Smart Darting and Basin Hopping Monte Carlo

In some instances, we have prior knowledge of states of the system that are thermodynamically meaningful. Then we can take advantage of such information and generate the proper samples that allow, for instance, the calculation of the relative free energy of such states. Let us reconsider the partition function for the ensemble of states for N distinguishable particles in three dimensions, 

These are exact expressions for the configuration integrals. Alternatively, we can write the partition function as 

choose a real number e. Then define an e-sphere around each element of the set Rj  

Accept or reject the move according to the Boltzmann criterion. 

For this algorithm, one can prove that detailed balance is guaranteed and the exact average of any configuration-dependent property over the accessible space is obtained. Two key issues determine the detailed balance. The first is the fact that the trial probability to pick the displacement vector Dfc to go from the fcth to the Zth e-sphere equals the trial probability to pick the displacement vector D fc for the reverse step. The second issue is that the trial probability for a local MC step that moves the walker from a point inside an e-sphere to a point outside that sphere is the same as for the reverse move i.e., (1 - / ) times what it would be in a walk restricted to local moves. 

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Smart Darting and Basin Hopping Monte Carlo... 

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