Monte Carlo force-bias

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. 

Rao M and B J Berne 1979. On the Force Bias Monte Carlo Simulation of Simple Liquids. Journal Chemical Physics 71 129-132. 

J Cao, BJ Berne. Monte Carlo methods for accelerating hamer crossing Anti-force-bias and variable step algorithms. J Chem Phys 92 1980-1985, 1990. 

Physisorption of the substrate molecules on the cavity walls can also be described successfully by intermolecular force fields. Energetics and siting of relatively large molecules in the confined space ofzeolite cavities and chaimels can be simulated by special simulation technics, for instance, the configiu-ational-bias Monte Carlo approach, which is specially well-suited for this purpose. ... 

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... 

Given that the two techniques in some ways complement each other in their ability to explore phase space, it is not surprising that there has been some effort to combine the two methods. Some of the techniques that we have considered in this chapter and in Chapter 7 incorporate elements of the Monte Carlo and molecular d)mamics techniques. Two examples are the stochastic collisions method for performing constant temperature molecular dynamics, and the force bias Monte Carlo method. More radical combinations of the two techniques are also possible. 

Abstract Configurational-bias Monte Carlo simulations in the Gibbs ensemble have been carried out to determine the vapor-liquid coexistence curve for a pentadecanoic acid Langmuir monolayer. Two different force fields were studied (i) the original monolayer model of Karaborni and Toxvaerd including anisotropic interactions between alkyl tails, and (ii) a modified version of this model which uses an isotropic united-atom description for the methylene and methyl groups and includes dispersive interactions between the tail segments and the water surface. 

Binder has written an introduction to the theory and methods of Monte Carlo simulation techniques in classical statistical mechanics that are capable of providing measurements of equilibrium properties and of simulating transport and relaxation phenomena. The standard Metropolis algorithm of system sampling has latterly been supplemented by the force bias, Brownian dynamics, and molecular dynamics techniques, and, as noted in the first report, with the aid of these the study has commenced of the behaviour of polymeric systems. 

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. 

Abstract In this contribution, we present an overview of the various techniques for combining atomistic molecular dynamics with Monte Carlo simulations, mainly in the context of condensed matter systems, as well as a brief summary of the main accelerated dynamics techniques. Special attention is given to the force bias Monte Carlo technique and its combination with molecular dynamics, in view of promising recent developments, including a definable timescale. Various examples of the application of combined molecular dynamics / Monte Carlo simulations are given, in order to demonstrate the enhanced simulation efficiency with respect to either pure molecular dynamics or Monte Carlo. 

In order to understand Monte Carlo simulations in general and force bias Monte Carlo in particular, it is useful to recall the crucially important condition of detailed balance. This condition can be expressed as... 

Note that most of these algorithms are used to generate a single trajectory, similar to the accelerated molecular dynamics techniques. However, the stochastic MC component does not allow to assign a timescale to the simulation, except in the case of so-called time stamped force bias Monte Carlo (tfMC, see below) [47]. Thus, a comparison in terms of a boost factor with the accelerated dynamics techniques cannot be made. 

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... 

Mees MJ, Pourtois G, Neyts EC, Thijsse BJ, Stesmans A (2012) Uniform-acceptance force-bias Monte Carlo method with time scale to study solid-state diffusion. Phys Rev B 85(13) 134301... 

Limonova M, Groenewegen J, Thijsse BJ (2010) Modeling diffusion and phase transitions by a uniform-acceptance force-bias Monte Carlo method. Phys Rev B 81 (14) 144107 Neyts EC, Thijsse BJ, Mees MJ, Bal KM, Pourtois G (2012) Establishing uniform acceptance in force biased Monte Carlo simulations. J Chem Theory Comput 8 1865-1869 Rossky P, Doll J, Eriedman H (1978) Brownian dynamics as smart Monte-Carlo simulation. J Chem Phys 69(10) 4628 633... 

In conclusion we note that Monte Carlo studies on highly structured fluids can be very misleading. At low temperatures this procedure leads to configurational bottlenecks. Although the force bias method is prefercible to the standard Metropolis scheme, it is important that any sampling method be thoroughly tested to ensure that a unique equilibrium state is reached. 

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