Research

277 KiB

Open AccessArticle

Deformed Algebras and Generalizations of Independence on Deformed Exponential Families

by Hiroshi Matsuzoe and Tatsuaki Wada

Entropy 2015, 17(8), 5729-5751; https://doi.org/10.3390/e17085729 - 10 Aug 2015

Cited by 17 | Viewed by 4755

A deformed exponential family is a generalization of exponential families. Since the useful classes of power law tailed distributions are described by the deformed exponential families, they are important objects in the theory of complex systems. Though the deformed exponential families are defined [...] Read more.

A deformed exponential family is a generalization of exponential families. Since the useful classes of power law tailed distributions are described by the deformed exponential families, they are important objects in the theory of complex systems. Though the deformed exponential families are defined by deformed exponential functions, these functions do not satisfy the law of exponents in general. The deformed algebras have been introduced based on the deformed exponential functions. In this paper, after summarizing such deformed algebraic structures, it is clarified how deformed algebras work on deformed exponential families. In fact, deformed algebras cause generalization of expectations. The three kinds of expectations for random variables are introduced in this paper, and it is discussed why these generalized expectations are natural from the viewpoint of information geometry. In addition, deformed algebras cause generalization of independences. Whereas it is difficult to check the well-definedness of deformed independence in general, the κ-independence is always well-defined on κ-exponential families. This is one of advantages of κ-exponential families in complex systems. Consequently, we can well generalize the maximum likelihood method for the κ-exponential family from the viewpoint of information geometry. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

399 KiB

Open AccessArticle

Asymptotic Description of Neural Networks with Correlated Synaptic Weights

by Olivier Faugeras and James MacLaurin

Entropy 2015, 17(7), 4701-4743; https://doi.org/10.3390/e17074701 - 06 Jul 2015

Cited by 9 | Viewed by 4950

Abstract

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network [...] Read more.

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The main result of this article is that the image law through the empirical measure satisfies a large deviation principle with a good rate function which is shown to have a unique global minimum. Our analysis of the rate function allows us also to characterize the limit measure as the image of a stationary Gaussian measure defined on a transformed set of trajectories. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

424 KiB

Open AccessArticle

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

by Bertrand Lods and Giovanni Pistone

Entropy 2015, 17(6), 4323-4363; https://doi.org/10.3390/e17064323 - 19 Jun 2015

Cited by 20 | Viewed by 5099

Abstract

Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set Ɛ of mutually absolutely continuous densities. We study [...] Read more.

Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set Ɛ of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if Ɛ is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell’s weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyvärinen divergence. This requires us to generalize our approach to Orlicz–Sobolev spaces to include derivatives. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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382 KiB

Open AccessArticle

Generalized Stochastic Fokker-Planck Equations

by Pierre-Henri Chavanis

Entropy 2015, 17(5), 3205-3252; https://doi.org/10.3390/e17053205 - 13 May 2015

Cited by 25 | Viewed by 6697

Abstract

We consider a system of Brownian particles with long-range interactions. We go beyond the mean field approximation and take fluctuations into account. We introduce a new class of stochastic Fokker-Planck equations associated with a generalized thermodynamical formalism. Generalized thermodynamics arises in the case [...] Read more.

We consider a system of Brownian particles with long-range interactions. We go beyond the mean field approximation and take fluctuations into account. We introduce a new class of stochastic Fokker-Planck equations associated with a generalized thermodynamical formalism. Generalized thermodynamics arises in the case of complex systems experiencing small-scale constraints. In the limit of short-range interactions, we obtain a generalized class of stochastic Cahn-Hilliard equations. Our formalism has application for several systems of physical interest including self-gravitating Brownian particles, colloid particles at a fluid interface, superconductors of type II, nucleation, the chemotaxis of bacterial populations, and two-dimensional turbulence. We also introduce a new type of generalized entropy taking into account anomalous diffusion and exclusion or inclusion constraints. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

349 KiB

Open AccessArticle

On the κ-Deformed Cyclic Functions and the Generalized Fourier Series in the Framework of the κ-Algebra

by Antonio Maria Scarfone

Entropy 2015, 17(5), 2812-2833; https://doi.org/10.3390/e17052812 - 04 May 2015

Cited by 10 | Viewed by 4817

Abstract

We explore two possible generalizations of the Euler formula for the complex \(\kappa\)-exponential, which give two different sets of \(\kappa\)-deformed cyclic functions endowed with different analytical properties. In a case, the \(\kappa\)-sine and \(\kappa\)-cosine functions take real values on \(\Re\) and are characterized [...] Read more.

