Exercises in Statistical Mechanics
Based on course by Doron Cohen, has to be proofed
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
This exercises pool is intended for a graduate course in “statistical mechanics”. Some of the
problems are original, while other were assembled from various undocumented sources. In partic-
ular some problems originate from exams that were written by B. Horovitz (BGU), S. Fishman
(Technion), and D. Cohen (BGU).
====== [Exercise 2353]
Tension of a stretched chain
A rubber band is modeled as a single chain of N1 massless non-interacting links, each of fixed length a. Consider
a one-dimensional model where the links are restricted to point parallel or anti-parallel to a given axis, while the
endpoints are constraint to have a distance X= (2n−N)a, where nis an integer. Later you are requested to use
approximations that allow to regard Xas a continuous variable. Note that the body of the chain may extend beyond
the length X, only its endpoints are fixed. In items (c,d) a spring is pushed between the two endpoints, such that the
additional potential energy −KX 2favors large X, and the system is released (i.e. Xis free to fluctuate).
(a) Calculate the partition function Z(X). Write the exact combinatorial expression. Explain how and why it is
related trivially to the entropy S(X).
(b) Calculate the force f(X) that the chain applies on the endpoints. Use the Stirling approximation for the
derivatives of the factorials.
(c) Determine the temperature Tcbelow which the X= 0 equilibrium state becomes unstable.
(d) For T < Tcwrite an equation for the stable equilibrium distance X(T). Find an explicit solution by expanding
f(X) in leading order.