Acceptance criterion Monte Carlo

Fig. 3.7. Schematic comparison of the multicanonical and iterative transition-matrix methods. The light gray box indicates a single iteration of several thousand or more individual Monte Carlo steps. The acceptance criterion includes the configuration-space density of the original ensemble, p, and is presented for symmetric moves such as single-particle displacements...

Molecular dynamics can be coupled to a heat bath (see below) so that the resulting ensemble asymptotically approaches that generated by the Metropolis Monte Carlo acceptance criterion (Eq. 10). Thus, molecular dynamics and Monte Carlo are in principle equivalent for the purpose of simulated annealing although in practice one implementation may be more efficient than the other. Recent comparative work (Adams, Rice, Brunger, in preparation) has shown the molecular dynamics implementation of crystallographic refinement by simulated annealing to be more efficient than the Monte Carlo one. 

The success and efficiency of simulated annealing depends on the choice of the annealing schedule [58], that is, the sequence of numerical values Ti > T2 > > T for the temperature. Note that multiplication of the temperature T by a factor s is formally equivalent to scaling the target E by 1/s. This applies to both the Monte Carlo as well as the molecular dynamics implementation of simulated annealing. This is immediately obvious upon inspection of the Metropolis Monte Carlo acceptance criterion (Eq. 10). For molecular dynamics this can be seen as follows. Let E be scaled by a factor 1/s while maintaining a constant temperature during the simulation. 

Fig. 2. Schematic of the Monte Carlo library design and redesign strategy (from Falcioni and Deem, 2000). (a) One Monte Carlo round with 10 samples an initial set of samples, modification of the samples, measurement of the new figures of merit, and the Metropolis criterion for acceptance or rejection of the new samples, (b) One parallel tempering round with five samples at and five samples at f>2- In parallel tempering, several Monte Carlo simulations are performed at different temperatures, with the additional possibility of sample exchange between the simulations at different temperatures.

The principles of the Monte Carlo approach are briefly described in Sec. 5.3. It should be mentioned that a true Monte Carlo simulation does not involve any energy minimization step. Each randomly generated conformation is evaluated by applying the Metropolis criterion and is rejected or accepted without any further minimization. Furthermore, no time-dependent movements or conformational changes are investigated as is the case in MD simulations. 

Monte Carlo techniques we call this procedure to create a new trajectory a trial move . Efficient methods for such generation of new trajectories by modifying existing ones will be discussed in the following sections. Next, we accept the newly generated path with a certain probability. There are many ways to construct an appropriate acceptance probability. The simplest is based on detailed balance of moves in trajectory space. This criterion requires that the frequency of accepted moves from to x " is... 

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