We explore two possible generalizations of the Euler formula for the complex \(\kappa\)-exponential, which give two different sets of \(\kappa\)-deformed cyclic functions endowed with different analytical properties. In a case, the \(\kappa\)-sine and \(\kappa\)-cosine functions take real values on \(\Re\) and are characterized by an asymptotic log-periodic behavior. In the other case, the \(\kappa\)-cyclic functions take real values only in the region \(|x|\leq1/|\kappa|\), while, for \(|x|>1/|\kappa|\), they assume purely imaginary values with an increasing modulus. However, the main mathematical properties of the standard cyclic functions, opportunely reformulated in the formalism of the \(\kappa\)-mathematics, are fulfilled by the two sets of the \(\kappa\)-trigonometric functions. In both cases, we study the orthogonality and the completeness relations and introduce their respective generalized Fourier series for square integrable functions. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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1510 KiB

Open AccessArticle

Kappa and q Indices: Dependence on the Degrees of Freedom

by George Livadiotis

Entropy 2015, 17(4), 2062-2081; https://doi.org/10.3390/e17042062 - 08 Apr 2015

Cited by 44 | Viewed by 6501

Abstract

The kappa distributions, or their equivalent, the q-exponential distributions, are the natural generalization of the classical Boltzmann-Maxwell distributions, applied to the study of the particle populations in collisionless space plasmas. A huge step in the development of the theory of kappa distributions [...] Read more.

The kappa distributions, or their equivalent, the q-exponential distributions, are the natural generalization of the classical Boltzmann-Maxwell distributions, applied to the study of the particle populations in collisionless space plasmas. A huge step in the development of the theory of kappa distributions and their applications in space plasma physics has been achieved with the discovery that the observed kappa distributions are connected with the solid statistical background of non-extensive statistical mechanics. Now that the statistical framework has been identified, it is straightforward to improve our understanding of the nature of the kappa index (or the entropic q-index) that governs these distributions. One critical topic is the dependence of the kappa index on the degrees of freedom. In this paper, we first show how this specific dependence is naturally emerged, using the formalism of the N-particle kappa distribution of velocities. Then, the result is extended in the presence of potential energies. It is shown that the kappa index is simply related to the kinetic and potential degrees of freedom. In addition, it is shown that various problems of non-extensive statistical mechanics, such as (i) the correlation dependence on the total number of particles; and (ii) the normalization divergence for finite kappa indices, are resolved considering the kappa index dependence on the degrees of freedom. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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315 KiB

Open AccessArticle

Ricci Curvature, Isoperimetry and a Non-additive Entropy

by Nikos Kalogeropoulos

Entropy 2015, 17(3), 1278-1308; https://doi.org/10.3390/e17031278 - 16 Mar 2015

Cited by 10 | Viewed by 5059

Abstract

Searching for the dynamical foundations of Havrda-Charvát/Daróczy/ Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an N-Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications [...] Read more.

Searching for the dynamical foundations of Havrda-Charvát/Daróczy/ Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an N-Ricci curvature or a Bakry-Émery-Ricci curvature in the configuration/phase space of a system. We explore some of the implications of this tensor and its associated curvature and present a connection with the non-additive entropy under investigation. We present an isoperimetric interpretation of the non-extensive parameter and comment on further features of the system that can be probed through this tensor. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

241 KiB

Open AccessArticle

Information Geometry on the \(\kappa\)-Thermostatistics

by Tatsuaki Wada and Antonio M. Scarfone

Entropy 2015, 17(3), 1204-1217; https://doi.org/10.3390/e17031204 - 12 Mar 2015

Cited by 12 | Viewed by 5164

Abstract

We explore the information geometric structure of the statistical manifold generated by the \(\kappa\)-deformed exponential family. The dually-flat manifold is obtained as a dualistic Hessian structure by introducing suitable generalization of the Fisher metric and affine connections. As a byproduct, we obtain the [...] Read more.

We explore the information geometric structure of the statistical manifold generated by the \(\kappa\)-deformed exponential family. The dually-flat manifold is obtained as a dualistic Hessian structure by introducing suitable generalization of the Fisher metric and affine connections. As a byproduct, we obtain the fluctuation-response relations in the \(\kappa\)-formalism based on the \(\kappa\)-generalized exponential family. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

905 KiB

Open AccessArticle

Weakest-Link Scaling and Extreme Events in Finite-Sized Systems

by Dionissios T. Hristopulos, Manolis P. Petrakis and Giorgio Kaniadakis

Entropy 2015, 17(3), 1103-1122; https://doi.org/10.3390/e17031103 - 09 Mar 2015

Cited by 10 | Viewed by 5968

Abstract

Weakest-link scaling is used in the reliability analysis of complex systems. It is characterized by the extensivity of the hazard function instead of the entropy. The Weibull distribution is the archetypical example of weakest-link scaling, and it describes variables such as the fracture [...] Read more.

