In this chapter we shall study the basic properties of the Boltzmann equation. It is clear from Eq. (7.22) of Chapter I that the left-hand side and the right-hand side are completely different in nature, both from a mathematical and a physical standpoint. The left-hand side contains a linear differential operator which acts on the space- and time-dependence of f; if we equate this side to zero, we obtain an equation for the time evolution in absence of collisions, and the differential operator is accordingly called the “free-streaming operator.”
KeywordsBoltzmann Equation Maxwellian Distribution Mass Velocity Collision Operator Equality Sign
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The content of this chapter is generally standard. References 6–10 of Chapter I can be consulted as general references. Section 1—For a mathematical proof that Eq. (1.18) gives all the solutions of Eq. (1.17) see ref 8 of Chapter I. Section 4—Maxwell’s discussion about boundary conditions can be found in:
- 1.J. C. Maxwell, Phil. Trans. Royal Soc. I, Appendix (1879) ; reprinted in : The Scientific Papers of J. C. Maxwell, Dover Publications (1965).Google Scholar
Section 5—For the dispute on the H-theorem see the papers by Boltzmann, Zermelo, and Poincaré in the collection
- 2.S. G. Brush (ed.), Kinetic Theory, Vol. II, Pergamon Press, New York and London (1966). For a modern proof of Poincaré’s theorem see 3. M. Kac, Probability and Related Topics in Physical Sciences, Interscience Publishers, New York (1959).Google Scholar