Stochastic Dynamics of Complex Systems

Transcription

1 Series on Complexity Science Vol. 2 Stochastic Dynamics of Complex Systems From Glasses to Evolution

2 Series on Complexity Science ISSN: Series Editor: Henrik Jeldtoft Jensen (Imperial College London, UK) Published Vol. 1: Vol. 2: A Complexity Approach to Sustainability: Theory and Application by Angela Espinosa & Jon Walker Stochastic Dynamics of Complex Systems: From Glasses to Evolution by Paolo Sibani & Henrik Jeldtoft Jensen

3 Series on Complexity Science Vol. 2 Stochastic Dynamics of Complex Systems From Glasses to Evolution Paolo Sibani University of Southern Denmark, Denmark Henrik Jeldtoft Jensen Imperial College London, UK ICP Imperial College Press

4 Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover designed from watercolours by Henrik Jeldtoft Jensen Series on Complexity Science Vol. 2 STOCHASTIC DYNAMICS OF COMPLEX SYSTEMS From Glasses to Evolution Copyright 2013 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN Typeset by Stallion Press enquiries@stallionpress.com Printed in Singapore.

5 Preface The term complex system appearing in the title of this book undoubtedly means different things to different people. If one conventionally describes a system in terms of interacting entities, i.e. particles or, more generally, degrees of freedom, many, including the authors of this book, would agree that its degree of complexity hinges on the number of degrees of freedom being very large and on the interactions lacking simple symmetries, such as the translational symmetry of a crystal. In this view, an amorphous material, e.g. a glass, is more complex than a crystal. Furthermore, complex systems cannot be completely isolated from their environment and are therefore subject to random influences beyond the observer s control. For this reason, a complex system is typically described by statistical properties which can be estimated from measurements or from computer simulations of suitable stochastic models. Chaotic systems are fully deterministic dynamical systems with only few degrees of freedom. They are hence not complex in the above sense, nor is a quantum mechanical system such as the hydrogen atom. Truly enough, quantum mechanical predictions are probabilistic in nature, but the uncertainty expressed by say, the Heisenberg relations is not rooted in our ignorance of the accidentals affecting the history of the system but reflects instead intrinsic limits to which all descriptions of nature at very small length scales are subject. v

6 vi Stochastic Dynamics of Complex Systems Many books exist about complex systems. Most are concerned with structure and other stationary properties, and when dynamical effects are considered, they do not lead to essential change of macroscopic properties. Consider for example the magnetic fluctuations of a material, kept exactly at its magnetic phase transition point. The average magnetization is equal to zero but the probability distribution of the size of the magnetic fluctuations has very interesting properties which reflect an incipient change of symmetry of the system, but do not change in time. The main focus of this book is on dynamics and especially on dynamics leading to change, i.e. on evolution and non-stationary states. We are concerned with slow but persistent changes observable at the macroscopic or systemic level. Unlike the magnet just mentioned, systems with these properties lack time translational symmetry. One may say the book s focus is narrow, or one may perhaps agree that dynamical evolution over long time scales is a prominent feature of all the systems we intuitively think of as complex, say ecosystems, the brain or the economy. In the physics literature the term ageing refers to slow changes of observable properties which occur over time scales much longer than the patience, or indeed the lifetime of the observer. In the book, we use and develop this term for complex systems outside the realm of physics. The book is divided into two parts. The material in the first part attempts to provide the necessary mathematical and computational tools and the intuition needed to deal with the systems described in the second part. Initially, the topics are of the standard sort, and are hence covered in several other books, but the emphasis and selection reflect both the author s interests and the overall theme of the book. Our discussion of Record Dynamics is new and is expected to be of relevance to a broad range of very different complex systems. The first part is suitable as teaching material for a one semester course dealing with the application of stochastic processes to natural science. The second part of the book contains an introduction to the scientific literature and deals in some detail with the description of complex phenomena of physical and biological nature, e.g,

