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For a single internal species of noninteracting particles with quadratic dispersion in three dimensions, expand the grand canonical partition function at small quantum gas fugacity. With $\eta =+1$ for bosons and $\eta =-1$ for fermions, $n{\lambda }^3=z+\eta z^2/2^{3/2}+O(z^3)$ and $\beta p{\lambda }^3=z+\eta z^2/2^{5/2}+O(z^3)$. Eliminating $z$ gives $p=n{k}_{B}T[1-\eta n{\lambda }^3/2^{5/2}+\cdots ]$. Bosonic bunching lowers the pressure at fixed density; fermionic exclusion raises it. This correction needs no interaction potential. For $g$ equally populated internal species, the coefficient in terms of total number density is reduced by $g$. In the dilute one-species limit the equal-time pair detection correlation is $g^{(2)}(r)=1+\eta e^{-2\pi r^2/{\lambda }^2}$. The exchange contribution to the two-particle cluster integral is therefore $-\frac{1}{2}\int [g^{(2)}(r)-1]d^{3}r=-\eta {\lambda }^3/2^{5/2}$, providing a direct pair-counting check of the coefficient.

For one internal species, the grand canonical partition function of a classical ideal gas satisfies $\ln \Xi =zV/{\lambda }^3$, with quantum gas fugacity $z=e^{\beta \mu }$ and thermal de Broglie wavelength $\lambda =\sqrt{2\pi {\hslash }^2/(m{k}_{B}T)}$. Differentiating the grand potential gives $N=zV/{\lambda }^3$, hence $\mu =k_{B}T\ln (n{\lambda }^3)$. An internal partition factor must be included consistently if internal states are available.

A lattice gas describes particle positions by occupations of fixed cells. In the single-occupancy model, $n_{\alpha }\in \{0,1\}$ and $N={\sum }_{\alpha }{n}_{\alpha }$. Its grand canonical partition function sums configurations with weight $\exp [-\beta (U-\tilde{\mu }N)]$. Discretizing a classical position integral into cells of volume $b$ gives $\tilde{\mu }=\mu +k_{B}T\ln (b/{\lambda }^3)$, with thermal de Broglie wavelength $\lambda$.

For energy $U=-4J{\sum }_{⟨\alpha {\alpha }^{\prime }⟩}{n}_{\alpha }{n}_{{\alpha }^{\prime }}$ on a regular lattice of coordination $q$, the substitution $n_{\alpha }=(1+{\sigma }_{\alpha })/2$ maps the model to the Ising model with coupling $J$ and effective field $h=qJ+\tilde{\mu }/2$. The mean Ising spin is a density imbalance, $⟨\sigma ⟩=2bn-1$. For $J>0$ below the critical temperature, the field-driven Ising transition becomes gas-liquid phase coexistence.

A localized orbital with two electron spin states has energies $E_0=0$, $E_1=ϵ$ and $E_2=2ϵ+U$. Its zero-temperature equilibrium occupation minimizes ${\mathcal{E}}_N=E_N-\mu N$, where $\mu$ is the chemical potential. For repulsion $U>0$, the occupation is successively $0,1,2$ as $\mu$ crosses $ϵ$ and $ϵ+U$. For attraction $U<0$, the singly occupied state is never the minimum: the occupation jumps from $0$ to $2$ at $\mu =ϵ+U/2$. Exactly at a crossing the ground-state occupation is not unique.

At positive temperature, its grand canonical partition function is $\Xi =1+2e^{-\beta (ϵ-\mu )}+e^{-\beta (2ϵ+U-2\mu )}$, where $\beta =1/(k_{B}T)$. The expected value of the occupation follows by differentiating: $⟨N⟩=k_{B}T{\partial }_{\mu }\ln \Xi$. The factor two counts the two singly occupied microstates, not an additional pair state.

The grand canonical partition function isThe probabilities in part (i) and their logarithms giveAt fixed volume and fixed $\mu$, differentiate the partition sum:Combining with the preceding identity yieldsThe variable held fixed is essential; this is the grand canonical derivative.

For occupation numbers zero or one, the energy and particle number are additive over modes. The grand canonical partition function factorizes:Dividing the product of Boltzmann weights by this product givesThe marginal mean occupation is thereforeThis is the Fermi-Dirac distribution of a noninteracting Fermi gas: the exclusion principle allows only zero or one fermion in each single-particle state, including its spin label.

The bosonic coherent-state resolution of identity gives the ordinary trace by insertion into a number-state basis. For fermions, the fermionic coherent-state trace requires a sign in the bra. In one mode, for an even operator $A$,Without that sign the result would be the supertrace $A_{00}-A_{11}$. Applying this identity mode by mode to the even operator $A=e^{-\beta (H-\mu N)}$ gives the grand canonical partition function

For each one-particle state $i$, a boson occupation number can be any $n_i\ge 0$. Its contribution to the grand canonical partition function is the geometric seriesThe noninteracting modes factorize, soConvergence requires $\mu <{\min }_i{ϵ}_i$. Since the lowest energy is zero, the limiting maximum chemical potential is

The chemical potential is the change in internal energy when one particle is added at fixed entropy and volume:Equivalently, $\mu =(\partial F/\partial N{)}_{T,V}$ for the Helmholtz free energy.

