Gibbs Measures in Biology and Physics. The Potts Model

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