Boltzmann-Gibbs Entropy Research Articles

Nonextensive statistical mechanics for robust physical parameter estimation: the role of entropic index

The problem of inferencing parameters of complex systems from measured data has been extensively studied based on the inverse problem theory. Classically, an inverse problem is formulated as an optimisation task in which the objective function is based on the maximisation of Boltzmann–Gibbs entropy under appropriate constraints. In this classical framework, errors are assumed to follow a Gaussian-behaviour. However, in many situations, the errors are non-Gaussian and therefore the classical approach is predestined to failure. To mitigate the inverse-problem sensitivity to non-Gaussian errors, we have considered in this study a q-generalised objective function within the context of nonextensive statistical mechanics. In this regard, the errors are assumed to follow the q-Gaussian distribution, which arises from the maximisation of the nonadditive Tsallis entropy. We study the robustness properties of the q-objective function, which is a generalisation of the classical objective function, through the so-called influence function. In this work, we analyse and discuss from an analytical and numerical perspective, the role of the entropic index (q) of Tsallis entropy on the effectiveness and robustness in the inference of physical parameters from with strongly noisy-data. In this perspective, we show that exists an optimum value for the entropic index at the limit $$q \rightarrow 3$$ , which implies in $$\delta _q = 3 - q \approx 0$$ in the q-objective function. To validate our proposal, we consider a classic geophysical inverse problem by employing a realistic geological model and a data set contaminated by spikes.

Thermodynamics of structure-forming systems

Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here, we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations the approach is equivalent to the grand-canonical ensemble; for small systems we find significant deviations. We derive the detailed fluctuation theorem and Crooks’ work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed. We apply the results to several physical systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition.

Black holes thermodynamics from a dual Kaniadakis entropy

Kaniadakis entropy, an important extension of the Boltzmann-Gibbs entropy, was proposed some years ago. Inspired by a dual Renyi entropy, in this paper we have suggested a dual Kaniadakis entropy and we applied this approach to analyze the thermodynamics of black holes. We have obtained an equipartition theorem and, after that, we derived the temperature and the heat capacity. The values of κ and mass M will show if the black hole can be thermally stable or not.

Universal Statistical Laws for the Velocities of Collective Migrating Cells.

Migratory dynamics of collective cells is central to the morphogenesis of biological tissues. The statistical distribution of cell velocities in 2D confluent monolayers is measured through large-scale and long-term experiments of various cell types lying on different substrates. A linear relation is discovered between the variability and the mean of cell speeds during the jamming process of confluent cell monolayers, suggesting time-invariant distribution profile of cell velocities. It is further found that the probability density function of cell velocities obeys the non-canonical q-Gaussian statistics, regardless of cell types and substrate stiffness. It is the Tsallis entropy, instead of the classical Boltzmann-Gibbs entropy, that dictates the universal statistical laws of collective cell migration. The universal statistical law stems from cell-cell interactions, as demonstrated by the wound healing experiments. This previously unappreciated finding provides a linkage between cell-level heterogeneity and tissue-level ensembles in embryonic development and tumor growth.

Generalized Fokker-Planck equations derived from nonextensive entropies asymptotically equivalent to Boltzmann-Gibbs.

We derive generalized Fokker-Planck equations (FPEs) based on two nonextensive entropy measures S_{±} that depend exclusively on the probability. These entropies have been originally obtained from the superstatistics framework, therefore they regard nonequilibrium systems outlined by a long-term stationary state in view of a spatiotemporally fluctuating intensive quantity. Moreover, entropies S_{±} as well as Boltzmann-Gibbs (BG) entropy S_{B} both pertain to the same asymptotical equivalence class, thus suggesting that S_{±} could depict a consistent thermodynamic generalization of BG. For these reasons, we assert that transport phenomena to be accounted for by our models shall coincide with the portrait given by the conventional FPEs for systems comprehending short-range interactions or a high number of accessible microstates, whereas, for systems composed of a small number of microstates, or those with long-range interactions, the governing equations of motion are to be the FPEs here derived, as long as the system fulfills the attributes mentioned above. We discuss the anomalous diffusion exhibited by the two generalized FPEs and also present some numerical applications. In particular, we find that there are models regarding biological sciences, for the study of congregation and aggregation behavior, the structure of which coincides with the one of our models.

