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Derivation

Grand-Canonical Ensemble and the Grand Potential

Statement

For a system in thermal and diffusive contact with a reservoir at temperature \(T\) and chemical potential \(\mu\), maximising the Gibbs entropy subject to fixed mean energy and mean particle number yields the grand-canonical distribution \(p_i \propto e^{-\beta(E_i-\mu N_i)}\). Its normalisation defines the grand partition function \(\Xi(T,V,\mu)=\sum_i e^{-\beta(E_i-\mu N_i)}\), and the grand potential is \(\Phi=-k_BT\ln\Xi\), from which \(\langle N\rangle=-\left(\partial\Phi/\partial\mu\right)_{T,V}\) and \(\operatorname{Var}(N)=k_BT\,(\partial\langle N\rangle/\partial\mu)_{T,V}\).

Why it matters

The grand-canonical ensemble is the natural language for any system that can exchange both energy and particles with its surroundings: adsorbed monolayers, electrons in a metal in contact with a lead, photons and phonons (whose number is unconstrained), and every quantum gas. It converts an awkward fixed-\(N\) combinatorial problem into an unconstrained sum over occupation numbers, which is why Fermi-Dirac and Bose-Einstein statistics fall out almost trivially once \(\Xi\) factorises over single-particle modes.

It also delivers thermodynamics for free. A single logarithm, \(\Phi=-k_BT\ln\Xi\), packages pressure, entropy, and particle number as first derivatives, and the second derivative of \(\Phi\) with respect to \(\mu\) measures density fluctuations, tying microscopic statistics directly to the measurable isothermal compressibility.

