Boltzmann-Gibbs Research Articles

On thermodynamic self-consistency of generic axiomatic-nonextensive statistics

Generic axiomatic-nonextensive statistics introduces two asymptotic properties, to each of which a scaling function is assigned. The first and second scaling properties are characterized by the exponents c and d, respectively. In the thermodynamic limit, a grand-canonical ensemble can be formulated. The thermodynamic properties of a relativistic ideal gas of hadron resonances are studied, analytically. It is found that this generic statistics satisfies the requirements of the equilibrium thermodynamics. Essential aspects of the thermodynamic self-consistency are clarified. Analytical expressions are proposed for the statistical fits of various transverse momentum distributions measured in most-central collisions at different collision energies and colliding systems. Estimations for the freezeout temperature (Tch) and the baryon chemical potential (μb) and the exponents c and d are determined. The earlier are found compatible with the parameters deduced from Boltzmann-Gibbs (BG) statistics (extensive), while the latter refer to generic nonextensivities. The resulting equivalence class (c,d) is associated with stretched exponentials, where Lambert function reaches its asymptotic stability. In some measurements, the resulting nonextensive entropy is linearly composed on extensive entropies. Apart from power-scaling, the particle ratios and yields are excellent quantities to highlighting whether the particle production takes place (non)extensively. Various particle ratios and yields measured by the STAR experiment in central collisions at 200, 62.4 and 7.7 GeV are fitted with this novel approach. We found that both c and d < 1, i.e. referring to neither BG- nor Tsallis-type statistics, but to (c,d)-entropy, where Lambert functions exponentially rise. The freezeout temperature and baryon chemical potential are found comparable with the ones deduced from BG statistics (extensive). We conclude that the particle production at STAR energies is likely a nonextensive process but not necessarily BG or Tsallis type.

A bag-of-paths framework for network data analysis

This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs–Boltzmann distribution is defined over a bag of paths in a network, that is, on a representation that considers all paths independently. We show that, under this distribution, the probability of drawing a path connecting two nodes can easily be computed in closed form by simple matrix inversion. This probability captures a notion of relatedness, or more precisely accessibility, between nodes of the graph: two nodes are considered as highly related when they are connected by many, preferably low-cost, paths. As an application, two families of distances between nodes are derived from the BoP probabilities. Interestingly, the second distance family interpolates between the shortest-path distance and the commute-cost distance. In addition, it extends the Bellman–Ford formula for computing the shortest-path distance in order to integrate sub-optimal paths (exploration) by simply replacing the minimum operator by the soft minimum operator. Experimental results on semi-supervised classification tasks show that both of the new distance families are competitive with other state-of-the-art approaches. In addition to the distance measures studied in this paper, the bag-of-paths framework enables straightforward computation of many other relevant network measures.

Generalised partition functions: inferences on phase space distributions

Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.

Refined two-index entropy and multiscale analysis for complex system

As a fundamental concept in describing complex system, entropy measure has been proposed to various forms, like Boltzmann–Gibbs (BG) entropy, one-index entropy, two-index entropy, sample entropy, permutation entropy etc. This paper proposes a new two-index entropy Sq,δ and we find the new two-index entropy is applicable to measure the complexity of wide range of systems in the terms of randomness and fluctuation range. For more complex system, the value of two-index entropy is smaller and the correlation between parameter δ and entropy Sq,δ is weaker. By combining the refined two-index entropy Sq,δ with scaling exponent h(δ), this paper analyzes the complexities of simulation series and classifies several financial markets in various regions of the world effectively.

Inter-occurrence times and universal laws in finance, earthquakes and genomes

A plethora of natural, artificial and social systems exist which do not belong to the Boltzmann–Gibbs (BG) statistical-mechanical world, based on the standard additive entropy SBG and its associated exponential BG factor. Frequent behaviors in such complex systems have been shown to be closely related to q-statistics instead, based on the nonadditive entropy Sq (with S1=SBG), and its associated q-exponential factor which generalizes the usual BG one. In fact, a wide range of phenomena of quite different nature exist which can be described and, in the simplest cases, understood through analytic (and explicit) functions and probability distributions which exhibit some universal features. Universality classes are concomitantly observed which can be characterized through indices such as q. We will exhibit here some such cases, namely concerning the distribution of inter-occurrence (or inter-event) times in the areas of finance, earthquakes and genomes.

