Finding Maximum Dispersion of Total Number (N)

[KFC]

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?

 

[KFC]

For fermi stat., the total number of particles is
$$N = \sum_r \overline{n}_r$$

so how to calculate $$\overline{N}$$ ?

 

[charng]

I would approach this question by first defining the terms and assumptions being made. The total number (N) refers to the total number of particles in a given system, and the dispersion refers to the variation or spread of this total number. The given statistics, such as Fermi-Dirac or Bose-Einstein, describe the distribution of particles among different energy states.

To find the maximum dispersion of the total number, we need to find the value of N that results in the highest possible value of \overline{(\Delta N)^2}. This can be achieved by finding the maximum value of \overline{(N-\overline{N})^2}.

One approach to finding this maximum could be through optimization techniques, such as maximizing the function \overline{(N-\overline{N})^2} with respect to N. This would involve taking the derivative of the function and setting it equal to zero, then solving for N. However, this approach may not be feasible or accurate for all systems.

Another approach could be to use statistical methods, such as Monte Carlo simulations, to generate a large number of samples and calculate the dispersion for each sample. The sample with the highest dispersion would then indicate the maximum dispersion of the total number.

Additionally, the specific system and its properties may also play a role in determining the maximum dispersion of the total number. For example, in a system with a fixed number of particles, the maximum dispersion would be reached when all particles are equally distributed among all energy states. However, in a system with varying particle numbers, the maximum dispersion may be influenced by factors such as temperature and energy levels.

In conclusion, finding the maximum dispersion of the total number in a system described by given statistics requires careful consideration of the system's properties and may involve mathematical or statistical approaches.