Dynamic Monte Carlo simulations method

A dynamic Monte Carlo simulation method was used to determine the time evolution of g t) and fcm by allowing all chains to move simultaneously. Here 5fcm is the me2in-square displacement of the center of mass, i.e.. 

Calculations of relative partition coefficients have been reported using the free energy perturbation method with the molecular dynamics and Monte Carlo simulation methods. For example, Essex, Reynolds and Richards calculated the difference in partition coefficients of methanol and ethanol partitioned between water and carbon tetrachloride with molecular dynamics sampling [Essex et al. 1989]. The results agreed remarkably well with experiment... 

As stated in Sec. 3.1, only ideal systems will be considered in this section. This definition implies that there is no intramolecular reaction, a condition which is satisfied in practice for very low concentrations of Af monomers (f >2), in the A2 + Af chainwise polymerization. To take into account intramolecular reactions it would be necessary to introduce more advanced methods to describe network formation, such as dynamic Monte Carlo simulations. 

Dynamic Monte Carlo simulations were first used by Verdier and Stockmayer (5) for lattice polymers. An alternative dynamical Monte Carlo method has been developed by Ceperley, Kalos and Lebowitz (6) and applied to the study of single, three dimensional polymers. In addition to the dynamic Monte Carlo studies, molecular dynamics methods have been used. Ryckaert and Bellemans (7) and Weber (8) have studied liquid n-butane. Solvent effects have been probed by Bishop, Kalos and Frisch (9), Rapaport (10), and Rebertus, Berne and Chandler (11). Multichain systems have been simulated by Curro (12), De Vos and Bellemans (13), Wall et al (14), Okamoto (15), Kranbu ehl and Schardt (16), and Mandel (17). Curro s study was the only one without a lattice but no dynamic properties were calculated because the standard Metropolis method was employed. De Vos and Belleman, Okamoto, and Kranbuehl and Schardt studies included dynamics by using the technique of Verdier and Stockmayer. Wall et al and Mandel introduced a novel mechanism for speeding relaxation to equilibrium but no dynamical properties were studied. These investigations indicated that the chain contracted and the chain dynamic processes slowed down in the presence of other polymers. 

Wicke et al. [3] were the first to apply a MC simulation to a catalytic reaction based on the Langmuir-Hinshelwood mechanism. They studied the importance of the formation of clusters of adsorbed molecules on a catalyst surface. Many microscopic mathematical models of heterogeneous catalytic systems have been developed since then. However, the time dependence of the reactions in real time could not be followed. Recently more refined MC methods have been developed, so that with these new dynamic Monte Carlo (DMC) methods, the behavior of catalytic systems in real time can be simulated. 

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]... 

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... 

The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

The simple approach discussed above is valid for steady-state Monte Carlo simulations, but dynamic simulations are also possible. In this case, the model probabilities must be updated frequently, generally after every iteration. A detailed discussion of these methods would be too lengthy to be included herein the most common algorithm for dynamic Monte Carlo simulation follows the approach proposed by Gillespie, which requires the discretization of the polymerization reactor with small control volumes and the conversion of the polymerization kinetic rates into molecular collision frequencies [97, 98],... 

R. J. Gelten, R. A. van Santen and A. P. J. Jansen, Dynamic Monte Carlo Simulations of Oscillatory Heterogeneous CatcJytic Reactions in Molecular Dynamics From Classical to Quantum Methods, Elsevier Amsterdam (1999). 

Finally, an entirely different approach to simulating gelation is the Dynamic Monte Carlo (DMC) method, in which chemical reactions are modeled by stochastic integration of phenomenological kinetic rate laws [23]. This has been used successfully to understand the onset of gel formation, first-shell substitution effects, and the influence of cyclization in silicon alkoxide systems [24—26]. However, this approach has not so far been extended to include the instantaneous positions and diffusion of each oUgomer, which would be necessary in order for the calculation to generate an actual model of an aerogel that could be used in subsequent simulations. 

The selection process is the place where the optimization in EC methods really takes place. This is where methods such as dynamic or kinetic Monte Carlo (DMC) simulations become important. They are used to compute the properties of a system or process. These properties are then converted to a fitness value. This fitness value is for satisfaction of a particular requirement of performance which is then operated on by the EC methods. The conversion is different for each system and property and also determines how effective the selection is. Dynamic Monte Carlo simulation, as we have already discussed, is a method to simulate elementary processes along with the actual rate. The method uses each individual reaction as an elementary event, which means that timescales comparable to actual experiments can be simulated. The reaction rate constants that it needs as input can be calculated using quantum chemical methods such as density functional theory, which results in what has been termed ab initio kinetics (see Chapter 3.10.4). 

Because of the large size of the systems studied, simple analytical potential energy functions must be used. Thus, almost all of the studies that simulate biological systems at the allatom level do so using molecular mechanics. In order to simulate these systems at a finite temperature, molecular dynamics and Monte Carlo simulation methods must be employed. 

The Kalman filter-based dynamic state estimation tools in combination with Monte Carlo simulation methods can be employed to estimate probability of failure in instrumented structures with performance functions encompassing unmeasured system states (Ching and Beck 2007). The variance reduction strategies developed in the context of reliability analysis when applied in conjunction with the dynamic state estimation techniques could be used to determine the updated probability of failure of the structural system. For example, the data-based extreme value analysis and the Girsanov transformation-based method can be used to determine the reliability of existing structures (Radhika and Manohar 2010 Sundar and Manohar 2013). 

The present chapter has centered on experimental efforts performed to study confined polymer crystallization. However, molecular dynamics simulations and dynamic Monte Carlo simulations have also been recently employed to study confined nucleation and crystallization of polymeric systems [99, 147]. These methods and their application to polymer crystallization are discussed in detail in Chapter 6. A recent reference by Hu et al. reviews the efforts performed by these researchers in trying to understand the effects of nanoconfinement on polymer crystallization mainly through dynamic Monte Carlo simulations of lattice polymers [147, 311]. The authors have performed such types of simulations in order to study homopolymers confined in ultrathin films [282], nanorods [312] and nanodroplets [147], and crystallizable block components within diblock copolymers confined in lamellar [313, 314], cylindrical [70,315], and spherical [148] MDs. 

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