Table of contents

In the recent increasing interests in power-law behaviors beyond the usual exponential ones, there have been some concrete attempts in statistical physics to generalize the standard Boltzmann-Gibbs statistics. Among such generalizations, nonextensive statistical mechanics has been well studied for about the last two decades with many modifications and refinements. The generalization has provided not only a theoretical framework but also many applications such as chaos, multi-fractal, complex systems, nonequilibrium statistical mechanics, biophysics, econophysics, information theory and so on. At the same time as the developments in the generalization of statistical mechanics, the corresponding mathematical structures have also been required and uncovered. In particular, some deep connections to mathematical sciences such as q-analysis, information geometry, information theory and quantum probability theory have been revealed recently. These results obviously indicate an existence of the generalized mathematical structure including the mathematical framework for the exponential family as a special case, but the whole structure is still unclear.

In order to make an opportunity to discuss the mathematical structure induced from generalized entropies by scientists in many fields, the international workshop 'Mathematical Aspects of Generalized Entropies and their Applications' was held on 7–9 July 2009 at Kyoto TERRSA, Kyoto, Japan. This volume is the proceedings of the workshop which consisted of 6 invited speakers, 14 oral presenters, 7 poster presenters and 63 other participants. The topics of the workshop cover the nonextensive statistical mechanics, chaos, cosmology, information geometry, divergence theory, econophysics, materials engineering, molecular dynamics and entropy theory, information theory and so on. The workshop was organized as the first attempt to discuss these mathematical aspects with leading experts in each area.

We would like to express special thanks to all the invited speakers, the contributors and the participants at the workshop. We are also grateful to RIMS (Research Institute for Mathematical Science) in Kyoto University and the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B), 18300003, 2009 for their support.

Organizing Committee

Editors of the Proceedings Hiroki Suyari (Chiba University, Japan) Atsumi Ohara (Osaka University, Japan) Tatsuaki Wada (Ibaraki University, Japan)

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

We briefly review central concepts concerning nonextensive statistical mechanics, based on the nonadditive entropy . Among others, we focus on possible realizations of the q-generalized Central Limit Theorem, including at the edge of chaos of the logistic map, and for quasi-stationary states of many-body long-range-interacting Hamiltonian systems.

We determine the limit distributions of sums of deterministic chaotic variables in unimodal maps assisted by a novel renormalization group (RG) framework associated to the operation of increment of summands and rescaling. In this framework the difference in control parameter from its value at the transition to chaos is the only relevant variable, the trivial fixed point is the Gaussian distribution and a nontrivial fixed point is a multifractal distribution with features similar to those of the Feigenbaum attractor. The crossover between the two fixed points is discussed and the flow toward the trivial fixed point is seen to consist of a sequence of chaotic band mergers.

The Boltzmann-Gibbs probability distribution, seen as a statistical model, belongs to the exponential family. Recently, the latter concept has been generalized. The q-exponential family has been shown to be relevant for the statistical description of small isolated systems. Two main applications are reviewed: 1. The distribution of the momentum of a single particle is a q-Gaussian, the distribution of its velocity is a deformed Maxwellian; 2. The configurational density distribution belongs to the q-exponential family.

The definition of the temperature of small isolated systems is discussed. It depends on defining the thermodynamic entropy of a microcanonical ensemble in a suitable manner. The simple example of non-interacting harmonic oscillators shows that Rényi's entropy functional leads to acceptable results.

Dynamical characterization and behavior of two types of nonlinear Fokker-Planck equations (NFPEs) are studied within the framework of generalized thermostatistics. On the basis of generalized entropies NFPEs are shown to be constructed so that H-theorems hold with Lyapunov functionals that are given by free energy functionals associated with the generalized entropies. In the case of the ordinary type of NFPEs such H-theorems ensure convergence of solutions to their uniquely determined equilibrium solutions. In the case of mean-field type NFPEs (DNFPEs) that may exhibit bifurcation phenomena the H-theorems are shown to still hold to ensure global stability of solutions. Systematic description is given of local stability analysis based on the second-order variations of the free energy functionals.

In the two-parameter generalization of thermostatistics based on the Sharma-Taneja-Mittal entropy not only the generalized entropic functional S_a,b but also a new functional _a,b plays a fundamental role. These functionals are related to the finite difference and averaging operators arising in finite difference calculus.

