Boltzmann-Gibbs Statistical Mechanics Research Articles

Astrophysical thermonuclear functions for Boltzmann–Gibbs statistics and Tsallis statistics

We present an analytic proof of the integrals for astrophysical thermonuclear functions which are derived on the basis of Boltzmann–Gibbs statistical mechanics. Among the four different cases of astrophysical thermonuclear functions, those with a depleted high-energy tail and a cut-off at high energies find a natural interpretation in q-statistics.

Ubiquity of metastable-to-stable crossover in weakly chaotic dynamical systems

We present a comparative study of several dynamical systems of increasing complexity, namely, the logistic map with additive noise, one, two and many globally coupled standard maps, and the Hamiltonian mean field model (i.e., the classical inertial infinitely ranged ferromagnetically coupled XY spin model). We emphasize the appearance, in all of these systems, of metastable states and their ultimate crossover to the equilibrium state. We comment on the underlying mechanisms responsible for these phenomena (weak chaos) and compare common characteristics. We point out that this ubiquitous behavior appears to be associated to the features of the nonextensive generalization of the Boltzmann–Gibbs statistical mechanics.

Nonextensive statistical mechanics: A brief introduction

Boltzmann-Gibbs statistical mechanics is based on the entropy $S_{BG}=-k \sum_{i=1}^W p_i \ln p_i$. It enables a successful thermal approach of ubiquitous systems, such as those involving short-range interactions, markovian processes, and, generally speaking, those systems whose dynamical occupancy of phase space tends to be ergodic. For systems whose microscopic dynamics is more complex, it is natural to expect that the dynamical occupancy of phase space will have a less trivial structure, for example a (multi)fractal or hierarchical geometry. The question naturally arises whether it is possible to study such systems with concepts and methods similar to those of standard statistical mechanics. The answer appears to be {\it yes} for ubiquitous systems, but the concept of entropy needs to be adequately generalized. Some classes of such systems can be satisfactorily approached with the entropy $S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$ (with $q \in \cal R$, and $S_1 =S_{BG}$). This theory is sometimes referred in the literature as {\it nonextensive statistical mechanics}. We provide here a brief introduction to the formalism, its dynamical foundations, and some illustrative applications. In addition to these, we illustrate with a few examples the concept of {\it stability} (or {\it experimental robustness}) introduced by B. Lesche in 1982 and recently revisited by S. Abe.

Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann–Gibbs statistical mechanics [based on S1≡−k∫du p(u)ln p(u)]. Similarly, other classes of models point toward nonextensive statistical mechanics [based on Sq≡k[1−∫du[p(u)]q]/[q−1], where the value of the entropic index q∈R depends on the specific model]. We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the type u̇=f(u)+g(u)ξ(t)+η(t), where ξ(t) and η(t) are independent zero-mean Gaussian white noises with respective amplitudes M and A. This leads to the Fokker–Planck equation ∂tP(u,t)=−∂u[f(u)P(u,t)]+M∂u{g(u)∂u[g(u)P(u,t)]}+A∂uuP(u,t). Whenever the deterministic drift is proportional to the noise induced one, i.e., f(u)=−τg(u)g′(u), the stationary solution is shown to be P(u,∞)∝{1−(1−q)β[g(u)]2}1/(1−q) [with q≡(τ+3M)/(τ+M) and β=(τ+M/2A)]. This distribution is precisely the one optimizing Sq with the constraint 〈[g(u)]2〉q≡{∫du [g(u)]2[P(u)]q}/{∫du [P(u)]q}=const. We also introduce and discuss various characterizations of the width of the distributions.

Fluxes of cosmic rays: a delicately balanced stationary state

The analysis of cosmic rays fluxes as a function of energy reveals a knee slightly below 10 16 eV and an ankle close to 10 19 eV. Their physical origins remain up to now quite enigmatic; in particular, no elementary process is known which occurs at energies close to 10 16 eV. We propose a phenomenological approach along the lines of nonextensive statistical mechanics, a formalism which contains Boltzmann–Gibbs statistical mechanics as a particular case. The knee then appears as a crossover between two fractal-like thermal regimes, the crossover being caused by process occurring at energies ten million times lower than that of the knee, in the region of the quark hadron transition (≃10 9 eV). This opens the door to an unexpected standpoint for further clarifying the phenomenon.

Nonextensive statistical mechanics and economics

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann–Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accommodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy S BG=−k∑ ip i ln p i , and typically provides exponential laws for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form S q=k(1−∑ i p i q)/(q−1) (with S 1= S BG ), and typically provides, for the same type of description, (asymptotic) power laws.

