Pair-Correlation Effects in Many-Body Systems: Towards a Complete Theoretical Description of Pair-Correlations in the Static and Kinetic Description of Many-Body Systems (Springer Theses) 9783031296116, 9783031296123, 3031296117

Table of contents :
Supervisor’s Foreword
Abstract
Acknowledgements
Contents
Acronyms
1 Introduction
1.1 The Ising Model: A Brief History
1.1.1 Spontaneous Magnetization in One Dimension (1920)
1.1.2 The Bragg-Williams Approximation (1934)
1.1.3 The Bethe-Guggenheim Approximation (1935)
1.1.4 Spontaneous Magnetization in Two Dimensions (1936)
1.1.5 Intermezzo: The Fixed-Magnetization Ensemble
1.1.6 The Lattice Gas (1952)
1.1.7 Kikuchi's Cluster Variation Approximation (1952)
1.1.8 Cahn-Hilliard Free Energy for Non-uniform Systems (1958)
1.1.9 Stochastic Dynamics of the Ising Model (1963)
1.1.10 Intermezzo: Continuous and Discrete-Time Dynamics
1.1.11 Kadanoff's Block Spin Method (1966)
1.1.12 The Ising Model at Present Day (2000–2022)
1.2 Biological Applications of the Ising Model
1.2.1 Cell Adhesion
1.3 Scope and Outline of the Thesis
References
2 Bethe-Guggenheim Approximation for Uniform Systems
2.1 Initial Setup and Definitions
2.2 Rewriting the Ising Hamiltonian
2.3 Rewriting the Partition Function
2.4 Pair Approximation Ansatz
2.5 Evaluating the Normalization Constant
2.6 Evaluating the Partition Sum
2.7 Free Energy Density
2.8 Phase Diagram
2.8.1 Binodal
2.8.2 Spinodal
2.8.3 Critical Point
2.9 Free Energy Minima
2.9.1 One-Dimensional Line with barz=2
2.9.2 Square and Bethe Lattice with barz=4
2.10 Mean Field Approximation
2.10.1 Fixed-Magnetization Partition Function
2.10.2 Free Energy Density
2.10.3 Phase Diagram
2.10.4 Free Energy Minima
2.11 Qualitative Differences Between MF and BG Approximation
2.11.1 Free Energy Barrier and Curvature
2.11.2 Error Analysis
References
3 Bethe-Guggenheim Approximation for Non-uniform Systems
3.1 Initial Setup and Definitions
3.1.1 Lattice Specification and the Thermodynamic Limit
3.1.2 Coarse-Grained Lattice Observables
3.2 Rewriting the Ising Hamiltonian over Spin Blocks
3.3 Rewriting the Partition Function over Spin Blocks
3.4 Pair Approximation Ansatz
3.5 Evaluating the Normalization Constant
3.6 Evaluating the Partition Sum
3.6.1 Optimization w.r.t. ij rightarrow limNbs[cdot]
3.6.2 limNbs[cdot] rightarrow Optimization w.r.t. (x,y)
3.7 Equilibrium Profile
3.7.1 Interface Steepness and Width
3.7.2 Interface Broadening for Infinite Coupling
3.8 Linear Stability Analysis
3.9 Mean Field Approximation
3.9.1 Partition Function
3.9.2 Free Energy Functional
3.9.3 Interface Steepness and Width
3.9.4 Absence of Interface Broadening
3.9.5 Spinodal Decomposition
3.10 Error Analysis
References
4 Delocalization-Induced Interface Broadening in Strongly Interacting Systems
4.1 Introduction and Motivation
4.2 Motivating Example: Interface Delocalization
4.3 Cahn-Hilliard Theory Including Pair-Correlations
4.3.1 From the Partition Function to the Field Theory
4.4 Equilibrium Profile and Exact Results
4.4.1 Comparison with Simulations
4.4.2 Comparison with Exact Results
4.5 Disentangling Interface Delocalization
4.6 Spinodal Decomposition
4.7 Nucleation
4.8 Concluding Remarks
References
5 Criticality in Cell Adhesion
5.1 Introduction
5.1.1 Outline of the Chapter
5.2 Interacting Adhesion Bonds Under Shared Force
5.2.1 Equilibrium
5.2.2 Kinetics
5.2.3 Strategy Roadmap
5.3 Equilibrium Properties of Adhesion Clusters
5.3.1 Small and Intermediate Clusters
5.3.2 Thermodynamic Limit
5.3.3 Phase Diagram and Critical Behavior
5.4 Kinetics of Cluster Formation and Dissolution
5.4.1 Small and Intermediate Clusters
5.4.2 Thermodynamic Limit
5.4.3 Dynamical Phase Transition and Critical Behavior
5.5 Mechanical Regulation of Cell Adhesion
5.5.1 Criticality at Equilibrium
5.5.2 Criticality in Kinetics
5.6 Criticality in the Ising Model
5.7 Concluding Remarks
5.7.1 Model Limitations
References
6 Global Speed Limit for Finite-Time Dynamical Phase Transition in Nonequilibrium Relaxation
6.1 Introduction and Motivation
6.2 Fundamentals
6.2.1 Equilibrium
6.2.2 Kinetics
6.2.3 Kinetics in the Thermodynamic Limit
6.3 Dynamical Phase Transition
6.4 Critical Time
6.5 Lower Bounds on the Critical Time
6.5.1 Lower Bound for Quenches in the Two-Phase Domain
6.5.2 Lower Bound for Quenches in the One-Phase Domain
6.6 Antiferromagnetic Speed Limit for Relaxation
6.7 Asymptotic Measure Equivalence
6.8 Dynamical Phase Diagram
6.9 Conclusion
References
7 Conclusion and Outlook
7.1 Summary and Conclusion
7.2 Discussion
7.3 Concluding Perspective
References
Appendix Curriculum Vitae
Kristian Blom