Assumptions
Weak coupling to the reservoir.If the interaction energy between system and reservoir were not negligible compared with each subsystem's energy, energy and particle number would not be additive and \(\Xi\) would not factor from the reservoir; the exponential Boltzmann-Gibbs form would fail.
Reservoir far larger than the system.Only if the reservoir's \(T\) and \(\mu\) are unchanged by the exchange can we Taylor-expand its entropy to first order; a finite reservoir would give \(T,\mu\)-dependent corrections and a non-exponential distribution.
Fixed volume and a well-defined chemical potential.If \(\mu\) were ill-defined (e.g. no unique particle species or a non-equilibrium gradient), the conjugate constraint \(\langle N\rangle\) would not close the maximisation and \(\Xi\) would not exist as written.
Additive, extensive spectrum with a common Fock space.If states of different \(N\) did not live in a single Hilbert space (indefinite particle number) the sum over \(i\) mixing sectors of different \(N\) would be meaningless; second quantisation is what makes \(\sum_i\) over all \(N\) legitimate.
Derivation
1
\[ S_G=-k_B\sum_i p_i\ln p_i,\qquad \sum_i p_i=1,\quad \sum_i p_i E_i=\langle E\rangle,\quad \sum_i p_i N_i=\langle N\rangle \]
Start from the Gibbs entropy and impose normalisation plus the two macroscopic constraints (mean energy, mean number) that define diffusive-plus-thermal contact; this is the maximum-entropy setup for an open system. A
2
\[ \mathcal{L}=-k_B\sum_i p_i\ln p_i-\lambda_0\!\left(\sum_i p_i-1\right)-\lambda_E\!\left(\sum_i p_iE_i-\langle E\rangle\right)-\lambda_N\!\left(\sum_i p_iN_i-\langle N\rangle\right) \]
Introduce Lagrange multipliers \(\lambda_0,\lambda_E,\lambda_N\), one per constraint, so the constrained extremisation becomes an unconstrained stationarity problem in the \(p_i\). A
3
\[ \frac{\partial\mathcal{L}}{\partial p_i}=-k_B(\ln p_i+1)-\lambda_0-\lambda_E E_i-\lambda_N N_i=0 \]
Set the derivative with respect to each \(p_i\) to zero. The entropy is strictly concave and the constraints are linear, so this stationary point is the unique global maximum, not merely an extremum. B
4
\[ p_i=\exp\!\left(-1-\frac{\lambda_0}{k_B}\right)\exp\!\left(-\frac{\lambda_E E_i+\lambda_N N_i}{k_B}\right) \]
Solve algebraically for \(p_i\). The distribution is a product of a constant (fixing normalisation) and an exponential linear in \(E_i\) and \(N_i\) — the generic exponential-family form of a max-entropy solution. B
5
\[ \frac{\lambda_E}{k_B}\equiv\beta=\frac{1}{k_BT},\qquad \frac{\lambda_N}{k_B}\equiv-\beta\mu \]
Identify the multipliers thermodynamically. Comparing \(dS=(1/T)\,dU-(\mu/T)\,dN\) (from \(S(U,N)\)) with \(\partial S_G/\partial\langle E\rangle=\lambda_E\), \(\partial S_G/\partial\langle N\rangle=\lambda_N\) fixes \(\lambda_E=k_B\beta\) and \(\lambda_N=-k_B\beta\mu\); this reuses temperature-from-entropy-maximisation and extends it to the particle channel. B
6
\[ p_i=\frac{1}{\Xi}\,e^{-\beta(E_i-\mu N_i)},\qquad \Xi(T,V,\mu)\equiv\sum_i e^{-\beta(E_i-\mu N_i)} \]
Absorb the leading constant into a normalisation \(\Xi\) fixed by \(\sum_i p_i=1\). This defines the grand partition function; the sum runs over all microstates of every particle-number sector. A
7
\[ S_G=-k_B\sum_i p_i\big[-\beta(E_i-\mu N_i)-\ln\Xi\big]=k_B\beta\big(\langle E\rangle-\mu\langle N\rangle\big)+k_B\ln\Xi \]
Substitute the solved \(p_i\) back into \(S_G\) and use \(\sum_i p_i=1\), \(\sum_i p_iE_i=\langle E\rangle\), \(\sum_i p_iN_i=\langle N\rangle\). This evaluates the entropy at the maximum in closed form. B
8
\[ \langle E\rangle-TS_G-\mu\langle N\rangle=-k_BT\ln\Xi \]
Multiply the previous line by \(T\) and rearrange. The left side is exactly the thermodynamic grand potential \(\Phi=U-TS-\mu N\), obtained by Legendre-transforming the internal energy in both the entropy and particle channels. B
9
\[ \boxed{\;\Phi(T,V,\mu)=-k_BT\ln\Xi\;} \]
Identify the statistical grand potential with its thermodynamic namesake. This is the bridge relation for the grand-canonical ensemble, the analogue of \(F=-k_BT\ln Z\) for the canonical ensemble. A
10
\[ \langle N\rangle=\frac{1}{\Xi}\sum_i N_i\,e^{-\beta(E_i-\mu N_i)}=\frac{1}{\beta}\frac{\partial\ln\Xi}{\partial\mu}=-\left(\frac{\partial\Phi}{\partial\mu}\right)_{T,V} \]
Differentiate \(\ln\Xi\) with respect to \(\mu\) at fixed \(T,V\): each term brings down a factor \(\beta N_i\), reproducing the ensemble average. Dividing by \(\beta\) and using \(\Phi=-k_BT\ln\Xi\) gives \(\langle N\rangle=-\partial\Phi/\partial\mu\). B
11
\[ \frac{\partial\langle N\rangle}{\partial\mu}\bigg|_{T,V}=\frac{1}{\beta}\frac{\partial^2\ln\Xi}{\partial\mu^2}=\beta\!\left(\langle N^2\rangle-\langle N\rangle^2\right) \]
Differentiate \(\langle N\rangle=\beta^{-1}\partial_\mu\ln\Xi\) once more. The second logarithmic derivative of a partition function is a variance: \(\partial_\mu\langle N\rangle=\beta(\langle N^2\rangle-\langle N\rangle^2)\), the fluctuation-response relation in the particle channel. C
12
\[ \boxed{\;\operatorname{Var}(N)=\langle N^2\rangle-\langle N\rangle^2=k_BT\left(\frac{\partial\langle N\rangle}{\partial\mu}\right)_{T,V}=-k_BT\left(\frac{\partial^2\Phi}{\partial\mu^2}\right)_{T,V}\;} \]
Rearrange for the variance. Number fluctuations are set by the susceptibility \(\partial\langle N\rangle/\partial\mu\); positivity of the variance forces \((\partial\langle N\rangle/\partial\mu)_{T,V}\ge0\), a thermodynamic stability condition. B
Result
\[ \Xi=\sum_i e^{-\beta(E_i-\mu N_i)},\qquad \Phi=-k_BT\ln\Xi,\qquad \langle N\rangle=-\left(\frac{\partial\Phi}{\partial\mu}\right)_{T,V},\qquad \operatorname{Var}(N)=k_BT\left(\frac{\partial\langle N\rangle}{\partial\mu}\right)_{T,V} \]

