Monte Carlo Method and Computer Simulation

Date: 04/04/2002 at 21:26:34
From: Alina
Subject: Monte Carlo Method *Probability*

What is the Monte Carlo method? I know it's for estimating  
probability, but what is the point of it? If it were on a test, what 
would be asked?

Date: 04/06/2002 at 06:24:51
From: Doctor Mitteldorf
Subject: Re: Monte Carlo Method *Probability*

For many problems in physics, it is impossible to calculate what 
actually happens. Sure, we know the laws governing the situation, but 
if there are more than two or three objects obeying these laws, the 
equations become so complicated that they're impossible to solve 
exactly.

In the 1940's, when calculation machines were crude and the electronic 
computer had not been invented, scientists were trying to calculate 
the amount of uranium they would need to make an atomic bomb. They ran 
into calculation problems where they understood the way atoms behave, 
and could calculate probabilities for any one atom, but when many 
atoms were placed together, the equations became too complicated to 
solve. So they had an idea: get a whole lot of people in a room, and 
let each one have the equations of one single atom. The person would 
be instructed to act in a way that was partly predictable and partly 
random, just the way real atoms do. In real atomic bombs, atoms 
release neutrons when they split in two, and the neutrons have some 
probability of hitting another atom, causing it to split in two as 
well. Inside the room, people would decide when they were "split" 
based on their equations, and if they did split, they would pass slips 
of paper to their neighbors that indicated how many neutrons were 
coming out of them, and that information was used to help calculate 
the probabilities that the neighbor would "split" as well.

This was a very complicated and very expensive way to calculate 
something, because it involved so many people with calculating 
machines, working for many days. Only atomic scientists hired by the 
government could afford it.

But once the computer was invented and became cheap, every scientist 
could afford to do this kind of thing. The computer is really good at 
doing many calculations quickly, keeping the information about what 
happens in different places, and moving it around to other parts of 
the calculation.  

So now there are many thousands of scientists who do this kind of 
calculation all the time. It is no longer called "Monte Carlo method," 
but is simply called "computer simulation" or sometimes "computer 
modeling." 

Computer simulation is a tool for calculating everything from weather 
forecasts to traffic flow to stock market fluctuations. In my own 
work, I use computer simulation to study biological evolution.

You can read more about the history of the Monte Carlo method at this 
site by Sabri Pllana:

   http://www.geocities.com/CollegePark/Quad/2435/ 

You can read more about simulations for students, and download 
examples that you can try yourself from the Center for Connected 
Learning and Computer-Based Modeling at Northwestern University:

   http://www.ccl.sesp.northwestern.edu/ 

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/