Physics 831 - 2002
Statistical Physics
Problem Set 4
1. Calculate the constant
Cn=ZPx2
i<1
n
Y
i=1
dxi
using mathematical induction and the fact that C2=πand C3= 4π/3. Compare with the
result in the textbook [in particular, calculate Γ(2) and Γ(5/2)]. (5 pt)
2. Problem 6.4 [for the case of different gases] (3 pt)
3. Problem 6.3 (a) - (c) (6 pt)
4. Consider equilibrium of a body in an external time-independent field, for example, in grav-
itational field. The system is no longer spatially uniform. Still equilibrium with respect to
particle exchange requires that µ= const, but now the chemical potential may depend on
coordinates. In a gravitational field, the energy of a particle of a mass mhas just an extra
term u(r) = mgz. Then the chemical potential per particle has the form
µ(P, T ) = µ0(P, T ) + u(r),
where µ0is the chemical potential in the absence of the field. Prove this relation (3 pt) and
derive the equation for Pas a function of coordinates for a compressible system (3 pt).
5. Problem 6.1 (4 pt)
6. Problem 6.2 (4 pt)
The problems are from Kerson Huang, Statistical Mechanics, 2nd edition, (Wiley, NY 1987).