Static Monte Carlo methods

Consider a system with state space (configuration space) S for notational simplicity, let us assume that 5 is discrete (i.e., finite or countably infinite). Now let 7T = be a probability distribution on S, and let 

A = be a real-valued observable. Our goal is to devise a Monte 

The most straightforward approach standard Monte Carlo) is to generate independent random samples X, .X from the distribution tt (if one canIX and use the sample mean 

There are two cases to consider, depending on how welt one knows the function W x)  

In the first case, we can use as our estimator the weighted sample mean 

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While static Monte Carlo methods generate a sequence of statistically independent configurations, dynamic MC methods are always based on some stochastic Markov process, where subsequent configurations X of the system are generated from the previous configuration X —X —X" — > with some transition probability IF(X —> X ). Since to a large extent the choice of the basic move X —X is arbitrary, various methods differ in the choice of the basic unit of motion . Also, the choice of transition probability IF(X — > X ) is not unique the only requirement is that the principle... 

STATIC MONTE CARLO METHODS FOR THE SAW Clearly the probability of surviving to length N—kr is... 

QUASI-STATIC MONTE CARLO METHODS FOR THE SAW75... 

Calculations on dynamics of solvation shells are still in their infancy. However, very recent papers on this subject, show that in most examples we cannot expect a realistic picture of solvent shells from a purely static approach. Most probably, molecular dynamics calculations and Monte Carlo methods will produce a variety of interesting data and will improve our knowledge on solvation of ions substantially. 

Contents 1. Introduction 176 2. Static NMR Spectra and the Description of Dynamic Exchange Processes 178 2.1. Simulation of static NMR spectra 178 2.2. Simulation of DNMR spectra with average density matrix method 180 3. Calculation of DNMR Spectra with the Kinetic Monte Carlo Method 182 3.1. Kinetic description of the exchange processes 183 3.2. Kinetic Monte Carlo simulation of DNMR spectra for uncoupled spin systems 188 3.3. Kinetic Monte Carlo simulation of coupled spin systems 196 3.4. The individual density matrix 198 3.5. Calculating the FID of a coupled spin system 200 3.6. Vector model and density matrix in case of dynamic processes 205 4. Summary 211 Acknowledgements 212 References 212... 

It has already been seen in Seetion 2.17 that computer simulation of structures in aqueous solution can give rise to calculations of some static (e.g coordination numbers) and dynamic (e.g., diffusion coefficients) properties of ions in aqueous and nonaqueous solutions. One such computer approach is the Monte Carlo method. In this method, imaginary movements of the particles present are studied, but only those movements that /ower the potential energy. Another technique is molecular dynamics. In this method, one takes a manageable number of atoms (only a few hundred because of the expense of the computer time) and works out their movements at femtosecond intervals by applying Newtonian mechanics to the particles under force laws in which it is imagined that only pairwise interactions count. The parameters needed to compute these movements numerically are obtained by assuming that the calculations are correct and that one needs to find the parameters that fit. 

By increasing pressure and/or decreasing temperature, ionic quantum effects can become relevant. Those effects are important for hydrogen at high pressure [7, 48]. Static properties of quantum systems at finite temperature can be obtained with the Path Integral Monte Carlo method (PIMC) [19]. We need to consider the ionic thermal density matrix rather than the classical Boltzmann distribution ... 

A static assembly of chains resulting from a selection of random walks through conformation space is taken to constitute a time-averaged structure. Since Monte Carlo methods are quite demanding on computing time, simulating a highly complex polymer system is beyond the power of current methods, and simulated chains have to be much shorter than those of a true polymer. Despite these caveats the method can still provide a useful description of a polymer sample and permit the evaluation of such structural parameters as the mean square end-to-end distance... 

The study of static and dynamic properties of polymer liquids and glasses is one of the central topics of research in polymer science. A variety of experimental and theoretical techniqi s have been mobilized for this purpose. In recent years the techniques of computer simulations are increasingly finding application toward this goal. AU the various methods of simulating polymer molecules on computers, such as Monte Carlo, molecular mechanics. Brownian dynamics, and molecular dynamics simulation techniques have been utilized. Early works relied mostly on Monte Carlo techniques apfdied to schematic models of polymer Uquid built on a lattice. With increased capabilities of computers available in recent years, the use of more computationally intensive methods has become feasible, allowing simulations of more realistic, off-latdce models of bulk polymers by Brownian and molecular dynamics simulation techniques as well as by Monte Carlo methods. 

Atomic and molecular simulation methods can generally be categorized as either equilibrated or dynamic. Static simulations attempt to determine the structural and thermodynamic properties such as crystal structure, sorption isotherms, and sorbate binding. Structural simulations are often carried out using energy minimization schemes that are similar to molecular mechanics. Elquihbrium prop>erties, on the other hand, are based on thermodynamics and thus rely on statistical mechanics and simulating the system state function. Monte Carlo methods are then used to simulate these systems stochastically. 

In a molecular simulation, a zeolite is described by its geometry and the interactions among atoms (namely, the force fields). This atomic or molecular level information then needs to be translated into measurable macroscopic quantities so that computations can be validated against experiments. Statistical-modeling techniques, such as the classical Monte Carlo method, can be used to accurately compute the static properties of zeolites, provided the force fields assigned to the system are accurate enough and are based on experimental data. Dynamic properties, such as thermal conductivity or mass diffusivity, are most readily computed using classical MD. 

Todorov, I. T., Allan, N. L., Lavrentiev, M. Y, Ereemair, C. L., Mohn, C. E. and Purton, J. A. 2004. Simulation of mineral solid solutions at zero and high pressure using lattice statics, lattice dynamics and Monte Carlo methods. J. Phys. Condens. Matter 16 S2751. 

The Monte-Carlo method is a static statistical simulation method. 

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