The thermodynamic identify for a one-dimensional system is d = dU fdl When f is the external force exerted on the line and dl

The thermodynamic identify for a one-dimensional system is

τdσ = dU – fdl

When f is the external force exerted on the line and dl is the extension of line. By analog with (32) we form the derivative to find

–f /τ = (∂σ/∂I)0

The direction of the force is opposite to the conventional direction of the pressure.
We consider a polymeric chain of N links each of length ρ, with each link equally likely to be directed to the right and to left.
(a) Show that the number of arrangements that give a head-to-tail length of l = 2|s|ρ is

g(N, – s) + g(N,s) = 2N! / (½N + s)! (½N – s)!

(b) For |s| << N show that

σ(l) = log[2g(N, 0)] – l2/2Nρ2

(c) Show that the force at extension l is

f = lτ/Nρ2

The force is proportional to the temperature. The force arises because the polymer wants to curl up: the entropy is higher in a random coil than in an uncoiled configuration. Warming a rubber band makes it contract: warming a steel wire makes it expand.