Exact Results for Bose-Einstein and Fermi-Dirac Statistics for Finite Generic Systems

Bose-Einstein and Fermi-Dirac statistics for finite systems have been developed using a simple generic entropy definition. The exact entropy derivatives (properties) were obtained with posterior bosons and fermions adjustment for finite systems. The usual thermodynamic limits are also considered for comparison. Some applications to two-level systems were considered to obtain energy, heat capacity, chemical potential, magnetization, and susceptibility for fermion finite systems. Energy and magnetization reach a minimum, and their derivatives, such as heat capacity, chemical potential, and susceptibility, become zero at a non-null temperature, a characteristic of finite systems. This non-null temperature is given by the inverse of the harmonic series as a function of the system size. The exact limit for low temperatures regarding finite systems was carefully considered to show the remarkable difference between finite and infinite (thermodynamic limit) systems.