More Graduate Statistical Mechanics Problems, Fall 2004

1. Calculate the partition function for a 2-D ideal Bose gas in the thermodynamic limit (area taken to infinity). Find the average number of particles per unit area as a function of the chemical potential and temperature. Prove that in 2-D the specific heat of an ideal gas of fermions is the same as for bosons for all values of N and T.
2. The van der Waals equation exhibits a critical point when both the 1^st and 2^nd derivatives of the pressure with respect to volume at fixed T vanish. From this condition, find Pc, Tc and Vc in terms of parameters a,b. Show that the equation of state can be written in an universal form in terms of Pc, Vc and Tc. Sketch the P-V phase diagram for various temperatures above, below and at Tc. Identify the sign of the isothermal compressibility in each region and identify any unphysical regions.
3. Calculate the ratio of the isothermal compressibility for a van der Waals gas and an ideal gas. Sketch its behavior for various values of T as a function of volume V. Identify all unphysical regions.
4. Mechanical model of a phase transition: Consider a mass m suspended between points P1 and P2 by two identical springs with spring constant k and equilibrium length l₀. Let the points P1 and P2 be separated by 2L. Assume Hooke’s law to hold and that the mass is always close to a symmetry point P0 that it is sufficient to expand the potential up to 4^th order in the distance of m from P0.
1. Show that for L>l₀ , P0 is the equilibrium point while for L<l₀ every point on a circle with radius r0 around P0 is an equilibrium position. Calculate and draw r0 as a function of L.
2. Calculate and discuss the eigenfrequencies and eigenvectors for small oscillations around the equilibrium positions as a function of L. Draw all eigenfrequencies as a function of L.
3. Same as (b) but now assume a constant force F acts in the same plane as the circle of equilibrium positions. Consider r0 and the eigenfrequencies now as functions of L and F.