Grand-Canonical Ensemble

The grand-canonical ensemble describes equilibrium when a system can exchange both energy and particles with a reservoir. Its standard control variables are

$$T,\qquad \mu,\qquad V,$$

where $\mu$ is the chemical potential. Particle number fluctuates.

For a Hamiltonian $H$ and conserved number operator $N$, define

$$K \equiv H-\mu N.$$

The grand-canonical density operator is

$$\rho_{\beta,\mu} = \frac{ e^{-\beta K} }{ \Xi } = \frac{ e^{-\beta(H-\mu N)} }{ \Xi },$$

with grand partition function

$$\Xi(\beta,\mu,V) = \operatorname{Tr}_{\mathcal F} e^{-\beta(H-\mu N)}.$$

The trace runs over the relevant Fock space or direct sum of particle-number sectors. Restricting it to one fixed-$N$ sector would return a canonical calculation.

What Is Fixed and What Fluctuates?

Quantity	Grand-canonical status
temperature $T$	externally fixed
chemical potential $\mu$	externally fixed
volume $V$ and other controls	externally fixed
energy $E$	fluctuates
particle number $N$	fluctuates
density operator	$\rho_{\beta,\mu}$
thermodynamic potential	grand potential $\Omega_{\mathrm G}$

The ensemble is useful for open containers, adsorption, quantum gases, leads in mesoscopic transport, and any calculation where summing over occupation numbers is simpler than enforcing one exact total number.

Conservation Requirement

The standard equilibrium construction assumes

$$[H,N]=0.$$

Then energy and particle number can be labeled simultaneously, and

$$K=H-\mu N$$

generates the exponential weights.

More generally, a chemical potential is associated with a conserved charge. If the microscopic Hamiltonian does not conserve the proposed $N$, treating $\mu$ as an equilibrium Lagrange multiplier requires justification.

Mean-field pairing Hamiltonians can appear not to commute with number because a symmetry-breaking approximation has been made. The underlying number-conserving theory and the role of $\mu$ must then be kept conceptually separate from the effective quasiparticle Hamiltonian.

Fock-Space and Sector Decomposition

Write Fock space as

$$\mathcal F = \bigoplus_{N=0}^{\infty} \mathcal H_N.$$

When $H$ preserves particle number,

$$H = \bigoplus_{N=0}^{\infty} H_N.$$

The grand partition function becomes

$$\Xi = \sum_{N=0}^{\infty} e^{\beta\mu N} \operatorname{Tr}_{\mathcal H_N} e^{-\beta H_N}.$$

Defining the canonical fixed-$N$ partition functions

$$Z_N = \operatorname{Tr}_{\mathcal H_N} e^{-\beta H_N},$$

gives the central relation

$$\Xi = \sum_{N=0}^{\infty} z^N Z_N,$$

where

$$z \equiv e^{\beta\mu}$$

is the fugacity.

Thus $\Xi$ is a generating function for canonical partition functions.

Convergence

The sum

$$\Xi = \sum_N z^N Z_N$$

must converge. This requirement constrains $\mu$ and the model.

For bosons in one-particle levels $\epsilon_i$, each mode sum converges only if

$$ze^{-\beta\epsilon_i} < 1.$$

Therefore

$$\mu < \epsilon_{\min}$$

for a finite ideal Bose system with lowest one-particle energy $\epsilon_{\min}$. The limit $\mu\to\epsilon_{\min}$ is tied to macroscopic ground-mode occupation and requires separate treatment.

For a finite number of fermionic modes, each mode has only occupations $0$ and $1$, so the finite-mode grand partition function converges for every finite real $\mu$. Infinite-volume and continuum limits still require regularization and density control.

Joint Probabilities

Let $|N,a\rangle$ be a simultaneous eigenstate:

$$\begin{aligned} N|N,a\rangle &= N|N,a\rangle, \\ H|N,a\rangle &= E_{N,a}|N,a\rangle. \end{aligned}$$

Its grand-canonical probability is

$$p_{N,a} = \frac{ e^{-\beta(E_{N,a}-\mu N)} }{ \Xi }.$$

The particle-number distribution is

$$\Pr(N) = \frac{ z^N Z_N }{ \Xi }.$$

Conditioned on a chosen $N$, the state within $\mathcal H_N$ is canonical:

$$\rho_{\beta,\mu}\big|_N \propto e^{-\beta H_N}.$$

The chemical potential changes the probability assigned to different sectors, not the relative Boltzmann weights inside one fixed sector.

