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A monatomic gas in a 2d Universe - multiplicity

  • Thread starter TFM
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  • #1
TFM
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Homework Statement



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:

[tex] \Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{\pi^N}{N!}(2MU)^N [/tex]

(b)

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

Homework Equations



N/A

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:

[tex] \Omega_{space} = (\frac{A}{(\Delta x)^2})^N [/tex]

The multiplicity of momentum:

[tex] \Omega_{mom} = (\frac{A_{hypercircle}}{(\Delta p_x)})^{2N} [/tex]

[tex] \Omega_{mom} = (\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})}) [/tex]

The total multiplicity is the multiple:

[tex] \Omega = \Omega_{space}\Omega_{mom} [/tex]

[tex] \Omega = ((\frac{A}{(\Delta x)^2})^N )(\frac{A_{hypercircle} ^{2N}}{(\Delta p_x^{2N})} ) [/tex]

This can be rearranged, and using Heisenberg's Uncertainty princeple, [tex] \Delta x \Delta p_x \approx h [/tex]

this gives:
[tex] \Omega = \frac{A^N A_{hypercircle}}{h^{2N}} [/tex]

For indistinguishable particles:

[tex] \Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}A_{hypercircle} [/tex]

Area of hypercircle:

[tex] \frac{2\pi^N}{(N - 1)!}{\sqrt{2mu}^{2N-1}} [/tex]

and using approximations:

[tex] \Omega = \frac{1}{N!}\frac{A^N}{h^{2N}}\frac{2\pi^N}{N!}{2mu^N} [/tex]

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

Any ideas how to get rid of it???

Thanks in advanced,

TFM
 

Answers and Replies

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

TFM
 

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  • #3
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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.
 

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