4.5: Grand Canonical Ensemble (GCE)
( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand\Dalpha
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dbeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dgamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Ddelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Depsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dvarepsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dzeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Deta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dtheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dvartheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Diota
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dkappa
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dlambda
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Dvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\DGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\DDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\DTheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vmu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vnu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vxi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vom
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vtau
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vupsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vchi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vpsi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\Vomega
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\VGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\VDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\newcommand\BI{\mib I}}
\)
\newcommand { M}
\newcommand { m}
}
\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)
1$#1_$
\newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\)
\newcommand\mutB{\tilde\mu}\ns_\ssr{B}
\( \newcommand\xhihOZ
Callstack: at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
\( \newcommand\labar
Callstack: at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.05:_Grand_Canonical_Ensemble_(GCE)), /content/body/p/span, line 1, column 23
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∪S being called the ‘universe’. We assume that the system’s volume V∗S is fixed, but otherwise it is allowed to exchange energy and particle number with W. Hence, the system’s energy E∗S and particle number N∗S will fluctuate. We ask what is the probability that S is in a state |n⟩ with energy E∗n and particle number N∗n. This is given by the ratio
P∗n=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}\ .