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

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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.

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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=Ld, 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 Tc, 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?