Weakest-link scaling is used in the reliability analysis of complex systems. It is characterized by the extensivity of the hazard function instead of the entropy. The Weibull distribution is the archetypical example of weakest-link scaling, and it describes variables such as the fracture strength of brittle materials, maximal annual rainfall, wind speed and earthquake return times. We investigate two new distributions that exhibit weakest-link scaling, i.e., a Weibull generalization known as the κ-Weibull and a modified gamma probability function that we propose herein. We show that in contrast with the Weibull and the modified gamma, the hazard function of the κ -Weibull is non-extensive, which is a signature of inter-dependence between the links. We also investigate the impact of heterogeneous links, modeled by means of a stochastic Weibull scale parameter, on the observed probability distribution. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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Graphical abstract

241 KiB

Open AccessArticle

Speed Gradient and MaxEnt Principles for Shannon and Tsallis Entropies

by Alexander L. Fradkov and Dmitry S. Shalymov

Entropy 2015, 17(3), 1090-1102; https://doi.org/10.3390/e17031090 - 06 Mar 2015

Cited by 20 | Viewed by 5112

Abstract

In this paper we consider dynamics of non-stationary processes that follow the MaxEnt principle. We derive a set of equations describing dynamics of a system for Shannon and Tsallis entropies. Systems with discrete probability distribution are considered under mass conservation and energy conservation [...] Read more.

In this paper we consider dynamics of non-stationary processes that follow the MaxEnt principle. We derive a set of equations describing dynamics of a system for Shannon and Tsallis entropies. Systems with discrete probability distribution are considered under mass conservation and energy conservation constraints. The existence and uniqueness of solution are established and asymptotic stability of the equilibrium is proved. Equations are derived based on the speed-gradient principle originated in control theory. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

687 KiB

Open AccessArticle

Fokker-Planck Equation and Thermodynamic System Analysis

by Umberto Lucia and Gianpiero Gervino

Entropy 2015, 17(2), 763-771; https://doi.org/10.3390/e17020763 - 09 Feb 2015

Cited by 6 | Viewed by 5927

Abstract

The non-linear Fokker-Planck equation or Kolmogorov forward equation is currently successfully applied for deep analysis of irreversibility and it gives an excellent approximation near the free energy minimum, just as Boltzmann’s definition of entropy follows from finding the maximum entropy state. A connection [...] Read more.

The non-linear Fokker-Planck equation or Kolmogorov forward equation is currently successfully applied for deep analysis of irreversibility and it gives an excellent approximation near the free energy minimum, just as Boltzmann’s definition of entropy follows from finding the maximum entropy state. A connection to Fokker-Planck dynamics and the free energy functional is presented and discussed—this approach has been particularly successful to deal with metastability. We focus our attention on investigating and discussing the fundamental role of dissipation analysis in metastable systems. The major novelty of our approach is that the obtained results enable us to reveal an appealing, and previously unexplored relationship between Fokker-Planck equation and the associated free energy functional. Namely, we point out that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

225 KiB

Open AccessArticle

A New Telegrapher’s-Poisson System in Semiconductor Theory: A Singular Perturbation Approach

by Alberto Rossani

Entropy 2015, 17(2), 528-538; https://doi.org/10.3390/e17020528 - 29 Jan 2015

Viewed by 4156

Abstract

In the theory of energy and momentum relaxation in semiconductor devices, the introduction of two temperatures and two mean velocities for electron and phonons is required. A new model, based on an asymptotic procedure for solving the kinetic equations of electrons and phonons [...] Read more.

In the theory of energy and momentum relaxation in semiconductor devices, the introduction of two temperatures and two mean velocities for electron and phonons is required. A new model, based on an asymptotic procedure for solving the kinetic equations of electrons and phonons is proposed, which naturally gives the displaced Maxwellian at the leading order. After that, balance equations for the electron number, energy densities and momentum densities are constructed, which constitute now a system of five equations for the chemical potential of electrons, the temperatures and the drift velocities. Moreover, Poisson’s equation is coupled, in order to calculate the self-consistent electric field. In Bloch’s approximation, we derive a telegrapher’s-Poisson system for the electron number density and the electric potential, which could allow simple semiconductor calculations, but still including wave propagation effects. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

182 KiB

Open AccessArticle

by Paolo Rossi

Entropy 2015, 17(1), 425-437; https://doi.org/10.3390/e17010425 - 19 Jan 2015

Cited by 3 | Viewed by 5344

Abstract

The frequency distribution of surnames turns out to be a relevant issue not only in historical demography but also in population biology, and especially in genetics, since surnames tend to behave like neutral genes and propagate like Y chromosomes. The stochastic dynamics leading [...] Read more.

