The 224-disc problem of Section 11.1 has a one-dimensional cousin that can be solved…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems
Error, Chance and the Machine, Ch Markov-chain Monte Carlo
Problem
The 224-disc problem of Section 11.1 has a one-dimensional cousin that can be solved exactly. Place N=20 rods, each of length a=0.03, on a ring of circumference 1, with the centres independent and uniform on the ring. Two rods overlap when their centres are less than a apart along the ring. The allowed configurations are those with no overlap, and the probability that a random placement is allowed is P=(1−Na)N−1, which you may take as given.
((a)) Evaluate P.
((b)) Estimate P as Section 11.1 estimated it for the discs: treat the N(N−1)/2 pairs as overlapping independently and use the Poisson form e−μ, where μ is the mean number of overlapping pairs. Two random centres are within a of each other with probability 2a. Compare the estimate with part (a). Which way does the independence assumption err, and by what factor?
((c)) By the rare-event rule of Chapter 10, how many random placements are needed to estimate P to a relative error of 10 per cent?
((d)) A Metropolis chain moves one rod at a time by a small random displacement and accepts the move when it creates no overlap. Why does this rule not need P? Give the ratio p(y)/p(x) for a move from one allowed configuration to another.