# A monatomic gas in a 2d Universe - multiplicity

1. Feb 2, 2009

### TFM

1. The problem statement, all variables and given/known data

Consider a monatomic ideal gas that lives in a two-dimensional universe (“flatland”), occupying an area A instead of a volume V.

(a)

By following the logic of the derivation for the three-dimensional case, show that the multiplicity of this gas can be written:

$$\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{\pi^N}{N!}(2MU)^N$$

(b)

Find an expression for the entropy of this two-dimensional gas. Express your result in terms of U, A and N.

2. Relevant equations

N/A

3. The attempt at a solution

I have practically finished part a, with one small exception. for the pi fraction, I have 2pi instead of pi???

Some of my working out:

The space multiplicity:

$$\Omega_{space} = (\frac{A}{(\Delta x)^2})^N$$

The multiplicity of momentum:

$$\Omega_{mom} = (\frac{A_{hypercircle}}{(\Delta p_x)})^{2N}$$

$$\Omega_{mom} = (\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})})$$

The total multiplicity is the multiple:

$$\Omega = \Omega_{space}\Omega_{mom}$$

$$\Omega = ((\frac{A}{(\Delta x)^2})^N )(\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})} )$$

This can be rearranged, and using Heisenberg's Uncertainty princeple, $$\Delta x \Delta p_x \approx h$$

this gives:
$$\Omega = \frac{A^N A_{hypercircle}}{h^{2N}}$$

For indistinguishable particles:

$$\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}A_{hypercircle}$$

Area of hypercircle:

$$\frac{2\pi^N}{(N - 1)!}{\sqrt{2mu}^{2N-1}}$$

and using approximations:

$$\Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{2\pi^N}{N!}{2mu^N}$$

As you can see, close, just that annoying 2...

Any ideas how to get rid of it???

TFM

2. Feb 4, 2009

### TFM

I have attached the entre workings out. any idewa how to get rid iof that pesky 2...?

TFM

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3. Oct 20, 2011

### Ambidext

Sorry for 2 year late reply. You're probably graduated by now. But according to Schroeder's book, for part (a),

you can simply throw away the 2 because the multiplicity is very large compared to the 2, and thus it won't really make a difference. He then apologizes below for sloppy working. But... oh well that's what I just read.