Gibbs Measures In Biology And Physics: The Potts Model

367 Pages·2022·8.649 MB·English

by  Utkir A. Rozikov

Gibbs Measures
in Biology
and Physics
The Potts Model
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Gibbs Measures
in Biology
and Physics
The Potts Model
Utkir A. Rozikov
V. I. Romanovskiy Institute of Mathematics,
Tashkent, Uzbekistan
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
1122669944 99778899881111225511223388__ttpp..iinndddd 22 1122//77//2222 99::0055 AAMM
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GIBBS MEASURES IN BIOLOGY AND PHYSICS
The Potts Model
Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd.
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To my teachers and collaborators
v
B1948 Governing Asia
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Contents
Preface xiii
About the Author xvii
Acknowledgments xix
Part I Introduction to thermodynamics and theory
of Gibbs measures 1
1. Introduction 3
1.1 What is a thermodynamic system . . . . . . . . . . . . . . 3
1.2 What we do for biology . . . . . . . . . . . . . . . . . . . 4
1.3 What we do for physics . . . . . . . . . . . . . . . . . . . 8
2. Gibbs measures and Potts model 15
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Lattice graphs . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Configuration space . . . . . . . . . . . . . . . . . 18
2.1.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . 19
2.1.4 The q-state Potts model. . . . . . . . . . . . . . . 19
2.1.5 Gibbs measure . . . . . . . . . . . . . . . . . . . . 20
2.2 A review of the Potts model on Zd . . . . . . . . . . . . . 24
2.3 Gibbs measures on trees . . . . . . . . . . . . . . . . . . . 28
2.3.1 Splitting Gibbs measure . . . . . . . . . . . . . . . 28
2.3.2 Boundary laws . . . . . . . . . . . . . . . . . . . . 33
2.3.3 Translation-invariant measures . . . . . . . . . . . 37
2.3.4 Extreme Gibbs measures . . . . . . . . . . . . . . 42
vii
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viii Gibbs Measures in Biology and Physics: The Potts Model
2.4 Applications of the Potts model . . . . . . . . . . . . . . . 47
2.4.1 Alloy behavior . . . . . . . . . . . . . . . . . . . . 48
2.4.2 Anomalies . . . . . . . . . . . . . . . . . . . . . . 48
2.4.3 Cell sorting . . . . . . . . . . . . . . . . . . . . . . 49
2.4.4 Financial markets . . . . . . . . . . . . . . . . . . 54
2.4.5 Flocking birds . . . . . . . . . . . . . . . . . . . . 56
2.4.6 Flowing foams . . . . . . . . . . . . . . . . . . . . 58
2.4.7 Image segmentation . . . . . . . . . . . . . . . . . 58
2.4.8 Medicine . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.9 Neural network. . . . . . . . . . . . . . . . . . . . 62
2.4.10 Phase transitions. . . . . . . . . . . . . . . . . . . 64
2.4.11 Political trends . . . . . . . . . . . . . . . . . . . . 65
2.4.12 Protein family . . . . . . . . . . . . . . . . . . . . 66
2.4.13 Protein folding . . . . . . . . . . . . . . . . . . . . 67
2.4.14 Signal reconstruction . . . . . . . . . . . . . . . . 69
2.4.15 Smectic phase . . . . . . . . . . . . . . . . . . . . 70
2.4.16 Sociology . . . . . . . . . . . . . . . . . . . . . . . 70
2.4.17 Spin glasses. . . . . . . . . . . . . . . . . . . . . . 71
2.4.18 Storage capacity . . . . . . . . . . . . . . . . . . . 73
2.4.19 Symmetric channels . . . . . . . . . . . . . . . . . 74
2.4.20 Technological processes . . . . . . . . . . . . . . . 75
2.4.21 Wetting transition . . . . . . . . . . . . . . . . . . 77
2.5 Comments and references for Part I. . . . . . . . . . . . . 78
Part II Thermodynamics in biology:
The Potts model 81
3. Hierarchy of DNAs and Holliday junctions 83
3.1 Models and Markov process of DNA sequence evolution . 84
3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . 84
3.1.2 A codon-based model . . . . . . . . . . . . . . . . 86
3.1.3 A 3-state model of the interacting DNAs . . . . . 92
3.2 Finite dimensional distributions and equations. . . . . . . 96
3.3 Translation-invariant Gibbs measures of the
set of DNAs . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Biological interpretations: Holliday junction and
branches of DNA . . . . . . . . . . . . . . . . . . . . . . . 102
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Contents ix
4. Thermodynamics in a system of interacting DNAs 109
4.1 The model and finite dimensional distributions . . . . . . 109
4.2 TIGMs of the set of DNAs . . . . . . . . . . . . . . . . . . 112
4.3 Markov chains of TIGM . . . . . . . . . . . . . . . . . . . 115
5. Holliday junctions for the Potts model of DNA 119
5.1 Definitions and system of functional equations . . . . . . . 119
5.2 TIGMs of the set of DNAs . . . . . . . . . . . . . . . . . . 122
5.3 Markov chains of TIGMs and Holliday junction of DNA . 127
6. Thermodynamics of DNA-RNA renaturation 131
6.1 Introduction and definitions . . . . . . . . . . . . . . . . . 131
6.2 Equations describing DNA-RNA renaturation . . . . . . . 134
6.3 Translation-invariant solutions. . . . . . . . . . . . . . . . 136
6.3.1 Solutions in the set M . . . . . . . . . . . . . . . . 136
6.3.2 Solutions in the set R2\M . . . . . . . . . . . . . 138
+
6.4 Gibbs measures: Conditions of DNA-RNA
renaturation . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7. On quadratic stochastic operators of Gibbs measures 147
7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Construction of QSO for finite E . . . . . . . . . . . . . . 149
7.3 Construction of QSO for continual case. . . . . . . . . . . 151
7.4 Measure of the Potts model . . . . . . . . . . . . . . . . . 155
7.5 Comments and references for Part II . . . . . . . . . . . . 156
Part III Thermodynamics in physics:
The Potts model 159
8. Full description of translation-invariant Gibbs measures
for the Potts model on trees 161
8.1 Translation-invariant SGMs . . . . . . . . . . . . . . . . . 161
8.2 Non-uniqueness of extreme Gibbs measure . . . . . . . . . 174
8.3 Boundary conditions for TISGMs . . . . . . . . . . . . . . 176
8.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.3.2 Boundary conditions for TISGMs . . . . . . . . . 182
8.3.3 Construction of boundary conditions. . . . . . . . 188