Inequality in energy consumption: statistical equilibrium or a question of accounting conventions?

Understanding inequality energy consumption at the global level delivers key insights for strategies to mitigate climate change. Recent contributions [4, 28, 48, 49] have studied energy inequality through the lens of maximum entropy. They claim a weighted international distribution of total primary energy demand should approach a Boltzmann-Gibbs maximum entropy equilibrium distribution in the form of an exponential distribution. This implies convergence to a Gini coefficient of 0.5 from above. The present paper challenges the validity of this claim and critically discusses the applicability of statistical equilibrium reasoning to economics from the viewpoint of social accounting. It is shown that the exponential distribution is only a robust candidate for a statistical equilibrium of energy inequality when employing one particular accounting convention for energy flows, the substitution method. But this method has become problematic with a higher renewable share in the international energy mix, and no other accounting method supports the claim of a convergence to a 0.5 Gini. We conclude that the findings based on maximum entropy reasoning are sensitive to accounting conventions and critically discuss the epistemological implications of this sensitivity for the use of maximum entropy approaches in social sciences.

Application of the nonlinear methods in pneumocardiogram signals.

In this work, the pneumocardiogram signals of nine rats were analysed by scale index, Boltzmann Gibbs entropy and maximum Lyapunov exponents. The scale index method, based on wavelet transform, was proposed for determining the degree of aperiodicity and chaos. It means that the scale index parameter is close to zero when the signal is periodic and has a value between zero and one when the signal is aperiodic. A new entropy calculation method by normalized inner scalogram was suggested very recently. In this work, we also used this method for the first time in an empirical data. We compared the both methods with maximum Lyapunov exponents and observed that using together the scale index and the entropy calculation method by normalized inner scalogram increases the reliability of the pneumocardiogram signal analysis. Thus, the analysis of the pneumocardiogram signals by those methods enables to compare periodical and/or nonlinear aspects for further understanding of dynamics of cardiorespiratory system.

Non-equilibrium statistical mechanical approach to the formation of non-Maxwellian electron distribution in space

Boltzmann-Gibbs (BG) entropy has additive and extensive properties, but for certain physical systems, such as those governed by long-range interactions – plasma or fully ionized gas being an example – it is speculated that the entropy must be non-additive and non-extensive. Because of the fact that Tsallis entropy possesses such characteristics, many spacecraft observations of charged particle distributions in space are interpreted with the conceptual framework based upon Tsallis statistical principles. This paper formulates the non-equilibrium statistical theory of space plasma, and it is shown that the steady state electrostatic turbulence in plasma coincides with the formation of non-Maxwellian electron distribution function known as the kappa distribution. The kappa distribution is equivalent to the q-Gaussian distribution in the Tsallis statistical theory, which represents the most probable state subject to Tsallis entropy. This finding represents an independent confirmation that the space plasma may indeed be governed by the Tsallis statistical principle.

Generalised asymptotic classes for additive and non-additive entropies

In the field of non-extensive statistical mechanics it is common to focus more attention on the family of parameter-dependent entropies rather than on those strictly depending only on the probability in which case there is no need to adjust a specific parameter. In particular, there exist two non-parametric entropy measures, , that resemble the Boltzmann-Gibbs (BG) entropy, SB, in the thermodynamic limit, whereas a difference between them arises inasmuch as the statistical system possesses a small number of microstates. The difference, although slight, accounts for meaningful physical consequences such as effective forces and inner interactions among constituents. Yet, in this letter we are to report some of the analytical attributes associated to entropies via the formulation introduced by Hanel and Thurner. These two functionals allow to construct a generalised classification of entropy measures in terms of their defining equivalence classes, which are determined by a pair of scaling exponents (c, d). As a result, it has been identified that and SB belong to the same asymptotic, equivalence class. The latter is an interesting fact since it does not occur for non-logarithmic, parameter-dependent entropies. Following this scheme, we also briefly discuss the features of the Sharma-Mittal, Rényi and Tsallis entropies in the asymptotic limit.

Derivation of density operators for generalized entropies with quantum analysis

We gave a simple derivation of density operator with the quantum analysis. We treated the functional of a density operator, and applied the maximum entropy principle. We obtained easily the density operators for the Tsallis entropy and the Rényi entropy with the q-expectation value (escort average), and also obtained easily the density operators for the Boltzmann–Gibbs entropy and the Burg entropy with the conventional expectation value. The quantum analysis works effectively in the calculation of the variation of the functional which includes trace.

