Configuration integrals particle simulations, Monte Carlo

An alternative to molecular dynamics based simulated annealing is provided by Metropolis importance sampling Monte Carlo (Metropolis et al., 1953) which has been widely exploited in the evaluation of configurational integrals (Ciccotti et al., 1987) and in simulations of the physical properties of liquids and solids (Allen and Tildesley, 1987). Here, as outlined in Chapters 1 and 2, a particle or variable is selected at random and displaced both the direction and magnitude of the applied displacement within standard bounds are randomly selected. The energy of this new state, new, is evaluated and the state accepted if it satisfies either of the following criteria ... 

Fi is the force on particle i caused by the other particles, the dots indicate the second time derivative and m is the molecular mass. The forces on particle i in a conservative system can be written as the gradient of the potential energy, V, C/, with respect to the coordinates of particle /. In most simulation studies, U is written as a sum of pairwise additive interactions, occasionally also three-particle and four-particle interactions are employed. The integration of Eq. (1) has to be done numerically. The simulation proceeds by repeated numerical integration for tens or hundreds of thousands of small time steps. The sequence of these time steps is a set of configurations, all of which have equal probability. The completely deterministic MD simulation scheme is usually performed for a fixed number of particles, iV in a fixed volume V. As the total energy of a conservative system is a constant of motion, the set of configurations are representative points in the microcanonical ensemble. Many variants of these two basic schemes, particularly of the Monte Carlo approach exist (see, e.g.. Ref. 19-23). 

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