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For density of states $AV{ϵ}^a$, $a>-1$, the grand canonical ensemble has $\log \mathcal{Z}=V{T}^{a+1}f(\mu /T)$. Strict monotonicity of Fermi gas entropy per particle makes $\mu /T$ constant at fixed entropy and particle number. Therefore $V{T}^{a+1}$ and $P{V}^{(a+2)/(a+1)}$ are constant along a reversible adiabatic process.

Let an isolated donor impurity admit an empty ionized state with statistical weight $g_0$ and a singly occupied neutral state with statistical weight $g_1$, with double occupation excluded. Its grand canonical ensemble weights are $g_0$ and $g_1{e}^{-\beta ({ϵ}_D-\mu )}$, so $r=g_1/g_0$ multiplies the occupied-to-empty ratio. A nondegenerate empty state and two allowed electron spin orientations give $r=2$; a simplified single-state occupation model uses $r=1$. With negligible electron holes and uncompensated donor impurities, charge neutrality and nondegenerate conduction occupation imply $n^2/(N_D-n)=(N_c/r)e^{-E_i/(k_{B}T)}$, where $E_i$ is the donor binding energy and $N_c$ the effective density of states.

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.

For a repulsive contact-interacting Bose-Einstein condensate, the classical Hamiltonian density generating evolution in a frame rotating at the chemical potential is $\mathcal{K}={\hslash }^2|\nabla \psi {|}^2/(2M)-\mu |\psi {|}^2+g|\psi {|}^4/2$. Its integral is the grand canonical ensemble generator $H-\mu N$, not the physical energy $H$ alone. For $g,\mu >0$ the uniform mean-field theory minima satisfy $|{\psi }_0{|}^2=\mu /g$, with a freely chosen constant phase. In the physical frame, the corresponding field has time factor $e^{-i\mu t/\hslash }$.

One noninteracting spatial orbital with two spin energies $ϵ\pm {\mu }_{B}B$, in the grand canonical ensemble with mean occupation one, has chemical potential $\mu =ϵ$ for every positive temperature. Its spin population difference is $-\tanh [{\mu }_{B}B/(2k_{B}T)]$, giving total-moment response $\chi ={\mu }_B^2/(2k_{B}T)$ at zero field. Empty and doubly occupied orbital states are permitted. Fixing exactly one electron instead leaves only the two singly occupied states and gives $\chi ={\mu }_B^2/(k_{B}T)$. The factor of two is an ensemble effect in this small system.

For a single impurity level in a crystal with allowed occupations zero and one and equal occupied/empty statistical weights, the Fermi-Dirac distribution givesSince $n_{\mathrm{full}}=N_D{f}_D$ and $n_{\mathrm{empty}}=N_D(1-f_D)$,The same result follows from the two grand canonical ensemble weights $1$ and $e^{-\beta ({ϵ}_D-\mu )}$.

The equal-statistical-weight convention is necessary for the printed formula when $n_{\mathrm{full}}$ and $n_{\mathrm{empty}}$ count whole donor impurities. More generally, donor charge-state degeneracy gives $n_{\mathrm{full}}/n_{\mathrm{empty}}=(g_{\mathrm{full}}/g_{\mathrm{empty}})e^{\beta (\mu -{ϵ}_D)}$. An ordinary singly occupied donor with two spin orientations and a nondegenerate empty state has ratio $2e^{\beta (\mu -{ϵ}_D)}$; at $\mu ={ϵ}_D$ its occupied-to-empty ratio is two, a concrete counterexample to treating the printed expression as universal. The following part uses the paper's ratio-one convention.

