From the Ergodic Hypothesis in Physics to the Ergodic Axiom in Economics Prepared for the 7
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Symmetry and Ergodicity Breaking, Mean-Field Study of the Ising Model
Symmetry and ergodicity breaking, mean-field study of the Ising model Rafael Mottl, ETH Zurich Advisor: Dr. Ingo Kirsch Topics Introduction Formalism The model system: Ising model Solutions in one and more dimensions Symmetry and ergodicity breaking Conclusion Introduction phase A phase B Tc temperature T phase transition order parameter Critical exponents Tc T Formalism Statistical mechanical basics sample region Ω i) certain dimension d ii) volume V (Ω) iii) number of particles/sites N(Ω) iv) boundary conditions Hamiltonian defined on the sample region H − Ω = K Θ k T n n B n Depending on the degrees of freedom Coupling constants Partition function 1 −βHΩ({Kn},{Θn}) β = ZΩ[{Kn}] = T r exp kBT Free energy FΩ[{Kn}] = FΩ[K] = −kBT logZΩ[{Kn}] Free energy per site FΩ[K] fb[K] = lim N(Ω)→∞ N(Ω) i) Non-trivial existence of limit ii) Independent of Ω iii) N(Ω) lim = const N(Ω)→∞ V (Ω) Phases and phase boundaries Supp.: -fb[K] exists - there exist D coulping constants: {K1,...,KD} -fb[K] is analytic almost everywhere - non-analyticities of f b [ { K n } ] are points, lines, planes, hyperplanes in the phase diagram Dimension of these singular loci: Ds = 0, 1, 2,... Codimension C for each type of singular loci: C = D − Ds Phase : region of analyticity of fb[K] Phase boundaries : loci of codimension C = 1 Types of phase transitions fb[K] is everywhere continuous. Two types of phase transitions: ∂fb[K] a) Discontinuity across the phase boundary of ∂Ki first-order phase transition b) All derivatives of the free energy per site are continuous -
Ergodicity, Decisions, and Partial Information
Ergodicity, Decisions, and Partial Information Ramon van Handel Abstract In the simplest sequential decision problem for an ergodic stochastic pro- cess X, at each time n a decision un is made as a function of past observations X0,...,Xn 1, and a loss l(un,Xn) is incurred. In this setting, it is known that one may choose− (under a mild integrability assumption) a decision strategy whose path- wise time-average loss is asymptotically smaller than that of any other strategy. The corresponding problem in the case of partial information proves to be much more delicate, however: if the process X is not observable, but decisions must be based on the observation of a different process Y, the existence of pathwise optimal strategies is not guaranteed. The aim of this paper is to exhibit connections between pathwise optimal strategies and notions from ergodic theory. The sequential decision problem is developed in the general setting of an ergodic dynamical system (Ω,B,P,T) with partial information Y B. The existence of pathwise optimal strategies grounded in ⊆ two basic properties: the conditional ergodic theory of the dynamical system, and the complexity of the loss function. When the loss function is not too complex, a gen- eral sufficient condition for the existence of pathwise optimal strategies is that the dynamical system is a conditional K-automorphism relative to the past observations n n 0 T Y. If the conditional ergodicity assumption is strengthened, the complexity assumption≥ can be weakened. Several examples demonstrate the interplay between complexity and ergodicity, which does not arise in the case of full information. -
On the Continuing Relevance of Mandelbrot's Non-Ergodic Fractional Renewal Models of 1963 to 1967
Nicholas W. Watkins On the continuing relevance of Mandelbrot's non-ergodic fractional renewal models of 1963 to 1967 Article (Published version) (Refereed) Original citation: Watkins, Nicholas W. (2017) On the continuing relevance of Mandelbrot's non-ergodic fractional renewal models of 1963 to 1967.European Physical Journal B, 90 (241). ISSN 1434-6028 DOI: 10.1140/epjb/e2017-80357-3 Reuse of this item is permitted through licensing under the Creative Commons: © 2017 The Author CC BY 4.0 This version available at: http://eprints.lse.ac.uk/84858/ Available in LSE Research Online: February 2018 LSE has developed LSE Research Online so that users may access research output of the School. