G25.2651: Statistical Mechanics

Problem set #1

Due: Feb. 3, 2003

1.

A one-dimensional simple harmonic oscillator is described by a Hamiltonian

Thus, the phase space is 2-dimensional consisting only of the momentum p and the coordinate q.

a.

Sketch the constant energy curves in this two-dimensional space corresponding to the condition H=E for different values of E.

b.

Derive Hamilton's equations for this sytem.

c.

Solve the equations p(t) and q(t) subject to the general initial condition , .

d.

By explicitly substituting the solutions back into the expression for the Hamiltonian, show that energy is conserved, i.e., that

e.

Next, consider the change of variables:

where J and are called the action and angle variables, respectively. Derive the new harmonic oscillator Hamiltonian in terms of J and

f.

Sketch the constant energy curves in the J- phase space.

g.

Derive Hamilton's equations for these new variables and solve for the motion of the system in terms of J and , and explain how the phase space of part f is mapped onto the phase space of part a.

2.

Consider an ensemble of systems evolving according to a equation of motion:

a.

Show that the equation of motion is non-Hamiltonian and find the phase space metric.

b.

Suppose that we begin with a distribution at t=0 (that is, a distribution of initial conditions to the equation of motion) given by

Describe qualitatively how you would expect this distribution to evolve as .

c.

Solve the Liouville equation

for the distribution f(x,t) at any time t such that the solution f(x,t) satisfies the above initial condition. Plot for various values of t.

Hint: Try taking f(x,0) and multiplying the by an arbitrary function of t, say g(t) that is required to satisfy g(0)=1 and . Use the Liouville equation to derive an equation that g(t) must satisfy and solve this equation.

d.

Show that your solution is properly normalized in the sense that

3.

Consider an equilibrium ensemble, called the ``uniform'' ensemble, for which the partition function is the total number of microscopic states satisfying , where is the phase space vector. This is also given by

where is the step function, for y;SPMlt;0 and for y;SPMgt;0. It is claimed that the entropy defined by

is equivalent to that of the standard microcanonical ensemble, S(N,V,E) This means that the thermodynamics generated by and S(N,V,E) are the same in the thermodynamic limit. Explain why this is so.

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