The chemical potential is $\mu =(\partial E/\partial N{)}_{S,V}$, equivalently $(\partial F/\partial N{)}_{T,V}$: it is the free-energy cost of adding a particle at fixed temperature and volume. With units $k_B=1$, a large reservoir has entropy $S_R(E_{\mathrm{tot}}-E,N_{\mathrm{tot}}-N)=S_R^0-E/T+\mu N/T$ to first order. Since the number of compatible reservoir states is proportional to its exponential, the grand canonical ensemble assignsFor independent fermions, a one-particle state of energy $\epsilon$ has occupation zero or one. Its partition factor is $1+e^{-(\epsilon -\mu )/T}$, so the mean occupation is the Fermi-Dirac distribution $[e^{(\epsilon -\mu )/T}+1{]}^{-1}$. Multiplication by the given density of states yields $dN=C{\epsilon }^{1/2}d\epsilon /[e^{(\epsilon -\mu )/T}+1]$.

At zero temperature the occupation is a step function, filled below $\mu$ and empty above it, so $\mu ={\epsilon }_F$ and $N=(2C/3){\epsilon }_F^{3/2}$. At fixed particle number, the low-temperature shift of $\mu$ is $O(T^2/{\epsilon }_F)$ and does not affect the leading estimate. In the narrow interval of width $T$ above ${\epsilon }_F$, replace the density of states by $C\sqrt{{\epsilon }_F}$ and put $u=(\epsilon -{\epsilon }_F)/T$. ThenEquivalently, the fraction above the Fermi energy is asymptotic to $(3\log 2/2)T/{\epsilon }_F$.

The microcanonical ensemble describes an isolated system at fixed energy, volume and particle number. The canonical ensemble describes fixed particle number and volume with heat exchange at a prescribed temperature. The grand canonical ensemble additionally allows particles to exchange with a reservoir, prescribing temperature, volume and chemical potential. Their respective equilibrium probabilities are uniform on the accessible energy shell, proportional to $e^{-\beta E_n}$, and proportional to $e^{-\beta (E_n-\mu N_n)}$. For stable short-range systems in the thermodynamic limit, concentration of relative energy and particle-number fluctuations and concavity of entropy make their bulk predictions equivalent; phase coexistence or nonadditive interactions require care.

For the microcanonical case restrict the allowed states to the fixed-energy shell of size $\Omega$, imposing $p_n=0$ outside it. Maximize the Gibbs entropy subject to $\sum p_n=1$. A Lagrange multiplier gives $-k_B(\log p_n+1)+\lambda =0$ on the shell, soFor the canonical case impose both normalization and $\sum p_n{E}_n=U$. Stationarity of $-k_B\sum p_n\log p_n-\lambda (\sum p_n-1)-k_B\beta (\sum p_n{E}_n-U)$ gives $p_n\propto e^{-\beta E_n}$. ThusThe last identification follows from $\partial S/\partial U=1/T$. Entropy is strictly concave in the probabilities, so these stationary distributions are maxima.

For the independent three-level particles, put $z=ϵ/(k_{B}T)$ and $Z_1=1+2\cosh z$. The full partition function is $Z_1^N$ because the particles occupy labeled lattice sites. Differentiation givesThe heat capacity also equals $N\mathrm{Var}(E_1)/(k_B{T}^2)$, confirming positivity. As $T\to +\infty$, the three occupations become equal:As $T\downarrow 0$, all particles occupy their unique lowest state:

A negative temperature is possible because the total energy is bounded above. An equilibrated population inversion has more particles at $+ϵ$ than at $-ϵ$. For example probabilities $(p_{-},p_0,p_{+})=(1,2,4)/7$ correspond to $T=-ϵ/(k_B\log 2)$ and mean energy $3Nϵ/7$. This is a macroscopic occupation pattern; an isolated individual microstate is not itself assigned a temperature. At the positive zero-temperature limit the ground state is unique, so the entropy tends to zero: the system obeys the third law of thermodynamics. The negative-temperature branch likewise approaches the unique highest-energy state as $T\to 0^{-}$; it does not introduce residual ground-state entropy.

Here $p(n)$ is the probability of microstate $n$, including all occupation data. In the grand canonical ensemble, with energy $E_n$, particle number $N_n$, chemical potential $\mu$ and $T>0$,Substitution in the Gibbs entropy gives $S=k\log \mathcal{Z}+(U-\mu N)/T$, where $U=⟨E⟩$ and $N=⟨N_n⟩$. Differentiating at fixed $\mu ,V$ gives ${\partial }_T\log \mathcal{Z}=(U-\mu N)/(k{T}^2)$. Therefore

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.