Reading. Openness to particle exchange adds a term \(-\mu N_i\) to the Boltzmann weight, controlled by the chemical potential just as \(-E_i\) is controlled by temperature. The single scalar \(\Phi=-k_BT\ln\Xi\) is a full thermodynamic potential in the variables \((T,V,\mu)\): its \(\mu\)-slope gives the mean number, its \(\mu\)-curvature gives the number fluctuations, and (since \(\Phi=-pV\) for a simple fluid) its \(V\)-behaviour gives the pressure. Fluctuations are not noise to be discarded — they are the linear response \(\partial\langle N\rangle/\partial\mu\) and are directly measurable through the compressibility.

Units check. \(\beta(E_i-\mu N_i)\) is \((\text{J})^{-1}\cdot\text{J}=\) dimensionless, so the exponent and \(\Xi\) are pure numbers. \(\Phi=-k_BT\ln\Xi\) has units \((\text{J K}^{-1})(\text{K})=\text{J}\), an energy. \(\partial\Phi/\partial\mu\) is J/(J) \(=\) dimensionless, matching \(\langle N\rangle\) (a pure count). \(\operatorname{Var}(N)=k_BT\,\partial\langle N\rangle/\partial\mu\) is \((\text{J})\cdot(\text{J}^{-1})=\) dimensionless, correct for a squared count.

Limiting cases
  • Fixed \(\mu\to\) fixed \(N\). When \(\operatorname{Var}(N)/\langle N\rangle^2\to0\) (large systems away from criticality), grand-canonical predictions coincide with canonical ones; the ensembles are equivalent in the thermodynamic limit.
  • Single mode, fermions. A level with occupancy \(0\) or \(1\) gives \(\Xi=1+e^{-\beta(\varepsilon-\mu)}\) and \(\langle n\rangle=[e^{\beta(\varepsilon-\mu)}+1]^{-1}\), the Fermi-Dirac distribution.
  • Single mode, bosons. Summing \(n=0,1,2,\dots\) gives \(\Xi=[1-e^{-\beta(\varepsilon-\mu)}]^{-1}\) and \(\langle n\rangle=[e^{\beta(\varepsilon-\mu)}-1]^{-1}\), the Bose-Einstein distribution (requires \(\mu<\varepsilon_{\min}\)).
  • Classical / dilute limit. For \(e^{\beta(\varepsilon-\mu)}\gg1\) both reduce to \(\langle n\rangle\approx e^{-\beta(\varepsilon-\mu)}\), Maxwell-Boltzmann, and \(\Xi=\exp(z\,Z_1)\) with fugacity \(z=e^{\beta\mu}\).
  • Photons/phonons (\(\mu=0\)). Particle number is unconstrained, so the number multiplier vanishes; \(\Xi\) reduces to a product of bosonic single-mode factors with \(\mu=0\).
Breaks when
  • Near a critical point. \((\partial\langle N\rangle/\partial\mu)_{T,V}\to\infty\), so \(\operatorname{Var}(N)\) diverges and relative fluctuations are \(O(1)\); ensemble equivalence fails and the naive thermodynamic-limit identification of \(\langle N\rangle\) with a sharp macroscopic \(N\) breaks down.
  • Bose-Einstein condensation. As \(\mu\to\varepsilon_0^-\) the ground-mode occupancy and its variance blow up; the grand-canonical variance \(\langle n_0^2\rangle-\langle n_0\rangle^2\approx\langle n_0\rangle^2\) is unphysically large (the "grand-canonical fluctuation catastrophe"), and one must fix \(N\) or treat the condensate separately.
  • Strong system-reservoir coupling / small systems. When interaction energy is comparable to subsystem energies, or the "reservoir" is not much larger than the system, energy and \(N\) are non-additive, the entropy expansion truncated at first order is invalid, and \(p_i\) is no longer exponential.
  • No well-defined \(\mu\). Non-equilibrium steady states with particle currents, or systems with unbounded-below spectra, have no single equilibrium chemical potential and \(\Xi\) either fails to exist or fails to describe the state.
Failure modes
  • Sign slip on the number term. Writing \(e^{-\beta(E_i+\mu N_i)}\); the correct weight is \(e^{-\beta(E_i-\mu N_i)}\), so increasing \(\mu\) raises the weight of high-\(N_i\) states.
  • Summing only within one \(N\)-sector. Treating \(\Xi\) as a canonical sum at fixed \(N\); the whole point is that \(\sum_i\) ranges over all particle numbers, which is what makes \(N_i\) a fluctuating variable.
  • Confusing \(\Phi\) with \(F\). Using \(\langle N\rangle=+\partial\Phi/\partial\mu\) (wrong sign) or \(\Phi=-k_BT\ln Z\); the grand potential carries a minus sign, \(\langle N\rangle=-\partial\Phi/\partial\mu\), and \(\Phi=F-\mu N\).
  • Fugacity book-keeping. Writing \(z=e^{-\beta\mu}\) instead of \(z=e^{+\beta\mu}\), so that \(\langle N\rangle=z\,\partial\ln\Xi/\partial z\) comes out with the wrong monotonicity.
  • Applying \(\mu<\varepsilon_{\min}\) to fermions. Imposing the bosonic bound on \(\mu\) for fermions; Fermi-Dirac allows any \(\mu\), including \(\mu>\varepsilon\), which simply drives \(\langle n\rangle\to1\).
  • Forgetting the Gibbs \(1/N!\). In the classical dilute limit, dropping indistinguishability so that \(\Xi\ne\exp(zZ_1)\); this reintroduces the Gibbs paradox.
Discussion

