the mean. As a result, the grand canonical ensemble can be highly inaccurate when applied to small systems of fixed particle number, such as atomic nuclei. Properties Grand potential, ensemble averages, and exact differentials The partial derivatives of the function give important grand canonical ensemble average quantities: Exact differential: From the above expressions, it can be seen that the function has the exact differential First law of thermodynamics: Substituting the above relationship for into the exact differential of , an equation similar to the first law of thermodynamics is found, except with average signs on some of the quantities: Thermodynamic fluctuations: The variances in energy and particle numbers are Correlations in fluctuations: The covariances of particle numbers and energy are Example ensembles The usefulness of the grand canonical ensemble is illustrated in the examples below. In each case the grand potential is calculated on the basis of the relationship which is required for the microstates' probabilities to add up to 1. Statistics of noninteracting particles Bosons and fermions (quantum) In the special case of a quantum system of many non-interacting particles, the thermodynamics are simple to compute. Since the particles are non-interacting, one can compute a series of single-particle stationary states, each of which represent a separable part that can be included into the total quantum state of the system. For now let us refer to these single-particle stationary states as orbitals (to avoid confusing these "states" with the total many-body state), with the provision that each possible internal particle property (spin or polarization) counts as a separate orbital. Each orbital may be occupied by a particle (or particles), or may be empty. Since the particles are non-interacting, we may take the viewpoint that each orbital forms a separate thermodynamic system. Thus each orbital is a grand canonical ensemble unto itself, one so simple that its statistics can be immediately derived here. Focusing on just one orbital labelled , the total energy for a microstate of particles in this orbital will be , where is the characteristic energy level of that orbital. The grand potential for the orbital is given by one of two forms, depending on whether the orbital is bosonic or fermionic: In each case the value gives the thermodynamic average number of particles on the orbital: the Fermi–Dirac distribution for fermions, and the Bose–Einstein distribution for bosons. Considering again the entire system, the total grand potential is