Number Superselection

The grand-canonical state is block diagonal:

$$\rho_{\beta,\mu} = \bigoplus_N \Pr(N)\rho_{N,\beta}.$$

It is a mixture over particle-number sectors. It does not require a coherent superposition of different total particle numbers.

This distinction matters when a particle-number superselection rule applies. Grand-canonical fluctuations are classical probabilities across sectors combined with quantum mixtures within sectors.

Symmetry-breaking descriptions can introduce states with indefinite number and a well-defined phase. Their relation to number-conserving finite systems and thermodynamic limits requires additional care.

Grand Potential

Define the grand potential

$$\Omega_{\mathrm G} = -k_{\mathrm B}T\ln\Xi.$$

Its equilibrium Legendre form is

$$\Omega_{\mathrm G} = U-TS-\mu\overline N,$$

where

$$\overline N \equiv \langle N\rangle.$$

For a simple homogeneous system,

$$d\Omega_{\mathrm G} = -S\,dT - P\,dV - \overline N\,d\mu.$$

Therefore

$$\begin{aligned} S &= -\left( \frac{\partial\Omega_{\mathrm G}}{\partial T} \right)_{V,\mu}, \\ P &= -\left( \frac{\partial\Omega_{\mathrm G}}{\partial V} \right)_{T,\mu}, \\ \overline N &= -\left( \frac{\partial\Omega_{\mathrm G}}{\partial\mu} \right)_{T,V}. \end{aligned}$$

In a homogeneous thermodynamic limit,

$$\Omega_{\mathrm G} = -PV.$$

This identity can receive surface and finite-size corrections and should not be imposed on an arbitrary trapped or inhomogeneous finite system.

Mean Particle Number

Differentiate at fixed $\beta$:

$$\frac{\partial}{\partial\mu} \ln\Xi = \beta\langle N\rangle.$$

Hence

$$\overline N = \frac{1}{\beta} \left( \frac{\partial\ln\Xi}{\partial\mu} \right)_{\beta,V}.$$

Using fugacity,

$$\overline N = z \frac{\partial}{\partial z} \ln\Xi.$$

The mean is generally not an integer, even though every number measurement returns an integer eigenvalue.

One chooses $\mu$ so that $\overline N$ has the desired value, often matching a fixed density in the thermodynamic limit.

Internal Energy and the β Derivative

At fixed $\mu$,

$$-\frac{\partial}{\partial\beta} \ln\Xi = \langle H-\mu N\rangle.$$

Therefore

$$U = -\left( \frac{\partial\ln\Xi}{\partial\beta} \right)_{\mu,V} + \mu\overline N.$$

The derivative produces the expectation of $K=H-\mu N$, not $H$ alone. Omitting the $\mu\overline N$ term is a common error.

An alternative parameterization uses

$$\alpha \equiv -\beta\mu.$$

Derivatives at fixed $\alpha$ and fixed $\mu$ are different. Every formula should state which variable is held fixed.

Entropy

Since

$$\ln\rho_{\beta,\mu} = -\beta(H-\mu N) - \ln\Xi,$$

the entropy is

$$S = k_{\mathrm B} \left[ \ln\Xi + \beta \left( U-\mu\overline N \right) \right].$$

Rearranging gives

$$\Omega_{\mathrm G} = U-TS-\mu\overline N.$$

This is the grand-canonical counterpart of $F=U-TS$.

Number Fluctuations

A second chemical-potential derivative gives

$$\frac{\partial^2}{\partial\mu^2} \ln\Xi = \beta^2 \left( \langle N^2\rangle - \langle N\rangle^2 \right).$$

Therefore

$$\operatorname{Var}(N) = \frac{1}{\beta} \left( \frac{\partial\overline N}{\partial\mu} \right)_{T,V}.$$

Equivalently,

$$\left( \frac{\partial\overline N}{\partial\mu} \right)_{T,V} = \beta\operatorname{Var}(N) \geq 0.$$

The average particle number is nondecreasing with chemical potential under the usual assumptions.