The microcanonical ensemble fixes energy, volume and particle number; the canonical ensemble fixes temperature, volume and particle number; and the grand canonical ensemble fixes temperature, volume and chemical potential while allowing particle exchange. For macroscopic systems with short-range interactions, away from phase coexistence and other nonconcave regimes, relative fluctuations vanish in the thermodynamic limit and the ensembles give equivalent local thermodynamics.

For bosons the occupation number is $n_{\mathbf{p}}=0,1,2,\dots$. Summing the geometric series for the single-mode grand canonical ensemble gives the Bose-Einstein distribution

A grand canonical ensemble describes a system that exchanges both energy and particles with a reservoir, at fixed $T,V,\mu$. If microstate $n$ has occupation $N_n$ and one-particle energy $E_n$, thenwith the allowed occupations chosen for the particle statistics.

The microcanonical ensemble describes an isolated system at fixed energy $E$, volume $V$, and particle number $N$, assigning equal weight to accessible states in a narrow energy shell.

The canonical ensemble describes a system that can exchange energy with a heat reservoir at fixed temperature $T$, while $V$ and $N$ remain fixed. A microstate $s$ has probability

The grand canonical ensemble allows exchange of both energy and particles at fixed $T$, $V$, and chemical potential $\mu$. Its state weights are

The choice is irrelevant for bulk thermodynamic observables in the thermodynamic limit when the system is additive, has sufficiently short-range interactions, and lies away from singular phase-coexistence regimes. Then relative energy and particle-number fluctuations vanish as the system size grows. These are the standard conditions for equivalence of statistical ensembles.

spin normalization in two-dimensional free-electron state counting. Treat the mobile electrons as independent particles in a constant interior potential, with parabolic dispersion $ϵ={\hslash }^2{k}^2/(2m)$, and impose confinement or periodic boundary conditions. electron–electron interactions and the detailed periodic ionic potential are neglected, while Fermi statistics are retained. Each spatial state occupies area $(2\pi {)}^2/A$ in two-dimensional wavevector space. For spin degeneracy $g_s$,The printed coefficient $Am/(2\pi {\hslash }^2)$ is the per-spin density of states. Unpolarized spin-one-half electrons have $g_s=2$, doubling it. Actual graphene near its Dirac points has approximately linear dispersion, so its energy-dependent density of states is not this parabolic-model result; the plane geometry alone does not establish the free-electron dispersion.^[ref]

Occupied disk and Fermi energy. At zero temperature fill the lowest states up to a disk of radius $k_F$. State counting givesThus the physical spin-degenerate result is ${ϵ}_F=\pi {\hslash }^{2}N/(Am)$. If $N$ instead counts one spin species, the per-spin convention gives $2\pi {\hslash }^{2}N/(Am)$. Each occupied wavevector can contain one electron of each allowed spin, never two in the same complete one-particle state.

Occupation probability and chemical potential of a two-dimensional ideal Fermi gas. A single fermion state permits occupation zero or one. In a grand canonical ensemble their relative weights are $1$ and $e^{-\beta (ϵ-\mu )}$. Normalizing givesThe chemical potential is fixed by the total particle number. Here the total density of states $D=g_{s}Am/(2\pi {\hslash }^2)$ is constant above the band bottom. Integrating the occupation gives $N=D{k}_{B}T\ln (1+e^{\mu /(k_{B}T)})$, and thereforeThe approach is exponentially close in this ideal two-dimensional constant-density-of-states model; a three-dimensional $T^2$ correction should not be inserted here. A narrow interval of width of order $k_{B}T$ around $\mu$ is thermally smeared.

A microstate specifies the microscopic state: classically the positions and momenta of all particles, or quantum mechanically an allowed many-body state. A macrostate specifies a few coarse variables such as energy, volume and particle number. Many microstates can give the same macrostate. A statistical ensemble is a probability assignment over hypothetical copies prepared with the same macroscopic constraints; it does not require a sample to be literally divided into many physical copies. An observable's ensemble mean is $⟨A⟩={\sum }_i{p}_i{A}_i$, with an integral replacing the sum in classical phase space.

