The grand canonical ensemble

Let a subsystem exchange both energy and particles with a large reservoir. Expand the reservoir entropy to first order in the energy and number transferred away from it. The thermodynamic identity $dS=(dE+p\,dV-\mu\,dN)/T$ at fixed volume gives the factor $e^{-\beta(E-\mu N)}$. Chemical potential has units of energy per particle and describes the free-energy cost of adding a particle under specified thermodynamic conditions. It need not equal the average energy carried by one particle.

The reservoir must hold its temperature and chemical potential approximately fixed despite exchanges. Weak interaction energy and equilibrium remain assumptions. Sum separately over all allowed number sectors and the states within each sector, with the correct quantum statistics or classical indistinguishability factors. A conserved total particle number for subsystem plus reservoir is compatible with a fluctuating subsystem number. For species that interconvert, independent chemical potentials are constrained by the conserved charges and reaction equilibria.

Differentiate the logarithm of $\Xi$ with respect to $\mu$ at fixed temperature and volume. The result is the mean particle number multiplied by $\beta$. A second derivative gives the number variance. Thus an equilibrium system whose occupation changes strongly when its chemical potential is nudged must also display substantial number fluctuations. The relation concerns a specified ensemble; a strictly fixed-number preparation has zero total-number variance even when local subregions fluctuate.

For a homogeneous extensive phase with only the usual volume work, the grand potential becomes $-pV$ in the thermodynamic limit. Surface free energies and finite boxes give corrections, so the pressure relation should not be imposed blindly on a few-site model. Grand canonical calculations often simplify independent-mode problems because the sum factorizes by mode. This algebraic convenience does not guarantee that number fluctuations are physically appropriate for an isolated condensate or an experiment that fixes particle number very precisely.

One spinless fermionic mode of energy $\epsilon$ allows occupations zero and one, so its partition factor is $1+e^{-\beta(\epsilon-\mu)}$. A noninteracting bosonic mode allows every nonnegative integer and instead produces a geometric series. That series converges only when $e^{-\beta(\epsilon-\mu)}<1$, requiring $\mu<\epsilon$ at positive temperature. This is a mathematical normalizability condition with direct physical meaning, not an optional convention about the zero of chemical potential.

For ideal bosons, the chemical potential approaches the lowest single-particle energy from below as a macroscopic ground-state population develops. Treating the ground mode grand canonically can produce very large number fluctuations. A real finite isolated gas with fixed total number needs the matching constraint, and interactions can change the fluctuation behavior. Shifting every single-particle energy and the chemical potential by the same constant leaves occupations unchanged; changing only one of them changes the exchange balance.