In the grand canonical ensemble, the variance in particle number is \(\left(N^{2}\right\rangle-\langle N\rangle^{2}=1 / \beta(\partial(N\rangle / \partial \mu)_{T, .}\) (a) Compute \(\left\langle N^{2}\right\rangle-\langle N\rangle^{2}\) for a classical ideal gas. (b) Compute \(\left\langle N^{2}\right\rangle-\langle N\rangle^{2}\) for a Bose-Einstein ideal gas at fairly high temperature (keep corrections to the classical result to first order in the particle density). How and why is your result different from the classical ideal gas result? (c) Compute \(\left\langle N^{2}\right\rangle-(N)^{2}\) for a Fermi-Dirac ideal gas at a fairly high temperature (keep corrections to the classical result to first order in the particle density). How and why is your result different from the classical ideal gas result?

The classical ideal gas provides a simple model where we can understand basic thermodynamic properties without quantum effects. In this context, we assume that gas particles do not interact and obey the laws of classical mechanics. The average number of particles \( \langle N \rangle \) in a given volume \( V \) at temperature \( T \) and chemical potential \( \mu \) is given by:

\[ \langle N \rangle = \frac{V}{\lambda^3} e^{\beta \mu} \]

Here, \( \lambda \) represents the thermal wavelength, and \( \beta \) is the inverse temperature. When differentiating \( \langle N \rangle \) with respect to the chemical potential \( \mu \):

\[ \left( \frac{\partial \langle N \rangle}{\partial \mu} \right)_{T} = \langle N \rangle \beta \]

Substituting back into the variance formula:

\[ \langle N^2 \rangle - \langle N \rangle^2 = \langle N \rangle \]

This result reflects that, for a classical ideal gas, the variance in particle number corresponds directly to the average number of particles. This is because particles are independently distributed.

In a Bose-Einstein ideal gas, particles obey Bose-Einstein statistics, leading to interesting quantum effects not seen in classical gases. At high temperatures, we can approximate the number of particles with first-order corrections to the particle density \( n = \frac{N}{V} \):

\[ \langle N \rangle = \frac{V}{\lambda^3} e^{\beta \mu}(1 + n\lambda^3) \]

Here, the correction term \( (1 + n\lambda^3) \) accounts for quantum statistics. Differentiating:

\[ \left( \frac{\partial \langle N \rangle}{\partial \mu} \right)_{T} \approx \langle N \rangle \beta \left(1 + n\lambda^3\right) \]

Substituting into the variance formula:

\[ \langle N^2 \rangle - \langle N \rangle^2 = \langle N \rangle \left(1 + n\lambda^3\right) \]

The variance is greater than in the classical case, due to the positive correction term. This increase in variance reflects the bunching effect of bosons, unlike classical particles. At low temperatures, this feature can lead to Bose-Einstein condensation, where a macroscopic number of particles occupy the same quantum state.

Conversely, in a Fermi-Dirac ideal gas, particles adhere to the Pauli exclusion principle, meaning no two fermions can occupy the same quantum state. For a Fermi-Dirac gas at high temperature, we use a similar correction approach:

\[ \langle N \rangle = \frac{V}{\lambda^3} e^{\beta \mu}(1 - n\lambda^3) \]

The negative correction term \( (1 - n\lambda^3) \) takes into account the exclusion principle. Differentiating:

\[ \left( \frac{\partial \langle N \rangle}{\partial \mu} \right)_{T} \approx \langle N \rangle \beta \left(1 - n\lambda^3\right) \]

Substituting into the variance formula:

\[ \langle N^2 \rangle - \langle N \rangle^2 = \langle N \rangle \left(1 - n\lambda^3\right) \]

The variance here is lower than in the classical case. This decrease happens because of fermion properties that restrict the number of particles occupying the same state. As a result, the variance reflects this anti-bunching behavior, which is fundamental to Fermi-Dirac statistics. This is crucial for understanding properties of systems like electron gases in metals, where electron behavior is governed by these statistics.