The frequency distribution of surnames turns out to be a relevant issue not only in historical demography but also in population biology, and especially in genetics, since surnames tend to behave like neutral genes and propagate like Y chromosomes. The stochastic dynamics leading to the observed scale-invariant distributions has been studied as a Yule process, as a branching phenomenon and also by field-theoretical renormalization group techniques. In the absence of mutations the theoretical models are in good agreement with empirical evidence, but when mutations are present a discrepancy between the theoretical and the experimental exponents is observed. Hints for the possible origin of the mismatch are discussed, with some emphasis on the difference between the asymptotic frequency distribution of a full population and the frequency distributions observed in its samples. A precise connection is established between surname distributions and the statistical properties of genealogical trees. Ancestors tables, being obviously self-similar, may be investigated theoretically by renormalization group techniques, but they can also be studied empirically by exploiting the large online genealogical databases concerning European nobility. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

923 KiB

Open AccessArticle

Entropy, Age and Time Operator

by Ilias Gialampoukidis and Ioannis Antoniou

Entropy 2015, 17(1), 407-424; https://doi.org/10.3390/e17010407 - 19 Jan 2015

Cited by 5 | Viewed by 6361

Abstract

The time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally [...] Read more.

The time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally introduced for quantum systems and highly unstable dynamical systems. Extending the time operator theory to regular Markov chains allows one to relate internal age with norm distances from equilibrium. The goal of this work is to express the evolution of internal age in terms of Lyapunov functionals constructed from entropies. We selected the Boltzmann–Gibbs–Shannon entropy and more general entropy functions, namely the Tsallis entropies and the Kaniadakis entropies. Moreover, we compare the evolution of the distance of initial distributions from equilibrium to the evolution of the Lyapunov functionals constructed from norms with the evolution of Lyapunov functionals constructed from entropies. It is remarkable that the entropy functionals evolve, violating the second law of thermodynamics, while the norm functionals evolve thermodynamically. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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213 KiB

Open AccessArticle

Tsallis Distribution Decorated with Log-Periodic Oscillation

by Grzegorz Wilk and Zbigniew Włodarczyk

Entropy 2015, 17(1), 384-400; https://doi.org/10.3390/e17010384 - 14 Jan 2015

Cited by 36 | Viewed by 5687

Abstract

In many situations, in all branches of physics, one encounters the power-like behavior of some variables, which is best described by a Tsallis distribution characterized by a nonextensivity parameter q and scale parameter T. However, there exist experimental results that can be described [...] Read more.

In many situations, in all branches of physics, one encounters the power-like behavior of some variables, which is best described by a Tsallis distribution characterized by a nonextensivity parameter q and scale parameter T. However, there exist experimental results that can be described only by a Tsallis distributions, which are additionally decorated by some log-periodic oscillating factor. We argue that such a factor can originate from allowing for a complex nonextensivity parameter q. The possible information conveyed by such an approach (like the occurrence of complex heat capacity, the notion of complex probability or complex multiplicative noise) will also be discussed. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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224 KiB

Open AccessArticle

Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle

by Tamás Sándor Biró, Péter Ván, Gergely Gábor Barnaföldi and Károly Ürmössy

Entropy 2014, 16(12), 6497-6514; https://doi.org/10.3390/e16126497 - 09 Dec 2014

Cited by 33 | Viewed by 6175

Abstract

Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and [...] Read more.

Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and \(q=1-1/\langle n \rangle + \Delta n^2/\langle n \rangle^2\). For the binomial distribution of \(n\) one obtains \(q=1-1/k\), for the negative binomial \(q=1+1/(k+1)\). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion \(\omega \ll E\). For general systems the average phase-space volume ratio \(\langle e^{S(E-\omega)}/e^{S(E)}\rangle\) to second order delivers \(q=1-1/C+\Delta \beta^2/\langle \beta \rangle^2\) with \(\beta=S^{\prime}(E)\) and \(C=dE/dT\) heat capacity. However, \(q \ne 1\) leads to non-additivity of the Boltzmann–Gibbs entropy, \(S\). We demonstrate that a deformed entropy, \(K(S)\), can be constructed and used for demanding additivity, i.e., \(q_K=1\). This requirement leads to a second order differential equation for \(K(S)\). Finally, the generalized \(q\)-entropy formula, \(K(S)=\sum p_i K(-\ln p_i)\), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, \(\Delta\beta^2\) we obtain a novel entropy formula. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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