Grand canonical ensemble

At fixed temperature, volume, and chemical potential, the grand canonical ensemble permits both energy and particle number to fluctuate and weights a state by $e^{-\beta (E-\mu N)}$.

The dimensionless quantum gas fugacity weights particle number in the grand canonical ensemble: a state with energy $E$ and particle number $N$ has weight $z^N{e}^{-\beta E}$. The chemical potential and one-particle energies must use the same reference. When the lowest one-particle energy is zero, $z\ll 1$ makes both fermion and boson occupations small throughout the low-energy spectrum. This dimensionless quantity must be distinguished from the pressure-valued fugacity used in chemical thermodynamics.

The grand canonical partition function is

For one independent fermion level, occupations 0 and 1 form a Bernoulli distribution and give $\mathrm{Var}n=\bar{n}(1-\bar{n})$. For one ideal boson level in contact with a reservoir, its geometric distribution gives $\mathrm{Var}n=\bar{n}(1+\bar{n})$. Both approach variance equal to mean at small occupation. Pauli exclusion suppresses fluctuations of nearly full fermion states, whereas large boson occupation gives variance of order mean squared. Fixed-total-number constraints change condensate fluctuation statements.

For a single one-particle state with $x_k\ll 1$, the empty and singly occupied contributions dominate. A fermion state gives ${\Xi }_k=1+x_k$, a boson state gives $(1-x_k{)}^{-1}$, and Maxwell–Boltzmann statistics give $e^{x_k}$. Thus all have the same leading dilute response, $\ln {\Xi }_k=x_k+O(x_k^2)$. This approximation does not impose an exclusion principle on classical particles.

The grand potential isFor a homogeneous extensive system, $\Phi =-pV$.

For noninteracting fermions with density of states $g$, define ${\Phi }_0(x)=-{\int }_0^x(x-y)g(y)dy$. The grand potential is ${\int }_0^{\infty }g(ϵ)F(ϵ)dϵ$, where $F=k_{B}T\ln (1-n_F)$ has derivative $F^{\prime }=n_F$. Integrating by parts twice gives the title expression, since ${\Phi }_0(0)={\Phi }_0^{\prime }(0)=0$ and upper boundary terms vanish for ordinary polynomially growing densities of states. The Fermi window therefore thermally averages the zero-temperature grand potential as a function of trial Fermi energy.

Differentiating the grand partition function at fixed temperature and volume gives $⟨N⟩=-{\partial }_{\mu }\Phi$ and $\mathrm{Var}N=-k_{B}T{\partial }_{\mu }^2\Phi =k_{B}T{\partial }_{\mu }⟨N⟩$. It follows because derivatives of the logarithm of the state sum generate cumulants of particle number. Nonnegative variance makes the grand potential concave in chemical potential in a stable grand-canonical description.