Grand Canonical Ensemble

The grand canonical ensemble is a probability distribution in statistical mechanics describing a system that can exchange both energy and particles with a reservoir. Its control variables are temperature $T$ , volume $V$ , and chemical potential $\mu$ ; its energy and particle number fluctuate. It differs from the canonical ensemble, which fixes particle number while allowing energy exchange, and the microcanonical ensemble, which fixes both energy and particle number. (web.mit.edu)

Physical interpretation and historical origin

An ensemble is a statistical collection of possible microscopic states, not necessarily a collection of physical systems. In the grand canonical ensemble, these states can contain different numbers of particles. A representative example is a fixed subvolume of a much larger gas: particles cross its boundary, carrying energy with them, while the surrounding gas maintains the temperature and chemical potential. (damtp.cam.ac.uk)

The reservoir must be sufficiently large that exchanges do not appreciably change its intensive variables. The system and reservoir are in thermal and chemical equilibrium, with matching temperature and chemical potential. Fixing $\mu$ therefore does not fix the instantaneous particle number; instead, it determines the distribution of particle numbers. (damtp.cam.ac.uk)

The ensemble framework was systematically developed by J. Willard Gibbs in Elementary Principles in Statistical Mechanics, published in 1902. Its final chapter extends the statistical treatment to systems whose particle composition varies. (en.wikisource.org)

Probability law and partition function

Let $i$ label a microscopic state with energy $E_i$ and particle number $N_i$ . Its probability is

p_i=\frac{\exp[-\beta(E_i-\mu N_i)]}{\Xi}, \qquad \beta=\frac{1}{k_{\mathrm B}T},

where $k_{\mathrm B}$ is the Boltzmann constant. The normalization factor

\Xi(T,V,\mu) =\sum_i \exp[-\beta(E_i-\mu N_i)]

is the grand partition function, a form of partition function. The sum includes all allowed particle-number sectors. (damtp.cam.ac.uk)

Grouping states by particle number gives

\Xi(T,V,\mu) =\sum_{N=0}^{\infty}e^{\beta\mu N}Z_N(T,V),

where $Z_N$ is the canonical partition function for exactly $N$ particles. Thus, the grand ensemble combines canonical ensembles with different particle numbers. Their probabilities are

P(N)=\frac{e^{\beta\mu N}Z_N}{\Xi}.

The dimensionless parameter $z=e^{\beta\mu}$ is often called the fugacity parameter in statistical mechanics; it should be distinguished from the pressure-dimensional fugacity commonly used in chemical thermodynamics. (damtp.cam.ac.uk)

The probability law follows by counting reservoir states compatible with each system state. To first order, removing energy $E$ and particles $N$ from the reservoir changes its entropy by

\Delta S_{\mathrm R}\simeq-\frac{E}{T}+\frac{\mu N}{T}.

Because reservoir multiplicity is proportional to $\exp(S_{\mathrm R}/k_{\mathrm B})$ , this produces the weight $\exp[-\beta(E-\mu N)]$ . The derivation neglects appreciable reservoir depletion and interaction energy across the system–reservoir boundary. (damtp.cam.ac.uk)

Grand potential and thermodynamic properties

The associated thermodynamic potential is the grand potential,

\Omega=-k_{\mathrm B}T\ln\Xi.

Its equilibrium thermodynamic expression is

\Omega=U-TS-\mu\overline N,

where $U=\langle E\rangle$ is the mean internal energy and $\overline N=\langle N\rangle$ . It replaces particle number by chemical potential as an independent variable through a Legendre transform of the Helmholtz free energy. (damtp.cam.ac.uk)

For a simple system with one particle species and no additional work variables,

d\Omega=-S\,dT-p\,dV-\overline N\,d\mu.

Consequently,

S=-\left(\frac{\partial\Omega}{\partial T}\right)_{V,\mu}, \qquad p=-\left(\frac{\partial\Omega}{\partial V}\right)_{T,\mu}, \qquad \overline N=-\left(\frac{\partial\Omega}{\partial\mu}\right)_{T,V}.

For a homogeneous, extensive bulk system, $\Omega=-pV$ , connecting the partition function directly to pressure and the equation of state. Surface contributions or nonextensive behavior require a more general treatment. (damtp.cam.ac.uk)

Useful partition-function identities are

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

The second identity emphasizes that differentiation at fixed $\mu$ yields the mean of $E-\mu N$ , rather than energy alone. (damtp.cam.ac.uk)

Particle-number fluctuations

Particle-number fluctuations are part of the ensemble’s definition, not measurement error. Their variance satisfies

\operatorname{Var}(N) =\langle N^2\rangle-\overline N^{\,2} =\frac{1}{\beta^2} \left(\frac{\partial^2\ln\Xi}{\partial\mu^2}\right)_{T,V} =k_{\mathrm B}T \left(\frac{\partial\overline N}{\partial\mu}\right)_{T,V}.

