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Statistical Physics and Information Theory Full Text Available At Full text available at: http://dx.doi.org/10.1561/0100000052 Statistical Physics and Information Theory Full text available at: http://dx.doi.org/10.1561/0100000052 Statistical Physics and Information Theory Neri Merhav Department of Electrical Engineering Technion | Israel Institute of Technology Haifa 32000 Israel [email protected] Boston { Delft Full text available at: http://dx.doi.org/10.1561/0100000052 Foundations and Trends R in Communications and Information Theory Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 USA Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is N. Merhav, Statistical Physics and Information Theory, Foundations and Trends R in Communications and Information Theory, vol 6, nos 1{2, pp 1{212, 2009 ISBN: 978-1-60198-406-7 c 2010 N. Merhav All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Cen- ter, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC). The `services' for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged. Authorization does not extend to other kinds of copy- ing, such as that for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. In the rest of the world: Permission to pho- tocopy must be obtained from the copyright owner. Please apply to now Publishers Inc., PO Box 1024, Hanover, MA 02339, USA; Tel. +1-781-871-0245; www.nowpublishers.com; [email protected] now Publishers Inc. has an exclusive license to publish this material worldwide. Permission to use this content must be obtained from the copyright license holder. Please apply to now Publishers, PO Box 179, 2600 AD Delft, The Netherlands, www.nowpublishers.com; e-mail: [email protected] Full text available at: http://dx.doi.org/10.1561/0100000052 Foundations and Trends R in Communications and Information Theory Volume 6 Issues 1{2, 2009 Editorial Board Editor-in-Chief: Sergio Verd´u Depart of Electrical Engineering Princeton University Princeton, New Jersey 08544 Editors Venkat Anantharam (UC. Berkeley) Amos Lapidoth (ETH Zurich) Ezio Biglieri (U. Torino) Bob McEliece (Caltech) Giuseppe Caire (U. Southern Neri Merhav (Technion) California) David Neuhoff (U. Michigan) Roger Cheng (U. Hong Kong) Alon Orlitsky (UC. San Diego) K.C. Chen (Taipei) Vincent Poor (Princeton) Daniel Costello (U. Notre Dame) Kannan Ramchandran (UC. Thomas Cover (Stanford) Berkeley) Anthony Ephremides (U. Maryland) Bixio Rimoldi (EPFL) Andrea Goldsmith (Stanford) Shlomo Shamai (Technion) Dave Forney (MIT) Amin Shokrollahi (EPFL) Georgios Giannakis (U. Minnesota) Gadiel Seroussi (MSRI) Joachim Hagenauer (TU Munich) Wojciech Szpankowski (Purdue) Te Sun Han (Tokyo) Vahid Tarokh (Harvard) Babak Hassibi (Caltech) David Tse (UC. Berkeley) Michael Honig (Northwestern) Ruediger Urbanke (EPFL) Johannes Huber (Erlangen) Steve Wicker (Cornell) Hideki Imai (Tokyo) Raymond Yeung (Hong Kong) Rodney Kennedy (Canberra) Bin Yu (UC. Berkeley) Sanjeev Kulkarni (Princeton) Full text available at: http://dx.doi.org/10.1561/0100000052 Editorial Scope Foundations and Trends R in Communications and Informa- tion Theory will publish survey and tutorial articles in the following topics: • Coded modulation • Multiuser detection • Coding theory and practice • Multiuser information theory • Communication complexity • Optical communication channels • Communication system design • Pattern recognition and learning • Cryptology and data security • Quantization • Data compression • Quantum information processing • Data networks • Rate-distortion theory • Demodulation and Equalization • Shannon theory • Denoising • Signal processing for • Detection and estimation communications • Information theory and statistics • Source coding • Information theory and computer • Storage and recording codes science • Speech and Image Compression • Joint source/channel coding • Wireless Communications • Modulation and signal design Information for Librarians Foundations and Trends R in Communications and Information Theory, 2009, Volume 6, 6 issues. ISSN paper version 1567-2190. ISSN online version 1567- 2328. Also available as a combined paper and online subscription. Full text available at: http://dx.doi.org/10.1561/0100000052 Foundations and Trends R in Communications and Information Theory Vol. 6, Nos. 1{2 (2009) 1{212 c 2010 N. Merhav DOI: 10.1561/0100000052 Statistical Physics and Information Theory Neri Merhav