REMARKS ON THE NONUNIVERSALITY OF BOLTZMANN-GIBBS STATISTICAL MECHANICS

How general are Boltzmann-Gibbs statistical mechanics and standard Thermodynamics? What classes of systems have thermostatistical properties, in particular equilibrium properties, that are correctly described by these formalisms? The answer is far from trivial. It is, however, clear today that these relevant and popular formalisms are not universal, this is to say that they have a domain of validity that it would be important to define precisely. A few remarks on these questions are herein presented, in particular in relation to nonextensive statistical mechanics, introduced a decade ago with the aim to cover at least some (but most probably not all) of the natural systems that are out of this domain of validity.

Nonextensive Statistical Mechanics: Some Links with Astronomical Phenomena

A variety of astronomical phenomena appear to not satisfy the ergodic hypothesis in the relevant stationary state, if any. As such, there is no reason for expecting the applicability of Boltzmann-Gibbs (BG) statistical mechanics. Some of these phenomena appear to follow, instead, nonextensive statistical mechanics. In the same manner that the BG formalism is based on the entropy $S_{BG}=-k \sum_i p_i \ln p_i$, the nonextensive one is based on the form $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$). The stationary states of the former are characterized by an {\it exponential} dependence on the energy, whereas those of the latter are characterized by an (asymptotic) {\it power-law}. A brief review of this theory is given here, as well as of some of its applications, such as the solar neutrino problem, polytropic self-gravitating systems, galactic peculiar velocities, cosmic rays and some cosmological aspects. In addition to these, an analogy with the Keplerian elliptic orbits {\it versus} the Ptolemaic epicycles is developed, where we show that optimizing $S_q$ with a few constraints is equivalent to optimizing $S_{BG}$ with an infinite number of constraints.

Statistical mechanics in the context of special relativity.

In Ref. [Physica A 296, 405 (2001)], starting from the one parameter deformation of the exponential function exp(kappa)(x)=(sqrt[1+kappa(2)x(2)]+kappax)(1/kappa), a statistical mechanics has been constructed which reduces to the ordinary Boltzmann-Gibbs statistical mechanics as the deformation parameter kappa approaches to zero. The distribution f=exp(kappa)(-beta E+betamu) obtained within this statistical mechanics shows a power law tail and depends on the nonspecified parameter beta, containing all the information about the temperature of the system. On the other hand, the entropic form S(kappa)= integral d(3)p(c(kappa) f(1+kappa)+c(-kappa) f(1-kappa)), which after maximization produces the distribution f and reduces to the standard Boltzmann-Shannon entropy S0 as kappa-->0, contains the coefficient c(kappa) whose expression involves, beside the Boltzmann constant, another nonspecified parameter alpha. In the present effort we show that S(kappa) is the unique existing entropy obtained by a continuous deformation of S0 and preserving unaltered its fundamental properties of concavity, additivity, and extensivity. These properties of S(kappa) permit to determine unequivocally the values of the above mentioned parameters beta and alpha. Subsequently, we explain the origin of the deformation mechanism introduced by kappa and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter kappa which results to depend on the light speed c and reduces to zero as c--> infinity recovering in this way the ordinary statistical mechanics and thermodynamics. The statistical mechanics here presented, does not contain free parameters, preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural, economic, and social sciences. Finally, in order to test the correctness and predictability of the theory, as working example we consider the cosmic rays spectrum, which spans 13 decades in energy and 33 decades in flux, finding a high quality agreement between our predictions and observed data.

Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics.

We uncover the dynamics at the chaos threshold mu infinity of the logistic map and find that it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for mu<mu infinity. We corroborate this structure analytically via the Feigenbaum renormalization-group (RG) transformation and find that the sensitivity to initial conditions has precisely the form of a q exponential, of which we determine the q index and the q-generalized Lyapunov coefficient lambda(q). Our results are an unequivocal validation of the applicability of the nonextensive generalization of Boltzmann-Gibbs statistical mechanics to critical points of nonlinear maps.

Mixing and equilibration: protagonists in the scene of nonextensive statistical mechanics

After a brief review of the present status of nonextensive statistical mechanics, we present a conjectural scenario where mixing (characterized by the entropic index q mix ⩽1) and equilibration (characterized by the entropic index q eq ⩾1) play central and inter-related roles, and appear to determine a priori the values of the relevant indices of the formalism. Boltzmann–Gibbs statistical mechanics is recovered as the q mix = q eq =1 particular case.

Tsallis' Nonextensive Statistical Mechanics and Pure-Electron Plasma.