For an isolated system at fixed $U,V,N$, the microcanonical ensemble assigns equal probability to accessible microstates in a narrow energy shell. If their number is $\Omega (U,V,N)$, the Boltzmann entropy is $S=k_B\ln \Omega$. Thermal contact between two initially isolated systems allows energy to be shared while their sum stays fixed. The most probable division maximizes $S_1(U_1)+S_2(U-U_1)$, so equilibrium requires equal $\partial S/\partial U$. This gives the statistical temperature definitionFor positive temperature, energy tends to flow from lower entropy slope to higher entropy slope until those slopes agree. The narrow energy-shell width represents macroscopic resolution; it must not exclude the physically accessible states.

For a small subsystem exchanging energy with a much larger bath, the bath has energy $U_{\mathrm{tot}}-E_i$ when subsystem microstate $i$ has energy $E_i$. Expanding its entropy gives $S_b(U_{\mathrm{tot}}-E_i)=S_b(U_{\mathrm{tot}})-E_i/T+\cdots$. Counting compatible bath states therefore yields the Boltzmann distribution and canonical ensemble:The exponential weights individual microstates. An energy level of degeneracy $g_j$ has total probability $g_j{e}^{-\beta E_j}/Z$; a more degenerate excited level can outweigh one ground-state level even though each of its microstates is less probable. The Gibbs entropy $-k_B{\sum }_i{p}_i\ln p_i$ gives $S=k_B\ln Z+U/T$. The equilibrium energy variance is $⟨(\Delta E{)}^2⟩=k_B{T}^2{c}_V$, linking measurable fluctuations to a macroscopic response.

If particle exchange is also allowed, a reservoir fixes $T$ and chemical potential $\mu$. The grand canonical ensemble weights a state by $e^{-\beta (E_i-\mu N_i)}/\Xi$, with $\Xi ={\sum }_{N,i}{e}^{-\beta (E_i-\mu N)}$. Mean particle number is $k_{B}T\partial \ln \Xi /\partial \mu$, while volume remains a constraint in this version. Thus the ensemble follows the physical exchange conditions: fixed energy, thermal contact, or thermal and particle contact.

In ordinary large short-range systems, relative fluctuations often decrease as $N^{-1/2}$ and the ensembles give the same bulk thermodynamics away from singular limits. Finite systems, critical fluctuations and long-range interactions require care. Interpreting a time average as an ensemble average additionally requires appropriate sampling or mixing dynamics; it is not guaranteed solely by naming an ensemble. thermal equilibrium probability assignments do not describe every driven or relaxing state.

The appropriate quantity to minimize is the energy including exchange of particles with the electron reservoir, $\mathcal{E}=E-\mu N$. The occupation of an interacting localized electron level follows fromFor repulsion $U>0$, the first electron costs $ϵ$ and the second costs $ϵ+U$. Comparing all three grand energies givesFor attraction $U<0$, the singly occupied state never minimizes the grand energy. At the crossing of the empty and doubly occupied states, ${\mu }_{\ast }=ϵ+U/2$, its grand energy is $-U/2>0$, while both competing states have zero grand energy. HenceFor $U=0$ the two thresholds coincide at $\mu =ϵ$, giving a direct $0\to 2$ step. At a threshold, distinct occupations are degenerate; zero-temperature minimization alone does not select a unique occupation. A grand canonical ensemble at small positive temperature smooths the steps, withIn particular, the equal-weight empty/pair limit for $U<0$ gives mean occupation one exactly at its crossing; this is not a finite-width singly occupied plateau.