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may freely distribute the URL (http://eprints.lse.ac.uk) of the LSE Research Online website. Eur. Phys. J. B (2017) 90: 241 DOI: 10.1140/epjb/e2017-80357-3 THE EUROPEAN PHYSICAL JOURNAL B Regular Article On the continuing relevance of Mandelbrot's non-ergodic fractional renewal models of 1963 to 1967? Nicholas W. Watkins1,2,3 ,a 1 Centre for the Analysis of Time Series, London School of Economics and Political Science, London, UK 2 Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry, UK 3 Faculty of Science, Technology, Engineering and Mathematics, Open University, Milton Keynes, UK Received 20 June 2017 / Received in final form 25 September 2017 Published online 11 December 2017 c The Author(s) 2017. This article is published with open access at Springerlink.com Abstract. -
Ergodicity and Metric Transitivity
Chapter 25 Ergodicity and Metric Transitivity Section 25.1 explains the ideas of ergodicity (roughly, there is only one invariant set of positive measure) and metric transivity (roughly, the system has a positive probability of going from any- where to anywhere), and why they are (almost) the same. Section 25.2 gives some examples of ergodic systems. Section 25.3 deduces some consequences of ergodicity, most im- portantly that time averages have deterministic limits ( 25.3.1), and an asymptotic approach to independence between even§ts at widely separated times ( 25.3.2), admittedly in a very weak sense. § 25.1 Metric Transitivity Definition 341 (Ergodic Systems, Processes, Measures and Transfor- mations) A dynamical system Ξ, , µ, T is ergodic, or an ergodic system or an ergodic process when µ(C) = 0 orXµ(C) = 1 for every T -invariant set C. µ is called a T -ergodic measure, and T is called a µ-ergodic transformation, or just an ergodic measure and ergodic transformation, respectively. Remark: Most authorities require a µ-ergodic transformation to also be measure-preserving for µ. But (Corollary 54) measure-preserving transforma- tions are necessarily stationary, and we want to minimize our stationarity as- sumptions. So what most books call “ergodic”, we have to qualify as “stationary and ergodic”. (Conversely, when other people talk about processes being “sta- tionary and ergodic”, they mean “stationary with only one ergodic component”; but of that, more later. Definition 342 (Metric Transitivity) A dynamical system is metrically tran- sitive, metrically indecomposable, or irreducible when, for any two sets A, B n ∈ , if µ(A), µ(B) > 0, there exists an n such that µ(T − A B) > 0. -
Josiah Willard Gibbs
GENERAL ARTICLE Josiah Willard Gibbs V Kumaran The foundations of classical thermodynamics, as taught in V Kumaran is a professor textbooks today, were laid down in nearly complete form by of chemical engineering at the Indian Institute of Josiah Willard Gibbs more than a century ago. This article Science, Bangalore. His presentsaportraitofGibbs,aquietandmodestmanwhowas research interests include responsible for some of the most important advances in the statistical mechanics and history of science. fluid mechanics. Thermodynamics, the science of the interconversion of heat and work, originated from the necessity of designing efficient engines in the late 18th and early 19th centuries. Engines are machines that convert heat energy obtained by combustion of coal, wood or other types of fuel into useful work for running trains, ships, etc. The efficiency of an engine is determined by the amount of useful work obtained for a given amount of heat input. There are two laws related to the efficiency of an engine. The first law of thermodynamics states that heat and work are inter-convertible, and it is not possible to obtain more work than the amount of heat input into the machine. The formulation of this law can be traced back to the work of Leibniz, Dalton, Joule, Clausius, and a host of other scientists in the late 17th and early 18th century. The more subtle second law of thermodynamics states that it is not possible to convert all heat into work; all engines have to ‘waste’ some of the heat input by transferring it to a heat sink. The second law also established the minimum amount of heat that has to be wasted based on the absolute temperatures of the heat source and the heat sink. -
Université Du Québec À Montréal Edwin B. Wilson Aux Origines De L
