Simulated annealing on a nineteen-atom cluster is run as in this chapter: the…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

Simulated annealing on a nineteen-atom cluster is run as in this chapter: the temperature starts at T0=0.5 ϵ/kBT_0=0.5\,\epsilon/k_B and is multiplied by a cooling factor γ\gamma after every sweep until it falls to 0.01 ϵ/kB0.01\,\epsilon/k_B, and a BFGS quench follows. From 4040 seeds each, a fast schedule, γ=0.99\gamma=0.99, reaches the lowest known energy 99 times, and a slow one, γ=0.999\gamma=0.999, reaches it 1717 times. ((a)) How many sweeps does each schedule take? ((b)) Find the difference between the two success rates and its standard error, combining the two binomial errors in quadrature. Is the slow schedule shown to be better? ((c)) Counting sweeps only, which schedule costs less per success? At the measured rate, how many independent fast runs give a 95 per cent chance of at least one success? ((d)) Twelve fast runs are made; three reach the target and none goes lower. A colleague writes that the cluster's global minimum has been found. What has the search shown, and what has it not?

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