Finding Maximum Dispersion of Total Number (N)

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To find the maximum dispersion of the total number of particles (N) in a statistical mechanics context, one can start by calculating the average number of particles, \(\overline{N} = \sum_r \overline{n}_r\), where \(\overline{n}_r\) represents the average number of particles in state r. The dispersion of N can be expressed as \(\overline{(\Delta N)^2} = \overline{(N - \overline{N})^2}\). The maximum dispersion occurs when the distribution of particles across states is maximized, which may involve using the properties of Fermi-Dirac or Bose-Einstein statistics. Understanding the relationship between the average number of particles and their energy states is crucial for calculating this maximum dispersion accurately.
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For some given statistics (e.g. Fermi-Dirac or Bose-Einstein), once we know the average number of particles at state r, it is easy to calculate the dispersion by calculating

\overline{(\Delta n_r)^2} = -\frac{1}{\beta}\frac{\partial \bar{n}_r}{\partial \epsilon_r}

and the total number of particle is just the sum of all average number of particles.

My question is: how do you find maximum dispersion of TOTAL NUMBER (N)?

I know
\overline{(\Delta N)^2} = \overline{(N-\overline{N})^2}

but how do you find the maximum?
 
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For fermi stat., the total number of particles is
N = \sum_r \overline{n}_r

so how to calculate \overline{N} ?
 
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