Active Particles, Volume 3: Advances in Theory, Models, and Applications (Modeling and Simulation in Science, Engineering and Technology)

✍ Scribed by Nicola Bellomo (editor), José Antonio Carrillo (editor), Eitan Tadmor (editor)

• Publisher: Birkhäuser
• Year: 2022
• Tongue: English
• Leaves: 230
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

This edited volume collects six surveys that present state-of-the-art results on modeling, qualitative analysis, and simulation of active matter, focusing on specific applications in the natural sciences. Following the previously published Active Particles volumes, these chapters are written by leading experts in the field and reflect the diversity of subject matter in theory and applications within an interdisciplinary framework. Topics covered include:

• Variability and heterogeneity in natural swarms
• Multiscale aspects of the dynamics of human crowds
• Mathematical modeling of cell collective motion triggered by self-generated gradients
• Clustering dynamics on graphs
• Random Batch Methods for classical and quantum interacting particle systems
• The consensus-based global optimization algorithm and its recent variants

Mathematicians and other members of the scientific community interested in active matter and its many applications will find this volume to be a timely, authoritative, and valuable resource.

✦ Table of Contents

Preface
Contents
Variability and Heterogeneity in Natural Swarms: Experiments and Modeling
1 Introduction
2 Sources of Variability in Nature
2.1 Development as a Source of Variation
2.2 Transient Changes in the Behavior of Individuals
2.3 Environmentally Induced Variations
2.4 Social Structure
2.5 Inherent/Intrinsic Properties and Animal Personality
2.6 Variability in Microorganisms
3 Experiments with Heterogeneous Swarms
3.1 Fish
3.2 Mammals
3.3 Insects
3.4 Microorganisms
4 Modeling Heterogeneous Collective Motion
4.1 Continuous Models
4.2 Agent-Based Models
4.3 Specific Examples: Locust
4.4 Specific Examples: Microorganisms and Cells
5 Summary and Concluding Remarks
References
Active Crowds
1 Introduction
2 Models for Active Particles
2.1 Continuous Random Walks
2.1.1 Excluded-Volume Interactions
2.2 Discrete Random Walks
2.3 Hybrid Random Walks
3 Models for Externally Activated Particles
3.1 Continuous Models
3.2 Discrete Models
4 General Model Structure
4.1 Wasserstein Gradient Flows
4.2 Entropy Dissipation
5 Boundary Effects
5.1 Mass Conserving Boundary Conditions
5.2 Flux Boundary Conditions
5.3 Other Boundary Conditions
6 Active Crowds in the Life and Social Science
6.1 Pedestrian Dynamics
6.2 Transport in Biological Systems
7 Numerical Simulations
7.1 One Spatial Dimension
7.2 Two Spatial Dimensions
References
Mathematical Modeling of Cell Collective Motion Triggered by Self-Generated Gradients
1 Introduction
2 The Keller–Segel Model and Variations
2.1 The Construction of Waves by Keller and Segel
2.2 Positivity and Stability Issues
2.3 Variations on the Keller–Segel Model
2.4 Beyond the Keller–Segel Model: Two Scenarios for SGG
3 Scenario 1: Strongest Advection at the Back
4 Scenario 2: Cell Leakage Compensated by Growth
5 Conclusion and Perspectives
References
Clustering Dynamics on Graphs: From Spectral Clustering to Mean Shift Through Fokker–Planck Interpolation
1 Introduction
1.1 Mean Shift-Based Methods
1.1.1 Lifting the Dynamics to the Wasserstein Space
1.2 Spectral Methods
1.2.1 Normalized Versions of the Graph Laplacian
1.2.2 More General Spectral Embeddings
1.3 Outline
2 Mean Shift and Fokker–Planck Dynamics on Graphs
2.1 Dynamic Interpretation of Spectral Embeddings
2.2 The Mean Shift Algorithm on Graphs
2.2.1 Mean Shift on Graphs as Inspired by Wasserstein Gradient Flows
2.2.2 Quickshift and KNF
3 Fokker–Planck Equations on Graphs
3.1 Fokker–Planck Equations on Graphs via Interpolation
3.2 Fokker–Planck Equation on Graphs via Reweighing and Connections to Graph Mean Shift
4 Continuum Limits of Fokker–Planck Equations on Graphs and Implications
4.1 Continuum Limit of Mean Shift Dynamics on Graphs
4.2 Continuum Limits of Fokker–Planck Equations on Graphs
4.3 The Witten Laplacian and Some Implications for Data Clustering
5 Numerical Examples
5.1 Numerical Method
5.2 Simulations
5.2.1 Graph Dynamics as Density Dynamics
5.2.2 Comparison of Graph Dynamics and PDE Dynamics
5.2.3 Clustering Dynamics
5.2.4 Effect of the Kernel Density Estimate on Clustering
5.2.5 Effect of Data Distribution on Clustering
5.2.6 Blue Sky Problem
5.2.7 Density vs. Geometry
References
Random Batch Methods for Classical and Quantum Interacting Particle Systems and Statistical Samplings
1 Introduction
2 The Random Batch Methods
2.1 The RBM Algorithms
2.2 Convergence Analysis
2.3 An Illustrating Example: Wealth Evolution
3 The Mean-Field Limit
4 Molecular Dynamics
4.1 RBM with Kernel Splitting
4.2 Random Batch Ewald: An Importance Sampling in the Fourier Space
5 Statistical Sampling
5.1 Random Batch Monte Carlo for Many-Body Systems
5.2 RBM-SVGD: A Stochastic Version of Stein Variational Gradient Descent
6 Agent-Based Models for Collective Dynamics
6.1 The Cucker–Smale Model
6.2 Consensus Models
7 Quantum Dynamics
7.1 A Theoretical Result on the N-Body Schrödinger Equation
7.1.1 Mathematical Setting and Main Result
7.2 Quantum Monte Carlo Methods
7.2.1 The Random Batch Method for VMC
7.2.2 The Random Batch Method for DMC
References
Trends in Consensus-Based Optimization
1 Introduction
1.1 Notation and Assumptions
1.1.1 The Weighted Average
2 Consensus-Based Global Optimization Methods
2.1 Original Statement of the Method
2.1.1 Particle Scheme
2.1.2 Mean-Field Limit
2.1.3 Analytical Results for the Original Scheme Without Heaviside Function
2.1.4 Numerical Methods
2.2 Variant 1: Component-Wise Diffusion and Random Batches
2.2.1 Component-Wise Geometric Brownian Motion
2.2.2 Random Batch Method
2.2.3 Implementation and Numerical Results
2.3 Variant 2: Component-Wise Common Diffusion
2.3.1 Analytical Results
2.3.2 Numerical Results
3 Relationship of CBO and Particle Swarm Optimization
3.1 Variant 4: Personal Best Information
3.1.1 Performance
4 CBO with State Constraints
4.1 Variant 5: Dynamics Constrained to Hyper-Surfaces
4.1.1 Analytical Results
5 Overview of Applications
5.1 Global Optimization Problems: Comparison to Heuristic Methods
5.2 Machine Learning
5.3 Global Optimization with Constrained State Space
5.4 PDE Versus SDE Simulations
6 Conclusion, Outlook and Open problems
References