For an ordinary extensive phase,

$$\overline N \sim V, \qquad \operatorname{Var}(N) \sim V,$$

so

$$\frac{ \sqrt{\operatorname{Var}(N)} }{ \overline N } \sim V^{-1/2}.$$

Relative number fluctuations vanish in the bulk even though absolute fluctuations grow.

Compressibility

Let

$$n = \frac{\overline N}{V}.$$

For a homogeneous system, the isothermal compressibility can be written

$$\kappa_T = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_T.$$

Using the fluctuation identity,

$$\kappa_T = \frac{ \beta\operatorname{Var}(N) }{ Vn^2 } = \frac{ \beta V\operatorname{Var}(N) }{ \overline N^2 }.$$

A divergent compressibility is associated with enhanced number fluctuations, but finite-size, phase-separation, stability, and ensemble qualifications must be checked.

Fluctuations of the Grand Hamiltonian

At fixed $\mu$,

$$\frac{\partial^2}{\partial\beta^2} \ln\Xi = \operatorname{Var}(H-\mu N).$$

Energy and number fluctuations are generally correlated:

$$\operatorname{Cov}(H,N) = \langle HN\rangle - \langle H\rangle\langle N\rangle.$$

Thus $\operatorname{Var}(H-\mu N)$ contains energy variance, number variance, and their covariance:

$$\begin{aligned} \operatorname{Var}(H-\mu N) ={}& \operatorname{Var}(H) \\ &+ \mu^2\operatorname{Var}(N) \\ &- 2\mu\operatorname{Cov}(H,N). \end{aligned}$$

One should not identify this directly with $\operatorname{Var}(H)$.

Relation to the Canonical Ensemble

Using

$$Z_N = e^{-\beta F_N},$$

the grand partition function becomes

$$\Xi = \sum_N \exp \left[ -\beta \left( F_N-\mu N \right) \right].$$

For a large system, the dominant sector approximately minimizes

$$F_N-\mu N.$$

The stationarity condition is

$$\left( \frac{\partial F}{\partial N} \right)_{T,V} = \mu.$$

This is the Legendre relation between Helmholtz free energy and grand potential.

At finite size, the grand-canonical state remains a distribution over integer $N$. Canonical and grand-canonical predictions can differ visibly when the mean population is small or when addition energies are large.

Recovering Fixed-N Information

Because

$$\Xi(z) = \sum_{N=0}^{\infty} z^N Z_N,$$

the canonical partition function $Z_N$ is the coefficient of $z^N$.

Formally,

$$Z_N = \frac{1}{2\pi i} \oint \frac{ \Xi(z) }{ z^{N+1} } \,dz,$$

where the contour encloses the origin inside a domain of analyticity.

In large systems, the contour integral can be estimated by a saddle point. The saddle chemical potential is chosen so that the grand-canonical mean equals the target $N$.

This coefficient extraction makes precise that the ensembles are related transforms, not interchangeable definitions.

Independent Modes

For noninteracting modes,

$$H = \sum_i \epsilon_i n_i, \qquad N = \sum_i n_i,$$

so

$$H-\mu N = \sum_i (\epsilon_i-\mu)n_i.$$

The grand partition function factorizes:

$$\Xi = \prod_i \xi_i.$$

The one-mode factor depends on the allowed occupations.

Statistics	Allowed $n_i$	One-mode factor $\xi_i$
boson	$0,1,2,\ldots$	$[1-e^{-\beta(\epsilon_i-\mu)}]^{-1}$
fermion	$0,1$	$1+e^{-\beta(\epsilon_i-\mu)}$

The corresponding mean occupations are

$$\begin{aligned} \overline n_i^{\mathrm B} &= \frac{1}{ e^{\beta(\epsilon_i-\mu)}-1 }, \\ \overline n_i^{\mathrm F} &= \frac{1}{ e^{\beta(\epsilon_i-\mu)}+1 }. \end{aligned}$$

These formulas are previews. Their derivations, limits, and gas thermodynamics belong in the Bose–Einstein and Fermi–Dirac statistics pages.