Nonextensive Nambu-Jona-Lasinio Model of QCD matter

We present a revisited version of the nonextensive QCD-based Nambu - Jona-Lasinio (NJL) model describing the behavior of strongly interacting matter proposed by us some time ago. As before, it is based on the nonextensive generalization of the Boltzmann-Gibbs (BG) statistical mechanics used in the NJL model to its nonextensive version based on Tsallis statistics, but this time it fulfils the basic requirements of thermodynamical consistency. Different ways in which this can be done, connected with different possible choices of the form of the corresponding nonextensive entropies, are presented and discussed in detail. The corresponding results are compared, discussed and confronted with previous findings.

A method that reveals the multi-level ultrametric tree hidden in p -spin-glass-like systems

.In the study of disordered models like spin glasses the key object of interest is the rugged energy hypersurface defined in configuration space. The statistical mechanics calculation of the Gibbs–Boltzmann partition function gives the information necessary to understand the equilibrium behavior of the system as a function of the temperature but is not enough if we are interested in the more general aspects of the hypersurface: it does not give us, for instance, the different degrees of ruggedness at different scales. In the context of the replica symmetry breaking (RSB) approach we discuss here a rather simple extension that can provide a much more detailed picture. The attractiveness of the method relies on the fact that it is conceptually transparent and the additional calculations are rather straightforward. We think that this approach reveals an ultrametric organisation with many levels in models like p-spin glasses when we include saddle points. In this first paper we present detailed calculations for the spherical p-spin glass model where we discover that the corresponding decreasing Parisi function q(x) codes this hidden ultrametric organisation.

Beyond the Shannon–Khinchin formulation: The composability axiom and the universal-group entropy

The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann–Gibbs (BG) entropy and could be applicable in thermodynamics, quantum mechanics and information theory. In Khinchin (1957), by extending previous ideas of Shannon (1948) and Shannon and Weaver (1949), Khinchin proposed a characterization of the BG entropy, based on four requirements, nowadays known as the Shannon–Khinchin (SK) axioms.The purpose of this paper is twofold. First, we show that there exists an intrinsic group-theoretical structure behind the notion of entropy. It comes from the requirement of composability of an entropy with respect to the union of two statistically independent systems, that we propose in an axiomatic formulation. Second, we show that there exists a simple universal family of trace-form entropies. This class contains many well known examples of entropies and infinitely many new ones, a priori multi-parametric. Due to its specific relation with Lazard’s universal formal group of algebraic topology, the new general entropy introduced in this work will be called the universal-group entropy. A new example of multi-parametric entropy is explicitly constructed.

Portevin–Le Chatelier effect and Tsallis nonextensive statistics

In this paper, we estimate Tsallis’ entropic index q≡qstat (stat denotes stationary states) for various stress serration time series of Cu–15%Al alloys corresponding to different deformation temperatures and types (A, B, C) of Portevin–Le Chatelier (PLC) bands. Tsallis statistical analysis is also conducted on experimental data derived from correlation diagrams focusing on PLC type C bands’ spatial organization at the surface of various specimens. The results concerning the stress serrations analysis reveal the non-Gaussian (Tsallis q-Gaussian), nonextensive, sub-additive, character of the underlying dynamics of the PLC type-A bands (q>1) indicating dynamics at the edge of chaos, with global long range correlations and power law scaling. Similarly, for PLC type B bands, a Tsallis q-Gaussian, non-extensive, super-additive, statistical profile was found with q<1, while for the PLC type C bands, a near Boltzmann–Gibbs (BG) statistical character was verified indicating Gaussian dynamics with short (weak) range correlations. Finally, Tsallis q-index for the spatial data of PLC type C bands was found above unity (q>1) indicating that type C bands are not randomly nucleated in the specimen (e.g. yielding a Gaussian distribution) but there exist spatial coupling and long range deformation correlations in the PLC type C bands spatial localization.

Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics

We study a long-range–interaction generalisation of the one-dimensional Fermi-Pasta-Ulam (FPU) β-model, by introducing a quartic interaction coupling constant that decays as (with strength characterised by b > 0). In the limit we recover the original FPU model. Through molecular dynamics we show that i) for the maximal Lyapunov exponent remains finite and positive for an increasing number of oscillators N, whereas, for , it asymptotically decreases as ; ii) the distribution of time-averaged velocities is Maxwellian for α large enough, whereas it is well approached by a q-Gaussian, with the index monotonically decreasing from about 1.5 to 1 (Gaussian) when α increases from zero to close to one. For α small enough, a crossover occurs at time tc from q-statistics to Boltzmann-Gibbs (BG) thermostatistics, which defines a “phase diagram” for the system with a linear boundary of the form with and , in such a way that the q = 1 (BG) behaviour dominates in the ordering, while in the ordering q > 1 statistics prevails.

