Tsallis Entropy, Likelihood, and the Robust Seismic Inversion
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Luis Gregorio Moyano
Luis Gregorio Moyano Rep´ublicadel L´ıbano 150, Mendoza, 5500 Mendoza, Argentina [email protected] Current position Facultad de Ciencias Exactas y Naturales Mendoza, Argentina Universidad Nacional de Cuyo Adjunt Professor May 2016 - Today CONICET Adjunt Researcher May 2016 - Today Education and training Universidad Carlos III de Madrid Madrid, Spain Postdoctoral Position at 2007 - 2008 Mathematics Department & GISC Centro Brasileiro de Pesquisas F´ısicas Rio de Janeiro, Brazil PhD in Physics 2001 - 2006 Supervisor: Prof. Constantino Tsallis. PhD Thesis: \Nonextensive statistical mechanics in complex systems: dynamical foundations and applications" Instituto Balseiro Bariloche, Argentina MSc in Physics 1997- 2000 Supervisors: Prof. Dami´anZanette and Prof. Guillermo Abramson. Msc. Thesis: \Learning in coupled dynamical systems" Research Interests • Machine Learning applications on networks, Natural Language Processing, Big Data applications. • Complex Networks, Social Networks, Complex Systems. • Statistical mechanics, applications to economical, social and biological systems. • Bayesian inference, Markovian systems, congestion and information transfer dynamics in networks. • Evolutionary game theory, emergence of cooperation in complex networks. • Nonlinear dynamics, coupled chaotic systems, synchronization. Luis G. Moyano - Curriculum Vitæ 1 Positions, fellowships and awards • IBM Research Brazil, Rio de Janeiro, Brazil - Staff Research Member December, 2013 - April, 2016 Staff Research Member at the Social Data Analytics -
Associating an Entropy with Power-Law Frequency of Events
entropy Article Associating an Entropy with Power-Law Frequency of Events Evaldo M. F. Curado 1, Fernando D. Nobre 1,* and Angel Plastino 2 1 Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil; [email protected] 2 La Plata National University and Argentina’s National Research Council (IFLP-CCT-CONICET)-C. C. 727, La Plata 1900, Argentina; plastino@fisica.unlp.edu.ar * Correspondence: [email protected]; Tel.: +55-21-2141-7513 Received: 3 October 2018; Accepted: 23 November 2018; Published: 6 December 2018 Abstract: Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy Sq. The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy #0, in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature Tq with kTq ∝ #0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of Tq are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures. -
Quantum Entanglement Inferred by the Principle of Maximum Tsallis Entropy
Quantum entanglement inferred by the principle of maximum Tsallis entropy Sumiyoshi Abe1 and A. K. Rajagopal2 1College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan 2Naval Research Laboratory, Washington D.C., 20375-5320, U.S.A. The problem of quantum state inference and the concept of quantum entanglement are studied using a non-additive measure in the form of Tsallis’ entropy indexed by the positive parameter q. The maximum entropy principle associated with this entropy along with its thermodynamic interpretation are discussed in detail for the Einstein- Podolsky-Rosen pair of two spin-1/2 particles. Given the data on the Bell-Clauser- Horne-Shimony-Holt observable, the analytic expression is given for the inferred quantum entangled state. It is shown that for q greater than unity, indicating the sub- additive feature of the Tsallis entropy, the entangled region is small and enlarges as one goes into the super-additive regime where q is less than unity. It is also shown that quantum entanglement can be quantified by the generalized Kullback-Leibler entropy. quant-ph/9904088 26 Apr 1999 PACS numbers: 03.67.-a, 03.65.Bz I. INTRODUCTION Entanglement is a fundamental concept highlighting non-locality of quantum mechanics. It is well known [1] that the Bell inequalities, which any local hidden variable theories should satisfy, can be violated by entangled states of a composite system described by quantum mechanics. Experimental results [2] suggest that naive local realism à la Einstein, Podolsky, and Rosen [3] may not actually hold. Related issues arising out of the concept of quantum entanglement are quantum cryptography, quantum teleportation, and quantum computation. -
On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State
entropy Article On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State Lu Wei Department of Electrical and Computer Engineering, University of Michigan, Dearborn, MI 48128, USA; [email protected] Received: 26 April 2019; Accepted: 25 May 2019; Published: 27 May 2019 Abstract: The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy. Keywords: entanglement entropy; quantum information theory; random matrix theory; variance 1. Introduction Classical information theory is the theory behind the modern development of computing, communication, data compression, and other fields. As its classical counterpart, quantum information theory aims at understanding the theoretical underpinnings of quantum science that will enable future quantum technologies. One of the most fundamental features of quantum science is the phenomenon of quantum entanglement. Quantum states that are highly entangled contain more information about different parts of the composite system. As a step to understand quantum entanglement, we choose to study the entanglement property of quantum bipartite systems. The quantum bipartite model, proposed in the seminal work of Page [1], is a standard model for describing the interaction of a physical object with its environment for various quantum systems. -
