Monte Carlo method finite-temperature studies

Phase transitions in small clusters have been studied extensively using both analytical and numerical (molecular dynamics and Monte Carlo) methods. The results of these studies are the main subject of this chapter and are discussed in detail in the forthcoming sections. Preempting this discussion, we want to draw attention here to one particular feature. In agreement with the prediction of our analytical model and in accord with Hill s picture, the results of our numerical studies " clearly bore out the fact that there exists a finite range of temperatures over which the solidlike and liquidlike... 

The finite temperature studies of Lennard-Jones lattice gas systems have been performed for the square [105,116], rectangular [106] and triangular [100,111,112] lattices using different approaches, including the simple mean-field theory, the renormalization group method, Monte Carlo simulation and Monte Carlo version of the coherent anomaly method. 

Effects of the gas - solid potential corrugation on the behaviour of monolayers formed on the (100) face of an fee crystal at finite temperatures have been recently studied by Patrykiejew et al. [163] with the help of Monte Carlo method. They have considered three-dimensional systems of constant volume and containing fixed number of particles interacting via the Lennard-Jones potential (1). The gas - solid interaction potential has been assumed to be represented by the two-fold Fourier series [88]... 

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. ... 

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. 

Because of the large size of the systems studied, simple analytical potential energy functions must be used. Thus, almost all of the studies that simulate biological systems at the allatom level do so using molecular mechanics. In order to simulate these systems at a finite temperature, molecular dynamics and Monte Carlo simulation methods must be employed. 

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