The grand-canonical ensemble is the second step in a ladder of Legendre transforms. The microcanonical ensemble fixes \((E,N,V)\); trading fixed energy for contact with a heat bath at temperature \(T\) gives the canonical ensemble and the Helmholtz free energy \(F=-k_BT\ln Z\); trading fixed particle number for contact with a particle reservoir at chemical potential \(\mu\) gives the grand-canonical ensemble and \(\Phi=-k_BT\ln Z_{\rm gc}\equiv-k_BT\ln\Xi\). Each transform swaps an extensive variable for its intensive conjugate, and each conjugation appears in the statistics as a new linear term in the exponent of the Boltzmann weight. This is why the derivation above is structurally identical to the canonical one: only a second constraint and a second multiplier are added.

The decisive practical advantage is factorisation. Because states of different total \(N\) all live in one Fock space, the sum over microstates becomes an independent sum over the occupation number of each single-particle mode: \(\Xi=\prod_k\Xi_k\), with \(\Xi_k=\sum_{n_k}e^{-\beta(\varepsilon_k-\mu)n_k}\). The intractable fixed-\(N\) constraint that couples all modes in the canonical ensemble simply dissolves. Fermi-Dirac and Bose-Einstein statistics then emerge from a two-term or geometric single-mode sum, and \(\ln\Xi=\sum_k\ln\Xi_k\) makes \(\Phi\), \(\langle N\rangle\), and pressure additive over modes.

The fluctuation relation \(\operatorname{Var}(N)=k_BT(\partial\langle N\rangle/\partial\mu)_{T,V}\) is a specific instance of the fluctuation-dissipation theorem: a spontaneous equilibrium fluctuation (the variance of \(N\)) equals a linear-response coefficient (how \(\langle N\rangle\) responds to a small change in \(\mu\)). Rewriting it in terms of measurable quantities, \(\operatorname{Var}(N)/\langle N\rangle=(N/V)k_BT\,\kappa_T\) with \(\kappa_T\) the isothermal compressibility, shows that microscopic counting statistics are fixed by a bulk mechanical response. That relative fluctuations scale as \(1/\sqrt{\langle N\rangle}\) is precisely why the ensembles agree in the thermodynamic limit and why thermodynamics looks deterministic.

At the level of rigour, the identification \(\Phi=-k_BT\ln\Xi\) is exact for the model Hamiltonian but its physical relevance rests on ensemble equivalence, which is a theorem, not a definition. Equivalence holds when the free energy per particle is a smooth (analytic) function of the intensive variables — precisely where the Legendre transform between \(F(T,V,N)\) and \(\Phi(T,V,\mu)\) is invertible. At a first-order transition the transform develops a flat segment (coexistence), and at a critical point it develops a cusp; in both cases \(\partial\langle N\rangle/\partial\mu\) is singular, the variance is extensive or larger, and grand-canonical averages no longer track sharp canonical values. The condensate fluctuation catastrophe of the ideal Bose gas is the cleanest textbook symptom: the grand-canonical ensemble is internally consistent but describes an ensemble of systems whose particle number fluctuates wildly, which is not the fixed-\(N\) laboratory sample.

