the grand potential
Some systems are open at both ends: they exchange not only energy but also particles with a huge reservoir — think of a small region of a gas that can trade molecules with the rest, or electrons hopping between a metal and its surroundings. The natural control variables are then temperature T and chemical potential mu, and the potential tailored to them is the grand potential. It is what a system minimizes when T, V and mu are fixed.
It is defined as Omega = U - T S - mu N = F - mu N, with natural variables T, V and mu, and differential dOmega = -S dT - p dV - N dmu. It is the Legendre transform of the Helmholtz free energy that trades particle number N for the chemical potential mu. Because the potential is extensive and its only extensive natural variable is V, it collapses to the simple form Omega = -p V for a bulk system.
The grand potential is the workhorse of the grand canonical ensemble, where Omega = -k_B T ln Xi and Xi is the grand partition function. It is the natural language for quantum ideal gases — deriving the Fermi-Dirac and Bose-Einstein distributions, degeneracy pressure, and Bose-Einstein condensation all start from Omega. Caveat: mu is fixed by the reservoir, and N here is only the average; the particle number genuinely fluctuates.
For an ideal gas the grand potential works out to Omega = -k_B T ln Xi = -p V, and differentiating gives back everything: N = -(dOmega/dmu)_{T,V} recovers the mean particle number and S = -(dOmega/dT)_{V,mu} the entropy. This single function encodes the full equation of state.
From Omega alone, one derivative each returns pressure, particle number and entropy.
In the grand canonical description the particle number is not fixed but fluctuates about a mean set by mu; Omega = -pV holds for the average, and relative fluctuations shrink as 1/sqrt(N) for large systems.