Thermodynamic, Non-Extensive, or Turbulent Quasi-Equilibrium for the Space Plasma Environment

The Boltzmann–Gibbs (BG) entropy has been used in a wide variety of problems for more than a century. It is well known that BG entropy is additive and extensive, but for certain systems such as those dictated by long-range interactions, it is speculated that the entropy must be non-additive and non-extensive. Tsallis entropy possesses these characteristics, and is parameterized by a variable q ( being the classic BG limit), but unless q is determined from microscopic dynamics, the model remains a phenomenological tool. To this day, very few examples have emerged in which q can be computed from first principles. This paper shows that the space plasma environment, which is governed by long-range collective electromagnetic interaction, represents a perfect example for which the q parameter can be computed from microphysics. By taking the electron velocity distribution function measured in the heliospheric environment into account, and considering them to be in a quasi-equilibrium state with electrostatic turbulence known as quasi-thermal noise, it is shown that the value corresponding to , or alternatively , may be deduced. This prediction is verified against observations made by spacecraft, and it is shown to be in excellent agreement. This paper constitutes an overview of recent developments regarding the non-equilibrium statistical mechanical approach to understanding the non-extensive nature of space plasma, although some recent new developments are also discussed.

Thermodynamical properties by neglecting thermal noise in quantum systems: Effects of ground-state perturbations

A quantum system defined by a Hamiltonian Hˆ0, represented by a diagonal matrix, presents zero Boltzmann–Gibbs entropy (SBG=0) at zero temperature (T=0). Herein it is shown that by introducing a perturbation λVˆ (λ>0), characterized by off-diagonal contributions, the Hamiltonian Hˆ=Hˆ0+λVˆ may present a nontrivial ground state, described by a new entropic form S (S≠SBG). The perturbation λVˆ is analyzed, and particularly its relation to an effective temperature θ, identified as the parameter thermodynamically conjugated to the entropy S. Two paradigmatic quantum-mechanical systems are considered, namely, the harmonic oscillator and the free particle in an infinite potential well, which are shown to display nontrivial thermodynamic properties by varying θ. Recent experiments on superconductor samples at very low temperatures, where the specific heat does not present a clear tendency to zero as T→0, is explained by means of a crossover between these two entropic forms.

Loop quantum gravity Immirzi parameter and the Kaniadakis statistics

In this letter we show that a possible connection between the LQG Immirzi parameter and the area of a punctured surface can emerge depending on the thermostatistics theory previously chosen. Starting from the Boltzmann–Gibbs entropy, the Immirzi parameter can be reobtained. Using the Kaniadakis statistics, which is an important non-Gaussian statistics, we derive a new relation between the Immirzi parameter, the kappa parameter and the area of a punctured surface. After that, we compare our result with the Immirzi parameter previously obtained in the literature within the context of Tsallis’ statistics. We demonstrate in an exact way that the LQG Immirzi parameter can also be used to compare both Kaniadakis and Tsallis statics.

A comparison of Shannon, Kullback–Leibler and renormalized entropies within successive bifurcations

An issue in the context of self-organization is the existence of bifurcation processes that are often observed in dissipative dynamical systems. The first bifurcation occurs when a stable fixed point becomes unstable as the parameter set of the system slightly changes. Then the system eventually paves their way to new stable branches. When the most ordered spatial patterns emerge, the system implies a high level of self-organization in the phase space. Exchange of energy/matter with its surroundings is allowed for in such “open” systems. Open systems cannot be analyzed in terms of usual H-theorem of Boltzmann since it is only valid for isolated systems. Thereby, the Boltzmann–Gibbs (BG) entropy (or Shannon) is not able to explain such systems which increase their order as a signature of self-organization. However, Klimontovich showed that the BG entropy can still be used in such systems after a renormalization procedure and he also introduced the renormalized entropy as a measure of the relative degree of order by using his S-theorem. In this paper, we analyze the Hénon map and the Rössler oscillator as high dimensional dissipative systems. We show that the renormalized entropy detects transition points of the systems and explains changes in relative order through successive bifurcations or chaotic band-mergings. The correlation analysis also shows that the renormalized entropy is a better measure than the Shannon or Kullback–Leibler (KL) entropies.