Common misconceptions. The chemical potential is not "the energy to add a particle" in general — it is \((\partial U/\partial N)_{S,V}=(\partial F/\partial N)_{T,V}\), a free-energy cost that includes entropy, and for an ideal Fermi gas at \(T=0\) it is the (positive) Fermi energy while for a classical ideal gas it is large and negative. A second frequent error is to think the grand-canonical ensemble describes one system with a literally varying particle count; it describes an ensemble (or a subvolume of a large system) whose members have different \(N\), and any single realisation still has a definite integer \(N\).

Worked examples
1
Single adsorption site (Langmuir isotherm). A surface site is either empty (\(E=0,N=0\)) or holds one molecule with binding energy \(-\varepsilon_b\) (\(E=-\varepsilon_b,N=1\)). Find the occupancy \(\langle n\rangle\) at \(T\) and gas chemical potential \(\mu\).
Set up the two-state grand sum. A
2
\[ \Xi=e^{-\beta(0-\mu\cdot0)}+e^{-\beta(-\varepsilon_b-\mu\cdot1)}=1+e^{\beta(\varepsilon_b+\mu)} \]
Sum over the two microstates. A
3
\[ \langle n\rangle=\frac{1}{\beta}\frac{\partial\ln\Xi}{\partial\mu}=\frac{e^{\beta(\varepsilon_b+\mu)}}{1+e^{\beta(\varepsilon_b+\mu)}}=\frac{1}{e^{-\beta(\varepsilon_b+\mu)}+1} \]
Differentiate \(\ln\Xi\); this is the Fermi-Dirac form (one site is a two-state fermionic mode). B
4
Numbers: \(\varepsilon_b=0.30\ \text{eV}\), \(T=300\ \text{K}\Rightarrow k_BT=0.02585\ \text{eV}\), gas at \(\mu=-0.20\ \text{eV}\). Then \(\varepsilon_b+\mu=0.10\ \text{eV}\), \(\beta(\varepsilon_b+\mu)=0.10/0.02585=3.868\).
Insert numbers with consistent eV units. A
\[ \langle n\rangle=\frac{1}{e^{-3.868}+1}=\frac{1}{1.0209}=0.980 \]

Reading. With binding beating the chemical-potential deficit by \(0.10\ \text{eV}\approx4\,k_BT\), the site is essentially always occupied (98%). Lowering \(\mu\) (dilute gas) empties it; raising it saturates it — the Langmuir isotherm.

Units check. The exponent \(\beta(\varepsilon_b+\mu)\) is (eV\(^{-1}\))(eV) = dimensionless; \(\langle n\rangle\) is a pure fraction between 0 and 1. Correct.

1
Number fluctuations in a classical ideal gas. For an ideal classical gas \(\ln\Xi=zZ_1\) with fugacity \(z=e^{\beta\mu}\) and \(Z_1=V/\lambda_T^3\). Find \(\langle N\rangle\), \(\operatorname{Var}(N)\), and the relative fluctuation for \(N=1.0\times10^{20}\) atoms.
Use the factorised grand potential of the classical ideal gas. A
2
\[ \langle N\rangle=\frac{1}{\beta}\frac{\partial\ln\Xi}{\partial\mu}=z\frac{\partial(zZ_1)}{\partial z}\cdot\frac{1}{z}\cdot z=zZ_1=\ln\Xi \]
Since \(\partial/\partial\mu=\beta z\,\partial/\partial z\) and \(\ln\Xi=zZ_1\) is linear in \(z\), \(\langle N\rangle=zZ_1\). B
3
\[ \operatorname{Var}(N)=\frac{1}{\beta}\frac{\partial\langle N\rangle}{\partial\mu}=z\frac{\partial(zZ_1)}{\partial z}=zZ_1=\langle N\rangle \]
The variance equals the mean because \(\ln\Xi\) is linear in \(z\): this is Poisson statistics, a hallmark of the classical ideal gas. B
4
\[ \frac{\sqrt{\operatorname{Var}(N)}}{\langle N\rangle}=\frac{\sqrt{\langle N\rangle}}{\langle N\rangle}=\frac{1}{\sqrt{\langle N\rangle}},\qquad \langle N\rangle=1.0\times10^{20} \]
Form the relative fluctuation and insert the number. A
\[ \frac{\sqrt{\operatorname{Var}(N)}}{\langle N\rangle}=\frac{1}{\sqrt{1.0\times10^{20}}}=1.0\times10^{-10} \]

Reading. The gas has Poissonian number statistics, \(\operatorname{Var}(N)=\langle N\rangle\), and the relative spread is one part in \(10^{10}\). This vanishingly small fluctuation is exactly why the grand-canonical and canonical ensembles are indistinguishable for macroscopic samples.

