Monte Carlo-type simulations numerical modeling

The Monte Carlo method therefore simulates by means of a system model an individual sampling value. For evaluation of the results, the known procedures of mathematical statistics can be used. A very important instrument in Monte-Carlo-type simulation is the randomizer. It generates random values within the numerical interval (0,1). The allocation of the random number to a specific value of the random variables is effected via a given distribution function in accordance with Figure 3.20. 

What are the limits of the approximated expression Eq. 4 Mainly those due to the mean-field nature of PB. For, say, 99 % of the studied systems, the ions are monovalent, ion-ion correlations in water can be safely ignored, and the standard expression is valid. This is no more the case in presence of multivalent counterions (or monovalent ions in solvent of low e). That opens to the fascinating concept of electrostatic attraction between hke-charged colloids, subject of numerous false analyses, debates, and controversies in the literature for 30 years. Figure 1 presents Monte Carlo (MC) simulations data for the force vs. separation law within the primitive model (two latex colloids and ions in continuous solvent) in presence of counterions of increasing valence. While the PB/DLVO prediction remains everywhere repulsive, the exact MC behavior deviates at intermediate separation and develops an attractive well deeper and deeper as the valence increases above 3. This non mean-field effect is due to the repulsions and correlations among the counterions localized in the intersticial region (discreteness of the condensed layer). The same type of colloidal attraction is responsible for a liquid-gas (concentrated solution-dilute solution) phase separation, observed... 

Another example is from the numerical study of phase transitions. Renormalization theory has proved accurate for the basic scaling properties of simple transitions. The attention of the research community is now shifting to corrections to scaling, and to more complex models. Very long simulations (also of the MCMC type) are done to investigate this effect, and it has been discovered that the random number generator can influence the results [3-6]. As computers become more powerful, and Monte Carlo methods become more commonly used and more central to scientific progress, the quality of the random number sequence becomes more important. 

The master equation, however, can only be solved analytically for very simple systems such as the gas-phase reaction A—>B. The analysis of these systems typically requires numerical simulation of a lattice-based kinetic Monte Carlo model. The lattice gas model can then be used to formulate the respective transition probabilities in order to solve the master equationThe groups of both Zhdanov[ ° ° ] and Kreuzerl ° l have been instrumental in demonstrating the application of lattice gas models to solve adsorption and desorption processed from surfaces. Once a lattice model has been formulated there are three types of solution ... 

ABSTRACT Paper presents a numerical method for determining changes of availability of web applications implemented in virtual environment. It takes into account the reliability and performance aspects of software and hardware elements of the web system. The revenue process model takes into account working hours of administrators and a time of repair (or reconfiguration) for each type of failures. The described method was a basis for the development of a Monte-Carlo simulator that allows calculating variability of web application availability over a week. The paper contains the numerical results for a test case web application implemented in virtualised environment. 

The goal of the present chapter is to describe some methods and approaches developed in the framework of the thermodynamic theory of adsorption. We confine ourselves to the thermodynamic approach, because this approach allows for direct engineering applications. The simulations in the framework of the thermodynamic approach are relatively simple and well repeatable, so that the algorithm for numerical solutions of the corresponding equilibrium problem may be generated on the basis of a relatively short and informal description. An alternative may be provided by ab initio calculations, direct appfication of the statistical mechanics (Monte Carlo) or other types of molecular simulations. These computations are much more complicated and they have not yet reached the stage where they may be directly used for modeling a wide variety of the practically important cases of mixed adsorption. 

When one tries to account for real polymer systems in terms of models of the type of Eqs. (1.1)-(1.8) the situation is rather unsatisfactory however, when one fits data on the coexistence curve or on (0 (AF/kBT)/0(() )j., the latter quantity being experimentally accessible via small angle scattering, one finds that one typically needs an effective y-parameter that does not simply scale proportional to inverse temperature, as Eq. (1.5) suggests. Moreover, there seems to be a pronounced (])-dependence of x, in particular for (]) — 1. Near (]) = (]) ", on the other hand, there are critical fluctuations (which have been intensely studied by Monte Carlo simulations [11-13,15] and also in careful experiments of polymer blends [16-18] and polymer solutions [19]). Sometimes in the literature a dependence of the x parameter on pressure [18] or even chain length is reported, too. Thus, there is broad consensus that the Flory-Huggins theory and its closely related extensions [20] are too crude as models to provide predictive descriptions of real polymer solutions and blends. A more promising approach is the lattice cluster approach of Freed and coworkers [21-23], where effective monomers block several sites on the lattice and have complicated shapes to somehow mimic the local chemical structure. However, this approach requires rather cumbersome numerical calculations, and is still of a mean-field character, as... 

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