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed by our former works [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008] and [K. Tadaki, Proceedings of LFCS'09, Springer's LNCS, vol. 5407, pp. 422–440, 2009], where we introduced the notion of thermodynamic quantities, such as partition function Z(T), free energy F(T), energy E(T), statistical mechanical entropy S(T), and specific heat C(T), into AIT. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature T itself, which is one of the most typical thermodynamic quantities. Namely, we showed that, for each of the thermodynamic quantities Z(T), F(T), E(T), and S(T) above, the computability of its value at temperature T gives a sufficient condition for T (0,1) to satisfy the condition that the partial randomness of T equals to T. In this paper, based on a physical argument on the same level of mathematical strictness as normal statistical mechanics in physics, we develop a total statistical mechanical interpretation of AIT which actualizes a perfect correspondence to normal statistical mechanics. We do this by identifying a microcanonical ensemble in the framework of AIT. As a result, we clarify the statistical mechanical meaning of the thermodynamic quantities of AIT.

Tsallis entropy, which is one of nonextensive entropies, gives a q-normal distribution as an equilibrium probability density function. Although a q-normal distribution is popular, there exists a problem what it means by calculating an expectation value with a corresponding escort distribution not a q-normal distribution itself. But we have an amazing property such that an escort distribution obtained by a q-normal distribution with a parameter q and a variance is another q-normal distribution with a different value of q and a scaled variance. Therefore calculating an expectation value with an escort distribution corresponds to calculating the expectation value with another q-normal distribution, that is to say, an escort distribution is nothing but another q-normal distribution. However it still remains the question why an expectation value should be calculated by another q-normal distribution. We call the procedure to get another q-normal distribution from a q-normal distribution through an escort distribution τ-transformation. This τ-transformation keeps a support of a q-normal distribution invariant, and makes the tails of a q-normal distribution thicker/thinner depending on a value of q. Thus a τ-transformation looks like a kind of multiresolutional analysis, if we consider a q-normal distribution as a window function.

This study considers q-Gaussian distributions and stochastic differential equations with both multiplicative and additive noises. In the M-dimensional case a q-Gaussian distribution can be theoretically derived as a stationary probability distribution of the multiplicative stochastic differential equation with both mutually independent multiplicative and additive noises. By using the proposed stochastic differential equation a method to evaluate a default probability under a given risk buffer is proposed.

To clarify the nonequilibrium processes of self-gravitating systems, we examine a system enclosed in a spherical container with reflecting walls, by N-body simulations. To simulate nonequilibrium processes, we consider loss of energy through the reflecting wall, i.e., a particle reflected at a non-adiabatic wall is cooled to mimic energy loss. We also consider quasi-equilibrium structures of stellar polytropes to compare with the nonequilibrium process, where the quasi-equilibrium structure is obtained from an extremum-state of Tsallis' entropy. Consequently, we numerically show that, with increasing cooling rates, the dependence of the temperature on energy, i.e., the -Ť curve, varies from that of microcanonical ensembles (or isothermal spheres) to a common curve. The common curve appearing in the nonequilibrium process agrees well with an -Ť curve for a quasi-equilibrium structure of the stellar polytrope, especially for the polytrope index n ∼ 5. In fact, for n > 5, the stellar polytrope within an adiabatic wall exhibits gravothermal instability [Taruya, Sakagami, Physica A, 322 (2003) 285]. The present study indicates that the stellar polytrope with n ∼ 5 likely plays an important role in quasi-attractors of the nonequilibrium process in self-gravitating systems with non-adiabatic walls.

We have studied the surface potential decays (SPD) of a variety of dielectric materials, and found that the SPD data of the samples with high charge retensions are well fitted by Tsallis q-exponential functions.

