New Forms of Ergodicity Breaking in Quantum Many-Body Systems
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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. -
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. -
Invariant Measures and Arithmetic Unique Ergodicity
Annals of Mathematics, 163 (2006), 165–219 Invariant measures and arithmetic quantum unique ergodicity By Elon Lindenstrauss* Appendix with D. Rudolph Abstract We classify measures on the locally homogeneous space Γ\ SL(2, R) × L which are invariant and have positive entropy under the diagonal subgroup of SL(2, R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproofofthe main result. 1. Introduction We recall that the group L is S-algebraic if it is a finite product of algebraic groups over R, C,orQp, where S stands for the set of fields that appear in this product. An S-algebraic homogeneous space is the quotient of an S-algebraic group by a compact subgroup. Let L be an S-algebraic group, K a compact subgroup of L, G = SL(2, R) × L and Γ a discrete subgroup of G (for example, Γ can be a lattice of G), and consider the quotient X =Γ\G/K. The diagonal subgroup et 0 A = : t ∈ R ⊂ SL(2, R) 0 e−t acts on X by right translation. In this paper we wish to study probablilty measures µ on X invariant under this action. Without further restrictions, one does not expect any meaningful classi- fication of such measures. For example, one may take L = SL(2, Qp), K = *The author acknowledges support of NSF grant DMS-0196124. -
Weak Ergodicity Breaking from Quantum Many-Body Scars
This is a repository copy of Weak ergodicity breaking from quantum many-body scars. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/130860/ Version: Accepted Version Article: Turner, CJ orcid.org/0000-0003-2500-3438, Michailidis, AA, Abanin, DA et al. (2 more authors) (2018) Weak ergodicity breaking from quantum many-body scars. Nature Physics, 14 (7). pp. 745-749. ISSN 1745-2473 https://doi.org/10.1038/s41567-018-0137-5 © 2018 Macmillan Publishers Limited. This is an author produced version of a paper accepted for publication in Nature Physics. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Quantum many-body scars C. J. Turner1, A. A. Michailidis1, D. A. Abanin2, M. Serbyn3, and Z. Papi´c1 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 2Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland and 3IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria (Dated: November 16, 2017) Certain wave functions of non-interacting quantum chaotic systems can exhibit “scars” in the fabric of their real-space density profile. -
Lecture Notes Or in Any Other Reasonable Lecture Notes
Ergodicity Economics Ole Peters and Alexander Adamou 2018/06/30 at 12:11:59 1 Contents 1 Tossing a coin4 1.1 The game...............................5 1.1.1 Averaging over many trials.................6 1.1.2 Averaging over time.....................7 1.1.3 Expectation value......................9 1.1.4 Time average......................... 14 1.1.5 Ergodic observables..................... 19 1.2 Rates................................. 23 1.3 Brownian motion........................... 24 1.4 Geometric Brownian motion..................... 28 1.4.1 Itˆocalculus.......................... 31 2 Decision theory 33 2.1 Models and science fiction...................... 34 2.2 Gambles................................ 34 2.3 Repetition and wealth evolution................... 36 2.4 Growth rates............................. 39 2.5 The decision axiom.......................... 41 2.6 The expected-wealth and expected-utility paradigms....... 45 2.7 General dynamics........................... 48 2.7.1 Why discuss general dynamics?............... 48 2.7.2 Technical setup........................ 50 2.7.3 Dynamic from a utility function.............. 54 2.7.4 Utility function from a dynamic.............. 56 2.8 The St Petersburg paradox..................... 59 2.9 The insurance puzzle......................... 65 2.9.1 Solution in the time paradigm............... 69 2.9.2 The classical solution of the insurance puzzle....... 71 3 Populations 72 3.1 Every man for himself........................ 73 3.1.1 Log-normal distribution................... 73 3.1.2 Two growth rates....................... 74 3.1.3 Measuring inequality..................... 75 3.1.4 Wealth condensation..................... 76 3.1.5 Rescaled wealth........................ 77 3.1.6 u-normal distributions and Jensen’s inequality...... 78 3.1.7 power-law resemblance.................... 79 3.2 Finite populations.......................... 80 3.2.1 Sums of log-normals.................... -
Recent Progress on the Quantum Unique Ergodicity Conjecture
