“Information Is All That Matters!”

Total Page:16

File Type:pdf, Size:1020Kb

“Information Is All That Matters!” “Information is all that matters!” The author1 is a doctor of physics completed at the Warsaw University of Technology in 2004. The subject of the doctorate were the method of estimation and noise reduction in the time series. The author is also a specialist in finance (especially in derivatives such as options), mathematics, informatics, and philosophy. At the beginning of his career for several years worked as a scientist at the Max-Planck Institute in Dresden and Warsaw University of Technology. He is the author of several scientific content items. For a long time he worked as a quant in financial institutions. He is currently a chief executive officer and a minority shareholder in the company Quant Technology2. For more information, please visit www.wonabru.com. The author is the founder and creator of Informationism. 1 E-mail: [email protected] 2 www.quant-technology.com Informationism from Philosophy to Quantitative Trading (4th edition)– Krzysztof Urbanowicz 1 “Logic is the way to independence” “To all that are above the schemes” - wonabru Informationism from Philosophy to Quantitative Trading (4th edition)– Krzysztof Urbanowicz 2 Introduction 5 What is Informationism? 7 Theory of the origin of the Universe 10 Question: Why Me? 13 Information theory - the foundation of Informationism 15 Principles of information 17 The generalization of the de'Broglie waves - information carrier 20 Experiment to proof Theory of Information 21 General description of idea of the experiment 21 Setup the experiment 24 25 Results 25 Conclusions 26 Principles of conservation of matter and information 27 The main principles of Informationism 27 Difficult questions that can be answered from the standpoint of Informationism 29 What is information? 29 Is there any absolute constant in our Universe in time and space? 29 How to explain gravity on the basis of Information Theory? 29 Synchronous communication: a higher level of thinking in the twenty-first century 30 Mysticism and Information Theory 31 What is energy? 31 Why is that information is spontaneously self-created? The three bodies problem. 32 What is the problem with the Second Law of Thermodynamics? Why do religious people claim that this is evidence for the existence of God? 33 Corpuscular-wave duality 34 Intuitive evidence that the matter can be converted to information. 34 Practical application of Informationism 35 The Theory of Objective Values 35 Objective Value Theory in finance 35 Normalization in Objective Value space 37 Examples 38 Objective Value (ObV) Option pricing model for power-law distribution 40 Put option 46 Greek letter Delta 47 Informationism from Philosophy to Quantitative Trading (4th edition)– Krzysztof Urbanowicz 3 Difference to Lisa Borland approach 47 Results for option pricing. 48 Delta Hedge Strategy based on the Objective Value Option Pricing theory 53 How Objective value theory works in trend forecasting of stochastic systems? 55 Statistical forecasting 58 Generalized gaussianity evaluation 59 The Wonabru investment method 60 Results based on WONABRU methodology 61 Introducing Shannon entropy into portfolio optimization 63 Equations of portfolio optimization 65 The explanation of introducing entropy to portfolio optimization 66 Stochastic processes in financial data – how to evaluate it? 73 Entropy estimation for a time series in the noise absence 75 Influence of noise on correlation integral 76 Noise estimation for financial time series 79 How quickly and properly calculate Value-at-Risk for the whole portfolio? 80 Examples of automated strategies 83 Strategy I 83 Strategy II 83 Strategy III 83 Strategy IV 84 Strategy V 84 Strategy VI 84 Bibliography 85 Informationism from Philosophy to Quantitative Trading (4th edition)– Krzysztof Urbanowicz 4 Introduction Someone may ask: Why am I introducing philosophy or even some kind of God to story about physics and finance? So I should refer to this question. One should start from the very beginning, if she/he wanted to build success. Success is created on strong basis, fundamentals, absolute Truth, objectives, fully logical proved and could not be discredited. Everybody should have the possibility to perform one’s Logic, in the same manner, with the understanding. I’ve started from looking something stable, absolute in time and space, in the Universe. The Theory of Creation of Universe shows how one may play with logic. I was looking also for universal principles which govern the Universe. Physics gives such a possibility. The Theory of Information is making huge unification of nowadays physical Laws. This is the basis. Without this, I could not make any progress in practical creation of investment strategies, etc. Automated investment strategies appears as the practical implementation and consequence of fundamentals. I must state that, there is an absolute in our Universe. This is Action (Logic). Action for physicists. Logic is for Informationists. I am Informationist, means Informationism is