COMPLEXITY, METASTABILITY, AND NONEXTENSIVITY: An International Conference

New axiomatics for statistical mechanics
View Description Hide DescriptionIt is here shown how to use pieces of macroscopic thermodynamics so as to generate microscopic probability distributions for generalized ensembles, thereby directly connecting macrostate‐axiomatics with microscopic results.

Nonextensive statistical mechanics and central limit theorems I—Convolution of independent random variables and q‐product
View Description Hide DescriptionIn this article we review the standard versions of the Central and of the Lévy‐Gnedenko Limit Theorems, and illustrate their application to the convolution of independent random variables associated with the distribution ( ), known as q‐Gaussian. This distribution emerges upon extremisation of the nonadditive entropy basis of nonextensive statistical mechanics. It has a finite variance for and an infinite one for We exhibit that, in the case of (standard) independence, the q‐Gaussian has either the Gaussian (if ) or the α‐stable Lévy distributions (if ) as its attractor in probability space. Moreover, we review a generalisation of the product, the q‐product, which plays a central role in the approach of the specially correlated variables emerging within the nonextensive theory.

Nonextensive statistical mechanics and central limit theorems II—Convolution of q‐independent random variables
View Description Hide DescriptionIn this article we review recent generalisations of the central limit theorem for the sum of specially correlated (or q‐independent) variables, focusing on Specifically, this kind of correlation turns the probability density function which emerges upon maximisation of the entropy into an attractor in probability space. Moreover, we also discuss a q‐generalisation of α‐stable Lévy distributions which can as well be stable for this special kind of correlation. Within this context, we verify the emergence of a triplet of entropic indices which relate the form of the attractor, the correlation, and the scaling rate, similar to the q‐triplet that connects the entropic indices characterising the sensitivity to initial conditions, the stationary state, and relaxation to the stationary state in anomalous systems.

On multivariate generalizations of the q‐central limit theorem consistent with nonextensive statistical mechanics
View Description Hide DescriptionThe q‐central limit theorem plays within nonextensive statistical mechanics the same key role that the classic central limit theorem plays within Boltzmann‐Gibbs statistical mechanics. It provides universality to some laws for strongly correlated systems described by q‐Gaussians. In the present note, two generalizations (direct and sequential), for the multivariate case, of the q‐central limit theorem are discussed.

Compatibility between dynamics and Tsallis statistics
View Description Hide DescriptionWe perform numerical simulations for the dynamics of a chain of weakly coupled particles, and consider the process of occupation of cells in phase space, in the spirit of paper [1]. A different behaviour is exhibited as the coupling constant is changed. It is discussed whether such a dynamical behaviour is compatible with the Tsallis statistics.

Extensive nonadditive entropy in quantum spin chains
View Description Hide DescriptionWe present details on a physical realization, in a many‐body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell‐Mann, Sato and one of us (C.T.), that the nonadditive entropy ( matrix; ) can conform, for an anomalous value of q (i.e., ), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L‐sized block of the Ising ferromagnetic chain at its critical transverse field, we obtain ). The present results suggest a new and powerful approach to measure entanglement in quantum many‐body systems. At the light of these results, and similar ones for a Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d‐dimensional many‐body Hamiltonians with short‐range interaction at we have that the additive entropy (i.e., ln L for and for ), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy i.e., it is extensive. The present discussion neatly illustrates that entropic additivity and entropic extensivity are quite different properties, even if they essentially coincide in the presence of short‐range correlations.

Correlations in superstatistical systems
View Description Hide DescriptionWe review some of the statistical properties of higher‐dimensional superstatistical stochastic models. As an example, we analyse the stochastic properties of a superstatistical model of 3‐dimensional Lagrangian turbulence, and compare with experimental data. Excellent agreement is obtained for various measured quantities, such as acceleration probability densities, Lagrangian scaling exponents, correlations between acceleration components, and time decay of correlations. We comment on how to proceed from superstatistics to a thermodynamic description.

Entropies for complex systems: generalized‐generalized entropies
View Description Hide DescriptionMany complex systems are characterized by non‐Boltzmann distribution functions of their statistical variables. If one wants to—justified or not—hold on to the maximum entropy principle for complex statistical systems (non‐Boltzmann) we demonstrate how the corresponding entropy has to look like, given the form of the corresponding distribution functions. By two natural assumptions that (i) the maximum entropy principle should hold and that (ii) entropy should describe the correct thermodynamics of a system (which produces non‐Boltzmann distributions) the existence of a class of fully consistent entropies can be deduced. Classical Boltzmann‐Gibbs entropy is recovered as a special case for the observed distribution being the exponential, Tsallis entropy is the special case for q‐exponential observations.

Tsallis distribution from minimally selected order statistics
View Description Hide DescriptionWe demonstrate that selection of the minimal value of ordered variables leads in a natural way to its distribution being given by the Tsallis distribution, the same as that resulting from Tsallis nonextensive statistics. The possible application of this result to the multiparticle production processes is indicated.

