Part 1 of the MMathPhys Kinetic Theory course (homepage)
"So one hundred and
forty-five years after Ludwig Eduard Boltzmann (1872) wrote down
his celebrated equation, the struggle to understand it goes on"
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 (with updates 16/10/17) as PDF (here).
These notes are still evolving and currently omit figures.
Chapters 1-6 and 8.1 of Harris (1971) are broadly similar to my
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 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
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
C. Cercignani, Slow Rarefied
Flows (Birkhäuser 2006) (link)
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)
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,
F. Golse, Boltzmann-Grad limit
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
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
F. G. King, BBGKY Hierarchy for Positive
Potentials (1975) PhD thesis, UC Berkeley (link to
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
in Philosophy of Physics
923-1074 (Elsevier 2007) (link
C. Villani, A Review of Mathematical Topics in Collisional
Handbook of Mathematical Fluid Mechanics
vol. 1 (2002,
eds S. Friedlander & D. Serre) (link
The links are to catalogue pages with full bibliographical
details. Older material is not covered by Oxford University's