The Monte Carlo Method

The Monte Carlo Method refers to a broad group of statistical procedures used to determine outcomes from a random set of variables. The method has applications
in numerous areas of science and engineering.

The term Monte Carlo was named after the Principality of Monaco and it’s use of the roulette wheel, a random number generator, in casinos. It came into use during the early 1940’s when many of the scientists and engineers working on the atomic bomb project referred to probabilistic methods for modeling random neutron diffusion as “Russian Roulette”.

True systematic development occurred in 1948 when Fermi, Ulam and Metropolis used Monte Carlo estimates for the eigenvalues (roots of a matrix equation) of the Schrodinger equation.

Procedures
The Method can be summarized in five steps:
1. Identify a model- something that can generate random outcomes: coins, dice, etc.
2. Define a trial- set parameters for each.
3. Was trial a success?
4. Repeat steps 1-3. 100 trials are acceptable.
5. Find estimated probability of a success.
Probability of success=number of successes/number of trials.

As can be seen, it is straightforward and can be taught to small children with some slight modifications. Numerous middle school teachers are now using Monte Carlo Methods to teach their students basic concepts of statistics and other sciences that use random events.

One random number experiment performed for fun in the nineteenth century and reproduced here is throwing darts at parallel lines on a board to estimate pi=3.14. based on the number of intersections between the dart and the line.
In this case we will be using darts on a quadrant of a circle. This makes the experiment simpler and does not require as many trials to reach an estimate.

The radius of the quadrant of the circle is the same as the side of the square.
If we throw enough darts for a trial, say 1,000, we can estimate the areas of the square and circle quadrant and use this data to estimate pi.
We can set up the equation as follows;
# darts hitting circle quadrant= area of circle quadrant (Ac)/# darts hitting inside square= area of square (As).
The area of a circle is pi*r2 divided by 4 for the quadrant and the area of a square is r2. Therefore, Ac/As= 1/4pi*r2/r2. A little algebra and we have 4Ac/As=pi.

Experiment courtesy of the University of Nebraska-Lincoln Physical Chemistry Lab(Chem 484) by TA Joy Woller, Spring 1996.

The Monte Carlo Method can be used in just about any situation where there is a large amount of trial data. Some of the more common are:
1. Classical- also called the Boltzman distribution, used for thermodynamic properties, energy structures, and rate coefficients.
2. Quantum- used to compute quantum-mechanical energies.
3. Path Integral-used to compute thermodynamic properties.
4. Volumetric- used for molecular volumes.
5. Simulation- an algorithm used to simulate a statistical process.

Economics, nuclear physics, population estimates, traffic flow, etc. etc., all benefit from this flexible method. After all, the number of random events in the universe is virtually infinite, and we need a method to make sense of this.

References: All Internet.