Force-bias displacements

The main difference between the force-bias and the smart Monte Carlo methods is that the latter does not impose any limit on the displacement that m atom may undergo. The displacement in the force-bias method is limited to a cube of the appropriate size centred on the atom. However, in practice the two methods are very similar and there is often little to choose between them. In suitable cases they can be much more efficient at covering phase space and are better able to avoid bottlenecks in phase space than the conventional Metropolis Monte Carlo algorithm. The methods significantly enhance the acceptance rate of trial moves, thereby enabling Icirger moves to be made as well as simultaneous moves of more than one particle. However, the need to calculate the forces makes the methods much more elaborate, and comparable in complexity to molecular dynamics. 

The force-bias Monte Carlo method [Pangali et al. 1978 Rao and Berne 1979] biases the movement according to the direction of the forces on it. Having chosen an atom or a molecule to move, the force on it is calculated. The force corresponds to the direction in which a real atom or molecule would move. In the force-bias Monte Carlo method the random displacement is chosen from a probability distribution function that peaks in the direction of this force. The smart Monte Carlo method [Rossky et al. 1978] also requires the forces on the moving atom to be calculated. The displacement of an atom or molecule in this method has two components one component is the force, and the other is a random vector... 

Some of the earliest attempts to devise efficient algorithms for MC simulations of fluids in a continuum are due to Rossky et al. and Pangali et al., who proposed the so-called Smart Monte Carlo and Force-Bias Monte Carlo methods, respectively. In a Force-Bias Monte Carlo simulation, the interaction sites of a molecule are displaced preferentially in the direction of the forces acting on them. In a Smart Monte Carlo simulation, individual-site displacements are also proposed in the direction of the forces in this case, however, a small stochastic contribution is also added to the displacements dictated by the forces. In both algorithms, the acceptance criteria for trial moves are modified to take into account the fact that displacements are not proposed at random but in the direction of intersite forces. 

Whereas in mixed MD/MC simulations, some of the atoms are moved by pure MD, and other particles are moved by pure MC, it is also possible to constmct algorithms in which the displacement itself is determined in part by a deterministic factor and in part by a stochastic factor. In this class, we can further distinguish essentially three techniques Langevin or stochastic dynamics, hybrid Monte Carlo, and force bias Monte Carlo and related techniques. 

A novel version of the uniform acceptance algorithm was recently published by Mees et al. [47]. In this version, which was termed time stamped force bias Monte Carlo (tfMC), the conditional probability for a displacement in the x-direction is given by ... 

Very similar to these force bias Monte Carlo algorithms is the Smart Monte Carlo technique by Rossky et al. [64]. This technique also requires the forces acting on the moving atom to be calculated. Also, the displacement is determined by two components, that is, the force, which acts as the deterministic component, and a random vector 5rf. The displacement is then written as... 

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