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PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu ---- Spring 2018

## Problem Set 3

Due Monday, March 5, in lecture

• Problem 1 [10 points total]

In lecture we derived the canonical partition function QN(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) = [ Vh3 (2πmE)3/2 ]N 1             [(3N/2) - 1]!  N! (Δ/E)

compute directly the Laplace transform,

 QN(T,V) = ∞∫0 dEΔ Ω(E,V,N)e-βE

and show that you get the same result as found in lecture using the factorization method.

• Problem 2 [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 2, you could compute the canoncial QN(T) by taking the Laplace transform of Ω(E,N) with respect to E. Don't do it this way! Instead, compute QN(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 3 [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 = NkBT.

(c) [5 pts] Show that the total average energy is, E = 3NkBT (and hence using (b) gives, E/V = 3p).

(d) [5 pts] Show that the ratio of specific heats is, Cp/CV = 4/3.

• Problem 4 [20 points total]

Consider a three dimensional classical ideal gas of atoms of mass m, moving in a potential energy

 V(x) = { ½ αx2 +∞ x≥0    x<0

The infinite potential for x < 0 may be viewed as a rigid wall filling the y-z plane at x=0. The atoms, therefore, are attracted to this wall, but they move freely in the y and z directions. Let T be the temperature and n = N/A be the total number of atoms per unit area of the wall.

a) [5 pts] Calculate the local atomic density n(x) (number of atoms per unit volume) as a function of the distance x from the wall. Note that

 n= ∞∫0 n(x) dx

b) [5 pts] Show that the ideal gas law (relating pressure, temperature, and density) continues to hold locally everywhere.

c) [5 pts] Find the pressure that the atoms exert on the wall. How does this pressure vary with temperature?

d) [5 pts] Calculate the energy and specific heat per unit area of the wall.