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Preface; Contents; Symbols; Introduction and Motivation; 1 Granular Fluids: From Everyday Life to the Lab; 1.1 The Granular ``States''; 1.2 Granular Flows; 1.2.1 Air Fluidization; 1.2.2 Shear; 1.2.3 Shakers; 1.3 Granular Versus Active Fluids; References; 2 Boltzmann Equation: A Gas of Grains; 2.1 Collisions; 2.1.1 Elastic Smooth Hard Spheres; 2.1.2 Statistics of Hard Spheres Collisions; 2.1.3 Inelasticity; 2.1.4 Inelastic Collapse; 2.2 The Boltzmann Equation; 2.2.1 Liouville and Pseudo-Liouville Equations; 2.2.2 The BBGKY Hierarchy; 2.2.3 The Boltzmann Hierarchy and the Boltzmann Equation
2.2.4 Collision Invariants and H-theorem2.2.5 The Maxwell Molecules; 2.2.6 The Enskog Correction; 2.3 The Boltzmann Equation for Granular Gases; 2.3.1 Average Energy Loss; 2.3.2 Sonine Polynomials; 2.3.3 The Homogeneous Cooling State; 2.3.4 Inelastic Maxwell Molecules; 2.3.5 Bulk Driving; 2.3.6 Looking for a ``Granular'' H-theorem; References; 3 Hydrodynamics: A Sea of Grains; 3.1 Granular Kinetic Theory; 3.1.1 A Sketch of the Chapman-Enskog Approximation Method; 3.1.2 Densities and Fluxes; 3.1.3 Equations for the Densities; 3.1.4 Chapman-Enskog Closure; 3.1.5 Inelastic Case
3.2 Critiques of Granular Hydrodynamics3.3 Applications of Granular Hydrodynamics; 3.3.1 Linear Stability Analysis of the Homogeneous Cooling State; 3.3.2 A Solvable Case: Granular Sedimentation in 2D; 3.3.3 Thermal Convection; 3.3.4 Other Instabilities of Granular Hydrodynamic; 3.4 Fluctuating Hydrodynamics; 3.4.1 Simple Models of Noise; 3.4.2 Deriving the Fluctuations from the Kinetic Equation; References; 4 Tracer's Diffusion: Swimming Through the Grains; 4.1 The Markovian Limit; 4.1.1 Decoupling the Gas from the Tracer; 4.1.2 The Transition Rate; 4.2 The Large Mass Limit
4.2.1 Langevin Equation for the Tracer4.3 Non-Markovian Tracer's Diffusion; 4.4 The Granular Brownian Ratchet; 4.4.1 Continuous Limit; 4.4.2 Other Methods, Models and Experiments; References; 5 The Arrow of Time: Past and Future of Grains; 5.1 Equilibrium from a Dynamical Perspective; 5.1.1 The Case of Markov Processes; 5.1.2 Entropy Production; 5.1.3 Observables Related to Entropy Production; 5.2 The Case of the Granular Intruder; 5.2.1 The Paradox of the Large Mass Limit; 5.2.2 Linear Response; 5.2.3 The Granular Motor; 5.2.4 Coupling with the Fluid: Non-equilibrium Re-established
5.3 Time-Reversal in Fluctuating HydrodynamicsReferences; Conclusion and Perspectives; Appendix A Expansion of the First Two Momentsof the Transition Rates for Large Massof the Tracer; Index
2.2.4 Collision Invariants and H-theorem2.2.5 The Maxwell Molecules; 2.2.6 The Enskog Correction; 2.3 The Boltzmann Equation for Granular Gases; 2.3.1 Average Energy Loss; 2.3.2 Sonine Polynomials; 2.3.3 The Homogeneous Cooling State; 2.3.4 Inelastic Maxwell Molecules; 2.3.5 Bulk Driving; 2.3.6 Looking for a ``Granular'' H-theorem; References; 3 Hydrodynamics: A Sea of Grains; 3.1 Granular Kinetic Theory; 3.1.1 A Sketch of the Chapman-Enskog Approximation Method; 3.1.2 Densities and Fluxes; 3.1.3 Equations for the Densities; 3.1.4 Chapman-Enskog Closure; 3.1.5 Inelastic Case
3.2 Critiques of Granular Hydrodynamics3.3 Applications of Granular Hydrodynamics; 3.3.1 Linear Stability Analysis of the Homogeneous Cooling State; 3.3.2 A Solvable Case: Granular Sedimentation in 2D; 3.3.3 Thermal Convection; 3.3.4 Other Instabilities of Granular Hydrodynamic; 3.4 Fluctuating Hydrodynamics; 3.4.1 Simple Models of Noise; 3.4.2 Deriving the Fluctuations from the Kinetic Equation; References; 4 Tracer's Diffusion: Swimming Through the Grains; 4.1 The Markovian Limit; 4.1.1 Decoupling the Gas from the Tracer; 4.1.2 The Transition Rate; 4.2 The Large Mass Limit
4.2.1 Langevin Equation for the Tracer4.3 Non-Markovian Tracer's Diffusion; 4.4 The Granular Brownian Ratchet; 4.4.1 Continuous Limit; 4.4.2 Other Methods, Models and Experiments; References; 5 The Arrow of Time: Past and Future of Grains; 5.1 Equilibrium from a Dynamical Perspective; 5.1.1 The Case of Markov Processes; 5.1.2 Entropy Production; 5.1.3 Observables Related to Entropy Production; 5.2 The Case of the Granular Intruder; 5.2.1 The Paradox of the Large Mass Limit; 5.2.2 Linear Response; 5.2.3 The Granular Motor; 5.2.4 Coupling with the Fluid: Non-equilibrium Re-established
5.3 Time-Reversal in Fluctuating HydrodynamicsReferences; Conclusion and Perspectives; Appendix A Expansion of the First Two Momentsof the Transition Rates for Large Massof the Tracer; Index