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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=WS being called the ‘universe’. We assume that the system’s volume VS is fixed, but otherwise it is allowed to exchange energy and particle number with W. Hence, the system’s energy ES and particle number NS will fluctuate. We ask what is the probability that S is in a state |n with energy En and particle number Nn. This is given by the ratio

Pn=lim

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}\ .


This page titled 4.5: Grand Canonical Ensemble (GCE) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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