The Entropy Universe
entropy Review The Entropy Universe Maria Ribeiro 1,2,* , Teresa Henriques 3,4 , Luísa Castro 3 , André Souto 5,6,7 , Luís Antunes 1,2 , Cristina Costa-Santos 3,4 and Andreia Teixeira 3,4,8 1 Institute for Systems and Computer Engineering, Technology and Science (INESC-TEC), 4200-465 Porto, Portugal; [email protected] 2 Computer Science Department, Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal 3 Centre for Health Technology and Services Research (CINTESIS), Faculty of Medicine University of Porto, 4200-450 Porto, Portugal; [email protected] (T.H.); [email protected] (L.C.); [email protected] (C.C.-S.); andreiasofi[email protected] (A.T.) 4 Department of Community Medicine, Information and Health Decision Sciences-MEDCIDS, Faculty of Medicine, University of Porto, 4200-450 Porto, Portugal 5 LASIGE, Faculdade de Ciências da Universidade de Lisboa, 1749-016 Lisboa, Portugal; [email protected] 6 Departamento de Informática, Faculdade de Ciências da Universidade de Lisboa, 1749-016 Lisboa, Portugal 7 Instituto de Telecomunicações, 1049-001 Lisboa, Portugal 8 Instituto Politécnico de Viana do Castelo, 4900-347 Viana do Castelo, Portugal * Correspondence: [email protected] Abstract: About 160 years ago, the concept of entropy was introduced in thermodynamics by Rudolf Clausius. Since then, it has been continually extended, interpreted, and applied by researchers in many scientific fields, such as general physics, information theory, chaos theory, data mining, and mathematical linguistics. This paper presents The Entropy -
Arxiv:1707.03526V1 [Cond-Mat.Stat-Mech] 12 Jul 2017 Eq
Generalized Ensemble Theory with Non-extensive Statistics Ke-Ming Shen,∗ Ben-Wei Zhang,y and En-Ke Wang Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China (Dated: October 17, 2018) The non-extensive canonical ensemble theory is reconsidered with the method of Lagrange multipliers by maximizing Tsallis entropy, with the constraint that the normalized term of P q Tsallis' q−average of physical quantities, the sum pj , is independent of the probability pi for Tsallis parameter q. The self-referential problem in the deduced probability and thermal quantities in non-extensive statistics is thus avoided, and thermodynamical relationships are obtained in a consistent and natural way. We also extend the study to the non-extensive grand canonical ensemble theory and obtain the q-deformed Bose-Einstein distribution as well as the q-deformed Fermi-Dirac distribution. The theory is further applied to the general- ized Planck law to demonstrate the distinct behaviors of the various generalized q-distribution functions discussed in literature. I. INTRODUCTION In the last thirty years the non-extensive statistical mechanics, based on Tsallis entropy [1,2] and the corresponding deformed exponential function, has been developed and attracted a lot of attentions with a large amount of applications in rather diversified fields [3]. Tsallis non- extensive statistical mechanics is a generalization of the common Boltzmann-Gibbs (BG) statistical PW mechanics by postulating a generalized entropy of the classical one, S = −k i=1 pi ln pi: W X q Sq = −k pi lnq pi ; (1) i=1 where k is a positive constant and denotes Boltzmann constant in BG statistical mechanics. -
Grand Canonical Ensemble of the Extended Two-Site Hubbard Model Via a Nonextensive Distribution
Grand canonical ensemble of the extended two-site Hubbard model via a nonextensive distribution Felipe Américo Reyes Navarro1;2∗ Email: [email protected] Eusebio Castor Torres-Tapia2 Email: [email protected] Pedro Pacheco Peña3 Email: [email protected] 1Facultad de Ciencias Naturales y Matemática, Universidad Nacional del Callao (UNAC) Av. Juan Pablo II 306, Bellavista, Callao, Peru 2Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos (UNMSM) Av. Venezuela s/n Cdra. 34, Apartado Postal 14-0149, Lima 14, Peru 3Universidad Nacional Tecnológica del Cono Sur (UNTECS), Av. Revolución s/n, Sector 3, Grupo 10, Mz. M Lt. 17, Villa El Salvador, Lima, Peru ∗Corresponding author. Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos (UNMSM), Av. Venezuela s/n Cdra. 34, Apartado Postal 14-0149, Lima 14, Peru Abstract We hereby introduce a research about a grand canonical ensemble for the extended two-site Hubbard model, that is, we consider the intersite interaction term in addition to those of the simple Hubbard model. To calculate the thermodynamical parameters, we utilize the nonextensive statistical mechan- ics; specifically, we perform the simulations of magnetic internal energy, specific heat, susceptibility, and thermal mean value of the particle number operator. We found out that the addition of the inter- site interaction term provokes a shifting in all the simulated curves. Furthermore, for some values of the on-site Coulombian potential, we realize that, near absolute zero, the consideration of a chemical potential varying with temperature causes a nonzero entropy. Keywords Extended Hubbard model,Archive Quantum statistical mechanics, Thermalof properties SID of small particles PACS 75.10.Jm, 05.30.-d,65.80.+n Introduction Currently, several researches exist on the subject of the application of a generalized statistics for mag- netic systems in the literature [1-3]. -
Generalized Molecular Chaos Hypothesis and the H-Theorem: Problem of Constraints and Amendment of Nonextensive Statistical Mechanics
Generalized molecular chaos hypothesis and the H-theorem: Problem of constraints and amendment of nonextensive statistical mechanics Sumiyoshi Abe1,2,3 1Department of Physical Engineering, Mie University, Mie 514-8507, Japan*) 2Institut Supérieur des Matériaux et Mécaniques Avancés, 44 F. A. Bartholdi, 72000 Le Mans, France 3Inspire Institute Inc., McLean, Virginia 22101, USA Abstract Quite unexpectedly, kinetic theory is found to specify the correct definition of average value to be employed in nonextensive statistical mechanics. It is shown that the normal average is consistent with the generalized Stosszahlansatz (i.e., molecular chaos hypothesis) and the associated H-theorem, whereas the q-average widely used in the relevant literature is not. In the course of the analysis, the distributions with finite cut-off factors are rigorously treated. Accordingly, the formulation of nonextensive statistical mechanics is amended based on the normal average. In addition, the Shore- Johnson theorem, which supports the use of the q-average, is carefully reexamined, and it is found that one of the axioms may not be appropriate for systems to be treated within the framework of nonextensive statistical mechanics. PACS number(s): 05.20.Dd, 05.20.Gg, 05.90.+m ____________________________ *) Permanent address 1 I. INTRODUCTION There exist a number of physical systems that possess exotic properties in view of traditional Boltzmann-Gibbs statistical mechanics. They are said to be statistical- mechanically anomalous, since they often exhibit and realize broken ergodicity, strong correlation between elements, (multi)fractality of phase-space/configuration-space portraits, and long-range interactions, for example. In the past decade, nonextensive statistical mechanics [1,2], which is a generalization of the Boltzmann-Gibbs theory, has been drawing continuous attention as a possible theoretical framework for describing these systems. -
Nonextensive Statistical Mechanics: a Brief Review of Its Present Status
Anais da Academia Brasileira de Ciências (2002) 74(3): 393–414 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc Nonextensive statistical mechanics: a brief review of its present status CONSTANTINO TSALLIS* Centro Brasileiro de Pesquisas Físicas, 22290-180 Rio de Janeiro, Brazil Centro de Fisica da Materia Condensada, Universidade de Lisboa P-1649-003 Lisboa, Portugal Manuscript received on May 25, 2002; accepted for publication on May 27, 2002. ABSTRACT We briefly review the present status of nonextensive statistical mechanics. We focus on (i) the cen- tral equations of the formalism, (ii) the most recent applications in physics and other sciences, (iii) the a priori determination (from microscopic dynamics) of the entropic index q for two important classes of physical systems, namely low-dimensional maps (both dissipative and conservative) and long-range interacting many-body hamiltonian classical systems. Key words: nonextensive statistical mechanics, entropy, complex systems. 1 CENTRAL EQUATIONS OF NONEXTENSIVE STATISTICAL MECHANICS Nonextensive statistical mechanics and thermodynamics were introduced in 1988 [1], and further developed in 1991 [2] and 1998 [3], with the aim of extending the domain of applicability of statistical mechanical procedures to systems where Boltzmann-Gibbs (BG) thermal statistics and standard thermodynamics present serious mathematical difficulties or just plainly fail. Indeed, a rapidly increasing number of systems are pointed out in the literature for which the usual functions appearing in BG statistics appear to be violated. Some of these cases are satisfactorily handled within the formalism we are here addressing (see [4] for reviews and [5] for a regularly updated bibliography which includes crucial contributions and clarifications that many scientists have given along the years). -
0802.3424 Property of Tsallis Entropy and Principle of Entropy
