PHY 418 Statistical Mechanics I Final Exam Spring '04

1) [30 points total]

Consider a system of particles in the grand canonical ensemble, with m the chemical potential and z the fugacity. If <N> is the average number of particles, and <E> is the average total energy,

a) [15 pts] show that

b) [15 pts] show that

__________________________________________________________

2) [35 points total]

Consider a non-interacting gas of extremely relativistic spin 1/2
fermions, whose energy-momentum relationship is well approximated by
e(**p**)=
c|**p**|. The density of the gas is n=N/V, and we are considering the
thermodynamic limit of VÆ∞.

a) [12 pts] The density of states, g(e), is defined as the number of single particle states with energy e per unit energy per unit volume. Compute g(e) for the gas.

b) [11 pts] Compute the Fermi energy of the gas.

c) [12 pts] Compute the pressure of the gas at T=0.

__________________________________________________________

3) [35 points total]

Consider a non-interacting gas of spin zero bosons, whose
energy-momentum relationship is given by e(**p**)=
A|**p**|^{s},
for some fixed positive numbers A and s. The dimensionality of the gas is
the number d, i.e. the "volume" of the gas is V=L^{d}, for a system of length L. In the
following parts, we are considering behavior in the thermodynamic limit of
VÆ∞.

a) [15 pts] For what values of s and d will there exist Bose-Einstein condensation at sufficient low temperatures?

b) [5 pts] For the case that there is Bose-Einstein condensation, write
an expression that gives how the condensate density no depends on the temperature
T, the condensation temperature T_{c}, and the total density n.

c) [10 pts] Show that the pressure is related to the energy density by

d) [5 pts] Can a gas of photons in d=3 undergo Bose-Einstein condensation?