Single Fermionic Mode

For one fermionic mode of energy $\epsilon$,

$$n\in\{0,1\}.$$

The grand partition function is

$$\Xi_{\mathrm F} = 1+e^{-\beta(\epsilon-\mu)}.$$

The occupation probability is

$$\overline n = \frac{ e^{-\beta(\epsilon-\mu)} }{ 1+e^{-\beta(\epsilon-\mu)} } = \frac{1}{ e^{\beta(\epsilon-\mu)}+1 }.$$

Since $n^2=n$,

$$\operatorname{Var}(n) = \overline n(1-\overline n).$$

At $\epsilon=\mu$, the mode is half occupied for every finite temperature.

Single Bosonic Mode

For one bosonic mode,

$$n\in\{0,1,2,\ldots\}.$$

If $\mu<\epsilon$,

$$\begin{aligned} \Xi_{\mathrm B} &= \sum_{n=0}^{\infty} e^{-\beta(\epsilon-\mu)n} \\ &= \frac{1}{ 1-e^{-\beta(\epsilon-\mu)} }. \end{aligned}$$

The mean occupation is

$$\overline n = \frac{1}{ e^{\beta(\epsilon-\mu)}-1 }.$$

The variance is

$$\operatorname{Var}(n) = \overline n \left( 1+\overline n \right).$$

As $\mu\to\epsilon^-$, both the mean and variance diverge for this ideal mode. In an ideal Bose gas, the lowest mode then requires explicit condensate treatment while excited-state capacity depends on dimension and density of states.

Chemical Potential Is Not a One-Particle Energy

Thermodynamically,

$$\mu = \left( \frac{\partial F}{\partial N} \right)_{T,V}$$

in the appropriate large-system description. It measures the free-energy cost of changing particle number, not necessarily the energy of a literal one-particle eigenstate.

In interacting systems, $\mu$ includes interaction and entropy effects. In finite systems, particle addition and removal energies can differ. In a band insulator, $\mu$ can lie in a gap where there is no one-particle state.

Chemical potentials equalize between subsystems that can exchange the corresponding conserved charge at equilibrium.

Energy-Zero and Chemical-Potential Conventions

Suppose every one-particle energy is shifted by a constant $C$. For a number-conserving many-particle Hamiltonian,

$$H' = H+CN.$$

If simultaneously

$$\mu' = \mu+C,$$

then

$$H'-\mu' N = H-\mu N.$$

The grand-canonical state and $\Xi$ are unchanged.

This joint shift differs from adding one global constant $C\mathbb I$ to the entire many-body Hamiltonian, which multiplies $\Xi$ by $e^{-\beta C}$ but leaves the normalized state unchanged.

Quoting $\mu$ without an energy-zero convention can therefore be misleading.

Chemical Potential gives the unified derivative, finite-sector, Lagrange-multiplier, fermionic, bosonic, electrochemical, and sign-convention account. This page retains the ensemble construction and number-sector statistics.

Systems with μ = 0

If particle number is not conserved and the system can freely create or destroy excitations in equilibrium, the associated chemical potential is typically zero.

Examples often include equilibrium photons and phonons:

$$\mu_\gamma=0, \qquad \mu_{\mathrm{phonon}}=0.$$

This statement has qualifications. Driven photon gases, approximately conserved quasiparticle numbers, exciton populations, and nonequilibrium steady states can admit effective chemical potentials. One must identify the conserved or slowly relaxing quantity and the preparation protocol.

Open-System Interpretation

A particle reservoir can exchange quanta with the system. Under weak coupling and suitable detailed-balance conditions, an open-system generator may have

$$\rho_{\beta,\mu} \propto e^{-\beta(H-\mu N)}$$

as a stationary state.

The grand-canonical ensemble defines the equilibrium target. It does not by itself derive tunneling rates, master equations, currents, or relaxation times.

At strong coupling, the reduced equilibrium state can differ from the bare grand-canonical form. Nonequilibrium leads with different $(T_\alpha,\mu_\alpha)$ generate currents and do not define one global equilibrium ensemble.

