- #1

zhuang382

- 10

- 2

- Homework Statement
- Use Grand canonical partition function to find n particles in a small volume v of the ideal gas with total volume ##V >> v##. In the limit ##\overline{N} >> 1## and ##n >> 1##to show that ##P(n) ##is a Gaussian curve.

- Relevant Equations
- $$\lambda = e^{\mu/\tau}$$ (absolute activity)

$$P(n) = \frac{1}{\mathcal{Z}} \lambda ^n Exp(-E/\tau)$$

My attempt : $$P(n) = \frac{1}{\mathcal{Z}} Exp[(n\mu -E)/\tau]$$, use $$\lambda = e^{\mu/\tau}$$, then the distribution can be written as $$P(n) = \frac{1}{\mathcal{Z}} \lambda^nExp[-E/\tau]$$

Note that the average number of particle can be written as $$<N>= \lambda \partial \lambda ( log \mathcal{Z}) = \lambda Z_1$$, and $$\mathcal{Z} \approx e^{\lambda Z_1} \text{ when }N \to \infty$$ and $$Z_1 = n_Q V$$, which is the partition function of ideal gas of a single particle.

To kill of ##\lambda## and ##Z_1##, use the above expression for ##<N>##:

$$P(n) = \frac{1}{e^{\lambda Z_1}}\lambda^n exp(-E/\tau)= \lambda ^n e^{-<n>} exp(-E/\tau)$$

Use ##<n> = \lambda Z_1##, we get $$P(n) = (\frac{<n>}{Z_1})^n e^{-<N>}e^{-E/\tau}$$

This is what I got so far... The hint is to use Stirling Approximation, but I think I need a term something like ##N^Ne^{-N}##...

Note that the average number of particle can be written as $$<N>= \lambda \partial \lambda ( log \mathcal{Z}) = \lambda Z_1$$, and $$\mathcal{Z} \approx e^{\lambda Z_1} \text{ when }N \to \infty$$ and $$Z_1 = n_Q V$$, which is the partition function of ideal gas of a single particle.

To kill of ##\lambda## and ##Z_1##, use the above expression for ##<N>##:

$$P(n) = \frac{1}{e^{\lambda Z_1}}\lambda^n exp(-E/\tau)= \lambda ^n e^{-<n>} exp(-E/\tau)$$

Use ##<n> = \lambda Z_1##, we get $$P(n) = (\frac{<n>}{Z_1})^n e^{-<N>}e^{-E/\tau}$$

This is what I got so far... The hint is to use Stirling Approximation, but I think I need a term something like ##N^Ne^{-N}##...