Thermostatistics of small systems: exact results in the microcanonical formalism

Several approximations are made to study the microcanonical formalism that are valid in the thermodynamics limit. Usually it is assumed that: (1) Stirling's approximation can be used to evaluate the number of microstates; (2) the surface entropy can be replaced by the volume entropy; and (3) derivatives can be used even if the energy is not a continuous variable. It is also assumed that the results obtained from the microcanonical formalism agree with those from the canonical one. However, it is not clear if these assumptions are right for very small systems (10–100 particles). To answer this question, two systems with exact solutions (the Einstein model of the solid and the two-level system) have been solved with and without these approximations.