Probability of the polymer chain

1. Aug 3, 2017

Kelly Lin

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

2. Relevant equations
I want to check if I think it right!

3. The attempt at a solution
If
N=1: ← or → (2 configurations/ each length is l)
N=2: ← or → or ←← or ←→ or →← or →→
[COLOR=#black]------[/COLOR]→[COLOR=#black]----[/COLOR]←
(6 configurations/ folded polymer's length is l/2 andunfolded polymer's length is l)
Thus, the configuration written in the function of x:
$$2^{N} \hspace{1cm}\text{ for } x=l \\ (l/x)! \hspace{1cm}\textrm{ otherwise}$$

2. Aug 3, 2017

Staff: Mentor

If N is the number of links of length l in the chain and x is the total length in a given configuration, how many links are in one direction and how many are in the other direction? How many ways are there are selecting the combinations of links that do this? (This is like a coin flipping problem).

3. Aug 3, 2017

Kelly Lin

oh!So the length l doesn't mean the total length?

4. Aug 3, 2017

Staff: Mentor

NO, it's the length of each link.

5. Aug 3, 2017

6. Aug 3, 2017

Kelly Lin

Actually, I have another question. How come the second question asks about the total number of configurations? In my opinion, there are infinite configurations depend on unlimited x. Does the question ask about the average configuration number?
Thanks!

7. Aug 3, 2017

Staff: Mentor

For a specified x, there are not an infinite number of configurations (at least not in 1D).

8. Aug 4, 2017

Kelly Lin

But, I think the first question asks the total number of specific x. Or, can the polymer be folded? This really make the answers of two questions different.

9. Aug 4, 2017

Staff: Mentor

It seems to me the implication is that it is folded along a straight line.

10. Aug 6, 2017

Kelly Lin

Is it what the question means?

11. Aug 6, 2017

Staff: Mentor

I don't think so. You need to figure out for a chain of identical joined links that can be folded along the x axis. So, for x = L, and N= 2, there are zero configurations. Leave off the arrows. For x = 2L and N = 2, there is only one configuration. For x = L and N = 1, there is only one configuration.

12. Aug 6, 2017

Kelly Lin

But the question mentions that each segment can be orientated in positive or negative directions. Don't we consider the direction (arrow) in different cases?

13. Aug 6, 2017

Kelly Lin

Also, why can't N=2 chain be folded? Thanks!!

14. Aug 6, 2017

Staff: Mentor

No. It's just if there is a fold or not at each jumction. The focus should be in the junctions, not the links.

15. Aug 6, 2017

Staff: Mentor

Yes If N=2, you can have x=1 with a fold.

16. Aug 6, 2017

Kelly Lin

So you mean if N=2 there are 3 configurations?

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17. Aug 6, 2017

Staff: Mentor

If N is 2 and x is 2, then there is 1 configuration. If N is 2 and x is 1, I would have to decide whether there are 2 configurations or 1. My inclination would be to count it as 1 configuration.

18. Aug 12, 2017

Kelly Lin

I think this is more viable!

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19. Aug 12, 2017

Kelly Lin

For the entropy in the system,
since
$$S=-k<lnP_{r}>=-k\sum_{r}{P_{r}lnP_{r}}$$
we get
$$S\approx -k\int{P(x)lnP(x)}dx=...=(-\frac{k}{2})(1-ln(\frac{2}{\pi N}))\approx -\frac{k}{2} \\ A=-\int S dT = \frac{1}{2}kT \\ U=A+TS=\frac{1}{2}kT-\frac{1}{2}kT=0$$
*A=free enegy; U=internal energy
So weird that U=0. Am I doing this wrong?? Thank you!!

20. Aug 12, 2017

haruspex

Suppose r of the N are oriented one way and the remaining N-r the other way. What would the total length be?

By the way, the expression on the right in b) cannot correct. The argument to exp() must be dimensionless, but x is a length. I would rewrite it by replacing x by x/l everywhere.