Physics 219

Homework 5

Do problems: Reif 7.2, 7.3(optional), 7.5, 7.9, 7.10, 7.14, 7.19, 7.24(optional)

and Sethna 6.6, and 6.11

and:

1. Consider the simulation shown here There's no gravity, and initially the little balls are at rest. The big ball is being pushed by a spring. After a long time, the system is in thermal equilibrium. Calculate the average position and rms displacement of the big ball. Assume the number of balls is large. If you prefer, you can take the big ball to instead be a piston with a flat surface.

2. Consider a closed cylindrical container of length $L$ with a piston in the middle dividing it up into two chambers. The whole system has constant energy $E$ and energy can be transfered between the two chambers. There are $N$ particles in the left chamber and $N$ particles in the right chamber, with $N >> 1$. The total volume is $V$. Treat the particles as an ideal gas, and ignore the momentum of the piston in your calculation.

What is the probability of finding the piston at position $x$?

3.Statistical Mechanics of a Lock. Consider a combination lock with $N$ digits. Each digit takes on the values $0 \dots 9$. If the correct combination is found, the lock opens and the energy is $-N\epsilon ~ < ~0$. Otherwise the energy of the lock is $0$. The lock is kept at a temperature $T$.

(a) Calculate the partition function $Z(T)$, and the free energy.

(b) Calculate the energy, entropy and free energy per digit in the limit of large N. Calculate the specific heat.

(c) The digits are simultaneously flipped randomly with a frequency of $\omega$. Estimate the average time it takes to find the correct combination.