Monte Carlo simulations cell theories

From the standpoint of informing effective Hamiltonians one seeks a series of microscopic realizations of the various states and uses microscopic analysis to determine the energetics of these states. Rabe and Waghmare (1995) elucidate the program of effective Hamiltonian construction as being to obtain a model system which reproduces the finite-temperature transition behavior of the original lattice Hamiltonian, while the simpler form and reduction in the number of degrees of freedom per unit cell makes it suitable for study by methods such as mean field theory and Monte Carlo simulation. ... 

The correspondence between a concentrated dispersion and an assembly of hard spheres has been pursued by several authors. " The Kirkwood-Alder hard-sphere transition is in qualitative agreement with experiment, but the coexistence region is in general too narrow. Introduction of attractive forces, in the Monte Carlo simulations and approximate perturbation-cell theories, leads to iijiprovement at high salt concentrations and large volume fractions. But at low salt concentrations there remains the fundamental problem that the particles are not in proper thermodynamic equilibrium with bulk electrolyte as Ninham and coauthors put it the diffuse double-layers of the particles fill up the entire volume of the system, and there is no place to be regarded as bulk . 

Figure 1.5 The probability for N particles to exit a wedge hopper has a broad power-law tail with P N) N (dashed line), seen in experiments, Monte Carlo simulation, and analytic theory. This is in distinct contrast to the distribution function found in conical hoppers, which decays exponentially. Shown are distribution functions for experimental hoppers of lengths L = 16.2-22.2 cm. Simulation and theory assume = 3 adjacent, statistically independent cells. (From Saraf, S. and Franklin, S.V., Physical Review E, 83(3), 030301, March 2011.)...

Sometimes simple models derived from continuum theory promise better results, but MD or Monte-Carlo (MC) simulations are still the preferable approaches for condensed-matter investigations [3,4], These methods were developed in the early 1950s [3, 4], They include a model-inherent dynamical description in the case of MD, and large samples can be accounted for. These methods allow working scientists to treat more than one molecule and make use of so-called periodic boundary conditions which mimic images around the central cell in such a way that problems due to surface effects can be overcome. 

In microfluid mechanics, the direct simulation Monte Carlo (DSMC) method has been applied to study gas flows in microdevices [2]. DSMC is a simple form of the Monte Carlo method. Bird [3] first applied DSMC to simulate homogeneous gas relaxation problem. The fundamental idea is to track thousands or millions of randomly selected, statistically representative particles and to use their motions and interactions to modify their positions and states appropriately in time. Each simulated particle represents a number of real molecules. Collision pairs of molecule in a small computational cell in physical space are randomly selected based on a probability distribution after each computation time step. In essence, particle motions are modeled deterministically, while collisions are treated statistically. The backbone of DSMC follows directly the classical kinetic theory, and hence the applications of this method are subject to the same limitations as kinetic theory. 

Numerical Results and Approximate Tlieories.— The phase diagram and structure of a model colloidal dispersion has been studied by computer simulation.The Monte Carlo results have been supplemented by predictions from two approximate analytic theories cell theoryand perturbation theory. 

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