Monte Carlo methods direct-space techniques

Direct Methods Direct-Space Techniques Patterson Methods Monte Carlo Simulated Annealing Genetic Algorithm Degree of Freedom Cost Function... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

An alternative computer simulation technique is the Monte Carlo (integration) method which evaluates directly the probabilities of finding the liquid in different configurations. It has the advantage over molecular dynamics in that it often explores the available configuration space more efficiently. It is also easier to implement since only the potential energy V q) is required rather than the forces derived from it. It has the disadvantage, however, in that it cannot model dynamical effects directly. 

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