- #1

Silviu

- 624

- 11

## Homework Statement

A polymer chain consist of a large number N>>1 segments of length d each. The temperature of the system is T. The segments can freely rotate relative to each other. A force f is applied at the ends of the chain. Find the mean distance ##\textbf{r}## between the ends.

## Homework Equations

## The Attempt at a Solution

So when there was no force the result was 0. Now I am a bit confused. I wrote the partition function as: $$Z=(2 \pi \int_0^\pi d \theta sin (\theta) e^{\beta f d cos (\theta)})^N = 4 \pi \frac{sinh(\beta f d)}{\beta f d}$$. Now the probability of a given state is $$p_{state}=\frac{e^{\beta f d \sum_1^N cos(\theta_i)}}{Z}$$ and I am thinking to write $$<\textbf{r}> = \frac{(2 \pi \int_0^\pi f(\theta) d \theta sin (\theta) e^{\beta f d cos (\theta)})^N}{Z}$$. But I am not sure what this ##f(\theta)## should be. I want it to represent the vectorial distance between 2 neighboring points as a function of the angle between the direction of the force and d, but I am not sure how to write it. Should I do it separately for x, y and z and add them up or is there a way to write it directly? Also the solution I have uses: $$F=-NTlog(Z)$$ and then: $$L = -\frac{\partial F}{\partial f}$$ where L is the answer required. I am not sure I understand why. First, it looks like a scalar not a vector and why would the distance be given by that derivative? Thank you!