Nonextensive Thermostatistics Research Articles

Nonextensive thermodynamics and glassy behaviour

An instructive and apparently simple model of fully-coupled rotators, the so-called Hamiltonian Mean Field (HMF) model, together with a generalized version with variable interaction range, have revealed a very complex out-of-equilibrium dynamics which can be considered paradigmatic for nonextensive systems. In this short paper we illustrate the interesting anomalous pre-equilibrium dynamics, focusing on the novel connections to the generalized nonextensive thermostatistics and the recent links to glassy systems.

Logarithmic diffusion and porous media equations: A unified description

In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a Lorentzian form, consequently this equation characterizes a superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.

Nonextensive triangle equality and other properties of Tsallis relative-entropy minimization

Kullback–Leibler relative-entropy has unique properties in cases involving distributions resulting from relative-entropy minimization. Tsallis relative-entropy is a one-parameter generalization of Kullback–Leibler relative-entropy in the nonextensive thermostatistics. In this paper, we present the properties of Tsallis relative-entropy minimization and present some differences with the classical case. In the representation of such a minimum relative-entropy distribution, we highlight the use of the q-product, an operator that has been recently introduced to derive the mathematical structure behind the Tsallis statistics. One of our main results is the generalization of triangle equality of relative-entropy minimization to the nonextensive case.

Nonextensive mean-field theory for ferroelectric nematic liquid crystal phases

Ferroelectric ordering in a nematic liquid crystal system is described using nonextensive thermostatistics. We use a mean-field potential, including effective potentials for both axial and polar interactions. By using a self-consistent numerical analysis, a complete q-dependent phase diagram is obtained as a function of the axial and polar interaction potential strengths. The phase transition behaviours and the generalized angular orientational distribution function of the molecules were investigated by studying the dependence of the polar and the axial order parameters on the reduced temperature for various values of the entropic index. This study can provide basic information to further detailed theoretical studies of molecular interactions.

Dark matter distribution function from non-extensive statistical mechanics

We present an analytical and numerical study of the velocity distribution function of self gravitating collisionless particles, which include dark matter and star clusters. We show that the velocity distribution derived through the Eddington’s formula is identical to the analytical one derived directly from the generalized entropy of non-extensive statistical mechanics. This implies that self gravitating collisionless structures are to be described by non-extensive thermo-statistics. We identify a connection between the density slope of dark matter structures, γ, from ρ∼r−γ, and the entropic index, q, from the generalized entropy, Sq. Our numerical result confirms the analytical findings of earlier studies and clarifies which is the correct connection between the density slope and the entropic index. We use this result to conclude that from a fundamental statistical mechanics point of view the central density slope of self gravitating collisionless dark matter structures is not constrained, and even cored dark matter structures are allowed with γ=0. We find that the outer density slope is bounded by γ=10/3.

Scaling relations in equilibrium nonextensive thermostatistics

The Euler and Gibbs–Duhem relations discussed in thermodynamics of ordinary extensive systems are shown to hold in the same forms also for nonextensive systems with long-range interactions with a novel interpretation of the entities appearing therein. In this way, Tsallis' conjecture for the scaling relations in equilibrium nonextensive thermostatistics is proved to be justified.

Cluster temperatures and non-extensive thermo-statistics

We propose a novel component to the understanding of the temperature structure of galaxy clusters which does not rely on any heating or cooling mechanism. The new ingredient is the use of non-extensive thermo-statistics which is based on the natural generalization of entropy for systems with long-range interactions. Such interactions include gravity and attraction or repulsion due to charges. We explain that there is growing theoretical indications for the need of this generalization for large cosmological structures. The observed pseudo temperature is generally different from the true thermodynamic temperature, and we clarify the connection between the two. We explain that this distinction is most important in the central part of the cluster where the density profile is most shallow. We show that the observed pseudo temperature may differ up to a factor 2/5 from the true thermodynamic temperature, either larger or smaller. In general the M–T and L–T relations will be affected, and the central DM slope derived through hydrostatic equilibrium may be either more shallow or steeper. We show how the true temperature can be extracted correctly either from the spectrum or from the shape of the Doppler broadening of spectral lines.

A nonextensive entropy path to probability distributions in solar wind turbulence

Abstract. The observed scale dependence of the probability distributions of the differences of characteristic solar wind variables is analyzed. Intermittency of the turbulent fluctuations at small-scale spatial separations is accompanied by strongly non-Gaussian distributions that turn into a normal distribution for large-scale separation. Conventional theoretical models are subject to insufficient physical justification since nonlocality in turbulence should be based on long-range interactions, provided recently by the bi-kappa distribution in the context of nonextensive thermo-statistics. Observed WIND and ACE probability distributions are accurately reproduced for different time lags by the one-parameter bi-kappa functional, a core-halo convolution, where kappa measures the degree of nonlocality or nonextensivity in the system. Gradual decoupling is obtained by enhancing the spatial separation scale corresponding to increasing kappa values, where a Gaussian is approached for infinite kappa. Consequently, long-range interactions introduced on the fundamental level of entropy generalization, are able to provide physically the source of the observed scale dependence of the turbulent fluctuations in the intermittent interplanetary medium.

