4.5: Grand Canonical Ensemble (GCE)

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Grand canonical distribution and partition function

Consider once again the situation depicted in Figure [universe], where a system $S$ is in contact with a world $W$, their union $U=W\cup S$ being called the ‘universe’. We assume that the system’s volume $V\ns_{\ssr{S}}$ is fixed, but otherwise it is allowed to exchange energy and particle number with $W$. Hence, the system’s energy $E\ns_{\ssr{S}}$ and particle number $N\ns_{\ssr{S}}$ will fluctuate. We ask what is the probability that $S$ is in a state $\sket{n}$ with energy $E\ns_n$ and particle number $N\ns_n$. This is given by the ratio

$$\begin{split} P\ns_n&=\lim_{\RDelta E\to 0}\ {D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n\, ,\, N\ns_{\ssr{U}}-N\ns_n)\,\RDelta E\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}},N\ns_{\ssr{U}})\,\RDelta E}\\ &={\hbox{\# of states accessible to $W$ given that $E\ns_{\ssr{S}}=E\ns_n$ and $N\ns_{\ssr{S}}=N\ns_n$}\over \hbox{total \# of states in $U$}}\ .\bvph \end{split}$$

Then

$$\begin{split} \ln P\ns_n&=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}}-E\ns_n\,,\, N\ns_{\ssr{U}}-N\ns_n) - \ln D\ns_{\ssr{U}} (E\ns_{\ssr{U}},N\ns_{\ssr{U}})\\ &=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}},N\ns_{\ssr{U}})- \ln D\ns_{\ssr{U}}(E\ns_{\ssr{U}},N\ns_{\ssr{U}})\bvph\\ &\qquad\qquad - E\ns_n\>{\pz\ln D\ns_{\ssr{W}}(E,N)\over\pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop N=N\ns_{\ssr{U}}} - N\ns_n\>{\pz\ln D\ns_{\ssr{W}}(E,N)\over\pz N}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop N=N\ns_{\ssr{U}}} \!\! +\ \ldots\\ &\equiv -\alpha-\beta E\ns_n+\beta\mu N\ns_n\ . \end{split}$$

The constants $\beta$ and $\mu$ are given by

$$\begin{aligned} \beta&={\pz\ln D\ns_{\ssr{W}}(E,N)\over\pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop N=N\ns_{\ssr{U}}} = {1\over \kT}\\ \mu&=-\kT\ {\pz\ln D\ns_{\ssr{W}}(E,N)\over\pz N}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop N=N\ns_{\ssr{U}}}\ .\end{aligned}$$

The quantity $\mu$ has dimensions of energy and is called the chemical potential. Nota bene: Some texts define the ‘grand canonical Hamiltonian’ $\HK$ as

$$\HK\equiv\HH-\mu\HN\ .$$

Thus, $P\ns_n=e^{-\alpha}\,e^{-\beta ( E\ns_n-\mu N\ns_n) }$. Once again, the constant $\alpha$ is fixed by the requirement that $\sum_n P\ns_n=1$:

$$P\ns_n={1\over \Xi}\, e^{-\beta (E\ns_n-\mu N\ns_n)}\quad,\quad \Xi(\beta,V,\mu)=\sum_n e^{-\beta (E\ns_n-\mu N\ns_n)}=\Tra e^{-\beta (\HH-\mu\HN) }=\Tra e^{-\beta\HK}\ .$$

Thus, the quantum mechanical grand canonical density matrix is given by

$$\vrhhat={e^{-\beta\HK}\over\Tra e^{-\beta\HK}}\ .$$

Note that $\big[\vrhhat,\HK\big]=0$. The quantity $\Xi(T,V,\mu)$ is called the grand partition function. It stands in relation to a corresponding free energy in the usual way:

$$\Xi(T,V,\mu)\equiv e^{-\beta\Omega(T,V,\mu)}\qquad\Longleftrightarrow\qquad\Omega=-\kT\,\ln\Xi\ ,$$

where $\Omega(T,V,\mu)$ is the grand potential, also known as the Landau free energy. The dimensionless quantity $z\equiv e^{\beta\mu}$ is called the fugacity.

If $\big[\HH,\HN\big]=0$, the grand potential may be expressed as a sum over contributions from each $N$ sector, viz.

$$\Xi(T,V,\mu)=\sum_N e^{\beta\mu N}\,Z(T,V,N)\ .$$

When there is more than one species, we have several chemical potentials $\{\mu\ns_a\}$, and accordingly we define

$$\HK=\HH-\sum_a\mu\ns_a\,\HN\ns_a\ ,$$

with $\Xi=\Tra e^{-\beta\HK}$ as before.

