Chapter 1
1 Introduction
2 Liouville Equation
2.1 Hamiltonian Systems
2.2 Frictional and Stochastic Forces
2.3 Impulsive Forces and Binary Collision Expansions
3 Boltzmann and Ring Kinetic Equations
4 Homogeneous Solution of the Boltzmann Equation
4.1 Homogeneous Cooling State
4.2 NESS for Heated Fluids
4.3 High Energy Tails
5 Conclusion
References
Chapter 2
1 Introduction
2 The Liouville Operator
2.1 Smooth Potentials
2.2 Elastic Hard-Core Interactions
2.3 Inelastic Collision
2.4 Time Evolution of the Distribution Function
3 Homogeneous Cooling State
3.1 Results for Spheres
3.2 Results for Needles
3.3 Breakdown of Homogeneit in Dense Systems of Needles
4 Non-Gaussian Distribution
5 Conclusion
Acknowledgements
Appendix
A Calculations for Spheres
B Calculations for Needles
References
Chapter 3
1 Introduction
2 Navier-Stokes Transport Coefficients
3 Self-Diffusion
4 Brownian Motion
5 Conclusion
Acknowledgements
References
Chapter 4
1 Introduction
2 Mesoscopic Nature of Granular Flows
3 Scale Dependence of Stresses and Fluctuations
4 Remarks on the Construction of a Kinetic Theory of Granular Gases
5 Boundary Conditions
6 Conclusion
Acknowledgements
References
Chapter 5
1 Introduction
2 Dependence of the Restitution Coe .cient on the Impact Velocity
3 Time-Evolution of Temperature and of the Velocity Distribution Function
4 Self-Diffusion in Granular Gases of Viscoelastic Particles
5 Conclusion
Acknowledgements
References
Chapter 6
1 Introduction
2 Boundaries with Frictional,Cylindrical Features
2.1 Preliminaries
2.2 Shear Stress
2.3 Slip Velocity
2.4 Energy Flux
3 A Boundary-Value Problem
3.1 Boundary Conditions
3.2 The Mean Velocity
3.3 Volume Fraction
4 Graphical Results
Acknowledgments
Appenidx
A Bumpiness for Spheres
B Boundaries with Spherical,Frictional Features
References
Chapter 7
1 Introduction
2 Choice of Boundary Conditions
2.1 Periodic Boundary Conditions
2.2 Reflecting Boundaries
3 Diffusion in an Elastic Gas
4 Forced Systems
4.1 Stationary State
4.2 Diffusion of One Particle
5 Conclusion
Acknowledgements
References
Chapter 8
1 Introduction
2 Experimental Apparatus and Procedures
2.1 Compound Pendulum Apparatus
2.2 Static Sticking Force Apparatus
3 Energy Loss in Collisions
3.1 Normal Incidence Collision Experiments,Smooth Surfaces
3.2 Normal Incidence Collision Experiments, Frost-Coated Surfaces
3.3 Energy Loss in Glancing Collisions
3.4 Dynamical Sticking Measurements in Normal Incidence Collision Experiments
4 Static Sticking Measurements
4.1 Static Sticking Measurements for Water Frost Layers
4.2 Other Surface Layers
5 Discussion of Measurements
5.1 Modeling the Coefficient of Restitution
5.2 Other Collision Experiments
5.3 Sticking Mechanisms
6 Application to Planetary Systems
6.1 Planetary Rings
6.2 Planetesimal Growth
7 Summary
Acknowledgements
References
Chapter 9
1 Introduction
2 Normal Impact of Elastoplastic Spheres
3 Oblique Impact of Elastic Spheres
4 Oblique Impact of Elastoplastic Spheres
References
Chapter 10
1 Introduction
2 Kinetic Equation and Boundary Conditions
3 The Quasielastic Regime
4 Results
Conclusions
References
Chapter 11
1 Introduction
2 Chains of Viscoelastic Particles
2.1 Optimal Mass-Distribution
2.2 The Maximum of the Optimal Mass-Distribution
3 Conclusion
References
Chapter 12
1 Introduction
2 Granular Temperature
