Biased Monte Carlo Methods

An alternative to the use of a fixed cutoff region is to relate the probability of choosing a solvent molecule to its distance from the solute, usually by some inverse power of the distance 

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 

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 an atom may undeigo. 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 larger 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. 

8 Tackling the Problem of Quasi-ergodicity J-walking and Multicanonical Monte Carlo 

In practice, it is found that this simple implementation is not the most effective approach. There are two particular problems First, when the two simulations are run in tandem then significant correlations can arise, which results in large systematic errors. There are a number of ways to avoid these correlations, such as moving the J-walker an extra number 

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For adsorption in zeolites, the biased Monte Carlo method as developed by Smit is an excellent method to determine the free energies of molecules adsorbed on zeolites [9bj. This method can be used to compute the concentration of molecules adsorbed on zeolites, as we discuss below. 

This condition satisfies the important pH = P- Biased Monte Carlo methods, although interesting, are beyond the scope of this text. 

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

The aim of the study is to investigate the influence of applied biases on electron transport properties in the n-channel of studied transistors. Ensemble Monte Carlo method has been chosen as a tool for electron transport simulation. 

The essence of the configurational bias Monte Carlo method is that a growing molecule is preferentially directed (i.e. biased) towards acceptable structures The effects of these biases can then be removed by modifying the acceptance rules The configurational bias methods are based upon work published in 1955 by Rosenbluth and Rosenbluth... 

Beyond the calculations discussed in the previous section, additional calculations were performed in order to verify that the results obtained were adequately modeled and completely converged. The stochastic nature of the Monte Carlo methods in KENO V.a can be biased by various parameter specifications. For the reactor criticals, the two parameters that were felt to have the most potential for bias or error in the calculated value of were the number of particle histories tracked... 

The basic techniques of the conventional Monte Carlo methods applied to fluid problems have changed little since their invention. Recently, however, there has been a good deal of experimentation seeking to tinker with the technique in order to get new kinds of information. This chapter examined a (no doubt biased) sample of such attempts. 

Sadanobu and Goddard (156) have used a Continuous Configuration Boltzmann Biased Monte Carlo calculation of the free energy of two independent imited atom Cioo chains from very dilute vapor up to about 70% of physical density. While these are longer chains than are typically attainable with CCB methods, the densities of the Sadanobu and Goddard systems are generally lower. 

They will be discussed in more detail later in this chapter. However, quantum Monte Carlo methods for fermions suffer from the notorious sign problem that originates in the antisymmetry of the many-fermion wave function and hampers the simulation severely. Techniques developed for dealing with the sign problem often reintroduce biases into the method, via, for instance. 

Recent work by Hassan et al. [43] has indicated that this method tends to increase the rank of the selected subset at the expense of spread. They used a Monte-Carlo method to maximize a diversity objective function based on the D-optimal criterion. The resulting sets of compounds were biased toward the periphery of the property space. This bias was especially evident when the number of compounds selected far exceeded the dimensionality of the space. It must be noted that the design matrix used in this study did not include higher-order combinations of properties, which may partially account for the redundancy of the results. 

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... 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

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