UNIVERSITÉ DU QUÉBEC À MONTRÉAL EDWIN B. WILSON AUX ORIGINES DE L’ÉCONOMIE MATHÉMATIQUE DE PAUL SAMUELSON: ESSAIS SUR L’HISTOIRE ENTREMÊLÉE DE LA SCIENCE ÉCONOMIQUE, DES MATHÉMATIQUES ET DES STATISTIQUES AUX ÉTATS-UNIS, 1900-1940 THÈSE PRÉSENTÉE COMME EXIGENCE PARTIELLE DU DOCTORAT EN ÉCONOMIQUE PAR CARVAJALINO FLOREZ JUAN GUILLERMO DÉCEMBRE 2016 UNIVERSITÉ DU QUÉBEC À MONTRÉAL EDWIN B. WILSON AT THE ORIGIN OF PAUL SAMUELSON'S MATHEMATICAL ECONOMICS: ESSAYS ON THE INTERWOVEN HISTORY OF ECONOMICS, MATHEMATICS AND STATISTICS IN THE U.S., 1900-1940 THESIS PRESENTED AS PARTIAL REQUIREMENT OF DOCTOR OF PHILOSPHY IN ECONOMIC BY CARVAJALINO FLOREZ JUAN GUILLERMO DECEMBER 2016 ACKNOWLEDGMENTS For the financial aid that made possible this project, I am thankful to the Université du Québec À Montréal (UQAM), the Fonds de Recherche du Québec Société et Culture (FRQSC), the Social Science and Humanities Research Council of Canada (SSHRCC) as well as the Centre Interuniversitaire de Recherche sur la Science et la Technologie (CIRST). I am also grateful to archivists for access to various collections and papers at the Harvard University Archives (Edwin Bidwell Wilson), David M. Rubenstein Rare Book & Manuscript Library, Duke University (Paul A. Samuelson and Lloyd Metzler), Library of Congress, Washington, D.C. (John von Neumann and Oswald Veblen) and Yale University Library (James Tobin). I received encouraging and intellectual support from many. I particularly express my gratitude to the HOPE Center, at Duke University, where I spent the last semester of my Ph.D. and where the last chapter of this thesis was written. In particular, I thank Roy Weintraub for all his personal support, for significantly engaging with my work and for his valuable research advice. -
Viscosity from Newton to Modern Non-Equilibrium Statistical Mechanics
Viscosity from Newton to Modern Non-equilibrium Statistical Mechanics S´ebastien Viscardy Belgian Institute for Space Aeronomy, 3, Avenue Circulaire, B-1180 Brussels, Belgium Abstract In the second half of the 19th century, the kinetic theory of gases has probably raised one of the most impassioned de- bates in the history of science. The so-called reversibility paradox around which intense polemics occurred reveals the apparent incompatibility between the microscopic and macroscopic levels. While classical mechanics describes the motionof bodies such as atoms and moleculesby means of time reversible equations, thermodynamics emphasizes the irreversible character of macroscopic phenomena such as viscosity. Aiming at reconciling both levels of description, Boltzmann proposed a probabilistic explanation. Nevertheless, such an interpretation has not totally convinced gen- erations of physicists, so that this question has constantly animated the scientific community since his seminal work. In this context, an important breakthrough in dynamical systems theory has shown that the hypothesis of microscopic chaos played a key role and provided a dynamical interpretation of the emergence of irreversibility. Using viscosity as a leading concept, we sketch the historical development of the concepts related to this fundamental issue up to recent advances. Following the analysis of the Liouville equation introducing the concept of Pollicott-Ruelle resonances, two successful approaches — the escape-rate formalism and the hydrodynamic-mode method — establish remarkable relationships between transport processes and chaotic properties of the underlying Hamiltonian dynamics. Keywords: statistical mechanics, viscosity, reversibility paradox, chaos, dynamical systems theory Contents 1 Introduction 2 2 Irreversibility 3 2.1 Mechanics. Energyconservationand reversibility . ........................ 3 2.2 Thermodynamics. -
Hamiltonian Mechanics Wednesday, ß November Óþõõ