7 Preface vii disordered magnetic materials, superconductors and glasses, and models of co-evolution in ecosystems and even of ant behaviour. These rather heterogeneous topics are all dealt with using similar techniques of analysis. For reasons of space, scope, bias and of intended readership, this part does not aspire to be a scientific review of the subjects covered. We believe that one or more of its chapters can be used as basic material in a graduate course on complex systems and that researchers in any field of science who want to enter the complexity arena might find it interesting as well. Throughout the book, each chapter begins with a qualitative introduction to the subject matter. It is our hope that these introductions will be useful not only for students and specialists following a course, but also for general readers with some basic knowledge of mathematics. Taken together, they can provide a background for the current hectic discussions about how complexity science can help to address some impelling problems in the world of today. May 2012, Odense and London Paolo Sibani and Henrik Jeldtoft Jensen

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9 Acknowledgements The present book is the outcome of a pleasant and long-standing collaboration between the authors. Beyond that, it reflects viewpoints on physics and complex systems which have evolved over the years through interactions with very many people. When it comes to the actual contents of the book, collaborations and discussions with colleagues and students have been of great importance. Paolo Sibani would like to thank Preben Alstrøm, Christian M. Andersen, Stefan Boettcher, Michael Brandt, Jesper Dall, John A. Hertz, Karl Heinz Hoffmann, Gregory G. Kenning, Peter Littlewood, Mark E.J. Newman, Richard G. Palmer, Andreas Pedersen, the late Jacob M. Pedersen, Jørgen B. Pedersen, Michel R. Schmidt, Christian Schön and Peter Salamon for their input on physics, science and all the rest. The many students who have been subjected to various drafts of the book and contributed with constructive criticism are commended for their patience and stamina in dealing with an unfinished work. A special thanks goes to Nikolaj Becker for carefully proof-reading parts of the manuscript and for delivering one of its figures. Henrik Jeldtoft Jensen is indebted to Tomas Alarcon, Kim Christensen, Simone A. di Collobiano, Matt Hall, Simon Laird, Daniel Lawson, Mario Nicodemi, Gunnar Pruessner, Tom O. Richardson and Ana Sendova-Franks. Both authors would like to thank Paul Anderson, Dominic Jones and Louis P. de Oliveira for their valuable input. ix

10 x Stochastic Dynamics of Complex Systems Families are important in all aspects of life, including the writing of a book. Paolo Sibani is indebted to his wife Karin who did his share of housework on top of her own and to our three kids, Nicolas, Camilla and Claudia, for cheering him up by doing well in all that they do. Henrik Jeldtoft Jensen is as always intrigued and overwhelmed by the persistent love and support from the world s three most wonderful women: his wife and daughters Vibeke N. Hansen, Barbara N.J. Jensen and Rebecca N.J. Jensen. May 2012, Odense and London, Paolo Sibani and Henrik Jeldtoft Jensen

11 Contents Preface Acknowledgements Index of Figures v ix xvii 1. Introduction 1 I Complex Dynamics: Tools and Applications Characterization of Collective Dynamics Introduction Basic Considerations Time Series and their Statistical Properties Stationary Time Series Non-stationary Time Series Stationary or Non-stationary? Markovian Stochastic Processes Introduction Langevin Equations Solution Techniques for the Langevin Equation Ballistic and Diffusive Limit 39 xi

12 xii Stochastic Dynamics of Complex Systems 3.3. General Properties of Markov Processes Master Equations and Markov Chains The Propagator and its Moments Fluctuation-dissipation Theorems Relaxation Properties Relaxation to a Prescribed Stationary State Eigenvalue Expansions of the Moments Markov Chains Six Examples of Master Equations Random Walks in Euclidean Space Continuous Time Random Walks Subordination Scaling Properties of CTRW Fokker Planck Equations Stationary Behaviour Monte Carlo Methods Introduction Importance Sampling The Metropolis Algorithm Event Driven Algorithms Standard KMC Methods Waiting Time Method Record Statistics and Extremal Statistics Introduction Records in White Noise Records and First Passage in Brownian Motion Extremal Value Statistics Relation between Record and Extreme Value Statistics Summary Complexity and Hierarchies Introduction 121

