  The microcanonical ensemble has long been a starting point for the
development of thermodynamics from statistical mechanics. However, this
approach presents two problems. First, it predicts that the entropy is only
defined on a discrete set of energies for finite, quantum systems, while
thermodynamics requires the entropy to be a continuous function of the energy.
Second, it fails to satisfy the stability condition ($\Delta S / \Delta U < 0$)
for first-order transitions with both classical and quantum systems. Swendsen
has recently shown that the source of these problems lies in the microcanonical
ensemble itself, which contains only energy eigenstates and excludes their
linear combinations. To the contrary, if the system of interest has ever been
in thermal contact with another system, it will be described by a probability
distribution over many eigenstates that is equivalent to the canonical ensemble
for sufficiently large systems. Novotny et al. have recently supported this
picture by dynamical numerical calculations for a quantum mechanical model, in
which they showed the approach to a canonical distribution for up to 40 quantum
spins. By simplifying the problem to calculate only the equilibrium properties,
we are able to extend the demonstration to more than a million particles.
