The kinetic theory of gases as we know it dates to the paper of Boltzmann in 1872. The justification and context of this equation has been clarified over the past half century to the extent that it comprises one of the most complete examples of many-body analyses exhibiting the contraction from a microscopic to a mesoscopic description. The primary result is that the Boltzmann equation applies to dilute gases with short ranged interatomic forces, on space and time scales large compared to the corresponding atomic scales. Otherwise, there is no a priori limitation on the state of the system. This means it should be applicable even to systems driven very far from its eqUilibrium state. However, in spite of the physical simplicity of the Boltzmann equation, its mathematical complexity has masked its content except for states near eqUilibrium. While the latter are very important and the Boltzmann equation has been a resounding success in this case, the full potential of the Boltzmann equation to describe more general nonequilibrium states remains unfulfilled. An important exception was a study by Ikenberry and Truesdell in 1956 for a gas of Maxwell molecules undergoing shear flow. They provided a formally exact solution to the moment hierarchy that is valid for arbitrarily large shear rates. It was the first example of a fundamental description of rheology far from eqUilibrium, albeit for an unrealistic system. With rare exceptions, significant progress on nonequilibrium states was made only 20-30 years later.

List of Figures. List of Tables. Preface. Acknowledgements. Introduction. Introduction Author.
1: Kinetic Theory of Dilute Gases. 1. Introduction. 2. Derivation of the Boltzmann equation. 3. Chapman-Enskog expansion. 4. The Boltzmann equation for gas mixtures. 5. Kinetic models.
2: Uniform Shear Flow in a Simple Gas. 1. Introduction. 2. The Boltzmann equation for uniform shear flow. 3. Moment equations for a gas of Maxwell molecules. 4. Third- and fourth-degree velocity movements. 5. Singular behavior of the velocity moments. 6. Perturbation expansion of the distribution function.
3: Kinetic Model for Uniform Shear Flow.1. Introduction. 2. The BKG equation for uniform shear flow. 3. Power-law repulsive potentials. Hard spheres. 4. The thermostatted state. 5. Small perturbations from the thermostatted uniform shear flow. 6. Heat transport under uniform shear flow. 7. Stability analysis of the thermostatted uniform shear flow.
5: Uniform Shear Flow in a Binary Mixture.
6: Planar Couette Flow in a Simple Gas.
7: Planar Couette Flow in a Binary Mixture.
Appendix A: Collisional moments for Maxwell molecules.