# Particle number fluctuations

- Page ID
- 5211

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

\[\Delta N = \sqrt{\langle N^2 \rangle - \langle N \rangle^2}\]

\( \zeta{\partial \over \partial \zeta}\zeta {\partial \over \partial \zeta}\ln {\cal Z}(\zeta,V,T)\) | \( {1 \over {\cal Z}}\sum_{N=0}^{\infty}N^2 \zeta^N Q(N,V,T) -{1 \over {\cal Z}^2} \left[\sum_{N=0}^{\infty} N \zeta^N Q(N,V,T)\right]^2\) | ||

\( \langle N^2 \rangle - \langle N \rangle^2 \) |

Thus,

\[\left(\Delta N\right)^2 =\zeta{\partial \over \partial \zeta} \zeta {\partial \over \partial \zeta} \ln {\cal Z} (\zeta, V, T) = ({KT}^2){\partial^2 \over \partial \mu^2}\ln {\cal Z}(\mu,V,T) = kTV{\partial^2 P \over \partial \mu^2}\]

\[a(v,T) = {1 \over N}A(N,V,T)\]

The chemical potential is defined by

\(\mu\) | \( {\partial A \over \partial N} =a(v,T) + N{\partial a \over \partial v}{\partial v \over \partial N}\) | ||

\( a(v,T) - v{\partial a \over \partial v}\) |

Similarly, the pressure is given by

\[P = -{\partial A \over \partial V} = -N{\partial a \over \partial v}{\partial v \over \partial V} = -{\partial a \over \partial v}\]

\[ {\partial \mu \over \partial v} = -v{\partial^2 a \over \partial v^2}\]

\[ {\partial P \over \partial \mu} = {\partial P \over \partial v}{\partial v \over \partial \mu} = {\partial^2 a \over \partial v^2} \left[v{\partial^2 a \over \partial v^2}\right]^{-1} = {1 \over v}\]

\[{\partial^2 P \over \partial \mu^2} = {\partial \over \partial v}{\partial P \over \partial \mu}{\partial v \over \partial \mu} = {1 \over v^2} \left[ v {\partial^2 a \over \partial v^2}\right]^{-1}= -{1 \over v^3 \partial P/\partial v}\]

\[\kappa_T = -{1 \over V}{\partial V \over \partial P}=-{1 \over v \partial p/\partial v}\]

Thus,

\[{\partial^2 P \over \partial \mu^2} = {1 \over v^2}\kappa_T\]

and

\[\Delta N = \sqrt{\frac{\langle N \rangle kT \kappa_T}{v}}\]

and the relative fluctuation is given by

\[{\Delta N \over N} = {1 \over \langle N \rangle}\sqrt{\frac{\langle N \rangle kT \kappa_T}{v}} \sim {1 \over \sqrt {\langle N \rangle }}\rightarrow 0\;\;{as}\langle N \rangle \rightarrow \infty\]

Therefore, in the thermodynamic limit, the particle number fluctuations vanish, and the grand canonical ensemble is equivalent to the canonical ensemble.