Intro; Supervisor's Foreword; Abstract; List of Publications; Acknowledgements; Contents; Part I Granular and Active Matter; 1 Introduction; 1.1 What is Granular Matter; 1.2 Granular States; 1.2.1 Granular Pressure, Internal Stresses, Jamming; 1.3 Granular Flows; 1.3.1 Rapid Granular Flows; 1.4 What is Active Matter; 1.4.1 Active Phenomenology; 1.4.2 Living Systems; 1.4.3 Non-living Systems; References; 2 Theoretical Models of Granular and Active Matter; 2.1 Kinetic Theory of Rapid Granular Flows; 2.1.1 From Liouville to Boltzmann Equation

2.1.2 Inelastic Collisions and Granular Boltzmann Equation2.1.3 The Homogeneous Cooling State and Haff's Law; 2.1.4 Steady State Representation; 2.1.5 Driven Granular Systems; 2.1.6 Inelastic Maxwell Molecules; 2.2 Active Models; 2.2.1 The Kuramoto Model; 2.2.2 The Vicsek Model; 2.2.3 Active Brownian Particles and Run-and-Tumble Dynamics; 2.2.4 Active Ornstein-Uhlenbeck Particles; 2.3 Grains and Active Particles; References; 3 Hydrodynamic Description and Lattice Models; 3.1 Hydrodynamics; 3.1.1 Conservative Interactions; 3.1.2 Granular Hydrodynamics

3.1.3 Hydrodynamic Instabilities and the HCS3.1.4 Active Hydrodynamics; 3.2 Fluctuating Hydrodynamics; 3.3 Lattice Models; 3.3.1 Conservative Models; 3.3.2 Dissipative Models; References; Part II Fluctuating Hydrodynamics of Granular and Active Matter: Lattice Models; 4 Granular Lattice: Fluctuating Hydrodynamics; 4.1 Definition of the Model; 4.1.1 Master Equation for the Lattice Model; 4.1.2 Physical Interpretation; 4.1.3 Evolution Equation for the One-Particle Distribution; 4.2 Hydrodynamics; 4.2.1 Microscopic Balance Equations; 4.2.2 Balance Equations in the Continuum Limit

4.3 Physically Relevant States4.3.1 The Homogeneous Cooling State; 4.3.2 The Uniform Shear Flow Steady State; 4.3.3 The Couette Flow Steady State; 4.3.4 Validity of the Hydrodynamic Description; 4.4 Fluctuating Hydrodynamics; 4.4.1 Definition of Fluctuating Currents; 4.4.2 Noise Correlations; 4.4.3 Cross-Correlations of the Noises and Gaussianity; 4.5 Numerical Results; 4.5.1 General Simulation Strategy; 4.5.2 Homogeneous and Non-homogeneous Cooling; 4.5.3 Uniform Shear Flow State; 4.5.4 The Couette Flow State; 4.5.5 Fluctuating Currents; References; 5 Granular Lattice: Beyond Molecular Chaos

5.1 Perturbative Solution for Temperature and Correlations5.2 Temperature and Correlations Evolution: Multiple-Scale Analysis; 5.3 Total Energy Fluctuations and Multiscaling; References; 6 Active Lattice Fluctuating Hydrodynamics; 6.1 Definition of the Model; 6.1.1 Microscopic Ingredients; 6.1.2 Physical Interpretation; 6.1.3 Microscopic Balance Equations; 6.2 Hydrodynamic Limit; 6.2.1 Homogeneous Fixed Points; 6.2.2 Stability Analysis of the Disordered State; 6.3 Fluctuating Hydrodynamics; 6.4 Numerical Results; 6.4.1 General Simulation Strategy; 6.4.2 Swarming Instability