Grained Monte Carlo Simulations

The main difference between Monte Carlo (MC) and Molecular Dynamics (MD) simulations is that we do not need to follow the physical trajectory of the system with MC, which, in turn, enables us to use unphysical moves to cover the relevant area of phase space more quickly. Such moves include chain breaking and reattachment,configurational bias, and reptation moves.  

Because we do not have to follow a physical trajectory in an MC simulation, we can also use models that are further removed from the true physical or chemical reality. Such models include lattice models (see, e.g.. Refs. 28,61,62). With lattice models, the space of our system is (typically) evenly divided into cells, each of which are represented by one lattice site. Lattices can be very simple cubes or they can be specially adapted, highly connected grids. Here again, we need super-atoms, which, however, can occupy only lattice sites. In most lattice models every site is either singly occupied or empty, meaning that the interaction sites have an impenetrable hard core, which contrasts to Lattice-Boltzmann models used in studies of hydrodynamics in which every lattice site is occupied by a density, in which case, one deals with a density-based field theory. In lattice models, there exist only a fixed number of distances that can be realized. It makes no sense to distinguish between, say, a 

In most lattice models, a super-atom can represent a monomer or a Kuhn segment of the chain.In most lattice models, only interactions of very close neighbors (first or second neighbors) are included such that the calculation of the energy, which is the computationally most expensive part of a Monte Carlo calculation, is a sum whose calculation scales linearly with the number of lattice sites. The actual mapping process, if done systematically, is easier than without using a lattice. With a lattice model, we have fewer points in the RDF that need to be reproduced otherwise, there is no fundamental difference between lattice and off-lattice models. 

Monte Carlo simulations can also be performed with an off-lattice model. In this case, the mapping is the same as described earlier for MD but no dynamic mapping is involved. Kreer et al. showed that the number of Monte Carlo moves can be mapped onto a pseudo-time. This mapping procedure can be used only if no nonphysical Monte Carlo moves are applied, i.e., only local physical moves are allowed. To accomplish this feat, we need to show that the simulation moves represent the true local dynamics of the model one includes only the moves that are possible and the relative abundance of 

In most cases, we are not interested in a dynamic mapping. In such circumstances, we can use all the advances of modem Monte Carlo technology, i.e., we can apply all conceivable physical and nonphysical moves to derive a correct representation of structure and thermodynamics with less computational effort than would be possible with either MD or MC using only local moves. 

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Collins, S., Stamatakis, M. Vlachos, D Adaptive coarse-grained Monte Carlo simulation of reaction and diffusion dynamics in heterogeneous plasma membranes. BMC Bioinformatics 11 (2010), pp. 218-218. 

Monte Carlo simulations, which include fluctuations, then yields Simulations of a coarse-grained polymer blend by Wemer et al find = 1 [49] in the strong segregation limit, in rather good... 

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. 

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

R. B. Pandey, A. Milchev, K. Binder. Semidilute and concentrated polymer solutions near attractive walls Dynamic Monte Carlo simulation of density and pressure profiles of a coarse-grained model. Macromolecules 50 1194-1204, 1997. 

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions Monte Carlo Simulation of the Bond Fluctuation Model... 

Polymers to Proteins, NIC Symposium Series, Jiilich, Germany, 2004, pp. 83—140. Monte Carlo Simulation of Polymers Coarse-Grained Models. 

Rhodium, incorporated in the silver halide grains, decreases sensitivity and increases contrast. This action has been attributed to depression of latent image formation because of deep electron trapping by the trivalent rhodium ion (183-185). Eachus and Graves (184) showed that rhodium, probably as a complex, acts as a deep trap for electrons at room temperature. Weiss and associates (186) concluded that the rhodium salts introduce deep traps for both electrons and holes. Monte Carlo simulation showed that the photographic properties could be accounted for in this way over a wide range of exposure times. 

Models for following the hydrogen diffusion process may be based on Monte Carlo simulation or network simulation incorporating both a regular lattice structure and irregular grain regions (Herrmann et ah, 2001). 

Katsoulakis, M.A., Majda, A.J. and Vlachos, D.G. (2003) Course-grained Stochastic Processes and Monte Carlo Simulations in Lattice Systems. 

Coarse-grained Stochastic Processes and Kinetic Monte Carlo Simulation for the Diffusion of Interacting Particles. J. Chem. Phys., 119, 9412-9427. 

Katsoulakis, M.A. Vlachos, D.G. Coarse-grained stochastic processes and kinetic Monte Carlo simulators for the diffusion of interacting particles. J. Chem. Phys. 2003, 119, 9412-9428. 

It was clearly demonstrated that the composite BN semiconductor polycrystalline bulk detectors with BN grains embedded in a polymer matrix operate as an effective detector of thermal neutrons even if they contain natural boron only (Uher et al. 2007). A reasonable signal-to-noise ratio was achieved with detector thickness of about 1 mm. A Monte Carlo simulation of neutron thermal reactions in the BN detector was done to estimate the detection efficiency and compare with widely used He-based detectors to prove advantages of BN detectors. They are found to be promising for neutron imaging and for large area sensors. 

In a staged multi-scale approach, the energetics and reaction rates obtained from these calculations can be used to develop coarse-grained models for simulating kinetics and thermodynamics of complex multi-step reactions on electrodes (for example see [25, 26, 27, 28, 29, 30]). Varying levels of complexity can be simulated on electrodes to introduce defects on electrode surfaces, composition of alloy electrodes, distribution of alloy electrode surfaces, particulate electrodes, etc. Monte Carlo methods can also be coupled with continuum transport/reaction models to correctly describe surfaces effects and provide accurate boundary conditions (for e.g. see Ref. [31]). In what follows, we briefly describe density functional theory calculations and kinetic Monte Carlo simulations to understand CO electro oxidation on Pt-based electrodes. 

Simulations on the effect of step free energy on grain growth behaviour have also been made. Figure 15.11 shows the result of a Monte Carlo simulation made by Cho. For the simulation, Cho assumed that the grain network was a set of grains with a Gaussian size distribution (standard deviation of 0.1) located on vertices of a two-dimensional square lattice. Deterministic rate equations, Eq. (15.15) for v/> and Eq. (15.29) for v j, were... 

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