Physics 219

Homework 2

Read Reif Chapter 3 and do Reif problems 3.2, 3.4, 3.5.

Read Sethna Chapter 3, then do:

1. Consider the simulation shown here of two circular disks of mass $m_1$ and $m_2$ and radii $r_1$ and $r_2$ that move in a two dimensional square container of length $L$. Initially the energy of the first one is $E$ and the second is $0$. There is no dissipation and no rotational motion of the disks. Calculate

$$\lim_{T->\infty} {1\over T} \int_0^T \vert{\bf v_1}(t)\vert dt.$$ (1)

Here ${\bf v_1}(t)$ is the velocity vector of the first particle as a function of time.

2. Consider an ideal gas in isolation with total energy $E$ of $N$ identical particles with mass $m$. The walls of the container are very heavy so that internal momentum of the gas particles is not conserved.

(a) What is the time average $\langle\dots\rangle$ of $\bf v_i\cdot v_j$? That is, calculate $\bf\langle v_i \cdot v_j \rangle$. Do this for $i = j$ and $i \neq j$. Symmetry arguments give the answers in a couple of lines.

(b) Now consider a situation where momentum is conserved. The walls could be very light (massless) compared to the mass of gas inside as with a balloon and the balloon is in a vacuum. Adding in conservation of momentum, to restrict the region of allowable motion in phase space, what happens to your answers for part (a)?

3. A classical gas of dumbbells. A dumbbell consists of 2 equal masses $m$ separated by a distance $d$ that rotate freely in three dimensions. Consider a dilute gas of $N$ of these in a volume $V$ and assume that the interaction between them is negligible.

(a) By starting with the kinetic energy of a dumbbell, obtain its Lagrangian, in terms of angular variables. Then obtain the canonical momenta and from that the Hamiltonian. When you do this it is easier to ignore the translation part and just consider the rotational part. You then add in the translational center of mass kinetic energy later.

(b) From the Hamiltonian, calculate the heat capacity using the basic assumption relating entropy to the number of accessible states, Reif eqn (3.3.12).