Several molecular dynamics techniques applying the Tsallis generalized distribution are presented. We have developed a deterministic dynamics to generate an arbitrary smooth density function ρ. It creates a measure-preserving flow with respect to the measure ρdω and realizes the density ρ under the assumption of the ergodicity. It can thus be used to investigate physical systems that obey such distribution density. Using this technique, the Tsallis distribution density based on a full energy function form along with the Tsallis index q ≥ 1 can be created. From the fact that an effective support of the Tsallis distribution in the phase space is broad, compared with that of the conventional Boltzmann-Gibbs (BG) distribution, and the fact that the corresponding energy-surface deformation does not change energy minimum points, the dynamics enhances the physical state sampling, in particular for a rugged energy surface spanned by a complicated system. Other feature of the Tsallis distribution is that it provides more degree of the nonlinearity, compared with the case of the BG distribution, in the deterministic dynamics equation, which is very useful to effectively gain the ergodicity of the dynamical system constructed according to the scheme. Combining such methods with the reconstruction technique of the BG distribution, we can obtain the information consistent with the BG ensemble and create the corresponding free energy surface. We demonstrate several sampling results obtained from the systems typical for benchmark tests in MD and from biomolecular systems.

This note studies geometrical structure of the manifold of escort probability distributions and proves that the resultant geometry is dually flat in the sense of information geometry. We use a conformal transformation that flattens the alpha-geometry of the space of the discrete probability distributions in order to realize escort probabilities in the framework of affine differential geometry. Dual pairs of potential functions and affine coordinate systems on the manifold are derived, and the associated canonical divergence is shown to be conformal to the alpha-divergence.

The driving force behind our study has been to overcome the difficulties you encounter when you try to extend the clear and convincing operational interpretations of classical Boltzmann-Gibbs-Shannon entropy to other notions, especially to generalized entropies as proposed by Tsallis. Our approach is philosophical, based on speculations regarding the interplay between truth, belief and knowledge. The main result demonstrates that, accepting philosophically motivated assumptions, the only possible measures of entropy are those suggested by Tsallis – which, as we know, include classical entropy. This result constitutes, so it seems, a more transparent interpretation of entropy than previously available. However, further research to clarify the assumptions is still needed. Our study points to the thesis that one should never consider the notion of entropy in isolation – in order to enable a rich and technically smooth study, further concepts, such as divergence, score functions and descriptors or controls should be included in the discussion. This will clarify the distinction between Nature and Observer and facilitate a game theoretical discussion. The usefulness of this distinction and the subsequent exploitation of game theoretical results – such as those connected with the notion of Nash equilibrium – is demonstrated by a discussion of the Maximum Entropy Principle.

We propose a definition for the entropy of a monotone set function defined on a lattice which are not necessarily the whole power set, but satisfy the condition of regularity. Our definition encompasses the classical definition of Shannon for probability measures, as well as the definition of Marichal for classical fuzzy measures and may have applicability to most fuzzy measures which appear in applications. We give also an axiomatization of this entropy. This axiomatization is in the spirit of Faddeev's axiomatization for the classical Shannon entropy. After that, using same idea we introduce a generalization of the Shapley value for a set function defined on a lattice and give two types of necessary and sufficient conditions.

We give a trace inequality related to the uncertainty relation of Wigner-Yanase-Dyson skew information. This inequality corresponds to a generalization of the uncertainty relation derived by S.Luo [8] for the quantum uncertainty quantity excluding the classical mixture. Finally we show a counter-example of the uncertainty relation related to Wigner-Yanase-Dyson skew information given in [6] recently.

We define q-Fisher informations and show q-Cramér-Rao type inequalities that the q-Gaussian distribution with special q-variances attains the minimum value of the q-Fisher informations.

The purpose of this presentation is to clarify the universal meaning of generalized non-extensive entropies and/or statistics based on Tsallis entropy from the viewpoint of "Micro-Macro duality". What plays here the most important roles is the notions of scales and scaling transformations which will be seen to be related closely to the main themes of this workshop, generalized non-extensive entropies and/or statistics and the information-geometrical structures of the classifying spaces.

This paper presents the mathematical reformulation for maximization of Tsallis entropy S_q in the combinatorial sense. More concretely, we generalize the original derivation of Maxwell-Boltzmann distribution law to Tsallis statistics by means of the corresponding generalized multinomial coefficient. Our results reveal that maximization of S_2−q under the usual expectation or S_q under q-average using the escort expectation are naturally derived from the combinatorial formulations for Tsallis statistics with respective combinatorial dualities, that is, one for additive duality and the other for multiplicative duality.