Continuum Monte Carlo

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

An answer therefore could be a less time consuming and novel mnltistep Continnnm Monte Carlo technique to solve problems created by multiphenomena characteristics of electrochemical processes and power sources. In the case shown for a lithium ion battery cathode material three different types of Continuum Monte Carlo codes are written to solve three different electrochemical phenomena. All the codes are based on fundamental electrochemical principles, therefore invaluable physics is not lost while deriving useable data. 

Shenoy et al. in 1999 proposed the quasi-continuum Monte Carlo (QCMC) method as a way to extend the quasi-continuum method to the study of equilibrium properties of materials at finite temperature. The objective of this treatment is to construct a computationally manageable expression for a temperature-dependent effective energy for a system maintained at fixed temperature. Such an energy would then be used instead of the zero-temperature effective energy (e.g., Eq. [10]) in a Monte Carlo formulation of the QC method. 

Similar to the quasi-continuum Monte Carlo approach (see above), the CG potential energy is the PMF for the constrained degrees of freedom ... 

Dynamic simulation approaches to model kinetic percolation are difficult to implement because of the inherent complexity of the problem, which requires intensive computation. As with any kinetic modd, the duration of the simulation must be commensruate with the critical timescales of the experiments. An early study to investigate the effeas of interactions on the percolation threshold was conducted by Bug et al. Here, a continuum Monte Carlo algorithm was used to modd a small system of 500 spherical particles undergoing Brownian motion. More recently, advanced simulation approaches such as Dissipative Particle Dynamics (DPD) have been applied to study kinetic percolation in composite sys-tems. " DPD is an off-lattice simulation technique similar to molecular dynamics, but applied to the supramolecular scale. Here, the larger-scale dynamics of a system are studied by monitoring the motion of particle clusters in response to pairwise, dissipative, and random forces. ... 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Pablo J J, M Laso, JI Siepmann and U W Suter 1993. Continuum-Configurational Bias Monte Carlo Simulations of Long-chain Alkanes. Molecular Physics 80 55-63. 

For reasons of space and because of their prime importance, we focus here on free energy calculations based on detailed molecular dynamics (MD) or Monte Carlo (MC) simulations. However, several other computational approaches exist to calculate free energies, including continuum dielectric models and integral equation methods [4,14]. 

The idea of a finite simulation model subsequently transferred into bulk solvent can be applied to a macromolecule, as shown in Figure 5a. The alchemical transformation is introduced with a molecular dynamics or Monte Carlo simulation for the macromolecule, which is solvated by a limited number of explicit water molecules and otherwise surrounded by vacuum. Then the finite model is transferred into a bulk solvent continuum... 

Lattice models have the advantage that a number of very clever Monte Carlo moves have been developed for lattice polymers, which do not always carry over to continuum models very easily. For example, Nelson et al. use an algorithm which attempts to move vacancies rather than monomers [120], and thus allows one to simulate the dense cores of micelles very efficiently. This concept cannot be applied to off-lattice models in a straightforward way. On the other hand, a number of problems cannot be treated adequately on a lattice, especially those related to molecular orientations and nematic order. For this reason, chain models in continuous space are attracting growing interest. 

Dodd, L. R. and Theodorou, D. N. Atomistic Monte Carlo Simulation and Continuum Mean Field Theory of the Structure and Equation of State Properties of Alkane and Polymer Melts. Vol 116,pp, 249-282,... 

Several remedies have been suggested for improving the PB based pKa prediction methods. Most of them are based on strategies that combine conformational flexibility with the PB calculation. You and Bashford included multiple conformers by systematically scanning the side chain torsion angles [107], Alexov and Gunner used Monte-Carlo protocol to sample positions of hydroxyl and other polar protons [1], This method, referred to as the multi-conformation continuum electrostatic (MCCE), was later extended to include rotamers for residues that have strong electrostatic... 

Baptista, A.M. Martel, P.J. Soares, C.M., Simulation of electron-proton coupling with a Monte-Carlo method application to cytochrome C3 using continuum electrostatics, Biophys. J. 1999, 76, 2978-2998... 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

The essence of Monte-Carlo models is to calculate the path of an ion as it penetrates a crystal. Early versions of these models used the binary collision approximation, i.e., they only treated collisions with one atom at a time. Careful estimates have shown that this is an accurate procedure for collisions with a single row of atoms (Andersen and Feldman, 1970). However, when the rows are assembled into a crystal the combined potentials of many neighboring atomic rows affect ion trajectories near the center of a channel. For this reason, the more sophisticated models used currently (Barrett, 1971, 1990 Smulders and Boerma, 1987) handle collisions with far-away atoms using the continuum string approximation,... 

Another aspect that has been theoretically studied109,124,129 is experimental evidence that Diels-Alder reactions are quite sensitive to solvent effects in aqueous media. Several models have been developed to account for the solvent in quantum chemical calculations. They may be divided into two large classes discrete models, where solvent molecules are explicitly considered and continuum models, where the solvent is represented by its macroscopic magnitudes. Within the first group noteworthy is the Monte Carlo study... 

This volume of Modem Aspects covers a wide spread of topics presented in an authoritative, informative and instructive manner by some internationally renowned specialists. Professors Politzer and Dr. Murray provide a comprehensive description of the various theoretical treatments of solute-solvent interactions, including ion-solvent interactions. Both continuum and discrete molecular models for the solvent molecules are discussed, including Monte Carlo and molecular dynamics simulations. The advantages and drawbacks of the resulting models and computational approaches are discussed and the impressive progress made in predicting the properties of molecular and ionic solutions is surveyed. 

Quantitative models of solute-solvent systems are often divided into two broad classes, depending upon whether the solvent is treated as being composed of discrete molecules or as a continuum. Molecular dynamics and Monte Carlo simulations are examples of the former 8"11 the interaction of a solute molecule with each of hundreds or sometimes even thousands of solvent molecules is explicitly taken into account, over a lengthy series of steps. This clearly puts a considerable demand upon computer resources. The different continuum models,11"16 which have evolved from the work of Bom,17 Bell,18 Kirkwood,19 and Onsager20 in the pre-computer era, view the solvent as a continuous, polarizable isotropic medium in which the solute molecule is contained within a cavity. The division into discrete and continuum models is of course not a rigorous one there are many variants that combine elements of both. For example, the solute molecule might be surrounded by a first solvation shell with the constituents of which it interacts explicitly, while beyond this is the continuum solvent.16... 

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