This book presents recently obtained mathematical results on Gibbs measures of the q-state Potts model on the integer lattice and on Cayley trees. It also illustrates many applications of the Potts model to real-world situations in biology, physics, financial engineering, medicine, and sociology, as well as in some examples of alloy behavior, cell sorting, flocking birds, flowing foams, and image segmentation. Gibbs measure is one of the important measures in various problems of probability theory and statistical mechanics. It is a measure associated with the Hamiltonian of a biological or physical system. Each Gibbs measure gives a state of the system. The main problem for a given Hamiltonian on a countable lattice is to describe all of its possible Gibbs measures. The existence of some values of parameters at which the uniqueness of Gibbs measure switches to non-uniqueness is interpreted as a phase transition. This book informs the reader about what has been (mathematically) done in the theory of Gibbs measures of the Potts model and the numerous applications of the Potts model. The main aim is to facilitate the readers (in mathematical biology, statistical physics, applied mathematics, probability and measure theory) to progress into an in-depth understanding by giving a systematic review of the theory of Gibbs measures of the Potts model and its applications.
Author(s): Utkir A. Rozikov
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 366
City: Singapore
Contents
Preface
About the Author
Acknowledgments
Part I Introduction to thermodynamics and theory of Gibbs measures
1. Introduction
1.1 What is a thermodynamic system
1.2 What we do for biology
1.3 What we do for physics
2. Gibbs measures and Potts model
2.1 Definitions
2.1.1 Lattice graphs
2.1.2 Configuration space
2.1.3 Hamiltonian
2.1.4 The q-state Potts model
2.1.5 Gibbs measure
2.2 A review of the Potts model on Zd
2.3 Gibbs measures on trees
2.3.1 Splitting Gibbs measure
2.3.2 Boundary laws
2.3.3 Translation-invariant measures
2.3.4 Extreme Gibbs measures
2.4 Applications of the Potts model
2.4.1 Alloy behavior
2.4.2 Anomalies
2.4.3 Cell sorting
2.4.4 Financial markets
2.4.5 Flocking birds
2.4.6 Flowing foams
2.4.7 Image segmentation
2.4.8 Medicine
2.4.9 Neural network
2.4.10 Phase transitions
2.4.11 Political trends
2.4.12 Protein family
2.4.13 Protein folding
2.4.14 Signal reconstruction
2.4.15 Smectic phase
2.4.16 Sociology
2.4.17 Spin glasses
2.4.18 Storage capacity
2.4.19 Symmetric channels
2.4.20 Technological processes
2.4.21 Wetting transition
2.5 Comments and references for Part I
Part II Thermodynamics in biology: The Potts model
3. Hierarchy of DNAs and Holliday junctions
3.1 Models and Markov process of DNA sequence evolution
3.1.1 Definitions
3.1.2 A codon-based model
3.1.3 A 3-state model of the interacting DNAs
3.2 Finite dimensional distributions and equations
3.3 Translation-invariant Gibbs measures of the set of DNAs
3.4 Biological interpretations: Holliday junction and branches of DNA
4. Thermodynamics in a system of interacting DNAs
4.1 The model and finite dimensional distributions
4.2 TIGMs of the set of DNAs
4.3 Markov chains of TIGM
5. Holliday junctions for the Potts model of DNA
5.1 Definitions and system of functional equations
5.2 TIGMs of the set of DNAs
5.3 Markov chains of TIGMs and Holliday junction of DNA
6. Thermodynamics of DNA-RNA renaturation
6.1 Introduction and definitions
6.2 Equations describing DNA-RNA renaturation
6.3 Translation-invariant solutions
6.3.1 Solutions in the set M
6.3.2 Solutions in the set R2+\M
6.4 Gibbs measures: Conditions of DNA-RNA renaturation
7. On quadratic stochastic operators of Gibbs measures
7.1 Definitions
7.2 Construction of QSO for finite E
7.3 Construction of QSO for continual case
7.4 Measure of the Potts model
7.5 Comments and references for Part II
Part III Thermodynamics in physics: The Potts model
8. Full description of translation-invariant Gibbs measures for the Potts model on trees
8.1 Translation-invariant SGMs
8.2 Non-uniqueness of extreme Gibbs measure
8.3 Boundary conditions for TISGMs
8.3.1 Setup
8.3.2 Boundary conditions for TISGMs
8.3.3 Construction of boundary conditions
9. Extremality of Gibbs measures
9.1 Eigenvalues of transition matrix
9.2 Non-extreme TISGMs
9.3 Extreme TISGMs
10. The model with external field
10.1 One non-zero coordinate
10.1.1 Criterion of uniqueness of TISGM
10.1.2 Number of TISGMs
10.1.3 Properties of the critical curves
10.2 An external field with non-zero coordinates
10.3 Potts model with competing interactions
10.3.1 The model and its ground states
10.3.2 Contours on trees
10.3.3 Non uniqueness of Gibbs measure
11. Periodic Gibbs measures
11.1 Definitions
11.2 Characterization of periodic Gibbs measures
11.3 Zero external field
11.3.1 Ferromagnetic case
11.3.2 Anti-ferromagnetic case
11.3.2.1 Case: k = 2.
11.3.2.2 Case: k ≥ 3.
11.4 Non-zero external field
11.4.1 The case k = 2, q = 3
11.4.2 The case of translation-invariant external field
11.4.3 A periodic external field
11.5 Weakly periodic measures
12. Free energies of the Potts model
12.1 Definitions and equations
12.2 Formula of free energy
12.3 Concrete free energie
12.3.1 Translation-invariant case
12.3.2 Non-translation-invariant BCs
12.3.3 Periodic BCs
12.3.4 Weakly periodic BCs
12.4 Comments and references for Part III
Bibliography
Index
