Introduction 1 Historical Introduction 1.1 What is a Gas? From the Billiard Table to Boyle's Law 1.2 Brief History of Kinetic Theory 2 Informal Derivation of the Boltzmann Equation 2.1 The Phase Space and the Liouville Equation 2.2 Boltzmann's Argument in a Modern Perspective 2.3 Molecular Chaos. Critique and Justification 2.4 The BBGKY Hierarchy 2.5 The Boltzmann Hierarchy and Its Relation to the Boltzmann Equation 3 Elementary Properties of the Solutions 3.1 Collision Invariants 33 3.2 The Boltzmann Inequality and the Maxwell Distributions 3.3 The Macroscopic Balance Equations 3.4 The H-Theorem 3.5 Loschmidt's Paradox 3.6 Poincare's Recurrence and Zermelo's Paradox 3.7 Equilibrium States and Maxwellian Distributions 3.8 Hydrodynamical Limit and Other Scalings 4 Rigorous Validity of the Boltzmann Equation 4.1 Significance of the Problem 4.2 Hard-Sphere Dynamics 4.3 Transition to L1. The Liouville Equation and the BBGKY Hierarchy Revisited 4.4 Rigorous Validity of the Boltzmann Equation 4.5 Validity of the Boltzmann Equation for a Rare Cloud of Gas in the Vacuum 4.6 Interpretation 4.7 The Emergence of Irreversibility 4.8 More on the Boltzmann Hierarchy Appendix 4.A More about Hard-Sphere Dynamics Appendix 4.B A Rigorous Derivation of the BBGKY Hierarchy Appendix 4.C Uchiyama's Example 5 Existence and Uniqueness Results 5.1 Preliminary Remarks 5.2 Existence from Validity, and Overview 5.3 A General Global Existence Result 5.4 Generalizations and Other Remarks Appendix 5.A 6 The Initial Value Problem for the Homogeneous Boltzmann Equation 6.1 An Existence Theorem for a Modified Equation 6.2 Removing the Cutoff: The L1-Theory for the Full Equation 6.3 The L∞-Theory and Classical Solutions 6.4 Long Time Behavior 6.5 Further Developments and Comments Appendix 6.A Appendix 6.B Appendix 6.C 7 Perturbations of Equilibria and Space Homogeneous Solutions 7.1 The Linearized Collision Operator 7.2 The Basic Properties of the Linearized Collision Operator 7.3 Spectral Properties of the Fourier-Transformed, Linearized Boltzmann Equation 7.4 The Asymptotic Behavior of the Solution of the Cauchy Problem for the Linearized Boltzmann Equation 7.5 The Global Existence Theorem for the Nonlinear Equation 7.6 Extensions: The Periodic Case and Problems in One and Two Dimensions 7.7 A Further Extension: Solutions Close to a Space Homogeneous Solution 8 Boundary Conditions 8.1 Introduction 8.2 The Scattering Kernel 8.3 The Accommodation Coefficients 8.4 Mathematical Models 8.5 A Remarkable Inequality 9 Existence Results for Initial-Boundary and Boundary Value Problems 9.1 Preliminary Remarks 9.2 Results on the Traces 9.3 Properties of the Free-Streaming Operator 9.4 Existence in a Vessel with Isothermal Boundary 9.5 Rigorous Proof of the Approach to Equilibrium 9.6 Perturbations of Equilibria 9.7 A Steady Problem 9.8 Stability of the Steady Flow Past an Obstacle 9.9 Concluding Remarks 10 Particle Simulation of the Boltzmann Equation 10.1 Rationale amd Overview 10.2 Low Discrepancy Methods 10.3 Bird's Scheme 11 Hydrodynamical Limits 11.1 A Formal Discussion 11.2 The Hilbert Expansion 11.3 The Entropy Approach to the Hydrodynamical Limit 11.4 The Hydrodynamical Limit for Short Times 11.5 Other Scalings and the Incompressible Navier-Stokes Equations 12 Open Problems and New Directions Author Index Subject Index