Peeter Joot's (OLD) Blog.

Math, physics, perl, and programming obscurity.

open system variance of N

Posted by peeterjoot on March 16, 2013

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Question: Variance of N in open system ([1] pr 3.14)

Show that for an open system

\begin{aligned}\text{var}(N) = \frac{1}{{\beta}} \left({\partial {\bar{N}}}/{\partial {\mu}}\right)_{{V, T}}.\end{aligned} \hspace{\stretch{1}}(1.0.1)

Answer

In terms of the grand partition function, we find the (scaled) average number of particles

\begin{aligned}\frac{\partial {}}{\partial {\mu}} \ln Z_{\mathrm{G}} &= \frac{\partial {}}{\partial {\mu}} \ln \sum_{r,s} e^{\beta \mu N_r - \beta E_s} \\ &= \frac{1}{{Z_{\mathrm{G}}}} \sum_{r,s} \beta N_r e^{\beta \mu N_r - \beta E_s} \\ &= \beta \bar{N}.\end{aligned} \hspace{\stretch{1}}(1.0.2)

Our second derivative provides us a scaled variance

\begin{aligned}\frac{\partial^2 {{}}}{\partial {{\mu}}^2} \ln Z_{\mathrm{G}} &= \frac{\partial {}}{\partial {\mu}} \left( \frac{1}{{Z_{\mathrm{G}}}} \sum_{r,s} \beta N_r e^{\beta \mu N_r - \beta E_s}  \right) \\ &= \frac{1}{{Z_{\mathrm{G}}}} \sum_{r,s} (\beta N_r)^2 e^{\beta \mu N_r - \beta E_s}-\frac{1}{{Z_{\mathrm{G}}^2}} \left( \sum_{r,s} \beta N_r e^{\beta \mu N_r - \beta E_s} \right)^2 \\ &= \beta^2 \left( \bar{N^2} - {\bar{N}}^2  \right)\end{aligned} \hspace{\stretch{1}}(1.0.3)

Together this gives us the desired result

\begin{aligned}\text{var}(N) &= \frac{1}{{\beta^2}}\frac{\partial {}}{\partial {\mu}} \left( \beta \bar{N}  \right) \\ &= \frac{1}{{\beta}}\frac{\partial {\bar{N}}}{\partial {\mu}}.\end{aligned} \hspace{\stretch{1}}(1.0.4)

References

[1] E.A. Jackson. Equilibrium statistical mechanics. Dover Pubns, 2000.

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