Algorithm Monte Carlo

Lee J 1993 New Monte Carlo algorithm—entropic sampling Phys. Rev.L 71 211-14... 

Tuckerman M, Berne B J, Martyna G J and Klein M L 1993 Efficient molecular dynamics and hybrid Monte Carlo algorithms for path integrals J. Chem. Phys. 99 2796-808... 

Leontidis E, Forrest B M, Widmann A FI and Suter U W 1995 Monte Carlo algorithms for the atomistio simulation of oondensed polymer phases J. Chem. Soc. Farad. Trans. 91 2355- 68... 

A configurational Monte Carlo algorithm based on uniform random trial moves and the acceptance probability... 

We have implemented the generalized Monte Carlo algorithm using a hybrid MD/MC method composed of the following steps. 

For a given potential energy function U r ), the corresponding generalized statistical probability distribution which is generated by the Monte Carlo algorithm is proportional to... 

L. Holm and C. Sander, Fast and simple Monte Carlo algorithm for side chain optimization in proteins. Proteins 14 (1992), 213-223. 

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. 

There are similar algorithms, also called simulated annealing, that are Monte Carlo algorithms in which the choice conformations obey a Gaussian distribution centered on the lowest-energy value found thus far. The standard deviation of this distribution decreases over the course of the simulation. 

The data structure organization described above has been implemented in the BFM as well as in a very efficient off-lattice Monte Carlo algorithm, discussed in detail in the next chapter, which was modified to handle EP and used to study shear rate effects on GM [57]. 

We can now take one of two approaches (1) construct a probabilistic CA along lines with the Metropolis Monte Carlo algorithm outlined above (see section 7.1.3.1), or (2) define a deterministic but reversible rule consistent with the microcanonical prescription. As we shall immediately see, however, neither approach yields the expected results. 

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

The availability of a phase space probability distribution for the steady state means that it is possible to develop a Monte Carlo algorithm for the computer simulation of nonequilibrium systems. The Monte Carlo algorithm that has been developed and applied to heat flow [5] is outlined in this section, following a brief description of the system geometry and atomic potential. 

The Monte Carlo algorithms require AE0, A ), and Attf. In attempting to move atom n in phase space, the n-dependent contribution to these formulas was identified and only the change in this was calculated for each attempted move. 

Figure 8 shows the r-dependent thermal conductivity for a Lennard-Jones fluid (p = 0.8, 7o = 2) [6]. The nonequilibrium Monte Carlo algorithm was used with a sufficiently small imposed temperature gradient to ensure that the simulations were in the linear regime, so that the steady-state averages were equivalent to fluctuation averages of an isolated system. 

For nonequilibrium statistical mechanics, the present development of a phase space probability distribution that properly accounts for exchange with a reservoir, thermal or otherwise, is a significant advance. In the linear limit the probability distribution yielded the Green-Kubo theory. From the computational point of view, the nonequilibrium phase space probability distribution provided the basis for the first nonequilibrium Monte Carlo algorithm, and this proved to be not just feasible but actually efficient. Monte Carlo procedures are inherently more mathematically flexible than molecular dynamics, and the development of such a nonequilibrium algorithm opens up many, previously intractable, systems for study. The transition probabilities that form part of the theory likewise include the influence of the reservoir, and they should provide a fecund basis for future theoretical research. The application of the theory to molecular-level problems answers one of the two questions posed in the first paragraph of this conclusion the nonequilibrium Second Law does indeed provide a quantitative basis for the detailed analysis of nonequilibrium problems. 

Lee, J., New Monte Carlo algorithm entropic sampling, Phys. Rev. Lett. 1993, 7L, 211-214... 

Hansmann, U. H. E. Okamoto, Y., New Monte Carlo algorithms for protein folding, Curr. Opin. Struct. Biol. 1999, 9, 177-183... 

Liang, F., Annealing contour Monte Carlo algorithm for structure optimization in an off-lattice protein model, J. Chem. Phys. 2004,120, 6756-6763... 

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