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.

Helpful background. Canonical ensembles supply the fixed-number states; Fock space combines their number sectors.

Particle exchange at fixed temperature and chemical potential

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 $\Omega_{\mathrm G}$

Assume $T>0$ here, a temperature-independent $H$, and a finite grand partition function with convergent derivatives. The ensemble describes equilibrium particle exchange; it can also approximate local properties of a large fixed-number system when ensemble equivalence applies.

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 noninteracting 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. The ideal-boson bound is not universal for interacting bosons: a repulsive on-site term $U n(n-1)/2$ with $U>0$ grows quadratically with $n$ and makes the one-site sum converge for every finite $\mu$ at $T>0$. The exercise below checks this explicitly.

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

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

With $H$ independent of temperature and chemical potential, and with other controls held fixed,

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

Therefore

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

For a homogeneous extensive bulk phase in the 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

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

At fixed $\alpha$ (equivalently fixed fugacity $z=e^{-\alpha}$),

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

This derivative changes $T$ while changing $\mu$ to keep $\beta\mu$ constant. It is a different path through the family of states from the fixed-$\mu$ derivative.

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:

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

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 at positive temperature, a dominant sector approximately minimizes

$$F_N-\mu N.$$

The stationarity condition is

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

This gives the thermodynamic Legendre relation when the chosen stationary point is a global minimum. If several sectors compete, the complete sum must be compared. At finite size, $Z_N$ can also be recovered as the coefficient of $z^N$ in $\Xi(z)$; see partition functions.

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.

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

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

Derivations and gas thermodynamics are developed in Bose–Einstein statistics and Fermi–Dirac statistics. Here the useful distinction is factorization: independent mode probabilities multiply only when no interaction or fixed-total-number constraint couples them.

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.

For addition/removal energies, electrochemical potentials, and nonconserved excitations, see chemical potential.

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.

Exercises

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.

Repulsion changes the bosonic convergence condition

One bosonic site has $H=\epsilon n+U n(n-1)/2$ with $U>0$. Show that its grand partition function converges for every finite real $\mu$ at $\beta>0$. Why does this not contradict the ideal-mode bound?

Solution

The $n$th term is $a_n=e^{-\beta[(\epsilon-\mu)n+Un(n-1)/2]}$. The ratio of consecutive terms is

$$\frac{a_{n+1}}{a_n}=e^{-\beta(\epsilon-\mu+Un)}\longrightarrow0.$$

Thus the positive series converges by the ratio test. At $U=0$ the ratio instead stays constant, so convergence requires $\mu<\epsilon$. The noninteracting bound cannot be applied to a repulsively interacting mode.

Conditioning independent modes

Two equal-energy fermionic modes are independent in a grand-canonical state. Condition on finding exactly one particle. Find $\langle n_1\rangle$, $\langle n_1n_2\rangle$, and the covariance in the conditioned state.

Solution

Equal energies give equal conditional weights to $\lvert10\rangle$ and $\lvert01\rangle$, irrespective of $\mu$. Thus each mean is $1/2$, $\langle n_1n_2\rangle=0$, and the covariance is $-1/4$. Conditioning enforces $n_1+n_2=1$ and destroys independent-mode factorization, even though the Hamiltonian has no interaction.

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• 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).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).