Magnetization dynamics: path-integral formalism for the stochastic Landau–Lifshitz–Gilbert equation

We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau–Lifshitz–Gilbert equation proposed by Brown (1963 Phys. Rev. 130 1677), with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized equation ensures the conservation of the magnetization modulus and the approach to the Gibbs–Boltzmann equilibrium in the absence of non-potential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.

Spatially adaptive stochastic methods for fluid–structure interactions subject to thermal fluctuations in domains with complex geometries

We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid–structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation–dissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs–Boltzmann distribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with O(N) computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications.

Generalized Statistical Mechanics at the Onset of Chaos

Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann–Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov exponent vanishes and ergodic and mixing properties cease to hold. There are three well-known routes to chaos in these deterministic dissipative systems, period-doubling, quasi-periodicity and intermittency, which provide the setting in which to explore the limit of validity of the standard BG structure. It has been shown that there is a rich and intricate behavior for both the dynamics within and towards the attractors at the onset of chaos and that these two kinds of properties are linked via generalized statistical-mechanical expressions. Amongst the topics presented are: (i) permanently growing sensitivity fluctuations and their infinite family of generalized Pesin identities; (ii) the emergence of statistical-mechanical structures in the dynamics along the routes to chaos; (iii) dynamical hierarchies with modular organization; and (iv) limit distributions of sums of deterministic variables. The occurrence of generalized entropy properties in condensed-matter physical systems is illustrated by considering critical fluctuations, localization transition and glass formation. We complete our presentation with the description of the manifestations of the dynamics at the transitions to chaos in various kinds of complex systems, such as, frequency and size rank distributions and complex network images of time series. We discuss the results.

A Fokker–Planck equation for a piecewise entropy functional defined in different space domains. An application to solute partitioning at the membrane–water interface

A nonlinear Fokker–Planck equation is proposed for a system subject to different statistics (in the present study, the Gibbs–Boltzmann and Fermi–Dirac statistics) defined in different contiguous regions of space. We solved the time-dependent mono-dimensional equation numerically, and solved the time-independent mono-dimensional equation analytically under the effect of a generic external potential equation. These zones are connected by a sharp but continuous transition region. Accurate numerical procedures ensure the convergence of the Fokker–Planck equation in the transition layer. We applied our general procedure to investigate both the stationary and the time-dependent kinetics of solute partitioning between aqueous and membrane phases. Because of the relative volumes of solute, water, and lipid (Vsolute≈Vwater<<Vlipid), the mixing entropy functional in water contains both solute and solvent contributions, while within the membrane only the solute entropy plays a significant role. Also, the potential term differs in space due to the solute interactions with different environments. Lastly, we added the effect of an electrostatic potential (occurring in all membrane systems) localized at the water interface which may deplete or increase the interfacial solute concentration. Rather surprisingly we found a strong coupling between the surface potential and the imposed asymmetric statistics. The effects are relevant when entropic and potential contributions are comparable; otherwise, the standard Boltzmannian behavior is recovered.

Black hole thermodynamical entropy

As early as 1902, Gibbs pointed out that systems whose partition function diverges, e.g. gravitation, lie outside the validity of the Boltzmann–Gibbs (BG) theory. Consistently, since the pioneering Bekenstein–Hawking results, physically meaningful evidence (e.g., the holographic principle) has accumulated that the BG entropy S BG of a (3+1) black hole is proportional to its area L 2 (L being a characteristic linear length), and not to its volume L 3. Similarly it exists the area law, so named because, for a wide class of strongly quantum-entangled d-dimensional systems, S BG is proportional to lnL if d=1, and to L d−1 if d>1, instead of being proportional to L d (d≥1). These results violate the extensivity of the thermodynamical entropy of a d-dimensional system. This thermodynamical inconsistency disappears if we realize that the thermodynamical entropy of such nonstandard systems is not to be identified with the BG additive entropy but with appropriately generalized nonadditive entropies. Indeed, the celebrated usefulness of the BG entropy is founded on hypothesis such as relatively weak probabilistic correlations (and their connections to ergodicity, which by no means can be assumed as a general rule of nature). Here we introduce a generalized entropy which, for the Schwarzschild black hole and the area law, can solve the thermodynamic puzzle.

