6.4: Particle Number Fluctuations
- Page ID
- 5211
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In the grand canonical ensemble, the particle number \(N\) is not constant. It is, therefore, instructive to calculate the fluctuation in this quantity. As usual, this is defined to be
\[\Delta N = \sqrt{\langle N^2 \rangle - \langle N \rangle^2} \nonumber \]
Note that
\[ \begin{align*} \zeta{\partial \over \partial \zeta}\zeta {\partial \over \partial \zeta}\ln {\cal Z}(\zeta,V,T) &= {1 \over {\cal Z}}\sum_{N=0}^{\infty}N^2 \zeta^N Q(N,V,T) -{1 \over {\cal Z}^2} \left[\sum_{N=0}^{\infty} N \zeta^N Q(N,V,T)\right]^2 \\[4pt] &= \langle N^2 \rangle - \langle N \rangle^2 \end{align*}\]
Thus,
\[ \begin{align*} \left(\Delta N\right)^2 &=\zeta{\partial \over \partial \zeta} \zeta {\partial \over \partial \zeta} \ln {\cal Z} (\zeta, V, T) \\[4pt] &= ({KT}^2){\partial^2 \over \partial \mu^2}\ln {\cal Z}(\mu,V,T) \\[4pt] &= kTV{\partial^2 P \over \partial \mu^2} \end{align*}\]
In order to calculate this derivative, it is useful to introduce the Helmholtz free energy per particle defined as follows:
\[a(v,T) = {1 \over N}A(N,V,T) \nonumber \]
where \(v={V \over N} = {1 \over \rho}\) is the volume per particle. The chemical potential is defined by
\[ \begin{align*} \mu &= {\partial A \over \partial N} \\[4pt] &=a(v,T) + N{\partial a \over \partial v}{\partial v \over \partial N} \\[4pt] &= a(v,T) - v{\partial a \over \partial v} \end{align*} \]
Similarly, the pressure is given by
\[P = -{\partial A \over \partial V} = -N{\partial a \over \partial v}{\partial v \over \partial V} = -{\partial a \over \partial v} \nonumber \]
Also,
\[ {\partial \mu \over \partial v} = -v{\partial^2 a \over \partial v^2} \nonumber \]
Therefore,
\[\begin{align*} \dfrac{\partial P}{\partial \mu} &= {\partial P \over \partial v}{\partial v \over \partial \mu} \\[4pt] &= {\partial^2 a \over \partial v^2} \left[v{\partial^2 a \over \partial v^2}\right]^{-1} \\[4pt] &= {1 \over v} \end{align*} \]
and
\[\begin{align*} \dfrac{\partial^2 P}{\partial \mu^2} &= {\partial \over \partial v}{\partial P \over \partial \mu}{\partial v \over \partial \mu} \\[4pt] &= {1 \over v^2} \left[ v {\partial^2 a \over \partial v^2}\right]^{-1} \\[4pt] &= -{1 \over v^3 \partial P/\partial v} \end{align*}\]
But recall the definition of the isothermal compressibility:
\[\kappa_T = -{1 \over V}{\partial V \over \partial P}=-{1 \over v \partial p/\partial v} \nonumber \]
Thus,
\[{\partial^2 P \over \partial \mu^2} = {1 \over v^2}\kappa_T \nonumber \]
and
\[\Delta N = \sqrt{\frac{\langle N \rangle kT \kappa_T}{v}} \nonumber \]
and the relative fluctuation is given by
\[{\Delta N \over N} = {1 \over \langle N \rangle}\sqrt{\frac{\langle N \rangle kT \kappa_T}{v}} \sim {1 \over \sqrt {\langle N \rangle }}\rightarrow 0\;\;{as}\langle N \rangle \rightarrow \infty \nonumber \]
Therefore, in the thermodynamic limit, the particle number fluctuations vanish, and the grand canonical ensemble is equivalent to the canonical ensemble.