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory.

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective.

Implications of the generalized entropy formalisms on the Newtonian gravity and dynamics

Employing the Verlinde's hypothesis, and considering two well-known generalized entropy formalisms, two modifications to the Newtonian gravity are derived. In addition, it has been shown that the generalized entropy measures may also provide theoretical basis for the Modified Newtonian Dynamics (MOND) theory and generate its modified forms. Since these entropy measures are also successful in describing the current accelerated universe, our results indicate that the origin of dark sectors of cosmos may be unified to meeting the generalized entropy measures instead of the Boltzmann–Gibbs entropy by the gravitational systems due to the long-range nature of gravity.

Relation between Boltzmann and Gibbs entropy and example with multinomial distribution

General relationship between mean Boltzmann entropy and Gibbs entropy is established. It is found that their difference is equal to fluctuation entropy, which is a Gibbs-like entropy of macroscopic quantities. The ratio of the fluctuation entropy and mean Boltzmann, or Gibbs entropy vanishes in the thermodynamic limit for a system of distinguishable and independent particles. It is argued that large fluctuation entropy clearly indicates the limit where standard statistical approach should be modified, or extended using other methods like renormalization group.

Exploring the gauge/gravity duality of a generalized von Neumann entropy

We consider a generalized von Neumann entropy that depends only on the density matrix. This is based on a modification of Boltzmann-Gibbs entropy by considering non-equilibrium systems on stationary states, and an entropy functional depending only on the probability. We propose a generalization of the replica trick and find that the resulting modified von Neumann entropy is precisely the previously mentioned entropy that was obtained by other assumptions. Then, we address the question whether alternative entanglement entropies can play a role in the gauge/gravity duality. Our focus are 2d CFT and their gravity duals. Our results show corrections to the von Neumann entropy S0. If the central charge is small/large they are correspondingly larger/smaller than the usual UV ones and also than the corrections to the corresponding minimal length in AdS3, which are comparable to the UV ones. This agrees with previous results indicating that the generalized entropy is more relevant for systems with fewer degrees of freedom. The correction terms due to the new entropy would modify the Ryu-Takayanagi identification between the CFT and the gravitational AdS3 minimal length. We discuss the ingredients for computing the corresponding gravity duals of generalized entanglement entropies.

Entropic nonadditivity, H theorem, and nonlinear Klein-Kramers equations.

We use the H theorem to establish the entropy and the entropic additivity law for a system composed of subsystems, with the dynamics governed by the Klein-Kramers equations, by considering relations among the dynamics of these subsystems and their entropies. We start considering the subsystems governed by linear Klein-Kramers equations and verify that the Boltzmann-Gibbs entropy is appropriated to this dynamics, leading us to the standard entropic additivity, S_{BG}^{(1∪2)}=S_{BG}^{1}+S_{BG}^{2}, consistent with the fact that the distributions of the subsystem are independent. We then extend the dynamics of these subsystems to independent nonlinear Klein-Kramers equations. For this case, the results show that the H theorem is verified for a generalized entropy, which does not preserve the standard entropic additivity for independent distributions. In this scenario, consistent results are obtained when a suitable coupling among the nonlinear Klein-Kramers equations is considered, in which each subsystem modifies the other until an equilibrium state is reached. This dynamics, for the subsystems, results in the Tsallis entropy for the system and, consequently, verifies the relation S_{q}^{(1∪2)}=S_{q}^{1}+S_{q}^{2}+(1-q)S_{q}^{1}S_{q}^{2}/k, which is a nonadditive entropic relation.

Economics and Finance: q-Statistical Stylized Features Galore

The Boltzmann–Gibbs (BG) entropy and its associated statistical mechanics were generalized, three decades ago, on the basis of the nonadditive entropy S q ( q ∈ R ), which recovers the BG entropy in the q → 1 limit. The optimization of S q under appropriate simple constraints straightforwardly yields the so-called q-exponential and q-Gaussian distributions, respectively generalizing the exponential and Gaussian ones, recovered for q = 1 . These generalized functions ubiquitously emerge in complex systems, especially as economic and financial stylized features. These include price returns and volumes distributions, inter-occurrence times, characterization of wealth distributions and associated inequalities, among others. Here, we briefly review the basic concepts of this q-statistical generalization and focus on its rapidly growing applications in economics and finance.