Grand Canonical Ensemble - Statistics of Bosons and Fermions

Statistics of Bosons and Fermions

For a quantum mechanical system, the eigenvalues (energies) and the corresponding eigenvectors (eigenstates) of the Hamiltonian (the energy function) completely describe the system. For a macroscopic system, the number of eigenstates (microscopic states) is enormous. Statistical mechanics provides a way to average all microscopic states to obtain meaningful macroscopic quantities.

The task of summing over states (calculating the partition function) appears to be simpler if we do not fix the total number of particles of the system because, for a noninteracting system, the partition function of grand canonical ensemble can be converted to a product of the partition functions of individual quantum states. This conversion makes the evaluation much easier. (However this conversion cannot be done in canonical ensemble, where the total number of particles is fixed. )

Each state is a spatial configuration for an individual particle. There may be none or some particles in each state. In quantum mechanics, all particles are either bosons or fermions. For fermions, no two particles can share a same state. But there is no such constraint for bosons. Therefore the partition function (of grand canonical ensemble) for each state can be written as

The is the energy of the state. For fermions, can be 0 or 1 (no particle or one particle in the state). For bosons, . The upper sign is for fermions and the lower sign is for bosons in the last step. The total partition function is then a product of the individual state partition functions.