Statistical Mechanics/Boltzmann and Gibbs factors and Partition functions/Gibbs Factors

Using the same method as we did with Boltzmann Factors, we consider a system in thermal and diffusive equilibrium with another system.

P(N₁,ε₁)/P(N₂,ε₂) = g(N₀ - N₁, U₀ - ε₁)/g(N₀ - N₂, U₀ - ε₂)

And by the exact same method as our derivation of Boltzmann factors (no surprises here):

P(N₁,ε₁)/P(N₂,ε₂) = exp((N₁μ - ε₁)/T)/exp((N₂μ - ε₂)/T)

Which we will use in the exact same way to create the Grand Partition Function.