The statistical mechanics of small clusters ( n ∼ 10–50 elements) of harmonic oscillators and
two-level systems is studied exactly, following the microcanonical, canonical and grand canonical
formalisms. For clusters with several hundred particles, the results from the three formalisms
coincide with those found in the thermodynamic limit. However, for clusters formed by a few tens of
elements, the three ensembles yield different results. For a cluster with a few tens of harmonic
oscillators, when the heat capacity per oscillator is evaluated within the canonical formalism, it
reaches a limit value equal to k B , as in the thermodynamic case, while within the microcanonical
formalism the limit value is k B (1 – 1/ n ). This difference could be measured experimentally. For
a cluster with a few tens of two-level systems, the heat capacity evaluated within the canonical and
microcanonical ensembles also presents dif…