Hamiltonian Mechanics Wednesday, ß November óþÕÕ Lagrange developed an alternative formulation of Newtonian me- Physics ÕÕÕ chanics, and Hamilton developed yet another. Of the three methods, Hamilton’s proved the most readily extensible to the elds of statistical mechanics and quantum mechanics. We have already been introduced to the Hamiltonian, N ∂L = Q ˙ − = Q( ˙ ) − (Õ) H q j ˙ L q j p j L j=Õ ∂q j j where the generalized momenta are dened by ∂L p j ≡ (ó) ∂q˙j and we have shown that dH = −∂L dt ∂t Õ. Legendre Transformation We will now show that the Hamiltonian is a so-called “natural function” of the general- ized coordinates and generalized momenta, using a trick that is just about as sophisticated as the product rule of dierentiation. You may have seen Legendre transformations in ther- modynamics, where they are used to switch independent variables from the ones you have to the ones you want. For example, the internal energy U satises ∂U ∂U dU = T dS − p dV = dS + dV (ì) ∂S V ∂V S e expression between the equal signs has the physics. It says that you can change the internal energy by changing the entropy S (scaled by the temperature T) or by changing the volume V (scaled by the negative of the pressure p). e nal expression is simply a mathematical statement that the function U(S, V) has a total dierential. By lining up the various quantities, we deduce that = ∂U and = − ∂U . T ∂S V p ∂V S In performing an experiment at constant temperature, however, the T dS term is awk- ward. -
Dynamical Systems and Ergodic Theory
MATH36206 - MATHM6206 Dynamical Systems and Ergodic Theory Teaching Block 1, 2017/18 Lecturer: Prof. Alexander Gorodnik PART III: LECTURES 16{30 course web site: people.maths.bris.ac.uk/∼mazag/ds17/ Copyright c University of Bristol 2010 & 2016. This material is copyright of the University. It is provided exclusively for educational purposes at the University and is to be downloaded or copied for your private study only. Chapter 3 Ergodic Theory In this last part of our course we will introduce the main ideas and concepts in ergodic theory. Ergodic theory is a branch of dynamical systems which has strict connections with analysis and probability theory. The discrete dynamical systems f : X X studied in topological dynamics were continuous maps f on metric spaces X (or more in general, topological→ spaces). In ergodic theory, f : X X will be a measure-preserving map on a measure space X (we will see the corresponding definitions below).→ While the focus in topological dynamics was to understand the qualitative behavior (for example, periodicity or density) of all orbits, in ergodic theory we will not study all orbits, but only typical1 orbits, but will investigate more quantitative dynamical properties, as frequencies of visits, equidistribution and mixing. An example of a basic question studied in ergodic theory is the following. Let A X be a subset of O+ ⊂ the space X. Consider the visits of an orbit f (x) to the set A. If we consider a finite orbit segment x, f(x),...,f n−1(x) , the number of visits to A up to time n is given by { } Card 0 k n 1, f k(x) A . -
Exploiting Ergodicity in Forecasts of Corporate Profitability
Exploiting ergodicity in forecasts of corporate profitability Philipp Mundt, Simone Alfarano and Mishael Milakovic Working Paper No. 147 March 2019 b B A M BAMBERG E CONOMIC RESEARCH GROUP 0 k* k BERG Working Paper Series Bamberg Economic Research Group Bamberg University Feldkirchenstraße 21 D-96052 Bamberg Telefax: (0951) 863 5547 Telephone: (0951) 863 2687 [email protected] http://www.uni-bamberg.de/vwl/forschung/berg/ ISBN 978-3-943153-68-2 Redaktion: Dr. Felix Stübben [email protected] Exploiting ergodicity in forecasts of corporate profitability∗ Philipp Mundty Simone Alfaranoz Mishael Milaković§ Abstract Theory suggests that competition tends to equalize profit rates through the pro- cess of capital reallocation, and numerous studies have confirmed that profit rates are indeed persistent and mean-reverting. Recent empirical evidence further shows that fluctuations in the profitability of surviving corporations are well approximated by a stationary Laplace distribution. Here we show that a parsimonious diffusion process of corporate profitability that accounts for all three features of the data achieves better out-of-sample forecasting performance across different time horizons than previously suggested time series and panel data models. As a consequence of replicating the empirical distribution of profit rate fluctuations, the model prescribes a particular strength or speed for the mean-reversion of all profit rates, which leads to superior forecasts of individual time series when we exploit information from the cross-sectional collection of firms. The new model should appeal to managers, an- alysts, investors and other groups of corporate stakeholders who are interested in accurate forecasts of profitability. -
