Franklin

Statistical Mechanics of Disordered Systems : A Mathematical Perspective.

Author/Creator:
Bovier, Anton.
Publication:
Cambridge : Cambridge University Press, 2006.
Format/Description:
Book
1 online resource (328 pages)
Series:
Cambridge Series in Statistical and Probabilistic Mathematics
Cambridge Series in Statistical and Probabilistic Mathematics ; v.18
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Subjects:
Statistical mechanics.
Form/Genre:
Electronic books.
Summary:
A self-contained graduate-level introduction to the statistical mechanics of disordered systems.
Contents:
Cover
Half-title
Series-title
Title
Copyright
Dedication
Contents
Preface
Nomenclature
Part I Statistical mechanics
1 Introduction
1.1 Thermodynamics
2 Principles of statistical mechanics
2.1 The ideal gas in one dimension
2.2 The micro-canonical ensemble
2.3 The canonical ensemble and the Gibbs measure
2.4 Non-ideal gases in the canonical ensemble
2.5 Existence of the thermodynamic limit
2.6 The liquid-vapour transition and the van der Waals gas
2.7 The grand canonical ensemble
3 Lattice gases and spin systems
3.1 Lattice gases
3.2 Spin systems
3.3 Subadditivity and the existence of the free energy
3.4 The one-dimensional Ising model
3.5 The Curie-Weiss model
4 The Gibbsian formalism for lattice spin systems
4.1 Spin systems and Gibbs measures
4.2 Regular interactions
4.2.1 Some topological background
4.2.2 Local specifications and Gibbs measures
4.3 Structure of Gibbs measures: phase transitions
4.3.1 Dobrushin's uniqueness criterion
4.3.2 The Peierls argument
4.3.3 The FKG inequalities and monotonicity
5 Cluster expansions
5.1 High-temperature expansions
5.2 Polymer models: the Dobrushin-KoteckĀ“y-Preiss criterion
5.3 Convergence of the high-temperature expansion
5.4 Low-temperature expansions
5.4.1 The Ising model at zero field
5.4.2 Ground-states and contours
Part II Disordered systems: lattice models
6 Gibbsian formalism and metastates
6.1 Introduction
6.2 Random Gibbs measures and metastates
6.3 Remarks on uniqueness conditions
6.4 Phase transitions
6.5 The Edwards-Anderson model
7 The random-field Ising model
7.1 The Imry-Ma argument
7.2 Absence of phase transitions: the Aizenman-Wehr method
7.2.1 Translation-covariant states
7.2.2 Order parameters and generating functions.
7.3 The Bricmont-Kupiainen renormalization group
7.3.1 Renormalization group and contour models
7.3.2 The ground-states
7.3.3 The Gibbs states at finite temperature
Part III Disordered systems: mean-field models
8 Disordered mean-field models
9 The random energy model
9.1 Ground-state energy and free energy
9.2 Fluctuations and limit theorems
9.3 The Gibbs measure
9.4 The replica overlap
9.5 Multi-overlaps and Ghirlanda-Guerra relations
10 Derrida's generalized random energy models
10.1 The standard GREM and Poisson cascades
10.1.1 Poisson cascades and extremal processes
10.1.2 Convergence of the partition function
10.1.3 The Gibbs measures
10.2 Models with continuous hierarchies: the CREM
10.2.1 Free energy
10.2.2 The empirical distance distribution
10.2.3 Multi-overlap distributions
10.3 Continuous state branching and coalescent processes
10.3.1 Genealogy of flows of probability measures
10.3.2 Coalescent processes
10.3.3 Finite N setting for the CREM
10.3.4 Neveu's continuous state branching process
10.3.5 Coalescence and Ghirlanda-Guerra identities
11 The SK models and the Parisi solution
11.1 The existence of the free energy
11.2 2nd moment methods in the SK model
11.2.1 Classical estimates on extremes
11.3 The Parisi solution and Guerra's bounds
11.3.1 Application of a comparison lemma
11.3.2 Computations with the GREM
11.3.3 Talagrand's theorem
11.4 The Ghirlanda-Guerra relations in the SK models
11.5 Applications in the p-spin SK model
12 Hopfield models
12.1 Origins of the model
12.2 Basic ideas: finite M
12.3 Growing M
12.3.1 Fluctuations of Phi
12.3.2 Logarithmic equivalence of lump-weights
12.4 The replica symmetric solution
12.4.1 Local convexity
12.4.2 The cavity method 1.
12.4.3 Brascamp-Lieb inequalities
12.4.4 The local mean field
12.4.5 Gibbs measures and metastates
12.4.6 The cavity method 2
13 The number partitioning problem
13.1 Number partitioning as a spin-glass problem
13.2 An extreme value theorem
13.3 Application to number partitioning
References
Index.
Notes:
Description based on publisher supplied metadata and other sources.
Contributor:
Gill, R.
Ripley, Brian D.
Ross, S.
Silverman, B.W.
Stein, M.
Other format:
Print version: Bovier, Anton Statistical Mechanics of Disordered Systems
ISBN:
9780511167690
9780521849913
OCLC:
173610031