Entropy and Gibbs-Duhem relation

In the GCE, the Boltzmann entropy is

$$\begin{split} S&=-\kB\sum_n P\ns_n\ln P\ns_n\\\ &=-\kB\sum_n P\ns_n\,\Big(\beta\Omega-\beta E\ns_n + \beta\mu N\ns_n\Big)\\ &=-{\Omega\over T} + {\langle \HH \rangle\over T} - {\mu\,\langle \HN \rangle\over T}\ , \end{split}$$

which says

$$\Omega=E-TS-\mu N\ ,$$

where

$$\begin{aligned} E&=\sum_n E\ns_n\,P\ns_n=\Tra\big(\vrhhat\,\HH\big)\\ N&=\sum_n N\ns_n\,P\ns_n=\Tra\big(\vrhhat\,\HN\big)\ .\end{aligned}$$

Therefore, $\Omega(T,V,\mu)$ is a double Legendre transform of $E(S,V,N)$, with

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

which entails

$$S=-\pabc{\Omega}{T}{V,\mu} \qquad,\qquad p=-\pabc{\Omega}{V}{T,\mu} \qquad,\qquad N=-\pabc{\Omega}{\mu}{T,V}\ .$$

Since $\Omega(T,V,\mu)$ is an extensive quantity, we must be able to write $\Omega=V\omega(T,\mu)$. We identify the function $\omega(T,\mu)$ as the negative of the pressure:

$$\begin{split} {\pz\Omega\over\pz V}&=-{\kT\over\Xi}\,\pabc{\Xi}{V}{T,\mu} ={1\over\Xi}\sum_n\,{\pz E\ns_n\over \pz V}\> e^{-\beta(E\ns_n-\mu N\ns_n)}\\ &=\pabc{E}{V}{T,\mu}=-p(T,\mu)\ . \end{split}$$

Therefore,

$$\Omega=-pV \qquad ,\qquad p=p(T,\mu)\quad\hbox{(equation of state)\ .}$$

This is consistent with the result from thermodynamics that $G=E-TS+pV=\mu N$. Taking the differential, we recover the Gibbs-Duhem relation,

$$d\Omega = -S\,dT - p\,dV - N\,d\mu =-p\,dV - V dp \quad \Rightarrow\quad S\,dT - V dp + N\,d\mu=0\ .$$

Generalized Susceptibilities in the GCE

We can appropriate the results from §5.8 and apply them, mutatis mutandis, to the GCE. Suppose we have a family of observables $\big\{\hat Q \ns_k\big\}$ satisfying $\big[{\hat Q}\ns_{k\ns}\,,\,{\hat Q}\ns_{k'}\big]=0$ and $\big[\HH\ns_0\,,\,{\hat Q}\ns_k\big]=0$ and $\big[\HN_a\,,\,{\hat Q}\ns_k\big]=0$ for all $k$, $k'$, and $a$. Then for the grand canonical Hamiltonian

$$\HK\ns(\Vlambda)=\HH\ns_0-\sum_a \mu\ns_a\,\HN\ns_a-\sum_k\lambda\ns_k\,{\hat Q}\ns_k\ ,$$

we have that

$$Q\ns_k({\Vlambda},T)=\langle\,{\hat Q}\ns_k\,\rangle=-\pabc{\Omega}{\lambda\ns_k}{T,\mu\ns_a,\,\lambda\ns_{k'\ne k}}$$

and we may define the matrix of generalized susceptibilities,

$$\xhi\ns_{kl}={1\over V}\,{\pz Q\ns_k\over\pz\lambda\ns_l}=-{1\over V}\,{\pz^2\Omega\over\pz\lambda\ns_k\,\pz\lambda\ns_l}\ .$$

Fluctuations in the GCE

Both energy and particle number fluctuate in the GCE. Let us compute the fluctuations in particle number. We have

$$N=\langle\,\HN\,\rangle={\Tra \HN\,e^{-\beta(\HH-\mu\HN)}\over \Tra e^{-\beta(\HH-\mu\HN)}}={1\over\beta}\,{\pz\over\pz\mu}\,\ln\Xi\ .$$

Therefore,

$$\begin{split} {1\over\beta} \,{\pz N\over\pz \mu}&={\Tra \HN^2\,e^{-\beta(\HH-\mu\HN)}\over \Tra e^{-\beta(\HH-\mu\HN)}}- \left({\Tra \HN\,e^{-\beta(\HH-\mu\HN)}\over \Tra e^{-\beta(\HH-\mu\HN)}}\right)^{\!\!2}\\ &=\blangle\HN^2\brangle - \blangle\HN\brangle^2\bvph\ . \end{split}$$

