Let particles exchange with a reservoir

The entropy differential for a reservoir at fixed volume contains energy divided by temperature minus chemical potential times particle number divided by temperature. If the subsystem gains energy and particles, the reservoir loses both. Expanding its entropy therefore yields a weight exponential in minus beta times subsystem energy minus chemical potential times subsystem number. The sign of the chemical-potential term follows from this exchange argument; it is not chosen by analogy with the energy term.

Sum that weight over every allowed particle-number sector and every physical microstate within it. If fixed-number canonical partition functions already include the Gibbs factorial, do not divide by another factorial in the outer number sum. Fugacity is the exponential of beta times chemical potential and is dimensionless. Changing the one-particle energy zero requires shifting chemical potential by the same amount to preserve occupations. Only differences between a level energy and chemical potential enter the one-state weights.

At fixed temperature and volume, differentiating the logarithm of the grand partition function with respect to chemical potential gives mean number times beta. A second derivative gives the number variance times beta squared. Equivalently, number variance is thermal energy times the response of mean number to chemical potential. It is nonnegative in a stable grand-canonical state. These derivatives must not be taken along a path where chemical potential is simultaneously adjusted to hold number fixed.

For a classical ideal gas with one internal state, summing its canonical functions produces an exponential grand partition function. Its number distribution is Poisson, with variance equal to mean. This differs from a binomial distribution of occupants in finitely many exclusion-limited states. Both can become sharply concentrated in relative terms for large mean populations, yet they describe different state spaces. The ensemble determines what fluctuates, while the particle statistics determine the allowed occupation count for each state.

A single fermionic state permits occupation zero or one, so its grand sum has two terms. A bosonic state permits every nonnegative occupation, so its sum is geometric. Differentiation gives denominators differing by a sign. A fermionic chemical potential may lie above or below the lowest level. At positive temperature, the bosonic grand sum instead requires chemical potential strictly below the lowest level for a finite system; approaching that boundary can generate a macroscopic ground occupation.

Independent equal-energy fermionic states give binomial total occupancy and connect directly to the probability lab. Bosonic occupation has a geometric distribution with variance larger than its mean, and cannot be represented by that same binomial instrument. Calling both outputs a generic quantum probability would hide the decisive difference. Derive the occupation probability first, then use the lab only for the fermionic set whose zero-or-one occupation assumptions actually match its exact binomial model.