Lecture notes and suggested reading for Part 1, kinetic theory of neutral particles, as lectured in MT17.

Lecture notes as PDF (here) and problem sheet (

These notes are still evolving and currently omit figures.

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 for hard spheres 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). His student King's 1975 thesis extended these results to compactly supported potentials. The current state of mathematical knowledge is described concisely in chapter 2 of Cercignani's

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 nd

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,

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

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,

H. Grad,

H. Grad,

S. Harris,

K. Huang

K. Huang

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

F. G. King, *BBGKY Hierarchy for Positive
Potentials* (1975) PhD thesis, UC Berkeley (link to
thesis on
ProQuest)

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)

J. Uffink,

C. Villani,

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