Note now that

$${\blangle\HN^2\brangle - \blangle\HN\brangle^2\over \blangle\HN\brangle^2}={\kT\over N^2}\,\pabc{N}{\mu}{T,V}={\kT\over V}\> \kappa\ns_T\ ,$$

where $\kappa\ns_T$ is the isothermal compressibility. Note:

$$\begin{split} \pabc{N}{\mu}{T,V}&={\pz(N,T,V)\over\pz(\mu,T,V)}=-{\pz(N,T,V)\over\pz(V,T,\mu)}\\ &=-{\pz(N,T,V)\over\pz(N,T,p)}\cdot{\pz(N,T,p)\over\pz(V,T,p)}\cdot\stackrel{1}{\overbrace{\pz(V,T,p)\over\pz(N,T,\mu)}} \cdot{\pz(N,T,\mu)\over\pz(V,T,\mu)}\bvph\\ &=-{N^2\over V^2}\pabc{V}{p}{T,N}={N^2\over V}\,\kappa\ns_T\ . \end{split}$$

Thus,

$${(\RDelta N)\ns_{\ssr{RMS}}\over N}=\sqrt{\kT\,\kappa\ns_T\over V}\ ,$$

which again scales as $V^{-1/2}$.

Gibbs ensemble

Let the system’s particle number $N$ be fixed, but let it exchange energy and volume with the world $W$. Mutatis mutandis, we have

$$P\ns_n=\lim_{\RDelta E\to 0}\ \lim_{\Delta V\to 0}\ {D\ns_{\ssr{W}}(E\ns_{\ssr{U}}-E\ns_n\, ,\, V\ns_{\ssr{U}}-V\ns_{\!n})\,\RDelta E\,\RDelta V\over D\ns_{\ssr{U}}(E\ns_{\ssr{U}},V\ns_{\ssr{U}})\,\RDelta E\,\RDelta V}.$$

Then

$$\begin{split} \ln P\ns_n&=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}}-E\ns_n\,,\, V\ns_{\ssr{U}}-V\ns_{\!n}) - \ln D\ns_{\ssr{U}} (E\ns_{\ssr{U}},V\ns_{\ssr{U}})\\ &=\ln D\ns_{\ssr{W}} (E\ns_{\ssr{U}},V\ns_{\ssr{U}})- \ln D\ns_{\ssr{U}}(E\ns_{\ssr{U}},V\ns_{\ssr{U}})\bvph\\ &\qquad - E\ns_n\>{\pz\ln D\ns_{\ssr{W}}(E,V)\over\pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop V=V\ns_{\ssr{U}}} - V\ns_{\!n}\>{\pz\ln D\ns_{\ssr{W}}(E,V)\over\pz V}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop V=V\ns_{\ssr{U}}} \!\! + \ldots\\ &\equiv -\alpha-\beta E\ns_n-\beta p\, V\ns_{\!n}\ . \end{split}$$

The constants $\beta$ and $p$ are given by

$$\begin{aligned} \beta&={\pz\ln D\ns_{\ssr{W}}(E,V)\over\pz E}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop V=V\ns_{\ssr{U}}} = {1\over \kT}\\ p&=\kT\ {\pz\ln D\ns_{\ssr{W}}(E,V)\over\pz V}\bigg|\nd_{E=E\ns_{\ssr{U}}\atop V=V\ns_{\ssr{U}}}\ .\end{aligned}$$

The corresponding partition function is

$$Y(T,p,N)=\Tra e^{-\beta(\HH+pV)}={1\over V\ns_0}\int\limits_0^\infty\!dV\,e^{-\beta pV}\,Z(T,V,N)\equiv e^{-\beta G(T,p,N)}\ ,$$

where $V\ns_0$ is a constant which has dimensions of volume. The factor $V_0^{-1}$ in front of the integral renders $Y$ dimensionless. Note that $G(V'_0)=G(V\ns_0)+\kT\ln(V'_0/V\ns_0)$, so the difference is not extensive and can be neglected in the thermodynamic limit. In other words, it doesn’t matter what constant we choose for $V\ns_0$ since it contributes subextensively to $G$. Moreover, in computing averages, the constant $V\ns_0$ divides out in the ratio of numerator and denominator. Like the Helmholtz free energy, the Gibbs free energy $G(T,p,N)$ is also a double Legendre transform of the energy $E(S,V,N)$, viz.

$$\begin{split} G&=E-TS+pV \\ dG &= -S\,dT + V dp + \mu\,dN\ , \end{split}$$

which entails

$$S=-\pabc{G}{T}{p,N} \qquad,\qquad V=+\pabc{G}{p}{T,N} \qquad,\qquad \mu=+\pabc{G}{N}{T,p}\ .$$