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Generalized temperature from single-particle statistics

It is possible to extend the definition of temperature even to systems of few particles, like in a quantum dot. The generalized temperature is obtained by considering time ensembles instead of configuration-space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small system of fermions (N even less than 10) with a single/double-occupancy system. The finite quantum grand canonical ensemble, obtained under the hypothesis of ergodicity and orthodicity, allows expressing the generalized temperature from the ratio of the average time of occupation τ 1 {\displaystyle \tau _{1}} and τ 2 {\displaystyle \tau _{2}} of the single/double-occupancy system:

where E is the Fermi energy. This generalized temperature tends to the ordinary temperature when N goes to infinity.