Rubber Band Boltzmann Statistics

[samjohnny]

Homework Statement

Homework Equations

$$ Z(1) = \sum_{i=1}^{} e^{\frac{E_i}{K_bT}} $$ where ##E_i## is each of the possible energy states available to a single link (in this case the right and the left states).

$$ P=\frac{\sum_{i=1}^{} e^{\frac{E_i}{K_bT}}}{Z} $$

The Attempt at a Solution

Hi all,

For part a) I obtained ## Z(1) = 2cosh(\frac{lF}{K_bT}) ## = the partition function for a single link.

For b) the probability of a single link to point to the right is: ## P=\frac{exp[\frac{lF}{K_bT}]}{2cosh(\frac{lF}{K_bT})} ##.

And for part c), the total partition function would be ##Z=[Z(1)]^N##, where ##Z(1)## is as given in the answer for part a).

Is this all correct thus far?

For part d) however I'm unsure on how to proceed. Any ideas?

 

[TSny]

Looks good so far. For (d), try to come up with an expression for ##\left< N_R \right>## based on your result for (b).

 

[samjohnny]

Looks good so far. For (d), try to come up with an expression for ##\left< N_R \right>## based on your result for (b).

Ah right, since we have the probability of a link pointing to the right, and we can yield the same for a link pointing to the left, we can simply compute the expectation/average value as ##\left< N_R \right> = NP_R## where ##P_R## is the result of part b). Doing the same for the ##\left< N_L \right>##, gives ##L=Nltanh(\frac{lF}{K_bT})## as required.

For part e), am I right in saying that as the temperature is increased (i.e. as the argument of tanh gets smaller), the total L decreases. And if we define the average energy as ##\left< E \right> = -LF##, then the average energy will also decrease as T increases. And since L decreases, there are more possible microstates of right/left pointing links, and so the entropy increases. Is this correct?

 

[TSny]

Ah right, since we have the probability of a link pointing to the right, and we can yield the same for a link pointing to the left, we can simply compute the expectation/average value as ##\left< N_R \right> = NP_R## where ##P_R## is the result of part b). Doing the same for the ##\left< N_L \right>##, gives ##L=Nltanh(\frac{lF}{K_bT})## as required.

Yes. Looks good.

For part e), am I right in saying that as the temperature is increased (i.e. as the argument of tanh gets smaller), the total L decreases.

Yes, heating the stretched rubber band tends to make the band contract.

And if we define the average energy as ##\left< E \right> = -LF##, then the average energy will also decrease as T increases.

##\left< E \right> = -LF## is a deduction rather than a definition. Are you sure the average energy decreases as T increases?

And since L decreases, there are more possible microstates of right/left pointing links, and so the entropy increases. Is this correct?

Yes.

 

[samjohnny]

##\left< E \right> = -LF## is a deduction rather than a definition. Are you sure the average energy decreases as T increases?

Ah right, where we use the fact that ##\left< E \right> = \left< N_R \right>E_R +\left< N_L \right>E_L ##. So, we've determined that L decreases as the stretched band is heated. So then ##LF## must also get smaller. However, the fact that there is a minus sign (##\left< E \right> = -LF##) indicate that the Energy is in fact increasing (i.e. becoming more positive). Is that it?

 

[TSny]

Yes, that's right. (Edit: Perhaps better to say that <E> becomes less negative.)