Let a microstate $s$ have energy $E_s$ and particle number $N_s$. Maximizing the Gibbs entropy subject to normalization and fixed mean values of $E$ and $N$ is the maximum-entropy derivation of equilibrium ensembles. With Lagrange multipliers $\alpha$, $\beta$, and $-\beta \mu$, varyThe stationarity equation is $-\log p_s-1-\alpha -\beta (E_s-\mu N_s)=0$. Normalization therefore gives the grand canonical ensembleHere $\mathcal{Z}$ is the grand canonical partition function and $k_B$ is the Boltzmann constant.

For one fermionic quantum state of energy $\epsilon$, the Pauli exclusion principle permits occupation numbers only $n=0,1$. Its two grand-canonical weights are $1$ and $e^{-\beta (\epsilon -\mu )}$, so its mean occupation is the Fermi-Dirac distribution

For a free nonrelativistic particle, $\epsilon ={\hslash }^2{k}^2/(2m)$. In a region of area $A$, the number of wave-vector states in the annulus from $k$ to $k+dk$, including the two spin angular momentum states, isSince $kdk=(m/{\hslash }^2)d\epsilon$, the two-dimensional free-electron density of states is constant:

We now use units in which $k_B=1$, as in the question. At zero temperature, the Fermi-Dirac distribution is a step function, and henceAt positive temperature the same fixed particle number satisfies the exact relationThusThis is the low-temperature particle-number cancellation for constant density of states. Therefore, for $T\ll {\epsilon }_F$, the mean number of particles in the energy interval $[\epsilon ,\epsilon +d\epsilon ]$ is

Finally, compare the internal energy with its zero-temperature value. The thermally excited particles above ${\epsilon }_F$ and holes below ${\epsilon }_F$ have equal leading particle numbers, so their terms proportional to ${\epsilon }_F$ cancel. With $x=|\epsilon -{\epsilon }_F|/T$,where the integral follows from the fermion-to-boson thermal integral ratio. Differentiation gives the linear low-temperature heat capacity of a two-dimensional Fermi gas:The requested power law is therefore linear in $T$, with exponent one.

For an ultrarelativistic particle, $\epsilon =pc$. Including the two spin states of an electron, counting wavevectors in a sphere gives the density of statesThe grand canonical partition function of the ideal Fermi gas is thereforeSince the grand potential satisfies $pV=k_{B}T\log \mathcal{Z}$, integration by parts givesThusThis is the equation of state of an ideal ultrarelativistic gas.

In the grand canonical ensemble, the system exchanges both energy and particles with a reservoir at fixed temperature $T$ and chemical potential $\mu$. The probability of a particular many-body microstate $i$, with energy $E_i$ and particle number $N_i$, is its Boltzmann weight $e^{-\beta (E_i-\mu N_i)}$ divided by the grand canonical partition function $\Xi$, where $\beta =1/(k_{B}T)$. The denominator normalizes ${\sum }_i{p}_i=1$; a degeneracy is accounted for by counting distinct microstates, rather than changing an individual state's probability.

For noninteracting particles, specify a microstate by the occupation numbers $n_r$ of one-particle levels of energy ${ϵ}_r$. Then $E-\mu N={\sum }_r({ϵ}_r-\mu )n_r$, so the Gibbs probability factorizes between levels. Put $x_r=e^{-\beta ({ϵ}_r-\mu )}$.

For fermions, the Pauli exclusion principle permits $n_r=0,1$ in each complete one-particle state, giving ${\Xi }_r=1+x_r$. Its mean occupancy isthe Fermi-Dirac distribution.

For bosons, $n_r=0,1,2,\dots$, so the geometric sum is ${\Xi }_r=1/(1-x_r)$ when $x_r<1$. Differentiating that sum givesthe Bose-Einstein distribution. Finite-system convergence requires $\mu <{ϵ}_{\min }$; condensation is the limiting approach to the lowest energy with macroscopic ground-state occupation. Both quantum distributions reduce to $e^{-\beta ({ϵ}_r-\mu )}$ when occupation is small. The quantities derived here are mean occupancies of states, rather than normalized probabilities over the set of one-particle energies.

For $N$ indistinguishable classical particles the canonical partition function is an integral over phase space,The factor $1/N!$ removes overcounting under particle permutations, and $h^{3N}$ makes the classical phase-space measure dimensionless. The pair potential depends only on positions, so all momentum integrations separate. The three-dimensional Gaussian integral for each particle givesThe grand canonical partition function is $\Xi ={\sum }_{N=0}^{\infty }{e}^{\beta \mu N}{Z}_N$, where chemical potential $\mu$ controls the mean number of particles. ThereforeFor $N=0$, the empty products and configurational integral equal one. The factor $1/2$ in the interaction prevents double counting of unordered particle pairs.