understandable for me and fully logical. How the quant trading can fetch values from Informationism? Trading is very tough staff. It needs very good understanding of science (not only physics). Well performing in programming. One also should disclaim intuition, emotions and feelings in trading, because thus make you lose. Apart above leaves just logic. Everything in this World, in this Universe, is logical, so can be understood by each person. Informationism gives you the way to build your own Absolut, because without an Absolute, you won't do anything successful. Absolut gives you strength not to change your mind, if your logic says that solution should be close, but emotions want you go back. Your Absolute makes you to be convinced and this is very important when things are starting to go badly. On the other hand, absolute value means objective, means really existing. On the Objective Value Theory (ObV) are based many strategies in this book. Why objective value is so important? It is, because, objective is additive. In the next step it is stated to be commutative. One can then, perform any arithmetic on such values like divisions, multiplications etc... About objective values are referring Gödel’s Theory of Incompleteness3. When values are additive, one can build principles, like principle of conservation. Principle, that, objective values are limited. Last one principle is that, on creation of the objective value the minimal dynamic is performed (minimal action or minimal dynamic entropy principle). I will talk about the last one in details later. You will see that, starting from nothing, vacuum, emptiness one can recognize principles ruling the Universe. This rules we can use for prediction. For example: if something is minimized or going to equilibrium, we can use it to found out the trend. Informationism is very helpful, because even if we stuck in dead 3 http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems Informationism from Philosophy to Quantitative Trading (4th edition)– Krzysztof Urbanowicz 5 point, like in developing some theory, every time we can refer to our Absolute and find solution. Quant Trading, means automated trading, also every aspects of success in your life, is based on solid backgrounds. Here we have Informationism. This is my grounds and also can be yours. In the next section we will explain the main ideas of Informationism and Theory of Information and you will acknowledge, why it is so important. Informationism from Philosophy to Quantitative Trading (4th edition)– Krzysztof Urbanowicz 6 What is Informationism? Informationism is the philosophy of science, based on Information Theory. Information Theory creates a binding of matter with the information. Information Theory is the foundation of Informationism and we devote to this theory entire chapter later in this book. In short, this theory says that, the action, as because it is discrete, can be converted to information. There is well-known corpuscular-wave duality found by de’Broglie. This duality is saying, that each moving mass generates wave. We generalized this theory also to not moving masses, but for which mass is changing. Mass can be diminished, when energy is released by e.g. nuclear reaction. Strong fields binds all neutron and protons together in kernel of atom. This energy, which keeps atom kernel together, increases the mass by the value given by Einstein well-known formula E = mc2. In the rest of this book, when we are saying about information, which interacts with matter, we are meaning, that the wave is interacting with matter. Information is just expression of this wave, which is countable and visible, especially in Fourier space of this wave. Such a waves we will be calling Information waves. Returning to main point of this book. Action is the energy change over time. Information is described simply by a number. The minimum value of the action h (Planck's constant h = 6.62e-34 [Js]) is one bit of information. Information is not static, but dynamic. Formed by doing something, in short, action. The matter, however, is discrete but static. Well-known Einstein theory describes the conversion of mass to energy, means E = mc2. This can also be rewritten in a more general way, that the change of mass produces change of energy, ie. ΔE = Δmc2. If we now multiply both sides by a changing of time, in which this action took place, we get ΔEΔt = ΔmΔtc2. As already mentioned, the energy change over time, gives the action, which is generates amount of information of the value: 퐼 ∙ ℎ = 푐2 ∙ Δ푚 ∙ Δ푡 , (1) where I is the number of bits. This is the basic formula for the conversion of matter to waves which carry information. Informationism is a complete foundation of the world of physics. There are only discrete values, countable, limited and finite. Energy, as because is described by continuous values, does not exist in reality. It helps in the description of some phenomena. The idea of energy explains the some physics laws, but in reality the production of information waves causes phenomena made by energy (such as the atomic bomb).
Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Electron Power-Law Spectra in Solar and Space Plasmas
    Space Science Reviews manuscript No. (will be inserted by the editor) Electron Power-Law Spectra in Solar and Space Plasmas M. Oka · J. Birn · M. Battaglia · C. C. Chaston · S. M. Hatch · G. Livadiotis · S. Imada · Y. Miyoshi · M. Kuhar · F. Effenberger · E. Eriksson · Y. V. Khotyaintsev · A. Retino` Received: date / Accepted: date Abstract Particles are accelerated to very high, non-thermal energies in solar and space plasma environments. While energy spectra of accelerated electrons often exhibit a power law, it remains unclear how electrons are accelerated to high energies and what processes determine the power-law index d. Here, we review previous observations of the power-law index d in a variety of different plasma environments with a particular focus on sub-relativistic electrons. It appears that in regions more closely related to magnetic reconnection (such as the ‘above-the-looptop’ solar hard X-ray source and the plasma sheet in Earth’s magnetotail), M. Oka · C. C. Chaston Space Sciences Laboratory, University of California Berkeley 7 Gauss Way, Berkeley, CA 94720 Tel.: +1-510-642-1350 E-mail: [email protected] J. Birn Space Science Institute, Boulder, Colorado, USA Los Alamos National Laboratory, Los Alamos, NewMexico, USA M. Battaglia · M. Kuhar Institute of 4D Technologies, School of Engineering, University of Applied Sciences and Arts Northwestern Switzerland, CH-5210 Windisch,Switzerland S. M. Hatch Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire, USA G. Livadiotis Southwest Research Institute, San Antonio, TX-78238, USA S. Imada · Y. Miyoshi Institute for Space-Earth Environmental Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464- 8601 Aichi, Japan F.
    [Show full text]
  • Information Theoretical Properties of Tsallis Entropies
    Information theoretical properties of Tsallis entropies Shigeru FURUICHI1∗ 1Department of Electronics and Computer Science, Tokyo University of Science, Yamaguchi, 756-0884, Japan Abstract. A chain rule and a subadditivity for the entropy of type β, which is one of the nonadditive entropies, were derived by Z.Dar´oczy. In this paper, we study the further relations among Tsallis type entropies which are typical nonadditive entropies. The chain rule is generalized by showing it for Tsallis relative entropy and the nonadditive entropy. We show some inequalities related to Tsallis entropies, especially the strong subadditivity for Tsallis type entropies and the subadditivity for the nonadditive entropies. The subadditivity and the strong subadditivity naturally lead to define Tsallis mutual entropy and Tsallis conditional mutual entropy, respectively, and then we show again chain rules for Tsallis mutual entropies. We give properties of entropic distances in terms of Tsallis entropies. Finally we show parametrically extended results based on information theory. Keywords : Chain rule, subadditivity, strong subadditivity, entropic distance, Fano’s inequality, entropy rate, Tsallis mutual entropy and nonadditive entropy 2000 Mathematics Subject Classification : 94A17, 46N55, 26D15 PACS number: 65.40.Gr,02.50.-r,89.70.+c 1 Introduction For the probability distribution p(x) ≡ p(X = x) of the random variable X, Tsallis entropy was defined in [1] : q Sq(X) ≡− p(x) lnq p(x), (q 6= 1) (1) x X with one parameter q as an extension of Shannon entropy, where q-logarithm function is defined x1−q−1 by lnq(x) ≡ 1−q for any nonnegative real number x and q.
    [Show full text]
  • Generalized Maxwell Relations in Thermodynamics with Metric Derivatives
    entropy Article Generalized Maxwell Relations in Thermodynamics with Metric Derivatives José Weberszpil 1 ID and Wen Chen 2,* 1 Instituto Multidisciplinar–Departamento de Tecnologias e Linguagens, Universidade Federal Rural do Rio de Janeiro, UFRRJ-IM/DTL, Governador Roberto Silveira n/n, Nova Iguaçú, Rio de Janeiro 23890-000, Brazil; [email protected] or [email protected] 2 College of Mechanics and Materials, Hohai University, Jiangning District, Fuchengxi Road 8, Nanjing 211100, China * Correspondence: [email protected] Received: 5 July 2017; Accepted: 27 July 2017; Published: 7 August 2017 Abstract: In this contribution, we develop the Maxwell generalized thermodynamical relations via the metric derivative model upon the mapping to a continuous fractal space. This study also introduces the total q-derivative expressions depending on two variables, to describe nonextensive statistical mechanics and also the a-total differentiation with conformable derivatives. Some results in the literature are re-obtained, such as the physical temperature defined by Sumiyoshi Abe. Keywords: deformed maxwell relations; metric derivatives; fractal continuum; generalized statistical mechanics; Legendre structure 1. Introduction Maxwell relations are thermodynamic equations which establish the relations between various thermodynamic quantities (e.g., pressure, P, volume, V, Entropy, S, and temperature, T) in equilibrium thermodynamics via other fundamental quantities known as thermodynamical potentials—the most important being internal energy, U, Helmholtz free energy, F, enthalpy, H, and Gibbs free energy, G. A Legendre transform converts from a function of one set of variables to another function of a conjugate set of variables. With the use of Legendre transforms, we can study different equations of state. For example, with the adequate change in variables, U(S, V) transforms to F = F(V, T).