13 Contents xiii 6.2. Kinetically Constrained Hierarchical Dynamics Dynamical Hierarchies in Thermal Relaxation Energy Landscapes Introduction Modelling Energy Landscapes Coarse-graining and the Kramers Approximation Applying the Kramers Approximation to a Hierarchical Landscape Model Exploring Energy Landscapes Record Dynamics and Marginal Stability Introduction Coarse-Graining Complex Dynamics Marginal Stability Homogeneity of Record Dynamics in Logarithmic Time Scaling Behaviour in Record Dynamics 158 II Complex Systems with Similar Dynamics Ageing of Spin Glasses Introduction Spin Glasses The Order Parameter Are Spin Glasses in Equilibrium? Ageing: Experimental and Numerical Observations Theoretical Considerations Theoretical Outlook Magnetic Relaxation in Superconductors Introduction Magnetic Fields and Type II Superconductors 196

14 xiv Stochastic Dynamics of Complex Systems Restricted Occupancy Model Intermittent Magnetic Relaxation Ageing of Colloids Introduction Hard Sphere Colloids The Critical Density Non-ergodic Ageing Behaviour A Heuristic Model Conclusion Evolving Biological Systems Introduction Models of Biological Evolution Quasi-Species Model Kauffman Models NK Model NKC Model Bak Sneppen Model Tangled Nature Model Model Description General Qualitative Behaviour The Time Arrow in the Tangled Nature Model The Evolution of the Fluctuations in Birth Rate The Offspring Probability and Adaptation Conclusions Non-stationary Ageing Dynamics in Ant Societies Introduction Model Description General Considerations 248

15 Contents xv Data Analysis Discussion 254 III Epilogue What is Complexity Science? 259 Bibliography 263 Index 273

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17 Index of Figures Figure 2.1. Surface Air Temperature (SAT) in the Arctic region 19 Figure 2.2. Spurious 1/f spectrum 32 Figure 3.1. Linear Response 48 Figure 4.1. Binder Cumulants 90 Figure 6.1. A regular tree 123 Figure 7.1. A tree model with inequivalent minima 137 Figure 7.2. Valleys in a bistable potential 143 Figure 7.3. LDOS for a TSP problem 146 Figure 7.4. Relaxation tree for the TSP 147 Figure 8.1. Link between marginal stability and record dynamics 153 Figure 9.1. AC susceptibiities 169 Figure 9.2. ZFC ageing curves 172 Figure 9.3. CoolingCurves 174 Figure 9.4. Memory behaviour in spin glasses 175 Figure 9.5. ZFC magnetization and energy fluctuations 178 Figure 9.6. Energy fluctuation PDFs at different ages 181 Figure 9.7. The phase diagram of the Sherrington and Kirkpatrick model. The relevance of this phase diagram for real spin glasses is hotly contested 185 Figure 9.8. Hierarchical structure of spin glass landscape 190 xvii

18 xviii Stochastic Dynamics of Complex Systems Figure Sketch of layered ROM 198 Figure The detailed time variation of the total number of vortices N(t) 201 Figure The time dependence of the density of flux lines 202 Figure PDF of the number of vortices vs. time 204 Figure Creep rate vs. temperature 205 Figure Intermediate Scattering Function 210 Figure Diffusion in hard sphere colloids 212 Figure Intermediate Scattering Function, dense colloids 213 Figure Diffusion in dense hard sphere colloids 214 Figure Model MSD data 217 Figure Quakes in colloids 217 Figure Intermittent evolution of the occupancy in type space 232 Figure Time dependence of the total population 233 Figure Distribution of interaction strengths 234 Figure Cumulated number of transitions vs. time 235 Figure PDF of population fluctuations 236 Figure The standard deviation of birth rate fluctuations decreases over time 238 Figure Total population vs. time with non-convex offspring function 240 Figure Correlation function for consecutive waiting times 249 Figure Utility function vs time 250