Physics 219

Homework 6

Do Sethna problems: 7.2, 7.4, 7.11, 7.16 (optional and discussed in many textbooks)

Reif 8.1 (optional), 8.5, 8.6, 8.8 (optional), 8.19 (parts g and h are optional).

1. Consider the simulation shown here Using the canonical ensemble, calculate the variance of the endpoints. Assume the balls are very small so they only serve to equilibrate the system.

2. A one dimensional linear chain of masses connected together by springs is hanging vertically. The whole system is connected to a heat bath at temperature $T$. The Hamiltonian of the system is

$$E~=~ U+K ~=~ \sum_{i=1}^N [{1 \over 2} k (z_{i} - z_{i-1} )^2 +mgz_i + {p_i^2\over 2m}]$$

The 0th mass is held fixed at $z_0 = 0$. The system is treated classically.

(a) Write down the partition function for the system.

(b) What change of variables can you make to enable the integrals to be decoupled?

(c) Calculate the free energy and the specific heat.

(d) Calculate the average position of the ith mass. Note how these results depend on temperature.