Quantum Chaos and Statistical Mechanics

We briefly review the well known connection between classical chaos and classical statistical mechanics, and the recently discovered connection between quantum chaos and quantum statistical mechanics. Talk given at the Conference on Fundamental Problems in Quantum Theory Baltimore, June 18–22, 1994 Consider a dilute gas of hard spheres in a box with hard walls. Give the spheres some arbitrary initial distribution of momenta (and positions). Classically, after a few mean free times have passed, we expect that the distribution of momenta will be given by the Maxwell–Boltzmann (MB) formula, fMB(p) = (2πmkT ) −3/2 exp(−p/2mkT ) , (1) where the temperature T is given in terms of the conserved total energy U by the ideal-gas relation U = 3 2NkT . To see why this should be so, first note that the hamiltonian is simply