The grand canonical ensemble in generalized thermostatistics

We show that one can formulate a grand canonical ensemble with an uncertain number ofdegrees of freedom without making specific assumptions about the canonicalensemble. Several choices of grand canonical entropy functional are considered.The ideal gas in non-extensive thermostatistics is discussed as an application.

Nonextensive thermostatistics and the H-theorem revisited

In this paper, we present a new derivation of the H-theorem and the corresponding collisional equilibrium velocity distributions, within the framework of Tsallis’ nonextensive thermostatistics. Unlike previous works, in our derivation we do not assume any modification on the functional form of Boltzmann's original “molecular chaos hypothesis”. Rather, we explicitly introduce into the collision scenario, the existence of statistical dependence between the molecules before the collision has taken place, through a conditional distribution f( v → 2| v → 1) . In this approach, different equilibrium scenarios emerge depending on the value of the nonextensive entropic parameter.

Why Tsallis’ statistics?

Non-extensive thermostatistics is a new paradigm statistical mechanics (SM) that has reached the rather mature age of 15 in 2003. It has considerably extended both SM formalism and its range of applicability. It constitutes one of physics’ new frontiers, with advances being made almost on a weekly basis. This is a good time to pause and reflect on its significance.

Nonextensivity and galaxy clustering in the Universe

We investigate two important questions about the use of the nonextensive thermostatistics (NETS) formalism in the context of nonlinear galaxy clustering in the Universe. Firstly, we define a quantitative criterion for justifying nonextensivity at different physical scales. Then, we discuss the physics behind the ansatz of the entropic parameter q( r). Our results suggest the approximate range where nonextensivity can be justified and, hence, give some support to the applicability of NETS to the study of large-scale structures.

The thermodynamic limit in the non-extensive thermostatistics

Thermodynamic limit is more than just: ‘the limit of the system when size tends to infinity’. This rough definition can be sufficient in the simplest cases, but precise definition of the thermodynamic limit involves subtle constraints in order to compare systems of various sizes N but in the similar thermodynamic state. We discuss in this paper how these constraints must be handled in the non-extensive thermostatistics, within the canonical ensemble. Role of the entropy index q on the proper definition of the thermodynamic limit, is emphasized. In particular, the only choice to get non-trivial thermodynamic limit is: N( q−1)= constant. It can lead to a steady state different from the Boltzmann–Gibbs thermodynamic limit. We show that the hypothesis of non-extensivity of the entropy is generally not so crucial for the infinite systems, with the notable exception of the critical systems.

CONTINUITY OF A CLASS OF ENTROPIES AND RELATIVE ENTROPIES

The present paper studies continuity of generalized entropy functions and relative entropies defined using the notion of a deformed logarithmic function. In particular, two distinct definitions of relative entropy are discussed. As an application, all considered entropies are shown to satisfy Lesche's stability condition. The entropies of Tsallis' non-extensive thermostatistics are taken as examples.

The need for alternative information measures

In 1948 Shannon advanced information theory (IT) as a new branch of mathematics and a powerful tool for understanding the intricacies of the communication process. Nine years later Jaynes’ conceived the maximum entropy principle and was able to shed with it much light onto statistical mechanics and thermodynamics. Shannon's logarithmic information measure, the MaxEnt protagonist, was successfully connected with the thermodynamic entropy. Since 1988 a new project, called nonextensive thermostatistics (NET), came into being after the pioneering work by Tsallis and collaborators. It has achieved today a remarkable degree of success, amid a not small degree of controversy. It is also a MaxEnt construction but uses a different information measure in the leading role. We advance here some epistemological considerations on the proper place of any information measure within a theoretical physics’ construct in order put NET into a more solid conceptual framework.

Description of quasi-equilibrium states of self-gravitating systems based on non-extensive thermostatistics

We present an application of Tsallis’ generalized entropy to analysis for thermodynamical nature of the self-gravitating stellar systems. Extremizing the Tsallis entropy, we obtain an equation of the state known as stellar polytrope. And a role of the extremes as a quasi-equilibrium is shown from results of N-body simulations.

The second law in nonextensive quantum thermostatistics for small systems

Nonextensive quantum thermostatistics is shown to be consistent with the second law of thermodynamics if the entropic index, q , is in the range (0,2]. Clausius’ inequality is derived by making use of the convexity property of the quantum q -divergence which is a nonadditive generalization of the ordinary quantum relative entropy.

A possible deformed algebra and calculus inspired in nonextensive thermostatistics

We present a deformed algebra related to the q-exponential and the q-logarithm functions that emerge from nonextensive statistical mechanics. We also develop a q-derivative (and consistently a q-integral) for which the q-exponential is an eigenfunction. The q-derivative and the q-integral have a dual nature, that is also presented.

Glassy dynamics in the HMF model

We discuss the glassy dynamics recently found in the meta-equilibrium quasi-stationary states (QSS) of the HMF model. The relevance of the initial conditions and the connection with Tsallis nonextensive thermostatistics is also addressed.

The second law in nonextensive quantum thermostatistics for small systems

Nonextensive quantum thermostatistics is shown to be consistent with the second law of thermodynamics if the entropic index, q, is in the range (0,2]. Clausius’ inequality is derived by making use of the convexity property of the quantum q-divergence which is a nonadditive generalization of the ordinary quantum relative entropy.