Order-parameter scaling Monte Carlo

Figure 15.9 Spectrum for an elongated network (A, = 3.41, N = 51, Q = 0°) obtained by Monte-Carlo simulation. The abscissa scale denotes the reduced interaction A/vq, i.e. the orientational order parameter S. Ordinates are in linear arbitrary units...

Figure 60. Scaling plot (5.9) of the head-tail order parameter susceptibility obtained from Monte Carlo simulations of complete monolayer (-J3 x yfi)R30° CO on graphite (6 = 0.13 A) with two-dimensional Ising exponents y = 7/4 and y = 1, and = 11.9 K from the cumulant intersection in Fig. 59. Only the scaling regime 1 - T/T L " 0 is shown, and the data above (asterisks) and below (circles) the transition are superimposed for all system sizes I = 18. .. 60 solid and dotted lines are the amplitude fits (5.11) of the data above and below the transition, respectively. (Adapted from Fig. 4c of Ref. 215.)...

Natural attenuation is controlled by numerous processes, which include sorption, intraparticle diffusion as weU as biological and chemical degradation. In order to be able to quantify respectively predict the fate and transport of contaminants, appropriate models that are able to deal with the complexity and interactions of the involved processes need to be developed. Due to insufficient information on the spatial distribution of transport parameters in the subsurface, stochastic methods are a preferred alternative to deterministic approaches. In the present paper a one-dimensional Lagrangian streamtube model is used to describe the reactive transport of acenaphthene as a sample organic compoimd at field scale. As the streamtube model does not consider the heterogeneity of hydrogeochemical parameters but only hydraubc heterogeneity, model results from the streamtube model are compared in a Monte Carlo approach to results of a two-dimensional Eulerian model. 

Various quantities, such as the fraction of repeating xmits (monomers) captured at the interface (which serves as an order parameter of the localization phase transition) and the components of the polymer radius of gyration parallel (I g ) and perpendicular (/ gj ) to the phase boundary between the immiscible hquids, can be then studied in order to verify the predictions of the pertinent scaling analysis by comparison with results from Monte Carlo simulations [36,45-47]. As an example, we show the changing degree of copolymer localization (Fig. 8a) and the ensuing... 

Figure 13Sensitivity plot for the probability of failure of the TE device using 94 Monte Carlo simulations. The significant parameters influencing the Pf in order of importance are Weibull modulus of the TE material, scale parameter of the TE material, and elastic modulus of the TE material. 

In order to achieve these goals, we have adopted a multi-scale approach that comprises molecular and mesoscopic models for the liquid crystal. The molecular description is carried out in terms of Monte Carlo simulations of repulsive ellipsoids (truncated and shifted Gay-Berne particles), while the mesoscopic description is based on a dynamic field theory[5] for the orientational tensor order parameter, Q. ... 

First, the problem is solved using Monte Carlo simulation. It is possible to directly generate samples of M and Z since they follow Gaussian distributions. However, in order to generate samples from the lognormally distributed Y, its distribution parameters (the mean and standard deviation of the corresponding normal distribution) need to be estimated. The location parameter ly is calculated to be equal to 3.632611 and the scale parameter Cr is calculated to be equal to 0.0997513. It is trivial to code Monte Carlo simulation is a programming environment such as MATLAB. For this numerical example, the MATLAB codes would be... 

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