Homework Set # 11 - Due November 24

1. Consider an ideal Bose gas confined to a region of area A in two dimensions. Express the number of particles in the excited states, N_e, and the number of particles in the ground state, N₀, in terms of z, T, and A, and show that the system does not exhibit Bose-Einstein condensation unless T->0K.

2. Draw all the diagrams connected or disconnected, representing terms in the configuration integral with four factors f_ij. You should find 11 diagrams in total, of which 5 are connected.

3. Draw all the connected diagrams containing four dots and indicate how many f_ij factors each diagram represents. There are 6 diagrams in total.

4. Show that the nth virial coefficient depends on the diagrams in the expression for Z_U given in class that have n dots. Write the third virial coefficient B₃(T) in terms of an integral of f-functions.

5. Consider a gas of particles in d-dimensional space interacting through a pairwise central potential V(r) given by V(r)=∞ for 0<r<a, V(r)=-ε for a<r<b, and V(r)=0 for b<r<∞. a) Calculate the second virial coefficient B₂(T), and find its high- and low-temperature behavior. Hint: the volume of a sphere in d-space is given by V_d(r)=2π^d/2/[d(d/2-1)!]r^d; use that f_ij=e^-βV(r)-1. b) Calculate the first correction to the isothermal compressibility κ_T=-(1/V)(∂V/∂P)__T,N. c) In the high temperature limit reorganize the equation of state into the van der Waals form, and identify the van der Waals parameters. Hint: Use the high T expression for the virial coefficient found in (a).d) For b=a (a hard sphere), and d=1, calculate the third virial coefficient B₃(T).

6. Problem 10.9