    [Show full text]
  • Physica a X-Ray Binary Systems and Nonextensivity
    Physica A 389 (2010) 854–858 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa X-ray binary systems and nonextensivity Marcelo A. Moret a,b,∗, Valter de Senna a, Gilney F. Zebende a,b, Pablo Vaveliuk a a Programa de Modelagem Computational - SENAI - CIMATEC 41650-010 Salvador, Bahia, Brazil b Departamento de Física, Universidade Estadual de Feira de Santana, CEP 44031-460, Feira de Santana, Bahia, Brazil article info a b s t r a c t Article history: We study the x-ray intensity variations obtained from the time series of 155 light curves Received 9 April 2009 of x-ray binary systems collected by the instrument All Sky Monitor on board the satel- Received in revised form 14 August 2009 lite Rossi X-Ray Timing Explorer. These intensity distributions are adequately fitted by q- Available online 20 October 2009 Gaussian distributions which maximize the Tsallis entropy and in turn satisfy a nonlinear Fokker–Planck equation, indicating their nonextensive and nonequilibrium behavior. From PACS: the values of the entropic index q obtained, we give a physical interpretation of the dynam- 05.10.Gg ics in x-ray binary systems based on the kinetic foundation of generalized thermostatistics. 05.90.+m 89.75.Da The present findings indicate that the binary systems display a nonextensive and turbulent 97.80.-d behavior. ' 2009 Elsevier B.V. All rights reserved. Keywords: Astrophysical sources Tsallis statistics In recent years, the generalized thermostatistical formalism (GTS) proposed by Tsallis [1] has received increasing attention due its success in the description of certain phenomena exhibiting atypical thermodynamical features.
    [Show full text]
  • A Maximum Entropy Framework for Nonexponential Distributions
    A maximum entropy framework for nonexponential distributions Jack Petersona,b, Purushottam D. Dixitc, and Ken A. Dillb,1 aDepartment of Mathematics, Oregon State University, Corvallis, OR 97331; bLaufer Center for Physical and Quantitative Biology, Departments of Physics and Chemistry, State University of New York, Stony Brook, NY 11794; and cDepartment of Systems Biology, Columbia University, New York, NY 10032 Contributed by Ken A. Dill, November 7, 2013 (sent for review June 26, 2013) P Probability distributions having power-law tails are observed in functional S½fpkg = − kpklog pk subject to constraints, such as a broad range of social, economic, and biological systems. We the known value of the average energy hEi. This procedure gives −βE describe here a potentially useful common framework. We derive the exponential (Boltzmann) distribution, pk ∝ e k ,whereβ distribution functions {pk} for situations in which a “joiner particle” is the Lagrange multiplier that enforces the constraint. This var- k pays some form of price to enter a community of size k − 1, where iational principle has been the subject of various historical justifi- costs are subject to economies of scale. Maximizing the Boltzmann– cations. It is now commonly understood as the approach that Gibbs–Shannon entropy subject to this energy-like constraint chooses the least-biased model that is consistent with the known predicts a distribution having a power-law tail; it reduces to the constraint(s) (39). Boltzmann distribution in the absence of economies of scale. We Is there an equally compelling principle that would select fat- show that the predicted function gives excellent fits to 13 different tailed distributions, given limited information? There is a large distribution functions, ranging from friendship links in social net- literature that explores this.
    [Show full text]