Metropolis Monte Carlo dynamic sampling

Instead of using MD, the X variables may also be sampled stochastically. In the hybrid CMC/MD approach, Metropolis Monte Carlo69 is used to evolve the X variables and molecular dynamics is used to evolve the atomic coordinates. The Metropolis Monte Carlo criteria leads to the generation of a canonical ensemble of the ligands when the following transition probability is used... 

When applying a SA approach to crystal structure prediction, a Metropolis Monte Carlo scheme [20], rather than molecular dynamics [28], is usually chosen to sample the configurational space (different possible candidate structures). In practice, this scheme proceeds by comparing the quality (value of the cost function) of a new candidate structure with the current candidate structure. The new candidate is either rejected or used to replace the current candidate struc-... 

The first part of this chapter contains a short introduction to statistical mechanics of continuum models of fluids and macromolecules. The next section presents a discussion of basic sampling theory (importance sampling) and the Metropolis Monte Carlo and molecular dynamics methods. The remainder of the chapter is devoted to descriptions of methods for calculating F and S, including those that were mentioned above as well as others. 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

The output of simulations is a list of microscopic states in phase space. These are either a sample of points generated with Metropolis Monte Carlo, or a succession of points in time generated by molecular dynamics. Particle velocities, as well as higher time derivatives of particle positions, can also be saved during a molecular dynamics simulation. 

The first molecular simulations were performed almost five decades ago by Metropolis et al. (1953) on a system of hard disks by the Monte Carlo (MC) method. Soon after, hard spheres (Rosenbluth and Rosenbluth, 1954) and Lennard-Jones (Wood and Parker, 1957) particles were also studied by both MC and molecular dynamics (MD). Over the years, the simulation techniques have evolved to deal with more complex systems by introducing different sampling or computational algorithms. Molecular simulation studies have been made of molecules ranging from simple systems (e.g., noble gases, small organic molecules) to complex molecules (e.g., polymers, biomolecules). 

Molecular model-building (conformational search) methods fall into two general classes systematic and random. - Systematic methods search all possible combinations of torsional angles, whereas random methods usually involve a Monte Carlo (with Metropolis sampling ) or molecular dynamics trajectory. Both approaches attempt to search large areas of conformational space and eventually converge on the desired conformation or structure. Dis-... 

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