Mathematical Statistical Physics

Mathematical Statistical Physics

Lecture Notes of the Les Houches Summer School 2005

1st Edition - June 27, 2006

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  • Editors: Anton Bovier, François Dunlop, Aernout Van Enter, Frank Den Hollander, Jean Dalibard
  • eBook ISBN: 9780080479231
  • Hardcover ISBN: 9780444528131

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Description

The proceedings of the 2005 les Houches summer school on Mathematical Statistical Physics give and broad and clear overview on this fast developing area of interest to both physicists and mathematicians.

Key Features

  • Introduction to a field of math with many interdisciplinary connections in physics, biology, and computer science
  • Roadmap to the next decade of mathematical statistical mechanics
  • Volume for reference years to come

Readership

Libraries of mathematics and physics, Individual scientists

Table of Contents

  • Lecturers who Contributed to this volume

    ÉCOLE D’ÉTÉ DE PHYSIQUE DES HOUCHES

    École de Physique des Houches

    Previous sessions

    Organizers

    Lecturers

    Participants

    Preface

    Informal Seminars

    Random Matrices and Determinantal Processes

    1 Introduction

    2 Point processes

    3 Non-intersecting paths and the Aztec diamond

    4 Asymptotics

    5 The corner growth model

    Acknowledgement

    Some Recent Aspects of Random Conformally Invariant Systems

    Overview

    Conformal Random Geometry

    1 Preamble

    2 Introduction

    3 Intersections of random walks

    4 Mixing random & self-avoiding walks

    5 Percolation clusters

    Random Motions in Random Media

    1 Introduction

    2 RWRE

    3 RCM

    4 Back to RWRE

    Effective criterion:

    5 Diffusions in random environment

    An Introduction to Mean Field Spin Glas Theory: Methods and Results

    1 Introduction

    2 The mean field ferromagnetic model. Convexity and cavity methods

    3 The mean field spin glass model. Basic definitions

    4 The interpolation method and its generalizations

    5 The thermodynamic limit and the variational bounds

    6 The Parisi representation for the free energy

    7 Conclusion and outlook for future developments

    Acknowledgments

    Short-Range Spin Glasses: Selected Open Problems

    1 Introduction

    2 The Fortuin-Kasteleyn random cluster representation and phase transitions

    3 Spin glass ground states and invasion percolation

    4 Ground state multiplicity in the 2D EA spin glass

    Acknowledgment

    Computing the Number of Metastable States in Infinite-Range Models

    Acknowledgments

    Dynamics of Trap Models

    1 Introduction

    2 Definition of the Bouchaud trap model

    2.2. Assumption 2.2.

    2.3. Assumption 2.3.

    2.1 Examples of trap models

    2.2 Natural questions on trap models

    2.3 References

    3 The one-dimensional trap model

    3.3 Aging results

    3.4 Subaging results

    3.5 Behaviour of the aging functions on different time scales

    4 The trap model in dimension larger than one

    4.3 Aging results

    4.4 The coarse-graining procedure

    4.5 References

    5 The arcsine law as a universal aging scheme

    6 Applications of the arcsine law

    Appendix A Subordinators

    Quantum Entropy and Quantum Information

    1 Introduction

    2 Rudiments of Classical Information Theory

    3 Introduction to Quantum Information Theory

    4 Open systems

    5 Quantum entropy

    6 Data compression in Quantum Information Theory

    7 Quantum channels and additivity

    Acknowledgments

    Two Lectures on Iterative Coding and Statistical Mechanics

    1 Introduction

    2 Codes on graphs

    3 A simple-minded bound and belief propagation

    4 Density evolution a.k.a. distributional recursive equations

    5 The area theorem and some general questions

    6 Historical and bibliographical note

    Evolution in Fluctuating Populations

    1 Introduction

    Acknowledgement

    2 Some classical coalescent theory

    4 Spatial structure and the Malécot formula

    5 Spatial models

    Multi-Scale Analysis of Population Models

    1 Spatial diffusion models of population genetics

    2 Duality and representation via coalescent processes

    Elements of Nonequilibrium Statistical Mechanics

    1 Elements of introduction

    2 Elements of an H-theorem

    3 Elements of heat conduction

    4 Lagrangian approach

    5 A little entropology

    6 Closed systems

    7 Open systems

    8 Why is it useful?

    9 What is missing — among other things?

    Mathematical Aspects of the Abelian Sandpile Model

    Open question

    Gibbsianness and Non-Gibbsianness in Lattice Random Fields

    1 Historical remarks and purpose of the course

    2 Setup, notation, and basic notions

    3 Probability kernels, conditional probabilities, and statistical mechanics

    4 What it takes to be Gibbsian

    5 What it takes to be non-Gibbsian

    Acknowledgments

    Simulation of Statistical Mechanics Models

    1 Overview

    2 The Swendsen–Wang algorithm: some recent progress

    Acknowledgments

Product details

  • No. of pages: 848
  • Language: English
  • Copyright: © Elsevier Science 2006
  • Published: June 27, 2006
  • Imprint: Elsevier Science
  • eBook ISBN: 9780080479231
  • Hardcover ISBN: 9780444528131

About the Series Volume Editors

Anton Bovier

François Dunlop

Aernout Van Enter

Frank Den Hollander

Jean Dalibard

Jean Dalibard works in the field of atomic physics and quantum optics. His recent activities is centered on the physics of cold quantum gases, in particular Bose-Einstein condensation.

Affiliations and Expertise

Laboratoire Kastler Brossel, ENS, Paris, France

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