Problem Set 4

Problem Set 4

Due Monday, March 4, in lecture

Problem 1 [10 points total]
Consider the same situation as in problem 3 of the last Problem Set, i.e. a system of N distinguishable non-interacting objects, each of which can be in one of two possible states, "up" and "down", with energies +ε and -ε. Assume that N is large.
(a) Working in the canonical ensemble, find the Helmholtz free energy A(T, N) as a function of temperature T and number N.
[Note: Having found Ω(E,N) in problem 3 of HW 3, you could compute the canoncial Q_N(T) by taking the Laplace transform of Ω(E,N) with respect to E. Don't do it this way! Instead, compute Q_N(T) by directly summing the Boltzmann factor over all states in the phase space.
(b) Starting from A(T, N) of part (a), find the canonical entropy and express it as a function of the average energy E and number N. Show that your result agrees with your answer for the entropy in the microcanonical ensemble, as computed in problem 3, in the large N limit.
Problem 2 [15 points total]
Consider a classical gas of N indistinguishable non-interacting particles with ultrarelativistic energies, i.e. their kinetic energy - momentum relation is given by ε = pc, with c the speed of light and p the magnitude of the particle's momentum. The gas is confined to a box of volume V.
(a) Compute the canonical partition function for this system. [5 points]
(b) Show that this system obeys the usual ideal gas law, pV = Nk_BT. [5 points]
(c) Show that the total average energy is, E = 3Nk_BT (and hence using (b) gives, E/V = 3p). [5 points]
(d) Show that the ratio of specific heats is, C_p/C_V = 4/3. [5 points]
Problem 3 [15 points total]
Consider a classical ideal gas of indistinguishable, non-interacting, non-relativistic, particles confined to a region of three dimensional space by a harmonic potential V(r) rather than the walls of a box. This might be a model for a gas of atoms in a magnetic trap (we will talk more about this later in the semester). The single particle Hamiltonian is then,

H⁽¹⁾(r, p) = |p|²
2m + 1
2 mω_o²|r|²

Working in the canonical ensemble for a gas of N particles,
a) Compute the average energy E of the gas as a function of temperature T.
b) Compute the density of particles n(r) as a function of the radial distance r=|r| from the origin. n(r) should be normalized so that ∫d³r n(r) = N.
c) What is the average radial distance <r> of particles from the origin?
d) What is the pressure of the gas p(r) as a function of the radial distance r from the orign? To do this part, it might help to think of only the fraction of the gas that is within a spherical shell between radii r and r+Δr, for small Δr. Treat this shell as your system of interest, and find the pressure as a function of the density of the gas in this shell.
Problem 4 [20 points total]
In lecture we discussed the canonical ensemble, in which the temperature T, volume V, and number of particle N of a system are fixed, while the energy E is allowed to fluctuate. Suppose now that you wish to describe a system in which the temperature T, number of particles N, and pressure p are fixed, while the volume V is allowed to fluctuate. This would describe a system in contact with a thermal and mechanical reservior, in which the wall separating the system and the reservior is heat conducting and moveable. We can call this case the constant pressure ensemble.
a) Define the appropriate partition function Z(T, p, N) of the system in this new constant pressure ensemble. [5 points]
b) If you defined Z properly in part (a), then the Gibbs free energy should be given by
G(T, p, N) = -k_BT ln Z(T, p, N)
To demonstrate this, show that using G defined from Z as above, the average volume of the system is correctly given by,
<V> = (∂G/∂p)_T,N [5 points]
c) Derive a relation, in this constant pressure ensemble, between the isothermal compressibility κ_T and fluctuations in the volume V of the system. Show from this relation that the relative fluctuation in V vanishes in the thermodynamic limit. [5 points]
d) Consider an ideal gas of indistinguishable, non-relativistic, non-interacting, point particles of mass m. Explicitly compute the partition function Z(T,p,N) of this gas. Use Z to compute G(T,p,N), and then from G compute the specific heat at constant pressure, C_p. Show that you get the correct answer for the ideal gas. [5 points]