Part 1 of the MMathPhys Kinetic Theory course (homepage)

"So one hundred and three fifty four years after Ludwig Eduard Boltzmann (1872) wrote down his celebrated equation, the struggle to understand it goes on" [King 1975]

Lecture notes and suggested reading for Part 1, kinetic theory of neutral particles, as lectured in MT26.

The class will be 3–5pm on Monday 9th November 2026 in the Lindemann Lecture Theatre (the same as lectures)

The hand-in deadline is Thursday 5th November 2026 to Lucas Enrico Martinez via Canvas.

Problem sheet as PDF (here)

Lecture notes as PDF (here) last updated 26th October 2023. These notes are still evolving and currently omit some figures.

Handwritten notes from live Zoom lectures given in MT20. A "+" lecture was 1½ hours.

Notes from Monday week 2 morning lecture (lecture 1+)
Notes from Monday week 2 afternoon lectures (lectures 2 & 3)
Notes from Tuesday week 2 lecture (lecture 4)
Notes from Monday week 3 morning lecture (lecture 5+)
Notes from Monday week 3 afternoon lectures (lectures 6 & 7)
Notes from Tuesday week 3 lecture (lecture 8)

Suggested reading

Chapters 1-6 and 8.1 of Harris (1971) are broadly similar to my approach.

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)

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.

Chapter 1 of Cercignani's Slow Rarefied Flows is a good summary of most of this part of the course. 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.

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

The following definitely constitute further reading...

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 concise technical sections (especially the appendices). It is free to read online within the Oxford University network.

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 near-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).

NEW Yu Deng received a 2026 Fields Medal for completing Lanford's and King's programme for hard spheres and arbitrarily long times. See Deng's ICM lecture, and arXiv preprint with Zaher Hani and Xiao Ma.

There is a very large philosophy of physics literature about the emergence of irreversibility, see Uffink (2007).

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) (ProQuest)

C. Cercignani, The Boltzmann Equation and Its Applications (Springer 1988) (link) (read online in Oxford) NEW

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

S. Harris, An Introduction to the Theory of the Boltzmann Equation (1971) (2004 Dover reprint, link)

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

F. G. King, BBGKY Hierarchy for Positive Potentials (1975) PhD thesis, University of California 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 Physics vol. 38 (link) Online companion article.

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.