A rooted tree whose lower bound of average description length is given by Tsallis entropy
View Description Hide DescriptionShannon additivity, one of the Shannon‐Khinchin axioms, determines a lower bound of average code length of a D‐ary code tree. As its generalization, the generalized Shannon additivity is applied to determining a lower bound of average description length of a rooted tree which we call “q‐generalized D‐ary code tree”. The generalized Shannon additivity is one of the generalized Shannon‐Khinchin axioms for Tsallis entropy. This reveals that Tsallis entropy is a lower bound of average description length for the q‐generalized D‐ary code tree.

Generalised thermostatistics using hyperensembles
View Description Hide DescriptionThe hyperensembles, introduced by Crooks in a context of non‐equilibrium statistical physics, are considered here as a tool for systems in equilibrium. Simple examples like the ideal gas, the mean‐field model, and the Ising interaction on small square lattices, are worked out to illustrate the concepts.

Projections maximizing Tsallis entropy
View Description Hide DescriptionWe consider a small closed physical systems. It has fixed energy and if the energy is the sum of squares of its generale coordinates then the distribution of any single coordinate will follow a distribution that maximizes Tsallis entropy. For many aspects of entropy can be discussed for such small systems without going to the thermodynamical limit. By letting the number of degrees of freedom tend to infinity the ordinary entropy formalism is recovered.

Combinatorial entropy for distinguishable entities in indistinguishable states
View Description Hide DescriptionThe combinatorial basis of entropy by Boltzmann can be written where H is the dimensionless entropy of a system, per unit entity, N is the number of entities and W is the number of ways in which a given realization of the system can occur, known as its statistical weight. Maximizing the entropy (“MaxEnt”) of a system, subject to its constraints, is then equivalent to choosing its most probable (“MaxProb”) realization. For a system of distinguishable entities and states, W is given by the multinomial weight, and H asymptotically approaches the Shannon entropy. In general, however, W need not be multinomial, leading to different entropy measures. This work examines the allocation of distinguishable entities to non‐degenerate or equally degenerate, indistinguishable states. The non‐degenerate form converges to the Shannon entropy in some circumstances, whilst the degenerate case gives a new entropy measure, a function of a multinomial coefficient, coding parameters, and Stirling numbers of the second kind.

Exponential families and MaxEnt calculations for entropy measures of statistical physics
View Description Hide DescriptionFor a wide range of entropy measures, easy calculation of equilibria is possible using a principle of Game Theoretical Equilibrium related to Jaynes Maximum Entropy Principle. This follows previous work of the author and relates to Naudts [1], [2], and, partly, Abe and Bagci [3].

Some aspects of the dynamics towards the supercycle attractors and their accumulation point, the Feigenbaum attractor
View Description Hide DescriptionRecently, arXiv:0706.4415, arXiv:0706.4422, we reported novel details about the dynamical properties of the family of periodic superstable cycles in unimodal maps and of their accumulation point, the Feigenbaum attractor. Here we append results regarding these properties that offer further information about the intricate features that these special attractors generate.

The Hamiltonian Mean Field model: anomalous or normal diffusion?
View Description Hide DescriptionWe consider the out‐of‐equilibrium dynamics of the Hamiltonian Mean Field (HMF) model, by focusing in particular on the properties of single‐particle diffusion. As we shall here demonstrate analytically, if the autocorrelation of momenta in the so‐called quasi‐stationary states can be fitted by a q‐exponential, then diffusion ought to be normal for at variance with the interpretation of the numerical experiments proposed in Ref. [1].

Anomalous diffusion and quasistationarity in the HMF model
View Description Hide DescriptionWe explore the quasistationary regime of the Hamiltonian Mean Field Model (HMF) showing that at least three different classes of events exist, with a different diffusive behavior and with a relative frequency which depends on the size of the system. Along the same line of a recent work [1], these results indicate that one must be very careful in exchanging time averages with ensemble averages during the non‐ergodic metastable regime and at the same time they emphasize the role of finite size effects in the evaluation of the diffusive properties of the system.

Generalized canonical formalism for the Fokker‐Planck equation with scale‐invariant solutions
View Description Hide DescriptionA canonical formalism is developed for implementing the dilatation symmetry to the Fokker‐Planck equation. A condition is derived, under which the equation admits a nonstationary scale‐invariant solution.

General properties of nonlinear mean field Fokker‐Planck equations
View Description Hide DescriptionRecently, several authors have tried to extend the usual concepts of thermodynamics and kinetic theory in order to deal with distributions that can be non‐Boltzmannian. For dissipative systems described by the canonical ensemble, this leads to the notion of nonlinear Fokker‐Planck equation (T.D. Frank, Non Linear Fokker‐Planck Equations , Springer, Berlin, 2005). In this paper, we review general properties of nonlinear mean field Fokker‐Planck equations, consider the passage from the generalized Kramers to the generalized Smoluchowski equation in the strong friction limit, and provide explicit examples for Boltzmann, Tsallis and Fermi‐Dirac entropies.