Every single-particle state is of a fixed energy, ε ε . As the sub-ensemble associated with a single-particle state varies by the number of particles only, it is clear that the total energy of the sub-ensemble is also directly proportional to the number of particles in the single-particle state; where N is the number of particles, the total energy of the sub-ensemble will then be N ε ε . Beginning with the standard expression for a grand partition function and replacing E with N ε ε , the grand partition function takes the form Z = ∑ ∑ N exp ⁡ ⁡ ( ( N μ μ − − N ε ε ) / k B T ) = ∑ ∑ N exp ⁡ ⁡ ( N ( μ μ − − ε ε ) / k B T )