Boltzmann-Gibbs statistical mechanics is known to exhibit fundamental difficulties when a system under consideration contains long-range interactions. Tsallis' nonextensive statistical mechanics offers a consistent theoretical framework for treatment of such a system. In this article, an approach to statistically describing pure-electron plasma based on the Tsallis entropy is reviewed and related problems are discussed.

Phase transitions in nonextensive spin systems.

The spherical spin model with infinite-range ferromagnetic interactions is investigated analytically in the framework of nonextensive thermostatics generalizing the Boltzmann-Gibbs statistical mechanics. We show that for repulsive correlations, a weak-ferromagnetic phase develops. There is a tricritical point separating para-, weak-ferro, and ferro regimes. The transition from paramagnetic to weak-ferromagnetic phase is an unusual first-order phase transition in which a discontinuity of the averaged order parameter appears, even for finite number of spins. This result puts in a different way the question of the stability of critical phenomena with respect to the long-ranged correlations.

On mixing and metaequilibrium in nonextensive systems

The analytical and computational studies of various isolated classical Hamiltonian systems including long-range interactions suggest that the N→∞ and t→∞ limits do not commute for entire classes of initial conditions. This is, for instance, the case for inertial planar rotators whenever the time evolution is started with the so-called waterbag distribution for velocities and full parallelism for the angles. For fixed N, after a transient, a long and robust anomalous plateau can emerge as time goes on whose velocity distribution is not the Maxwellian one; at later times, the system eventually crosses over onto the usual, Maxwellian distribution. The duration of the plateau diverges with N. This plateau can be considered as a metaequilibrium (or metastable) state, and its description might be in the realm of nonextensive statistical mechanics (for which the entropic index q≠1), whereas at later times the description is well done by the usual Boltzmann–Gibbs statistical mechanics ( q=1). The purpose of these lines is to present a scenario for the mixing properties (i.e., sensitivity to the initial conditions) which is consistent with the observations just mentioned.

A nonextensive critical phenomenon scenario for quantum entanglement

We discuss the paradigmatic bipartite spin- 1 2 system having the probabilities (1+3 x)/4 of being in the Einstein–Podolsky–Rosen fully entangled state |Ψ −〉≡1/ 2 (|↑〉 A|↓〉 B−|↓〉 A|↑〉 B) and 3(1− x)/4 of being orthogonal. This system is known to be separable if and only if x⩽ 1 3 (Peres criterion). This critical value has been recently recovered by Abe and Rajagopal through the use of the nonextensive entropic form S q≡(1−Tr ρ q)/(q−1) (q∈ R; S 1=−Tr ρ ln ρ) which has enabled a current generalization of Boltzmann–Gibbs statistical mechanics. This result has been enrichened by Lloyd, Baranger and one of the present authors by proposing a critical-phenomenon-like scenario for quantum entanglement. Here, we further illustrate and discuss this scenario through the calculation of some relevant quantities.

Nuclear multifragmentation in nonextensive statistics: canonical formulation

We apply the canonical quantum statistical model of nuclear multifragmentation generalized in the framework of recently proposed Tsallis nonextensive thermostatistics for the description of the nuclear multifragmentation process. The test calculation in the system with A = 197 nucleons shows strong modification of the "critical" behavior associated with the nuclear liquid-gas phase transition for small deviations from the conventional Boltzmann-Gibbs statistical mechanics.

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1 Ȣ q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law 1/(1 − q) = 1/α min − 1/α max, where α min and α max are the extreme values appearing in the multifractal f(α) function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d f equals unity for all z in contrast with q which does depend on z, it becomes clear that d f plays no major role in the sensitivity to the initial conditions.

Nonextensive statistics: theoretical, experimental and computational evidences and connections

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present effort, after introducing some historical background, we briefly describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. In addition to these, here and there we point out various (possibly) relevant questions, whose answer would certainly clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.

Departure from Boltzmann-Gibbs statistics makes the hydrogen-atom specific heat a computable quantity

The specific heat of the (nonionized) hydrogen atom in free space cannot be calculated within Boltzmann-Gibbs statistical mechanics essentially because its partition function diverges. We show that a recently introduced generalized formalism can overcome this difficulty for q1 (the index q characterizes the statistics; Boltzmann-Gibbs corresponds to q=1).

Infinite-range Ising ferromagnet: Thermodynamic limit within Generalized Statistical Mechanics

We first discuss, for a variety of similar systems, the physical need for departure from Boltzmann-Gibbs statistical mechanics and thermodynamics. Then, we numerically discuss the infinite-range spin-1/2 Ising ferromagnet within the recently generalized statistical mechanics (canonical ensemble). Through the specific heat, we exhibit (for the first time, as far as we know, for an interacting system) that the thermodynamic limit is well defined.