Finite Systems and the Thermodynamic Limit

For a finite system, $\overline N$ can be noninteger and number fluctuations can be large relative to the mean. This is not a contradiction because $\overline N$ is an ensemble average.

For ordinary bulk systems,

$$\frac{ \sqrt{\operatorname{Var}(N)} }{ \overline N } \longrightarrow 0.$$

Canonical and grand-canonical local observables can then agree when $\mu$ is chosen to match the density.

Exceptions and caveats include:

• small systems;
• phase coexistence;
• long-range nonadditive interactions;
• constrained sectors;
• condensate fluctuations;
• observables explicitly sensitive to global $N$;
• mesoscopic charging energies.

Ensemble equivalence must be demonstrated for the regime and observable at hand.

Canonical Boundaries

This page owns:

• equilibrium at fixed $T,\mu,V$;
• the Fock-space trace and $\Xi=\sum_N z^N Z_N$;
• fugacity and convergence;
• the grand potential and its derivatives;
• mean number, number fluctuations, and compressibility;
• the relation to fixed-$N$ canonical ensembles;
• chemical-potential conventions and conservation caveats;
• independent-mode factorization as a bridge to quantum statistics.

Other pages own:

• the general Gibbs-operator structure: Thermal Density Operators;
• trace factorization, number-generating functions, and coefficient extraction: Partition Functions;
• the full potential network, Legendre transforms, Euler relations, and stability: Thermodynamic Potentials;
• grand-canonical entropy in the broader entropy context: Entropy in Quantum Statistical Mechanics;
• the general energy-and-number constrained-entropy derivation: Maximum Entropy Principle;
• the cross-ensemble fluctuation–response dictionary and static-limit caveats: Fluctuations and Susceptibilities;
• the low-fugacity emergence of Maxwell–Boltzmann statistics: Classical Limit of Quantum Statistics;
• fixed-$N$ thermodynamics and Helmholtz free energy: Canonical Ensemble;
• occupation-number and Fock-space construction: Composite Systems and Entanglement;
• Bose–Einstein and Fermi–Dirac distributions in depth: Quantum Statistics and Ideal Gases;
• the model-specific use of $H-\mu N$ to organize Mott lobes: Bose–Hubbard Model;
• particle-exchange dynamics and transport: Open Systems and Quantum Matter;
• full finite-temperature field theory: QFT.org.

Common Mistakes

• Taking the trace over one fixed-$N$ sector and calling it grand canonical.
• Forgetting the chemical-potential term in the exponential.
• Using a chemical potential for a nonconserved quantity without justification.
• Omitting the convergence condition for bosonic mode sums.
• Treating $\overline N$ as an eigenvalue that must be an integer.
• Interpreting number fluctuations as coherent superpositions across superselection sectors.
• Writing $-\partial_\beta\ln\Xi=U$ at fixed $\mu$ and forgetting $\mu\overline N$.
• Failing to state whether $\mu$, $\beta\mu$, or fugacity is fixed during differentiation.
• Confusing $\Omega_{\mathrm G}$ with an energy-level degeneracy or state-counting symbol.
• Assuming $\Omega_{\mathrm G}=-PV$ for every finite inhomogeneous system.
• Calling $\mu$ the energy of one particle in an interacting many-body system.
• Forgetting the joint shift of one-particle energy zero and $\mu$.
• Setting every bosonic chemical potential to zero.
• Assuming canonical and grand-canonical fluctuations agree at finite size.
• Inferring reservoir dynamics from the equilibrium density operator alone.

Exercises

Generate canonical sectors

Starting from the block decomposition of Fock space, derive

$$\Xi = \sum_{N=0}^{\infty} z^N Z_N.$$

Solution

When $[H,N]=0$, both operators are block diagonal in fixed-$N$ sectors:

$$H = \bigoplus_N H_N, \qquad N = \bigoplus_N N\mathbb I_N.$$

Therefore

$$\begin{aligned} \Xi &= \operatorname{Tr}_{\mathcal F} e^{-\beta(H-\mu N)} \\ &= \sum_N \operatorname{Tr}_{\mathcal H_N} e^{-\beta(H_N-\mu N)} \\ &= \sum_N e^{\beta\mu N} Z_N \\ &= \sum_N z^N Z_N. \end{aligned}$$

Derive number fluctuations

Show that

$$\left( \frac{\partial\overline N}{\partial\mu} \right)_{T,V} = \beta\operatorname{Var}(N).$$

Solution

First,

$$\overline N = \frac{1}{\beta} \frac{\partial\ln\Xi}{\partial\mu}.$$

Differentiating again,

$$\begin{aligned} \frac{\partial^2\ln\Xi}{\partial\mu^2} &= \beta^2 \left( \langle N^2\rangle - \langle N\rangle^2 \right). \end{aligned}$$

Thus

$$\frac{\partial\overline N}{\partial\mu} = \frac{1}{\beta} \frac{\partial^2\ln\Xi}{\partial\mu^2} = \beta\operatorname{Var}(N).$$

Compare one bosonic and fermionic mode

For one mode of energy $\epsilon$, compute $\Xi$, $\overline n$, and $\operatorname{Var}(n)$ for bosons and fermions.

Solution

Let

$$q = e^{-\beta(\epsilon-\mu)}.$$

For fermions,

$$\Xi_{\mathrm F} = 1+q, \qquad \overline n_{\mathrm F} = \frac{q}{1+q},$$

and because $n^2=n$,

$$\operatorname{Var}_{\mathrm F}(n) = \overline n_{\mathrm F} \left( 1-\overline n_{\mathrm F} \right).$$

For bosons, convergence requires $q<1$:

$$\Xi_{\mathrm B} = \frac{1}{1-q}, \qquad \overline n_{\mathrm B} = \frac{q}{1-q}.$$

The geometric distribution gives

$$\operatorname{Var}_{\mathrm B}(n) = \overline n_{\mathrm B} \left( 1+\overline n_{\mathrm B} \right).$$

Shift all one-particle energies

Show that shifting $H\to H+CN$ leaves the grand-canonical state unchanged only if $\mu$ is shifted appropriately.

Solution

Choose

$$\mu' = \mu+C.$$

Then

$$\begin{aligned} H'-\mu'N &= H+CN-(\mu+C)N \\ &= H-\mu N. \end{aligned}$$

The exponential, grand partition function, and normalized state are unchanged.

If $\mu$ is not shifted, the relative weights of particle-number sectors acquire factors $e^{-\beta CN}$ and the physical density changes.

Compressibility from fluctuations

For a homogeneous system with density $n=\overline N/V$, derive

$$\kappa_T = \frac{ \beta V\operatorname{Var}(N) }{ \overline N^2 }.$$

Solution

Start from

$$\kappa_T = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_T.$$

At fixed volume,

$$\frac{\partial n}{\partial\mu} = \frac{1}{V} \frac{\partial\overline N}{\partial\mu} = \frac{\beta}{V} \operatorname{Var}(N).$$

Since $n^2=\overline N^2/V^2$,

$$\kappa_T = \frac{V^2}{\overline N^2} \frac{\beta}{V} \operatorname{Var}(N) = \frac{ \beta V\operatorname{Var}(N) }{ \overline N^2 }.$$

Cross-Links

• Statistical Ensembles Overview
• Quantum Statistics Overview
• Maxwell–Boltzmann Limit
• Thermal Density Operators
• Canonical Ensemble
• Partition Functions
• Thermodynamic Potentials
• Entropy in Quantum Statistical Mechanics
• Maximum Entropy Principle
• Chemical Potential
• Ensemble Equivalence
• Ensemble Formula Sheet
• Fock Space and Occupation Number
• Number Operators
• Bose–Einstein Statistics
• Bose–Einstein Distribution Formula Card
• Ideal Bose Gas
• Fermi–Dirac Statistics
• Ideal Fermi Gas
• Green Functions in Many-Body QM
• Fermi–Dirac Distribution Formula Card
• Baths and Reservoirs
• Mesoscopic Transport
• Thermodynamic Limit

References

• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
• K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987).
• M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
• L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Butterworth–Heinemann (1980).
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).