Department of Electrical Engineering, Technion | Israel Institute of Technology, Haifa 32000, Israel, [email protected] Abstract This monograph is based on lecture notes of a graduate course, which focuses on the relations between information theory and statistical physics. The course was delivered at the Technion during the Spring of 2010 for the first time, and its target audience consists of EE graduate students in the area of communications and information theory, as well as graduate students in Physics who have basic background in infor- mation theory. Strong emphasis is given to the analogy and parallelism between information theory and statistical physics, as well as to the insights, the analysis tools and techniques that can be borrowed from statistical physics and `imported' to certain problem areas in informa- tion theory. This is a research trend that has been very active in the last few decades, and the hope is that by exposing the students to the meeting points between these two disciplines, their background and perspective may be expanded and enhanced. This monograph is sub- stantially revised and expanded relative to an earlier version posted in arXiv (1006.1565v1 [cs.iT]). Full text available at: http://dx.doi.org/10.1561/0100000052 Contents 1 Introduction 1 2 Basic Background in Statistical Physics 7 2.1 What is Statistical Physics? 7 2.2 Basic Postulates and the Microcanonical Ensemble 8 2.3 The Canonical Ensemble 17 2.4 Properties of the Partition Function and the Free Energy 21 2.5 The Energy Equipartition Theorem 32 2.6 The Grand-Canonical Ensemble 34 3 Physical Interpretations of Information Measures 39 3.1 Statistical Physics of Optimum Message Distributions 40 3.2 Large Deviations and Physics of Coding Theorems 42 3.3 Gibbs' Inequality and the Second Law 57 3.4 Boltzmann's H-Theorem and the DPT 73 3.5 Generalized Temperature and Fisher Information 88 4 Analysis Tools and Asymptotic Methods 95 4.1 Introduction 95 4.2 The Laplace Method 97 4.3 The Saddle-Point Method 101 ix Full text available at: http://dx.doi.org/10.1561/0100000052 4.4 Extended Example: Capacity of a Disordered System 111 4.5 The Replica Method 116 5 Interacting Particles and Phase Transitions 123 5.1 Introduction | Sources of Interaction 124 5.2 Models of Interacting Particles 125 5.3 A Qualitative Discussion on Phase Transitions 131 5.4 Phase Transitions of the Rate{Distortion Function 135 5.5 The One-Dimensional Ising Model 139 5.6 The Curie{Weiss Model 142 5.7 Spin Glasses and Random Code Ensembles 147 6 The Random Energy Model and Random Coding 153 6.1 REM without a Magnetic Field 153 6.2 Random Code Ensembles and the REM 159 6.3 Random Coding Exponents 166 7 Extensions of the REM 177 7.1 REM Under Magnetic Field and Source{Channel Coding 178 7.2 Generalized REM (GREM) and Hierarchical Coding 187 7.3 Directed Polymers in a Random Medium and Tree Codes 198 8 Summary and Outlook 203 Acknowledgments 205 References 207 Full text available at: http://dx.doi.org/10.1561/0100000052 1 Introduction This work focuses on some of the relationships and the interplay between information theory and statistical physics | a branch of physics that deals with many-particle systems using probabilistic and statistical methods in the microscopic level. The relationships between information theory and statistical ther- modynamics are by no means new, and many researchers have been exploiting them for many years. Perhaps the first relation, or analogy, that crosses one's mind is that in both fields there is a fundamental notion of entropy. Actually, in information theory, the term entropy was coined in the footsteps of the thermodynamic entropy. The ther- modynamic entropy was first introduced by Clausius in 1850, and its probabilistic-statistical interpretation was established by Boltzmann in 1872. It is virtually impossible to miss the functional resemblance between the two notions of entropy, and indeed it was recognized by Shannon and von Neumann. The well-known anecdote on this tells that von Neumann advised Shannon to adopt this term because it would provide him with \. a great edge in debates because nobody really knows what entropy is anyway." But the relationships between the two fields go far beyond the fact that both share the notion of entropy. In fact, these relationships have 1 Full text available at: http://dx.doi.org/10.1561/0100000052
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