3 Experimental Details
4 Packing Fraction and Velocity Distributions
5 Local Density Fluctuations
6 Mean Squared Displacement
7 Granular Temperature Profiles
8 Self-Diffusion
9 Velocity Auto-correlation Function
10 Positron Emission Particle Tracking
11 Conclusions
Acknowledgements
References
Chapter 13
Acknowledgements
References
Chapter 14
1 Introduction
2 Experimental Setup
3 Pressure Measurements
4 Volume Measurements
5 Density Measurements
6 Towards a State Equation
7 Conclusion
References
Chapter 15
1 Introduction
2 Event Driven Simulations
3 Results
3.1 Dependence on the Sawtooth Shape
3.2 Dependence on the La er Width
3.3 Frequency Dependence
3.4 Dependence on the Coe .cient of Restitiution
4 Discussion
Acknowledgements
References
Chapter 16
1 Introduction
2 Hydrodynamic Model
3 Computer Simulation
4 Conclusions
Acknowledgements
References
Chapter 17
1 Introduction
2 A Simple Model for an Uncon .ned Disc
of Colliding Particles
2.1 The Standard Model
2.2 Further Models
3 Disc โSatellite Interactions
4 Planetary Rings and Arcs
4.1 Ring Radial Structure,Narrow Rings
and Con .nement Mechanisms
4.2 Planetary Arcs
4.3 Rings Origin
5 Other Discs and the Formation of Planets
6 Open Problems
References
Chapter 18
1 Introduction
2 Collisional Dynamics of Dense Rings
2.1 Impact Frequency
2.2 The Establishment of Local Energy Equilibrium
2.3 Radial Evolution
3 Simulations of Collisional Rings
3.1 Early Simulation Studies
3.2 Local Simulations
3.3 Results of Local Simulations
4 Self-Gravity
4.1 Expectations
4.2 Simulations of Self-Gravitating Systems
4.3 Numerical Methods in Self-Gravitating Simulations
5 Radial Evolution
6 Summary
References
Chapter 19
1 Introduction
2 Collisional Dynamics of Unperturbed Rings
3 Satellite Resonances
4 Angular Momentum Transfer
5 Scaling of Numerical Simulations
6 Ring Edges,Gaps and Narrow Ringlets
7 Summary
References
Chapter 20
Chapter 21
1 Introduction
2 Model System
2.1 Polydispersity
2.2 Particle Interactions
3 Simulation and Theory
3.1 Particle Correlations
3.2 Collision Rates and Energy Dissipation
3.3 Stress and the Equation of State
3.4 Accounting for the Dense,Ordered Phase
4 Pressure Gradient Due to Gravity
4.1 Densit Pro .le in Dilute Systems
4.2 Densit Pro .le for a Monodisperse Hard Sphere Gas
4.3 Comparison with Simulations
4.4 Bidisperse Systems with Gravitation
5 Summary and Outlook
Acknowledgements
References
Chapter 22
1 Introduction
2 Experimental Setup and Methods
3 Phase Diagram
4 Statistical Characterization of the Granular Gas
4.1 Pair Correlation Functions
4.2 Velocit Distribution Functions
4.3 Density-Velocit Cross-Correlations
5 Discussion
5.1 Clustering and Collapse
5.2 Two Dimensional Granular Gas
Acknowledgments
References
Chapter 23
1 Introduction
2 Configurational Statistics and Maximum Entropy Principle: Justification of a Thermodynamic Approach
3 Fermi Statistics of Weakly Excited Granular Materials
5 Test of Fermi Statistics by Molecular Dynamics Simulations
6 Condensation of Hard Spheres under Gravity
7 Conclusion
Acknowledgments
References
Chapter 24
1 Introduction
2 Examples for Simplified Models
2.1 Lattice Gas Automata
2.2 Direct Simulation Monte Carlo
3 Applications to Different Flows
3.1 Homogeneous Cooling
3.2 Bagnold Shear Flow
3.3 Clustering Instability
3.4 Heaps
4 Conclusion
References