MMathPhys Kinetic Theory course (homepage)

Lecture notes and suggested reading for Part 1, kinetic theory of neutral particles, as lectured in MT15.

Lecture notes as PDF (here) and problem sheet (with updates 30/10/15) as PDF (here).

These notes are still evolving and currently omit figures.

Suggested reading

My treatment of the BBGKY hierarchy follows chapter 3 of Kardar's book and lecture notes, and the second edition of Huang (also based on an MIT course)

The first rigorous results for the convergence of solutions of the BBGKY hierarchy to solutions of the Boltzmann equation were established by Lanford (1975). Lanford's result holds for
times less than 1/5 of a mean collision time (so 20% of the particles have collided).  The current state of mathematical knowledge is described concisely in chapter 2 of Cercignani's Slow Rarefied Flows, and in detail
in the book by Gallagher et al. (2014). There is a very large philosophy of physics literature about the emergence of irreversibility, see Uffink (2007) or David Wallace's reading list (here)

My derivation of the Boltzmann equation and the scattering cross-section B(θ,V) follows Grad (1949), Cercignani (1988) and the first edition of Huang.

Grad's 1949 and 1958 articles are both excellent reads. The latter first formalised the "Boltzmann–Grad" limit n → ∞, d → 0, with nd2 fixed.

Cercignani's scientific biography Ludwig Boltzmann: The Man Who Trusted Atoms is another worthwhile read. Focused mainly on the history and philosophy of Boltzmann's work,
it contains some very well written and compact technical sections (especially the appendices). It is free to read online (at least within the Oxford University network).

Cercignani's books, like much of the mathematical literature, focus on the BBGKY hierarchy and Boltzmann equation for "hard spheres" to avoid annoying technicalities caused by long-range interactions.

My 2007 paper describes different approaches to deriving the Navier–Stokes equations for slowly varying solutions of the Boltzmann equation via the Chapman-Enskog and other expansions.

The closest to a standard reference for the modern multiple-scales formulation of the Chapman–Enskog expansion is Cercignani (1990). A summary of what is known rigorously, primarily
for the incompressible limit of small Mach and Knudsen numbers, is in Saint-Raymond (2009).




C. Cercignani, Slow Rarefied Flows (Birkhäuser 2006) (link) (read online in Oxford) concise chapters 1 and 2 on the heuristic and rigorous derivations of the Boltzmann equation for hard spheres

C. Cercignani, Ludwig Boltzmann: The Man Who Trusted Atoms (OUP 2006) (Amazon) (read online in Oxford)

C. Cercignani, The Boltzmann Equation and Its Applications (Springer 1988) (link)

C. Cercignani, Mathematical Methods in Kinetic Theory (2nd edition, Springer 1990) (link)

P. J. Dellar Macroscopic descriptions of rarefied gases from the elimination of fast variables Phys. Fluids 19 107101

I. Gallagher, L. Saint-Raymond & B. Texier, From Newton to Boltzmann: Hard Spheres and Short-range Potentials (European Mathematical Society, 2014) (Amazon) (arXiv:1208.5753)

F. Golse, Boltzmann-Grad limit (2013), Scholarpedia 8(10):9141.

H. Grad, Principles of the Kinetic Theory of Gases in Handbuch der Physik, vol. 3, pp 205-294 (Springer 1958) (link)

H. Grad On the kinetic theory of rarefied gases (1949) Comm. Pure Appl. Math. 2 331-407

K. Huang, Statistical Mechanics, second edition (Wiley 1987)  author's page

K. Huang, Statistical Mechanics, first edition (Wiley 1963). Much more on the Boltzmann equation than the second edition, but does not cover BBGKY.

M. Kardar, Statistical Physics of Particles (CUP 2007) (Amazon) MIT OpenCourseWare lecture notes

O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems, Theory and Applications, pp. 1–111. (Springer 1975). Lecture Notes in Theoretical Physics vol. 38 (link) (eprint)

L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation (Springer 2009) Lecture Notes in Mathematics, vol. 1971 (link)

J. Uffink Compendium of the Foundations of Classical Statistical Physics in Philosophy of Physics pp. 923-1074 (Elsevier 2007) (link) (eprint)

C. Villani A Review of Mathematical Topics in Collisional Kinetic Theory in Handbook of Mathematical Fluid Mechanics vol. 1 (2002, eds S. Friedlander & D. Serre) (link) (eprint)

The links are to catalogue pages with full bibliographical details. Older material is not covered by Oxford University's online subscriptions.