# Thermal Physics / Math question

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Hey guys,

I'm practicing for a thermo test later this week. We have this question in the book:

Zipper Problem: A zipper has N links; each links has a state in which it is closed with energy 0 or open with energy $\varepsilon$. We require, however, that the zipper can only unzip from the left end, and that the link number s can only open if all links to the left (1,2,...s-1) are already open.

(a) Show that the partition function can be summed in the form:

$Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}$

Okay, so I approaced the problem by defining the possible energies as a sum of epsilon(n) where it goes from 0, 1, 2, 3 etc.

Hence, my partition equation:

$Z=\sum \exp{(\frac{-n\varepsilon}{\tau})} = \sum (\exp{(\frac{-\varepsilon}{\tau})} )^n$

I have been trying to manipulate this further forever. I know it should look like something starting with 1 + exp(..) since with n=0, the exponent will be =1, but I couldn't see how to transform it.

Finally, I resorted to the answer sheet, and it says that the above is right, and therefore it's obviously $Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}$

Obviously? What am I missing? How did they get from the ^n to that?

meh. Help!

~mooey

p.s

I see in another source that they rewrote the summation as:

$\sum_{s=0}^{N} x^s = \frac{1-x^{N+1}}{1-x}$ where $x=\exp{(\frac{-\varepsilon}{\tau})}$

Great... that looks like an expansion of a power series, and I guess it makes sense if you have the instinct to translate it like the above. Assuming I got stuck with this for an hour, is there any other way to do this, or is this simple a "remember your power series expansion" problem... ?

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Hey guys,

I'm practicing for a thermo test later this week. We have this question in the book:

Zipper Problem: A zipper has N links; each links has a state in which it is closed with energy 0 or open with energy $\varepsilon$. We require, however, that the zipper can only unzip from the left end, and that the link number s can only open if all links to the left (1,2,...s-1) are already open.

(a) Show that the partition function can be summed in the form:

$Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}$

Okay, so I approaced the problem by defining the possible energies as a sum of epsilon(n) where it goes from 0, 1, 2, 3 etc.

Hence, my partition equation:

$Z=\sum \exp{(\frac{-n\varepsilon}{\tau})} = \sum (\exp{(\frac{-\varepsilon}{\tau})} )^n$

I have been trying to manipulate this further forever. I know it should look like something starting with 1 + exp(..) since with n=0, the exponent will be =1, but I couldn't see how to transform it.

Finally, I resorted to the answer sheet, and it says that the above is right, and therefore it's obviously $Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}$

Obviously? What am I missing? How did they get from the ^n to that?

meh. Help!

~mooey

p.s

I see in another source that they rewrote the summation as:

$\sum_{s=0}^{N} x^s = \frac{1-x^{N+1}}{1-x}$ where $x=\exp{(\frac{-\varepsilon}{\tau})}$

Great... that looks like an expansion of a power series, and I guess it makes sense if you have the instinct to translate it like the above. Assuming I got stuck with this for an hour, is there any other way to do this, or is this simple a "remember your power series expansion" problem... ?

$\sum_{s=0}^{N} x^s = 1 +x \sum_{s=0}^{N} x^s - x^{N+1}$

$(1-x) \sum_{s=0}^{N} x^s = 1-x^{N+1}$

$\sum_{s=0}^{N} x^s = \dfrac {1-x^{N+1}}{1-x}$

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Thanks!

Sorry to be really slow here, but I'm not sure I get how you got to this step:

$\sum_{s=0}^{N} x^s = 1 +x \sum_{s=0}^{N} x^s - x^{N+1}$

Shouldn't the sum in the right hand be s=1 to N now? I'm just trying to understand the steps. The 1+ came from n=0, right?

(Sorry, this is obviously "low level" series manipulation, but sadly, I've never really taken any real math classes in those, so it's really a lot of hard work and guessing and tryingt o figure it out on my own here. Sorry if the questions sound stupid.)

~mooey

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Thanks!

Sorry to be really slow here, but I'm not sure I get how you got to this step:

Shouldn't the sum in the right hand be s=1 to N now?

No, the "x" in front takes care of that, but adds an "N+1" term that is subtracted out in the last term.

I'm just trying to understand the steps. The 1+ came from n=0, right?

right

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I think your book assumed that you realize the term is the geometric series. Note that it should be $Z=\frac{1-e^{-(N+1)\varepsilon/\tau}}{1-e^{ -\varepsilon / \tau}}$, not $Z=\frac{1-e^{-(N+1)\varepsilon/\tau}}{1-e^{ +\varepsilon / \tau}}$ (i.e. that you forgot the minus sign in the denominator).

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Okay, I see it now, though there's no chance in hell I could've seen it earlier without really.. knowing this trick...

But it helps knowing at least why it's like that, so thanks Dr Rocket!

And timo, you're right, oops, I forgot the minus - but only online. It came out okay on paper. LaTeX is annoying.

In any case, thanks a lot for helping.. I was very frustrating over the fact that it seems I got the answer but not quite the answer, and.. I had no clue why.

I guess I should go over a bit of series expansions.... If you have any suggestions on what to start from, that would be great. Otherwise, I guess I'm going google hunting.

Thanks!

~mooey

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