Problem Set 4

Problem Set 4

There will be no Discussion Question this week.

Problems -- Due Thursday, February 17, by 5pm.

Upload your solutions to Blackboard at this link: PS4

Problem 1 [25 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 number of states Ω(E,N), and then the entropy of the system S(E, N), as a function of fixed total energy E and number N (Hint: it may be useful to consider the numbers N₊ and N₋ of up and down objects). Sketch S(E,N) as a function of E for fixed N, and show that it is not a monotonic increasing function of E. What is the key feature of this system that causes the dependence of S on E to be so qualitatively different from that of an ideal gas? [10 points]
(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. Sketch T vs E for fixed N. [7 points]
(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? [8 points]
Problem 2 [10 points total]
In Notes Eq. (2.4.14) we derived the number of states Ω(E,V,N) for the idea gas of N point particles, which can be written as given below,

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

Note, compared to Eq. (2.4.14) the above expression has the factor N! in the denominator, as needed for indistinguishable particles, as described in Notes 2-7.
By taking the Laplace transform of Ω(E,V,N), directly compute the canonical partition function, as in Notes Eq. (2.8.6),

Q_N(T,V) = ∞
∫
0 dE
ΔE Ω(E,V,N)e^-βE

Using the resulting canonical partition function, compute the Helmholtz free energy A(T,V,N) and compare it to what was found in Problem Set 1.