STABLE ERGODICITY 1. Introduction a Dynamical System Is Ergodic If It
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 1, Pages 1{41 S 0273-0979(03)00998-4 Article electronically published on November 4, 2003 STABLE ERGODICITY CHARLES PUGH, MICHAEL SHUB, AND AN APPENDIX BY ALEXANDER STARKOV 1. Introduction A dynamical system is ergodic if it preserves a measure and each measurable invariant set is a zero set or the complement of a zero set. No measurable invariant set has intermediate measure. See also Section 6. The classic real world example of ergodicity is how gas particles mix. At time zero, chambers of oxygen and nitrogen are separated by a wall. When the wall is removed, the gasses mix thoroughly as time tends to infinity. In contrast think of the rotation of a sphere. All points move along latitudes, and ergodicity fails due to existence of invariant equatorial bands. Ergodicity is stable if it persists under perturbation of the dynamical system. In this paper we ask: \How common are ergodicity and stable ergodicity?" and we propose an answer along the lines of the Boltzmann hypothesis { \very." There are two competing forces that govern ergodicity { hyperbolicity and the Kolmogorov-Arnold-Moser (KAM) phenomenon. The former promotes ergodicity and the latter impedes it. One of the striking applications of KAM theory and its more recent variants is the existence of open sets of volume preserving dynamical systems, each of which possesses a positive measure set of invariant tori and hence fails to be ergodic. Stable ergodicity fails dramatically for these systems. But does the lack of ergodicity persist if the system is weakly coupled to another? That is, what happens if you have a KAM system or one of its perturbations that refuses to be ergodic, due to these positive measure sets of invariant tori, but somewhere in the universe there is a hyperbolic or partially hyperbolic system weakly coupled to it? Does the lack of egrodicity persist? The answer is \no," at least under reasonable conditions on the hyperbolic factor. -
Ergodic Theory in the Perspective of Functional Analysis
Ergodic Theory in the Perspective of Functional Analysis 13 Lectures by Roland Derndinger, Rainer Nagel, GÄunther Palm (uncompleted version) In July 1984 this manuscript has been accepted for publication in the series \Lecture Notes in Mathematics" by Springer-Verlag. Due to a combination of unfortunate circumstances none of the authors was able to perform the necessary ¯nal revision of the manuscript. TÄubingen,July 1987 1 2 I. What is Ergodic Theory? The notion \ergodic" is an arti¯cial creation, and the newcomer to \ergodic theory" will have no intuitive understanding of its content: \elementary ergodic theory" neither is part of high school- or college- mathematics (as does \algebra") nor does its name explain its subject (as does \number theory"). Therefore it might be useful ¯rst to explain the name and the subject of \ergodic theory". Let us begin with the quotation of the ¯rst sentence of P. Walters' introductory lectures (1975, p. 1): \Generally speaking, ergodic theory is the study of transformations and flows from the point of view of recurrence properties, mixing properties, and other global, dynamical, properties connected with asymptotic behavior." Certainly, this de¯nition is very systematic and complete (compare the beginning of our Lectures III. and IV.). Still we will try to add a few more answers to the question: \What is Ergodic Theory ?" Naive answer: A container is divided into two parts with one part empty and the other ¯lled with gas. Ergodic theory predicts what happens in the long run after we remove the dividing wall. First etymological answer: ergodhc=di±cult. Historical answer: 1880 - Boltzmann, Maxwell - ergodic hypothesis 1900 - Poincar¶e - recurrence theorem 1931 - von Neumann - mean ergodic theorem 1931 - Birkho® - individual ergodic theorem 1958 - Kolmogorov - entropy as an invariant 1963 - Sinai - billiard flow is ergodic 1970 - Ornstein - entropy classi¯es Bernoulli shifts 1975 - Akcoglu - individual Lp-ergodic theorem Naive answer of a physicist: Ergodic theory proves that time mean equals space mean.