Problem Set 3

Problem Set 3

Due Wednesday, March 4, by 4pm in the homework locker

Problem 1 [20 points total]
Consider 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 microcanonical ensemble, find the entropy of the system S(E, N) as a function of fixed total energy E and number N (Hint: it is useful to consider the numbers N₊ and N₋ of up and down objects).
(b) Using your result from (a) find the temperature T as a function of energy E and number N. Show that T will be negative if E>0.
(c) What happens if such a system (1) with T₁<0 comes into thermal contact with another such system (2) with T₂>0? Does T₁ increase or decrease? Does T₂ increase or decrease? In which direction does the heat flow?
Problem 2 [10 points total]
Consider the same system as in the previous problem. 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 the previous problem, 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 1, in the large N limit.
Problem 3 [10 points total]
In lecture we derived the canonical partition function Q_N(V,N) for the idea gas using a factorization method. One can also directly compute it by taking the Laplace transform of the microcanonical partition function Ω(E,V,N). Using our result from lecture,

Ω(E,V,N) = [ V
h³ (2πmE)^3/2 ]^N 1
[(3N/2) - 1]! N! (Δ/E)

compute directly the Laplace transform,

Q_N(T,V) = ∞
∫
0 dE
Δ Ω(E,V,N)e^-βE

and show that you get the same result for Q_N(T,V) as found in lecture using the factorization method.
Problem 4 [20 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) [5 pts] Compute the canonical partition function for this system.
(b) [5 pts] Show that this system obeys the usual ideal gas law, pV = Nk_BT.
(c) [5 pts] Show that the total average energy is, E = 3Nk_BT (and hence using (b) gives, E/V = 3p).
(d) [5 pts] Show that the ratio of specific heats is, C_p/C_V = 4/3.