fugacity
/ few-GASS-it-ee /
A convenient stand-in for the chemical potential, repackaged as something with the feel of a probability weight. Where the Boltzmann factor e^(-E/kT) governs the exchange of energy, the fugacity governs the exchange of particles: a dimensionless knob that measures how eager particles are to enter the system.
Precisely, fugacity is z = e^(mu / k_B T), where mu is the chemical potential and T the temperature. In the grand partition function Xi = sum over N of z^N Z_N it plays the role of an expansion variable that keeps track of particle number, and <N> = z d(ln Xi)/dz. In the quantum distributions the mean occupation of a single-particle state of energy epsilon is n = 1 / (z^-1 e^(epsilon/k_B T) -/+ 1), with the minus sign for bosons and the plus sign for fermions. For a classical ideal gas z = n lambda^3 = (N/V) lambda^3, small in the dilute limit.
It linearizes the algebra of the grand canonical ensemble and of quantum gases, and in a chemistry guise, carrying units of pressure, it measures departures from ideal-gas behaviour. An honest warning: 'fugacity' appears in two related but distinct conventions, the statistical-mechanics dimensionless z = e^(mu/kT), and the chemical-engineering fugacity with pressure units defined by mu = mu_0 + RT ln(f / P_0). Both encode the same chemical potential, but they are numerically different.
For a classical ideal gas the fugacity z = (N/V) lambda^3 equals the number of particles per thermal de Broglie volume; z << 1 is exactly the classical, non-degenerate regime, while z approaching 1 signals quantum degeneracy.
One dimensionless number tells you whether a gas is classical or quantum.
Watch the convention: statistical mechanics uses the dimensionless z = e^(mu/kT), while physical chemistry uses a fugacity with pressure units; both track the chemical potential but are numerically different.