This chapter provides the mathematical justification for the theory of equilibrium statistical mechanics. A Hamiltonian system which is ergodic is shown to have time-average behavior equal to the average behavior in the energy shell. Liouville’s theorem is used to justify the use of phase-space volume in taking this average. Exercises explore the breakdown of ergodicity in planetary motion and in dissipative systems, the application of Liouville’s theorem by Crooks and Jarzynski to non-equilibrium statistical mechanics, and generalizations of statistical mechanics to chaotic systems and to two-dimensional turbulence and Jupiter’s great red spot.
Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems
