Dynamic Monte Carlo algorithms

Korniss, G., Novotny, M.A., Rikvold, P.A. Parallelization of a dynamic Monte Carlo algorithm A partially rejection-free conservative approach. J. Comput. Phys. 1999,153, 488-508. 

An intricate problem is how to obtain initial configurations of the system of chains for the dynamic Monte Carlo algorithms. Several recipes have... 

The model discussed can easily be implemented as a dynamic Monte Carlo algorithm. A system of beads on the fee lattice is considered. The beads occupy all lattice sites. It is assumed that the beads vibrating with a certain frequency around the lattice sites attempt periodically to change their positions towards nearest neighboring sites. The attempts are represented by a field of unit vectors, assigned to beads and pointing in a direction of attempted motion, chosen randomly. Attempts of all beads are considered simultaneously. 

Proton transport in the gA channel was also simulated with a kinetic model (see section 16.3.5.3). " Each H2O molecule was allowed to take six orientations and each HsO molecule, four orientations, so that the chain of eleven H2O molecules could take 10 states. Three types of transitions were allowed rotation of H2O and HsO" proton transfer from HsO to a neighbouring H2O molecule when they form a hydrogen bond and proton uptake and release for water molecules located at the channel ends. The rate constants were taken of the TST type, with Ag values calculated by continuum electrostatics or deduced from data about proton transfer in water. For the proton uptake and release steps, the Ag value depended explicitly on the pH. The master equation was solved by a sequential dynamical Monte Carlo algorithm and the PMF was deduced from the probability of occupancy of the various sites. When no voltage was applied, the PMF was a symmetrical barrier with a maximum at 3.4 kcal mol . Stationary proton ffuxes calculated for various pH and voltages values were in reasonable agreement with the conductance data. Despite the simplified description of electrostatic interactions and the questionable... 

Fig. 1.4 Various examples of dynamic Monte Carlo algorithms for SAWs sites taken by beads are shown by dots, and bonds connecting the bead are shown by lines. Bonds that are moved are shown as a wavy line (before the move) or broken line (after the move), while bonds that are not moved are shown as full lines, (a) Generalized Verdier-Stockmayer algorithm on the simple cubic lattice showing three type of motions end-bond motion, kink-jump motion, 90° crankshaft rotation (b) slithering snake algorithm (c) pivot algorithm. (From Kremer and Binder )...

This subsection contains some exceedingly pedantic—but I hope useful— general considerations on dynamic Monte Carlo algorithms. 

To specify a dynamic Monte Carlo algorithm, we must specify three things ... 

Historically the earliest dynamic Monte Carlo algorithms for the SAW were local A-conserving algorithms they date back to the work of Delbriick"" and Verdier and Stockmayer , both published in 1962. During the subsequent two decades, mnnerous variants on this theme were proposed (see Table 2.4). Most of these algorithms employ some subset of moves A-F from Figs 2.1 and 2.2. 

Chains on a High Coordination Lattice Dynamic Monte Carlo Algorithm Details. 

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... 

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 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 ... 

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. 

In the original implementation of the replica-exchange method (REM) [67-69], Monte Carlo algorithm was used, and only the coordinates q (and the potential energy function E(q)) had to be taken into account. In molecular dynamics algorithm, on the other hand, we also have to deal with the momenta p. We proposed the following momentum assignment in (4.45) (and in (4.46)) [77] ... 

Our treatment of dynamic Monte Carlo differs in one very fundamental aspect from that of other authors the derivation of the algorithms and a large part of the interpretation of the results of the simulations are based on a master equation... 

The idea of the dynamic Monte Carlo method is not to compute probabilities Pa(t) explicitly, but to start with some particular configuration, representative for the initial state of the experiment one wants to simulate, and then generate a sequence of other configurations with the correct probability. The integral formulation directly gives us a useful algorithm to do this. 

The usual approach to dynamic Monte Carlo simulations is not based on the master equation, but starts with the definition of some algorithm. This generally starts, not with the computation of a time, but with a selection of a site and a reaction that is to occur at that site. We will show here that this can be extended to a method that also leads to a solution of the master equation, which we call the random-selection method (RSM). [31]... 

There are two other approaches that we want to mention here. The first is the collection of dynamic Monte Carlo methods that are defined using an algorithm. We include in this collection the method by Fichthorn and Weinberg. [34] The second approach consists of cellular automata (CA) in one form or another. 

The problem of real time in the algorithmic formulation of dynamic Monte Carlo has been solved by Fichthorn and Weinberg. [34] They replaced the reaction probabilities by rate constants, and assumed that the probability distribution Prx(t) of the time that a reaction occurs is a Poisson process i.e., it is given by... 

Fig. 6.3. Distributions of x in between two steady states (A, C), and dynamic fluctuations between these two states (B, D). The steady-state distributions (A and C) were calculated using (6.4) in the text. The fluctuations in x were calculated using a Gillespie-type Monte Carlo algorithm to the chemical master equation (Beard and Qian, 2008). Parameters Panels (a) and (b), fci = 2.7, = 0.6, ks = 0.25, fc4 =...

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