DIRECTED RANDOM MARKETS: CONNECTIVITY DETERMINES MONEY

Boltzmann–Gibbs (BG) distribution arises as the statistical equilibrium probability distribution of money among the agents of a closed economic system where random and undirected exchanges are allowed. When considering a model with uniform savings in the exchanges, the final distribution is close to the gamma family. In this paper, we implement these exchange rules on networks and we find that these stationary probability distributions are robust and they are not affected by the topology of the underlying network. We introduce a new family of interactions: random but directed ones. In this case, it is found the topology to be determinant and the mean money per economic agent is related to the degree of the node representing the agent in the network. The relation between the mean money per economic agent and its degree is shown to be linear.

A unified probabilistic framework for automatic 3D facial expression analysis based on a Bayesian belief inference and statistical feature models

Textured 3D face models capture precise facial surfaces along with the associated textures, making it possible for an accurate description of facial activities. In this paper, we present a unified probabilistic framework based on a novel Bayesian Belief Network (BBN) for 3D facial expression and Action Unit (AU) recognition. The proposed BBN performs Bayesian inference based on Statistical Feature Models (SFM) and Gibbs–Boltzmann distribution and feature a hybrid approach in fusing both geometric and appearance features along with morphological ones. When combined with our previously developed morphable partial face model (SFAM), the proposed BBN has the capacity of conducting fully automatic facial expression analysis. We conducted extensive experiments on the two public databases, namely the BU-3DFE dataset and the Bosphorus dataset. When using manually labeled landmarks, the proposed framework achieved an average recognition rate of 94.2% and 85.6% for the 7 and 16AU on face data from the Bosphorus dataset respectively, and 89.2% for the six universal expressions on the BU-3DFE dataset. Using the landmarks automatically located by SFAM, the proposed BBN still achieved an average recognition rate of 84.9% for the six prototypical facial expressions. These experimental results demonstrate the effectiveness of the proposed approach and its robustness in landmark localization errors.

QUANTUM FLUCTUATION DYNAMICS DURING THE TRANSITION OF A MESOSCOPIC BOSONIC GAS INTO A BOSE–EINSTEIN CONDENSATE

The condensate number distribution during the transition of a dilute, weakly interacting gas of N = 200 bosonic atoms into a Bose–Einstein condensate is modeled within number conserving master equation theory of Bose–Einstein condensation. Initial strong quantum fluctuations occuring during the exponential cycle of condensate growth reduce in a subsequent saturation stage, before the Bose gas finally relaxes towards the Gibbs–Boltzmann equilibrium.

An alternative to the conventional micro-canonical ensemble

The usual approach to the foundations of quantum statistical physics is based on the conventional micro-canonical ensemble as a starting point for deriving the Boltzmann–Gibbs (BG) equilibrium. It leaves, however, a number of conceptual and practical questions unanswered. Here we discuss these questions, thereby motivating the study of a natural alternative known as the quantum micro-canonical (QMC) ensemble. We present a detailed numerical study of the properties of the QMC ensemble for finite quantum systems, revealing good agreement with the existing analytical results for large quantum systems. We also propose the way to introduce analytical corrections accounting for finite-size effects. With the above corrections, the agreement between the analytical and the numerical results becomes very accurate. We demonstrate that the variance of energy fluctuations can be used to discriminate the QMC equilibrium from the BG equilibrium.

Rational Construction of Stochastic Numerical Methods for Molecular Sampling

In this article, we focus on the sampling of the configurational Gibbs–Boltzmann distribution, that is, the calculation of averages of functions of the position coordinates of a molecular N-body system modeled at constant temperature. Using the Baker–Campbell–Hausdorff expansion, we compare Langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property of a certain method in the high friction limit; this method, moreover, can be reduced to a simple modification of the Euler–Maruyama method for Brownian dynamics involving a non-Markovian (coloured noise) random process. In fully resolved (long run) molecular dynamics simulations, for our favored method, we observe up to two orders of magnitude improvement in the configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared with common alternatives.