Phys 516 Statistical Mechanics

HW Set I (due Tuesday, 11am, Sept 25, 2012)

Problem 1

Consider N-independent 1D harmonic oscillators, each of which has a frequency omega.

1) Find out the average energy of the N-oscillator system at temperature T.

Compare your result with the equipartition theorem: At temperature T, the average energy of each quadratic degree of freedom should be ½ kT.

2) What is the amplitude of fluctuations in the total energies?

Problem 2

During one of our lectures, we had studied what happens to fermions on M sites. Here you will work on bosons on M sites, each has an energy epsilon. Again the M-site system is in contact with a reservoir at chemical potential mu.

1) Find the number of all possible states with N bosons on M sites. Note N can be either smaller or equal or bigger than M.

2) Find the most probable number of particles in this system at temperature T.

3) Calculate the average number of bosons and fluctuations ?