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Applications of the Lace Expansion to Statistical-Mechanical Models

Akira Sakai

in Analysis and Stochastics of Growth Processes and Interface Models

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199239252
eISBN:
9780191716911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199239252.003.0006
Subject:
Mathematics, Probability / Statistics, Analysis

Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical ... More


5 Correlations and creating short‐range order

Reinhard B. Neder and Thomas Proffen

in Diffuse Scattering and Defect Structure Simulations: A cook book using the program DISCUS

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199233694
eISBN:
9780191715563
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233694.003.0005
Subject:
Physics, Crystallography: Physics

This chapter deals with the simulation of short range ordered crystals. Correlations are introduced as a convenient way to describe short-range order (SRO). The most common way to create structures ... More


The Ising model in two dimensions: summary of results

Barry M. McCoy

in Advanced Statistical Mechanics

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199556632
eISBN:
9780191723278
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199556632.003.0010
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz ... More


FLOWS AND FERROMAGNETS

Geoffrey Grimmett

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0008
Subject:
Mathematics, Probability / Statistics

The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety ... More


Exact solutions and related topics

Hidetoshi Nishimori and Gerardo Ortiz

in Elements of Phase Transitions and Critical Phenomena

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199577224
eISBN:
9780191722943
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199577224.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the ... More


Switching Paths for Ising Models with Long-Range Interaction

Nicolas Dirr

in Analysis and Stochastics of Growth Processes and Interface Models

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199239252
eISBN:
9780191716911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199239252.003.0011
Subject:
Mathematics, Probability / Statistics, Analysis

The chapter discusses a multiscale model for a two-phases material. The model is a stochastic process on the finest scale. The effective behaviour on larger scales is governed by deterministic ... More


Interacting systems on networks

Sergey N. Dorogovtsev

in Lectures on Complex Networks

Published in print:
2010
Published Online:
May 2010
ISBN:
9780199548927
eISBN:
9780191720574
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199548927.003.0013
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter considers systems of interacting agents placed on networks. The agents — spins, oscillators, interacting individuals, etc. — occupy the nodes of networks and interact with each other ... More


The Pfaffian solution of the Ising model

Barry M. McCoy

in Advanced Statistical Mechanics

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199556632
eISBN:
9780191723278
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199556632.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter presents the computation of the partition function of the two-dimensional Ising model in terms of a sum of Pfaffians using a combinatorial approach. This method is extended to derive the ... More


Implementation of the renormalization group

Hidetoshi Nishimori and Gerardo Ortiz

in Elements of Phase Transitions and Critical Phenomena

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199577224
eISBN:
9780191722943
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199577224.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the ... More


Calculation and computation

James P. Sethna

in Statistical Mechanics: Entropy, Order Parameters, and Complexity: Second Edition

Published in print:
2021
Published Online:
April 2021
ISBN:
9780198865247
eISBN:
9780191897610
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198865247.003.0008
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces Monte-Carlo techniques to simulate the equilibrium properties of complex systems, and perturbative techniques to calculate their behavior as an expansion about solvable ... More


Counting, Sampling, and Statistical Physics

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0013
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. ... More


Introduction: The simplest models of magnetism

Ralph Skomski

in Simple Models of Magnetism

Published in print:
2008
Published Online:
January 2010
ISBN:
9780198570752
eISBN:
9780191718816
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570752.003.0001
Subject:
Physics, Condensed Matter Physics / Materials

During the last 100 years, magnetism has made a giant step forward. However, many questions remained unanswered or were not even asked at that time. What is the atomic origin of the magnetisation, ... More


Phase Transitions and the Ising Model

Robert H. Swendsen

in An Introduction to Statistical Mechanics and Thermodynamics

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199646944
eISBN:
9780191775123
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646944.003.0030
Subject:
Physics, Condensed Matter Physics / Materials

The Ising model provides an extremely useful example for the investigation of phase transitions. This chapter provides both an introduction to the properties of the Ising model and an overview of the ... More


Mean-Field Theory of Phase Transitions

Hidetoshi Nishimori

in Statistical Physics of Spin Glasses and Information Processing: An Introduction

Published in print:
2001
Published Online:
January 2010
ISBN:
9780198509417
eISBN:
9780191709081
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509417.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test ... More


Statistical Theory, Soft Modes, And Phase Transitions

M. E. LINES and A. M. GLASS

in Principles and Applications of Ferroelectrics and Related Materials

Published in print:
2001
Published Online:
February 2010
ISBN:
9780198507789
eISBN:
9780191709944
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507789.003.0002
Subject:
Physics, Condensed Matter Physics / Materials

In order to acquire a simple physical picture of the dynamic mechanism of a phase transition it is necessary to use the simplest of many-body approximations. It is instructive, in particular, to ... More


The Arena of Conformal Models

Giuseppe Mussardo

in Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics

Published in print:
2020
Published Online:
May 2020
ISBN:
9780198788102
eISBN:
9780191830082
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198788102.003.0014
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 14 discusses how the identification of a class of universality is one of the central questions needing an answer for those in the field of statistical physics. This chapter discusses in ... More


Introduction

Giuseppe Mussardo

in Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics

Published in print:
2020
Published Online:
May 2020
ISBN:
9780198788102
eISBN:
9780191830082
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198788102.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 1 introduces various essential ideas on second-order phase transitions and the theoretical challenges that accompany them. Furthermore, it focuses on important issues, for example, ... More


Systems of Interacting Particles

Daniel V. Schroeder

in An Introduction to Thermal Physics

Published in print:
2021
Published Online:
March 2021
ISBN:
9780192895547
eISBN:
9780191915000
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780192895547.003.0008
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter presents two examples of the application of Boltzmann statistics to systems with nontrivial interactions between particles. The first example is a nonideal gas, treated approximately ... More


Exact S-Matrices

Giuseppe Mussardo

in Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics

Published in print:
2020
Published Online:
May 2020
ISBN:
9780198788102
eISBN:
9780191830082
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198788102.003.0018
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The Ising model in a magnetic field is one of the most beautiful examples of an integrable model. This chapter presents its exact S-matrix and the exact spectrum of its excitations, which consist of ... More


Duality in generalized Ising models

Franz J. Wegner

in Topological Aspects of Condensed Matter Physics: Lecture Notes of the Les Houches Summer School: Volume 103, August 2014

Published in print:
2017
Published Online:
March 2017
ISBN:
9780198785781
eISBN:
9780191827600
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198785781.003.0005
Subject:
Physics, Condensed Matter Physics / Materials

These lecture notes deal with Ising models with interactions containing products of more than two spins. After a number of examples have been given, some general statements are presented. Of ... More


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