Thermal Physics / Math question

[mooeypoo]

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 [math]\varepsilon[/math]. 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:

[math]Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}[/math]

 

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:

 

[math]Z=\sum \exp{(\frac{-n\varepsilon}{\tau})} = \sum (\exp{(\frac{-\varepsilon}{\tau})} )^n[/math]

 

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 [math]Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}[/math]

 

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:

 

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

 

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... ?

[DrRocket]

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 [math]\varepsilon[/math]. 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:

[math]Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}[/math]

 

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:

 

[math]Z=\sum \exp{(\frac{-n\varepsilon}{\tau})} = \sum (\exp{(\frac{-\varepsilon}{\tau})} )^n[/math]

 

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 [math]Z=\frac{1-\exp{(\frac{-(N+1)\varepsilon}{\tau})}}{1-exp{(\frac{\varepsilon}{\tau})}}[/math]

 

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:

 

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

 

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... ?

 

 

[math]\sum_{s=0}^{N} x^s = 1 +x \sum_{s=0}^{N} x^s - x^{N+1}[/math]

 

[math] (1-x) \sum_{s=0}^{N} x^s = 1-x^{N+1}[/math]

 

[math]\sum_{s=0}^{N} x^s = \dfrac {1-x^{N+1}}{1-x}[/math]

[mooeypoo]

Thanks!

 

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

[math]\sum_{s=0}^{N} x^s = 1 +x \sum_{s=0}^{N} x^s - x^{N+1}[/math]

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.)

 

And thanks for your time!

 

~mooey

 

 

[DrRocket]

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

[timo]

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

[mooeypoo]

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