The Monte Carlo Method

The Monte Carlo method is often referred to as a ‘computer experiment’. One might think of this as a way of conveying the fact that the output of simulations is not an equation, as in conventional theory. Instead, numbers appear on the computer screen in somewhat the same way that numbers appear on a measuring device in the laboratory. Thus there is the implication that somehow simulations are a bit of a ‘black box’ and that the use of the computer is hiding the underlying physics. The purpose of this note is partly to emphasize some of the mathematical rigor behind Monte Carlo: It is not a happy accident that the computer is generating configurations with the desired probability distribution! Indeed, the fundamental equations underlying simulations are the same as analytic theories, and one can view simulations as a way of solving the mathematics (differential equations) when it becomes too complicated for analytic techniques. With all that said, it is still useful to pursue the ‘Monte Carlo as experiment’ point of view. Consider the process of making a measurement in the laboratory. Nature prepares a ‘configuration’ i of the system, and the experimentalist takes that configuration and records a value for some quantity of interest. To get better statistics (or perhaps inevitably because of finite measuring time) nature actually produces many configurations, and the experimentalist averages over the values obtained. It is useful to emphasize that no matter how long the experimentalist measures, the configurations she sees are an incredibly small subset of those that the system is capable of exploring. Nature uses some very complex rules for time evolving the system from configuration to configuration, for example the many particle Newton or Schroedinger equations. These rules govern the states that the experimentalist sees, and hence the data she takes. Here’s one useful way to think about computer simulations: The goal of a computer simulation is to devise a method where the computer plays a similar role to that of nature for the experimentalist. That is, the computer generates configurations upon which we make measurements. This leaves us with the problem of devising instructions for the computer that replicate nature’s way of generating configurations. One approach to constructing a simulation would be actually coding up the microscopic equations governing the system’s time evolution. Simulations of classical systems going under the name ‘molecular dynamics’ are actually done precisely this way. One computes the force Fn on each particle n, uses the force to compute the acceleration an = Fn/m, and then moves the velocity and position forward a small time interval dt with, vn → vn + an dt; xn → xn + vn dt. But in the spirit of statistical mechanics, we really don’t care about the microscopic time evolution and the paths xn(t) and vn(t) the particles take in phase space. All we really need is to replicate the probability P ({xn, vn}) that nature uses to generate her configurations. If we can do that, we’ll get the same answers as the experimentalist!