  Gibbs and Boltzmann definitions of temperature agree only in the macroscopic
limit. The ambiguity in identifying the equilibrium temperature of a finite
sized `small' system exchanging energy with a bath is usually understood as a
limitation of conventional statistical mechanics. We interpret this ambiguity
as resulting from a stochastically fluctuating temperature coupled with the
phase space variables giving rise to a broad temperature distribution. With
this ansatz, we develop the equilibrium statistics and dynamics of small
systems. Numerical evidence using an analytically tractable model shows that
the effects of temperature fluctuations can be detected in equilibrium and
dynamical properties of the phase space of the small system. Our theory
generalizes statistical mechanics to small systems relevant to biophysics and
nanotechnology.
