The Monte Carlo Method

The method of molecular dynamics gives information about the time evolution of a microscopic system, and permits the evaluation of macroscopic properties as time averages. The alternative Monte Carlo method was developed at the end of 

In order to calculate an equilibrium property A of the system, we have to evaluate the integral 

We therefore have to generate a significant number of configurations, and we might (for example) envisage moving each of the particles in succession according to the prescription 

We do this as follows. The N particles are placed in a starting configuration, for example a regular lattice. Each particle is then tentatively moved at random. For each move, we calculate the change in the mutual potential energy, AU. If At/ is negative, then we allow the move. If At/ is positive, we allow the move with a probability of expi—U/kaT). 

To decide whether to allow the move or not, we generate a random number between 0 and 1. If this random number is less than cx i —U/k T), we allow the move. If the random number is greater than e.xp —U/k T) we leave the particle in its old position. 

---

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

Mbiler-Krumbhaar H and Binder K 1973 Dynamic properties of the Monte-Carlo method in statistical mechanics J. Stat. Phys. 8 1-24... 

Binder K (ed) 1995 The Monte Carlo Method in Condensed Matter Physics vol 71 Topics in Applied Physics 2nd edn (Berlin Springer)... 

Salsburg Z W, Jacobson J D, Fickett W and Wood W W 1959 Application of the Monte Carlo method to the lattice gas model. Two dimensional triangular lattice J. Chem. Phys. 30 65-72... 

Heermann D W and Burkitt A N 1995 Parallel algorithms for statistical physics problems The Monte Carlo Method In Condensed Matter Physios vol 71 Toplos In Applied Physios ed K Binder (Berlin Springer) pp 53-74... 

In this chapter we shall discuss some of the general principles involved in the two most common simulation techniques used in molecular modelling the molecular dynamics and the Monte Carlo methods. We shall also discuss several concepts that are common to both of these methods. A more detailed discussion of the two simulation methods can be found in Chapters 7 and 8. 

A sequence of successive configurations from a Monte Carlo simulation constitutes a trajectory in phase space with HyperChem, this trajectory may be saved and played back in the same way as a dynamics trajectory. With appropriate choices of setup parameters, the Monte Carlo method may achieve equilibration more rapidly than molecular dynamics. For some systems, then, Monte Carlo provides a more direct route to equilibrium structural and thermodynamic properties. However, these calculations can be quite long, depending upon the system studied. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution cui ve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probabihty of occurrence. The cumulative probabihty of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

The use of the Monte Carlo method in project appraisal was illustrated by Holland et al. [F. A. Holland, F. A. Watson, and J. K. Wilkinson, Chem. Eng., 81, 76-79 (Feb. 4, 1974)]. The cumulative-probability cui-ves of (DCFRR) and (NPV) can never be more accurate than the opinions on which they are based, and comparable accuracy can be obtained by the use of S-shaped cui ves with relatively small computational effort. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

N Metropolis, S Ulam. The Monte Carlo method. J Am Stat Assoc 44 335-341, 1949. 

A eomparison of the predieted erystallizer PSD performanee using the Monte Carlo method with data for both ealeium oxalate and ealeium earbonate preeipitation (Figure 8.27). 

K. Binder, ed.. Applications of the Monte Carlo Method in Statistical Physics. Berlin Springer, 1984. 

The study of surface chemical reaction processes using computer simulation techniques is quite an active field of research. Within this context the Monte Carlo method emerges as a powerful tool which contributes to the... 

The main notion of the percolation theory is the so-called percolation threshold Cp — minimal concentration of conducting particles C at which a continuous conducting chain of macroscopic length appears in the system. To determine this magnitude the Monte-Carlo method or the calculation of expansion coefficients of Cp by powers of C is used for different lattices in the knots of which the conducting par-... 

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

Levesque D, Weis JJ, Hansen JP (1984) In Binder K (ed) Topics in current physics 36 -applications of the Monte Carlo method in statistical physics, chap 2. Springer, Berlin Heidelberg New York... 

Figure 1.10. The figure demonstrates what is obtained when the Monte Carlo method (cf. Section 3.5.5) is used to simulate normally distributed values each histogram (cf. Section 1.8.1) summarizes 100 measurements obviously, many do not even come close to what one expects under the label bell curve. ...

---