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 48, Number 2, April 2011, Pages 211–228 S 0273-0979(2011)01323-4 Article electronically published on January 10, 2011 RECENT PROGRESS ON THE QUANTUM UNIQUE ERGODICITY CONJECTURE PETER SARNAK Abstract. We report on some recent striking advances on the quantum unique ergodicity, or QUE conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved manifold. The account falls naturally into two categories. The first concerns the general conjecture where the tools are more or less limited to microlocal analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. Our main emphasis is on the second category, especially where QUE has been proven. This note is not meant to be a survey of these topics, and the discussion is not chronological. Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context. 2. The general QUE conjecture In Figure 1 the domains ΩE, ΩS,andΩB are an ellipse,astadium and a Barnett billiard table, respectively. Superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions (states, modes) of the Laplacian. That is, they are solutions to ⎧ ⎪ ⎪ φj + λj φj = 0 in Ω , ⎪ ⎨⎪ φ = 0 (Dirichlet boundary conditions), (0) ∂Ω ⎪ ⎪ ⎪ 2 ⎩ |φj | dxdy =1. Ω ∂2 ∂2 ≤ ··· Here = divgrad = ∂x2 + ∂y2 , λ1 <λ2 λ3 are the eigenvalues, and the eigenfunctions are normalized to have unit L2-norm. -
Quantum Many-Body Scars Or Non-Ergodic Quantum Dynamics in Highly Excited States of a Kinematically Constrained Rydberg Chain
Quantum many-body scars or Non-ergodic Quantum Dynamics in Highly Excited States of a Kinematically Constrained Rydberg Chain Christopher J. Turner1, A. A. Michailidis1, D. A. Abanin2, M. Serbyn3, Z. Papi´c1 1School of Physics and Astronomy, University of Leeds 2Department of Theoretical Physics, University of Geneva 3IST Austria 15th December 2017 Lancaster, NQM2 arXiv:1711.03528 Outline What is a quantum scar? 1.0 L = 28 An experimental phenomena L = 32 0.8 2 i | ) t ( 0.6 2 Z | 2 0.4 4 Z Exact L Why is it happening? | h 2 0.2 FSA i | ψ 2 0.0 | n 0 10 20 30 t | h 0 What else is going on? 2 L 2 i | ψ 1 | n | h 0 0 10 20 30 n Quantum scars I First discussed by Heller 1984 in quantum stadium billiards. I Here, classically unstable periodic orbits of the stadium billiards (right) scarring a wavefunction (left). I One might expect unstable classical period orbits to be lost in the transition to quantum mechanics as the particle becomes \blurred". I This model is quantum ergodic but not quantum unique ergodic1. Think eigenstate thermalisation for all eigenstates vs. almost all eigenstates. 1Hassell 2010. ARTICLE doi:10.1038/nature24622 Probing many-body dynamics on a 51-atom quantum simulator Hannes Bernien1, Sylvain Schwartz1,2, Alexander Keesling1, Harry Levine1, Ahmed Omran1, Hannes Pichler1,3, Soonwon Choi1, Alexander S. Zibrov1, Manuel Endres4, Markus Greiner1, Vladan Vuletić2 & Mikhail D. Lukin1 Controllable, coherent many-body systems can provide insights into the fundamental properties of quantum matter, enable the realization of new quantum phases and could ultimately lead to computational systems that outperform existing computers based2 on classical approaches. -
Geometry and Dynamics in the Fractional Quantum Hall Effect: from Graviton Oscillation to Quantum Many-Body Scars
Geometry and dynamics in the fractional quantum Hall effect: from graviton oscillation to quantum many-body scars Zlatko Papic Anyons in Quantum Many-Body Systems, MPI PKS, Dresden, 21/01/2019 Historical timeline of quantum Hall effects Leinaas/Myrheim Moore and Read statistics in 2D Laughlin's theory introduce non-Abelian statistics of FQHE Thouless, [Moore, Read, Nucl. Phys. B. 360, 362 (1991)] Kosterlitz, Haldane Topological order IQHE/FQHE in graphene Integer QHE [K. Novoselov, P. Kim, E. Andrei] [Klitzing, Dorda, Pepper, X-G Wen PRL 45, 494 (1980)] 2005- 1982 mid 1980s 2003 2009 2012- 1977 1980 1983 1989 1991 2005- 2006 Majorana Fractional QHE Composite fermions zero modes (> 100 observed Jain [Delft, Princeton, states so far) TKNN Weizmann, [Tsui, Stormer, Gossard, Topological insulators Copenhagen...] PRL 48, 1559 (1982)] [Haldane; Kane, Mele; Bernevig, Hughes, Zhang; ...] FCIs, SPTs,... Numerical work Anyons (Haldane & Rezayi) confirms Kitaev's simpler models of Wilczek Laughlin's theory topological physics in lattice/spin systems Historical timeline of quantum Hall effects Leinaas/Myrheim Moore and Read statistics in 2D Laughlin's theory introduce non-Abelian statistics of FQHE Thouless, [Moore, Read, Nucl. Phys. B. 360, 362 (1991)] Kosterlitz, Haldane Topological order IQHE/FQHE in graphene Integer QHE [K. Novoselov, P. Kim, E. Andrei] [Klitzing, Dorda, Pepper, X-G Wen PRL 45, 494 (1980)] Q: FQHE is a mature subject (nearing 402005- years). mid 1980s 2009 2012- What1982 is there left to understand about2003 FQHE? 1977 1980 1983 1989 1991 2005- 2006 Majorana Fractional QHE Composite fermions zero modes (> 100 observed Jain [Delft, Princeton, states so far) TKNN Weizmann, [Tsui, Stormer, Gossard, Topological insulators Copenhagen...] PRL 48, 1559 (1982)] [Haldane; Kane, Mele; Bernevig, Hughes, Zhang; ...] FCIs, SPTs,..