  While the canonical ensemble has been tremendously successful in capturing
thermal statistics of macroscopic systems, deviations from canonical behavior
exhibited by small systems are not well understood. Here, using a small two
dimensional Ising magnet embedded inside a larger Ising magnet heat bath, we
characterize the failures of the canonical ensemble when describing small
systems. We find significant deviations from the canonical behavior for small
systems near and below the critical point of the two dimensional Ising model.
Notably, the agreement with the canonical ensemble is driven not by the system
size but by the statistical decoupling between the system and its surrounding.
A superstatistical framework wherein we allow the temperature of the small
magnet to vary is able to capture its thermal statistics with significantly
higher accuracy than the Gibbs-Boltzmann distribution. We discuss future
directions.
