1. 3 particles in a 1 dimenstional box:-
a. A 1D particle in a box has energy levels n with energy En = (h2Π2n2) / (2mL2) where n is an integer from 1 to ∞. For an open system at kbT = (h2Π2) / (2mL2), estimate what values of the chemical potential µ lead to on average three particles in the box for the case of fermions and bosons.
2. Critical temperature of a spin band
a. Consider a ferromagnetic Ising model on a rectangle with length N spins and width f(N) spins. Show that for f(N)=constant, that Tc =0
b. We know for f(N)=N that Tc > 0. For what m, such that f(N) = Nm, is Tc > 0? Explain why.
c. For what b, such that f(N) = (log(N))b, will Tc > 0. Explain why.
3. Damping vibrations
Find a copy of Tuckerman and Berne, J. Chem. Phys. 98, 7301 (1993)
a. Explain why the Generalized Langevin Equation and the Molecular Dynamics simulation have good agreement for this problem.
b. Table 1 shows that the vibrational relaxation rates decrease with frequency above ω=50. Estimate the relaxation rates for ω=10, 20, 30 and 40.
c. They note that for high frequencies < R4(0)> i is greater than 32(0)>2 which is the prediction for a Gaussian random process. Prove this prediction.
d. The authors conjecture that the non-Gaussian behavior is because the strong collisions should be treated as a Poisson process. Argue in favor or against this prediction.