Single-particle distribution function of a quantum dot system at finite temperature

In the present paper, we shall rigorously re-establish the result of the single-particle function of a quantum dot system at finite temperature. Unlike the proof given in our previous work (Phys. Rev. B 74 195414 (2006)), we take a different approach, which does not exploit the explicit expression of the Gibbs distribution function. Instead, we only assume that the statistical distribution function of the quantum dot system is thermodynamically stable. As a result, we are able to show clearly that the electronic structure in the quantum dot system is completely determined by its thermodynamic stability. Furthermore, the weaker requirements on the statistical distribution function also make it possible to apply the same method to the quantum dot systems in non-equilibrium states.