Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 34D / Solution

The ergodic hypothesis states that long-time averages of equilibrium observables along typical trajectories equal their ensemble averages on the accessible phase space with the same conserved quantities. It permits statistical mechanics to predict measured macroscopic quantities from equilibrium distributions rather than tracking a trajectory. In its microcanonical form the measure is on the constant-energy surface. It is an assumption about the relevant dynamics, not a consequence of particles being noninteracting; the calculations below use the equilibrium canonical ensemble at temperature $T$.