Thus, spontaneous fluctuations are linked to the response of particle number to chemical potential. (damtp.cam.ac.uk)

A transparent example is a three-dimensional, noninteracting, classical ideal gas of particles without internal degeneracy. With thermal wavelength

\lambda_T=\frac{h}{\sqrt{2\pi m k_{\mathrm B}T}},

its partition functions are

Z_N=\frac{1}{N!}\left(\frac{V}{\lambda_T^3}\right)^N, \qquad \Xi=\exp\left(\frac{zV}{\lambda_T^3}\right).

It follows that

\overline N=\frac{zV}{\lambda_T^3}, \qquad pV=\overline N k_{\mathrm B}T,

and particle number has a Poisson distribution:

P(N)=e^{-\overline N}\frac{\overline N^{\,N}}{N!}.

Hence $\operatorname{Var}(N)=\overline N$ , and the relative root-mean-square fluctuation is $1/\sqrt{\overline N}$ . These results apply in the dilute classical regime, where quantum degeneracy is negligible. (damtp.cam.ac.uk)

Quantum formulation

In quantum mechanics, the ensemble is represented by a density operator

\hat\rho =\frac{e^{-\beta(\hat H-\mu\hat N)}}{\Xi}, \qquad \Xi=\operatorname{Tr} e^{-\beta(\hat H-\mu\hat N)}.

Here $\hat H$ is the Hamiltonian and $\hat N$ is the particle-number operator. For particle-conserving Hamiltonians, $[\hat H,\hat N]=0$ , allowing states to be labeled by both energy and particle number. (damtp.cam.ac.uk)

For noninteracting particles, the grand partition function factorizes over single-particle states $r$ , with energies $\epsilon_r$ . For fermions, the Pauli exclusion principle permits occupations zero or one:

\Xi_{\mathrm F} =\prod_r\left[1+e^{-\beta(\epsilon_r-\mu)}\right], \qquad \langle n_r\rangle =\frac{1}{e^{\beta(\epsilon_r-\mu)}+1}.

For bosons, unrestricted nonnegative occupations give

\Xi_{\mathrm B} =\prod_r\frac{1}{1-e^{-\beta(\epsilon_r-\mu)}}, \qquad \langle n_r\rangle =\frac{1}{e^{\beta(\epsilon_r-\mu)}-1}.

These are the occupation laws of Fermi–Dirac statistics and Bose–Einstein statistics. For a finite ideal Bose system, convergence requires $\mu$ below the lowest single-particle energy. (damtp.cam.ac.uk)

Even when an experimental gas has fixed particle number, grand canonical calculations can simplify its thermodynamics. The chemical potential is then chosen so that the calculated mean particle number matches the specified number; it is not an additional independently fixed experimental parameter. (damtp.cam.ac.uk)

Computational applications

Grand canonical Monte Carlo methods sample configurations using particle insertions, deletions, and displacement moves. Acceptance rules incorporate the chemical potential and particle-number factors so that sampling obeys detailed balance with respect to the grand canonical distribution. For uniform insertion of a simple classical particle in three dimensions, with equally probable insertion and deletion attempts,

P_{\mathrm{acc}}^{\mathrm{ins}} =\min\left[ 1,\frac{V}{(N+1)\lambda_T^3} e^{-\beta(\Delta U-\mu)} \right].

Here $\Delta U$ is the change in configurational energy. More elaborate molecular models require corresponding proposal and internal-state factors. (pages.nist.gov)

These methods are used to study adsorption in porous materials, where a solid exchanges molecules with an external gas reservoir. Grand canonical transition-matrix methods can determine the particle-number distribution and use it to calculate thermophysical properties and investigate capillary phase transitions. (nist.gov)

Ensemble equivalence and limitations

In the thermodynamic limit, different ensembles often yield the same bulk thermodynamic properties when their control variables are matched. Nevertheless, equivalence of mean properties does not imply identical fluctuation distributions. Exact conservation constraints can affect scaled variances even when mean densities agree at infinite volume. (arxiv.org)

For a strictly fixed total particle number, $\operatorname{Var}(N)=0$ ; the grand canonical ensemble generally gives a nonzero value. Likewise, imposing exact charge conservation creates correlations absent from a treatment that fixes only the mean charge through a chemical potential. Ensemble choice therefore matters when predicting fluctuation observables, especially in small systems or systems with exact conservation constraints. (arxiv.org)

The standard reservoir construction also assumes negligible changes to reservoir temperature and chemical potential, and sufficiently weak system–reservoir coupling for an additive energy description. When these assumptions fail, the simple grand canonical weight need not describe the subsystem without corrections. (damtp.cam.ac.uk)

References

40 Chapter 10 – Statistical Mechanics, Molecular Methods, and Applicationsweb.mit.edu
Elementary Principles in Statistical Mechanicsen.wikisource.org
TrialComputeAdd — FEASST documentationpages.nist.gov
Use of the Grand Canonical Transition-Matrix Monte Carlo Method to Model Gas Adsorption in Porous Materialsnist.gov
Fluctuations in the Canonical Ensemblearxiv.org