  I present a theoretical model of a quantum statistical ensemble for which,
unlike in conventional physics, the total number of particles is extremely
small. The thermodynamical quantities are calculated by taking a small $N$ by
virtue of the orthodicity of canonical ensemble. The finite quantum grand
partition function of a Fermi-Dirac system is calculated. The model is applied
to a quantum dot coupled with a small two dimensional electron system. Such
system consists of an alternatively single and double occupied electron system
confined in a quantum dot, which exhanges one electron with a small $N$ two
dimensional electron reservoir. The analytic determination of the temperature
of a $(1\leftrightarrow 2)$ electron system and the role of ergodicity are
discussed. The generalized temperature expression in the small $N$ regime
recovers the usual temperature expression by taking the limit of $N\to\infty$
of the electron bath.