Units check. \(\operatorname{Var}(N)=\langle N\rangle\): both pure counts. The relative fluctuation is dimensionless, \(1/\sqrt{\text{count}}\to\) dimensionless number. Correct.

Problems
  1. Show from \(\Phi=-k_BT\ln\Xi\) that \(S=-(\partial\Phi/\partial T)_{V,\mu}\) and hence \(\Phi=U-TS-\mu N\).
    Solution Write \(\Xi=\sum_i e^{-\beta(E_i-\mu N_i)}\). Then \(\ln\Xi\) depends on \(T\) through \(\beta=1/k_BT\) and (at fixed \(\mu,V\)) \(\partial\ln\Xi/\partial T=(\partial\ln\Xi/\partial\beta)(d\beta/dT)\). Now \(\partial\ln\Xi/\partial\beta=-\langle E-\mu N\rangle=-(U-\mu N)\) and \(d\beta/dT=-1/(k_BT^2)\). So \(\partial\ln\Xi/\partial T=(U-\mu N)/(k_BT^2)\). Then \(-(\partial\Phi/\partial T)=k_B\ln\Xi+k_BT\,\partial\ln\Xi/\partial T=k_B\ln\Xi+(U-\mu N)/T\). Multiply the identity \(-k_BT\ln\Xi=\Phi\): \(k_B\ln\Xi=-\Phi/T\). Hence \(-(\partial\Phi/\partial T)=(-\Phi+U-\mu N)/T\equiv S\) provided \(\Phi=U-TS-\mu N\), which rearranges to \(S=(U-\mu N-\Phi)/T\), consistent. Thus \(S=-(\partial\Phi/\partial T)_{V,\mu}\) and \(\Phi=U-TS-\mu N\).
  2. A single localised orbital can hold \(0\), \(1\uparrow\), \(1\downarrow\), or \(2\) electrons; double occupancy costs Hubbard \(U\) on top of the single-particle energy \(\varepsilon\). Write \(\Xi\) and find \(\langle N\rangle\) at \(T,\mu\).
    Solution The four states: \((E,N)=(0,0),(\varepsilon,1),(\varepsilon,1),(2\varepsilon+U,2)\). So \(\Xi=1+2e^{-\beta(\varepsilon-\mu)}+e^{-\beta(2\varepsilon+U-2\mu)}\). With \(x\equiv e^{-\beta(\varepsilon-\mu)}\) and \(y\equiv e^{-\beta(2\varepsilon+U-2\mu)}\), \(\Xi=1+2x+y\). Then \(\langle N\rangle=\beta^{-1}\partial_\mu\ln\Xi=(2x\cdot1+y\cdot2)/\Xi=(2x+2y)/(1+2x+y)\). Check limits: \(U\to\infty\Rightarrow y\to0\Rightarrow\langle N\rangle=2x/(1+2x)\le1\) (double occupancy forbidden); \(U\to0\Rightarrow y=x^2\Rightarrow\Xi=(1+x)^2\), \(\langle N\rangle=2x/(1+x)=2/(e^{\beta(\varepsilon-\mu)}+1)\), two independent Fermi-Dirac spins.
  3. For an ideal Fermi gas, \(\ln\Xi=\sum_k\ln[1+e^{-\beta(\varepsilon_k-\mu)}]\). Derive \(\langle n_k\rangle\) and show \(\operatorname{Var}(n_k)=\langle n_k\rangle(1-\langle n_k\rangle)\).
    Solution Single mode: \(\Xi_k=1+e^{-\beta(\varepsilon_k-\mu)}\). \(\langle n_k\rangle=\beta^{-1}\partial_\mu\ln\Xi_k=e^{-\beta(\varepsilon_k-\mu)}/[1+e^{-\beta(\varepsilon_k-\mu)}]=1/[e^{\beta(\varepsilon_k-\mu)}+1]\). For the variance, \(\operatorname{Var}(n_k)=k_BT\,\partial\langle n_k\rangle/\partial\mu\). With \(f\equiv\langle n_k\rangle=1/(e^{\beta(\varepsilon_k-\mu)}+1)\), \(\partial f/\partial\mu=\beta\,e^{\beta(\varepsilon_k-\mu)}/(e^{\beta(\varepsilon_k-\mu)}+1)^2=\beta f(1-f)\). Hence \(\operatorname{Var}(n_k)=k_BT\cdot\beta f(1-f)=f(1-f)=\langle n_k\rangle(1-\langle n_k\rangle)\). It vanishes for \(f\to0\) or \(f\to1\) (empty or Pauli-blocked full mode), maximal at \(f=1/2\).
