Intro
Preface
Acknowledgment
Contents
Part I Basics of ADS and OV Models
1 Introduction to Asymmetric Dissipative Systems (ADS)
1.1 Background of ADS
1.1.1 Motivation
Basic Assumptions and Prospects
Non-equilibrium Dissipative Systems
Physical Properties in Emergence of Non-equilibrium Macroscopic Phenomena
1.1.2 Asymmetric Interactions in Dissipative System
Non-conservation of Momentum
Energy Flow Through a System and Non-equilibrium Stationary State
1.2 Minimal Model of ADS
1.2.1 Introducing the Optimal Velocity Model

1.2.2 Non-existence of Lagrangian
1.2.3 Examples of Phenomena Described by OV Model
1.3 ADS as a System of Active Matter
1.3.1 A Brief Review of Vicsek Model
Statistical Properties of Collective Motions
1.3.2 Comparing OV Model with Vicsek Model
Flocking Behaviours
Emergence of Solitary Objects
Macroscopic Approach and Statistical Properties
1.3.3 Importance of Asymmetry for Interactions in Flocking Behaviours
1.4 Overview of the Book
2 Optimal Velocity Model (OV Model)
2.1 Model
2.1.1 Single and Collective Motions

2.2 Homogeneous Flow Solution and its Stability
2.2.1 Linear Stability Analysis and Phase Transition
2.2.2 Dispersion Relation
2.3 Emergence of Instability Originating from Asymmetry
2.3.1 General OV Model
2.3.2 Linear Stability of General OV Model
2.3.3 Dispersion Relation for General OV Model
Coupled Oscillators with Viscous Force
Equivalence to Harmonic Oscillator System
Asymmetric Interaction without Dissipation
2.3.4 Particle Number Dependence in Linear Stability
2.4 Continuum System of OV Model
2.4.1 Continuation of Discrete System

2.4.2 Linear Stability Analysis
Note for Continuation of Asymmetry
2.4.3 Continuum System of General OV Model
2.5 Stability Change as Hopf Bifurcation
2.5.1 Hopf Bifurcation in Many-Particle System
2.5.2 Hopf Bifurcation Originating from Asymmetry
3 Cluster Flow Solutions
3.1 Moving Cluster
3.1.1 Spontaneous Emergence of a Moving Cluster
3.1.2 Non-equilibrium Stability of a Moving Cluster
Characteristic Time Interval
3.1.3 Profile of Cluster Flow Solution
Hysteresis Loop
Limit Cycle
3.1.4 Velocity of Moving Cluster

3.1.5 Statistical Properties of Cluster Flow
Distributions of Wave Numbers
Power Law Distributions for Sizes of Moving Clusters
3.2 Heaviside Step Function OV Model (Exactly Solvable Model 1)
3.2.1 Homogeneous Flow Solutions
Linear Stability Analysis
3.2.2 Construction of Cluster Flow Solution
Process of Escaping from a Cluster to a Fast Running Region
Process of Moving from a Fast Running Region to a Cluster
3.2.3 Exact Solutions of Cluster Flow
Induced Time Scale (Characteristic Time Interval)
Velocity of Moving Cluster
Hysteresis Loop and Limit Cycle
Process of Escaping from a Cluster to a Fast Running Region