Ensemble and applications. A grand canonical ensemble describes a system exchanging both energy and particles with a large reservoir. Its fixed control parameters are $T,V,\mu$, while microstates with different $N$ have probabilitiesWeak coupling allows the system's energy levels to remain meaningful; equilibrium fixes the reservoir temperature and chemical potential. The grand potential is $\Phi =-k_{B}T\ln \Xi =U-TS-\mu N$, with $N$ now the mean population. It generates entropy, pressure and mean particle number by differentiation. This ensemble is convenient for gases in contact with particle reservoirs, independently occupied quantum levels and surface adsorption; its particle-number fluctuations are part of the description, not an error in counting.

Classical single-level limit. For the single-level grand partition function in the dilute limit, put $x_k=e^{-\beta ({ϵ}_k-\mu )}$. Empty and singly occupied levels contribute $1$ and $x_k$. In the dilute classical limit $x_k\ll 1$, higher occupancies contribute only at order $x_k^2$ or beyond:This truncation does not impose the Pauli principle on a classical gas. A fermion state has exactly $1+x_k$, a boson state has $(1-x_k{)}^{-1}$, and Maxwell-Boltzmann occupancy weights give $e^{x_k}$; they all agree to this order. In particular, $\ln {\Xi }_k=x_k+O(x_k^2)$, with the classical Maxwell-Boltzmann result exactly $x_k$.

grand potential and chemical potential. With $z=e^{\beta \mu }$, summing the single-level logarithms and using the stated density of states givesSubstitute $ϵ=k_{B}T{x}^2$, for whichCombining this with the density-of-states prefactor gives $\ln \Xi =zV/{\lambda }^3$. ThusDifferentiating at fixed $T,V$,Similarly $p=-({\partial }_V\Phi {)}_{T,\mu }=k_{B}Tz/{\lambda }^3$, so $pV=N{k}_{B}T$. This is the grand-canonical chemical potential of a classical ideal gas. The condition $N{\lambda }^3/V\ll 1$ states the classical regime. The supplied density of states counts one internal state; extra internal degeneracies would have to be included consistently rather than inserted at only one stage.

One adsorption site. Take the free-vapour energy reference as zero and the binding energy as $-ϵ$. A site is either empty or singly occupied; the latter has vibrational partition function $z_s(T)$. Its adsorption-site grand partition function and mean occupation areThe chemical potential of an adsorbed species on equivalent sites follows by inverting the occupation ratioThe factor $1-\theta$ is the vacant-site probability. The binding reference and the energies included in $z_s$ must be chosen consistently, so that vibrational zero-point energy is not counted twice.

Isotherm. The vibrational factor in a Langmuir equilibrium constant is fixed by adsorption equilibrium, ${\mu }_s={\mu }_{\mathrm{vap}}$. Neglecting vapour internal degrees of freedom and writing its number density as $p/(k_{B}T)$ gives $e^{\beta {\mu }_{\mathrm{vap}}}=p{\lambda }^3/(k_{B}T)$. ThereforeThis is the Langmuir isotherm: initially linear, increasing monotonically and concave down, approaching unit coverage. Half coverage occurs at $p_{1/2}=1/b_s(T)$. The coefficient has inverse-pressure units. It assumes equivalent independent sites, at most one molecule per site and no lateral interaction or multilayer adsorption. The molecular mass appropriate to the vapour determines $\lambda$.

A real uptake measurement must separate adsorbed molecules from free gas in the apparatus. Langmuir's original mica experiment compared a mica-containing bulb with a similar blank and inferred uptake from calibrated gas volumes and pressures. His stated blank contribution and bath-temperature drift limited the inferred adsorption; a saturating curve alone does not establish ideal equivalent sites.^[ref]

An statistical ensemble specifies probabilities of microstates under particular macroscopic constraints. It represents statistical information about a system; the reservoir constraints of equilibrium ensembles specify what it can exchange with its surroundings.

The thermodynamic potentials generate mean quantities and response functions. For example, canonical energy variance is $⟨(\Delta E{)}^2⟩=k_B{T}^2{C}_V$. With short-range interactions and stable bulk phases, relative fluctuations usually vanish for large systems and these ensembles give equivalent bulk predictions after matching their mean variables. Finite systems, constraints and long-range interactions require greater care; the ensembles retain different fluctuating variables.^[ref]

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.