability distribution, done separately in two regimes. This yields acceptable outcomes but generates two sets of fitting parameters. The second approach is an entire fitting to all data points with the formula proposed by Tsallis et al. So far, this trial is not able to produce consistent results. In the third approach, we fit the data with two composite distributions which may lack theoretical support for the moment.

Fractal geometry, information growth and nonextensive thermodynamics

This is a study of the information evolution of complex systems by a geometrical consideration. We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness of the state number counting at any scale on fractal support, the incomplete normalization ∑ipiq=1 is applied throughout the paper, where q is the fractal dimension divided by the dimension of the smooth Euclidean space in which the fractal structure of the phase space is embedded. It is shown that the information growth is nonadditive and is proportional to the trace-form ∑ipi−∑ipiq which can be connected to several nonadditive entropies. This information growth can be extremized to give power-law distributions for these nonequilibrium systems. It can also be used for the study of the thermodynamics derived from Tsallis entropy for nonadditive systems which contain subsystems each having its own q. It is argued that, within this thermodynamics, the Stefan–Boltzmann law of blackbody radiation can be preserved.

Generalized blackbody distribution within the dilute gas approximation

We consider the application of Tsallis’ generalized statistical mechanics to systems of quantum particles (fermions or bosons). An approach due to Buyukkilic et al. to find analytic formulae for quantum distribution functions and its application to generalized blackbody radiation is discussed. Generalized blackbody laws (Stefan–Boltzmann law, Rayleigh–Jeans law and Wien empirical distribution law) are proposed.

MAXIMIZABLE INFORMATIONAL ENTROPY AS A MEASURE OF PROBABILISTIC UNCERTAINTY

In this work, we consider a recently proposed entropy S defined by a variational relationship

A mathematical structure for the generalization of conventional algebra

dI=d\bar{x}-\overline{dx}

Understanding Heavy Fermion from Generalized Statistics

as a measure of uncertainty of random variable x. The entropy defined in this way underlies an extension of virtual work principle

Incomplete normalization of probability on multifractals

\overline{dx}=0