arXiv: 0802.3424 Property of Tsallis entropy and principle of entropy increase Du Jiulin Department of Physics, School of Science, Tianjin University, Tianjin 300072, China E-mail: [email protected] Abstract The property of Tsallis entropy is examined when considering tow systems with different temperatures to be in contact with each other and to reach the thermal equilibrium. It is verified that the total Tsallis entropy of the two systems cannot decrease after the contact of the systems. We derived an inequality for the change of Tsallis entropy in such an example, which leads to a generalization of the principle of entropy increase in the framework of nonextensive statistical mechanics. Key Words: Nonextensive system; Principle of entropy increase PACS number: 05.20.-y; 05.20.Dd 1 1. Introduction In recent years, a generalization of Bltzmann-Gibbs(B-G) statistical mechanics initiated by Tsallis has focused a great deal of attention, the results from the assumption of nonadditive statistical entropies and the maximum statistical entropy principle, which has been known as “Tsallis statistics” or nonextensive statistical mechanics(NSM) (Abe and Okamoto, 2001). This generalization of B-G statistics was proposed firstly by introducing the mathematical expression of Tsallis entropy (Tsallis, 1988) as follows, k S = ( ρ q dΩ −1) (1) q 1− q ∫ where k is the Boltzmann’s constant. For a classical Hamiltonian system, ρ is the phase space probability distribution of the system under consideration that satisfies the normalization ∫ ρ dΩ =1 and dΩ stands for the phase space volume element. The entropy index q is a positive parameter whose deviation from unity is thought to describe the degree of nonextensivity. -
Q-Gaussian Approximants Mimic Non-Extensive Statistical-Mechanical Expectation for Many-Body Probabilistic Model with Long-Range Correlations
Cent. Eur. J. Phys. • 7(3) • 2009 • 387-394 DOI: 10.2478/s11534-009-0054-4 Central European Journal of Physics q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations Research Article William J. Thistleton1∗, John A. Marsh2† , Kenric P. Nelson3‡ , Constantino Tsallis45§ 1 Department of Mathematics, SUNY Institute of Technology, Utica NY 13504, USA 2 Department of Computer and Information Sciences, SUNY Institute of Technology, Utica NY 13504, USA 3 Raytheon Integrated Defense Systems, Principal Systems Engineer 4 Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro - RJ, Brazil 5 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA Received 3 November 2008; accepted 25 March 2009 Abstract: We study a strictly scale-invariant probabilistic N-body model with symmetric, uniform, identically distributed random variables. Correlations are induced through a transformation of a multivariate Gaussian distribution with covariance matrix decaying out from the unit diagonal, as ρ/rα for r =1, 2, …, N-1, where r indicates displacement from the diagonal and where 0 6 ρ 6 1 and α > 0. We show numerically that the sum of the N dependent random variables is well modeled by a compact support q-Gaussian distribution. In the particular case of α = 0 we obtain q = (1-5/3 ρ) / (1- ρ), a result validated analytically in a recent paper by Hilhorst and Schehr. Our present results with these q-Gaussian approximants precisely mimic the behavior expected in the frame of non-extensive statistical mechanics. -
Generalized Simulated Annealing Algorithms Using Tsallis Statistics: Application to Conformational Optimization of a Tetrapeptide
PHYSICAL REVIEW E VOLUME 53, NUMBER 4 APRIL 1996 Generalized simulated annealing algorithms using Tsallis statistics: Application to conformational optimization of a tetrapeptide Ioan Andricioaei and John E. Straub Department of Chemistry, Boston University, Boston, Massachusetts 02215 ~Received 18 December 1995! A Monte Carlo simulated annealing algorithm based on the generalized entropy of Tsallis is presented. The algorithm obeys detailed balance and reduces to a steepest descent algorithm at low temperatures. Application to the conformational optimization of a tetrapeptide demonstrates that the algorithm is more effective in locating low energy minima than standard simulated annealing based on molecular dynamics or Monte Carlo methods. PACS number~s!: 02.70.2c, 02.60.Pn, 02.50.Ey Finding the ground state conformation of biologically im- portant molecules has an obvious importance, both from the S52k( pilnpi ~3! academic and pragmatic points of view @1#. The problem is hard for biomolecules, such as proteins, because of the rug- when q 1. Maximizing the Tsallis entropy with the con- gedness of the energy landscape which is characterized by an straints → immense number of local minima separated by a broad dis- tribution of barrier heights @2,3#. Algorithms to find the glo- bal minimum of an empirical potential energy function for q ( pi51 and ( pi ei5const, ~4! molecules have been devised, among which a central role is played by the simulated annealing methods @4#. Once a cool- ing schedule is chosen, representative configurations of the where ei is the energy spectrum, the generalized probability allowed microstates are generated by methods either of the distribution is found to be molecular dynamics ~MD! or Monte Carlo ~MC! types.