Exercises in Statistical Mechanics

Exercises in Statistical Mechanics
Based on course by Doron Cohen, has to be proofed
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
This exercises pool is intended for a graduate course in “statistical mechanics”. Some of the
problems are original, while other were assembled from various undocumented sources. In particular some problems originate from exams that were written by B. Horovitz (BGU), S. Fishman
(Technion), and D. Cohen (BGU).
====== [Exercise 3240]
Bose gas in a uniform gravitational field
Consider an ideal Bose gas of particles of mass m in a uniform gravitational field of acceleration g.
(1) Show that the phenomenon of Bose-Einstein condensation in this gas sets in at a temperature Tc given by
s
#
"
8 1
πmgL
0
Tc ≈ Tc 1 +
9 ζ(3/2)
kTc0
where L is the height of the tank and mgL kTc0 , where Tc0 ≡ Tc0 (g = 0).
(2) Show that the condensation is accompanied by a discontinuity in the specific heat of the gas:
s
9
πmgL
(∆CV )T =Tc ≈ − ζ(3/2)N k
8π
kTc0
Hint: note the following expansion of the polylogarithmic function:
∞
Lν (e−α ) =
Γ(1 − ν) X (−1)i
+
ζ(ν − i)αi
α1−ν
i
i=0