Lecture notes as PDF (here) and problem sheet (

These notes are still evolving and currently omit figures.

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

in the book by Gallagher

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

K. Huang

K. Huang

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)

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.