  4. An ideal Bose mode of energy \(\varepsilon\) has \(\Xi=[1-e^{-\beta(\varepsilon-\mu)}]^{-1}\) (\(\mu<\varepsilon\)). Find \(\langle n\rangle\) and \(\operatorname{Var}(n)\), and comment on the ratio \(\operatorname{Var}(n)/\langle n\rangle^2\) as \(\mu\to\varepsilon\).
    Solution Let \(a=e^{-\beta(\varepsilon-\mu)}<1\). \(\ln\Xi=-\ln(1-a)\). \(\langle n\rangle=\beta^{-1}\partial_\mu\ln\Xi\); since \(\partial a/\partial\mu=\beta a\), \(\partial_\mu\ln\Xi=\beta a/(1-a)\), so \(\langle n\rangle=a/(1-a)=1/(e^{\beta(\varepsilon-\mu)}-1)\), Bose-Einstein. Variance: \(\operatorname{Var}(n)=k_BT\,\partial\langle n\rangle/\partial\mu\). With \(n=a/(1-a)\), \(\partial n/\partial a=1/(1-a)^2\) and \(\partial a/\partial\mu=\beta a\), so \(\partial n/\partial\mu=\beta a/(1-a)^2\), giving \(\operatorname{Var}(n)=a/(1-a)^2=\langle n\rangle(1+\langle n\rangle)\). Then \(\operatorname{Var}(n)/\langle n\rangle^2=(1+\langle n\rangle)/\langle n\rangle^2\to1\) as \(\langle n\rangle\to\infty\) (\(\mu\to\varepsilon\)): the standard deviation is as large as the mean — the grand-canonical condensate fluctuation catastrophe.
  5. A protein binds up to two identical ligands with binding energies \(-\varepsilon_1\) (first) and \(-\varepsilon_1-\varepsilon_2\) (both; \(\varepsilon_2\) includes cooperativity). Treating the ligand bath at chemical potential \(\mu\), write \(\Xi\) and the fractional saturation \(\theta=\langle N\rangle/2\), then evaluate for \(\varepsilon_1=0.15\ \text{eV}\), \(\varepsilon_2=0.25\ \text{eV}\), \(\mu=-0.10\ \text{eV}\), \(T=310\ \text{K}\).
    Solution States \((E,N)\): \((0,0)\), \((-\varepsilon_1,1)\) with degeneracy 2 (two equivalent sites), \((-\varepsilon_1-\varepsilon_2,2)\). Wait—for two distinct occupancies use \(\Xi=1+2e^{\beta(\varepsilon_1+\mu)}+e^{\beta(\varepsilon_1+\varepsilon_2+2\mu)}\) taking each bound state weight \(e^{-\beta(E-\mu N)}\). Here \(k_BT=8.617\times10^{-5}\times310=0.02671\ \text{eV}\). Exponents: first-bind \(\beta(\varepsilon_1+\mu)=(0.15-0.10)/0.02671=0.05/0.02671=1.872\Rightarrow e^{1.872}=6.50\); double \(\beta(\varepsilon_1+\varepsilon_2+2\mu)=(0.15+0.25-0.20)/0.02671=0.20/0.02671=7.488\Rightarrow e^{7.488}=1785\). Then \(\Xi=1+2(6.50)+1785=1+13.0+1785=1799\). \(\langle N\rangle=[2(6.50)\cdot1+1785\cdot2]/1799=(13.0+3570)/1799=3583/1799=1.992\). Saturation \(\theta=1.992/2=0.996\). The strong cooperative double-bound term (\(\varepsilon_2>0\)) dominates, saturating the protein (99.6%) despite the modestly negative ligand chemical potential.