How do Monte Carlo event generators work in general?

Question

Can anyone explain topologically how Monte Carlo event generators work? I'm struggling to find a source online which isn't super-specific to a certain area and instead discusses how they work at a basic/introductory level.

Answer 1

It is difficult to know at what level to pitch a useful answer, since you have not told us how much knowledge you already have. However, here is a basic answer that is not specific to particle physics.

A Monte Carlo simulation (named after the famous casino) is any algorithm that uses random sampling. Typically, a stochastic model of the subject is created, in which different events can happen at each time step - and these events may affect the set of events that is available at a later time step, or their probabilities. On each run of the simulation, the actual events that occur are chosen based on assumed or know probability distributions for each type of event. The initial state of the model may also be chosen at random. Each simulation run will produce a different sequence of events and so a different outcome, but what is usually of interest is not the actual outcome from any individual run, but the observed distribution of outcomes. To collect enough data on the distribution of outcomes, the simulation may be run hundreds, thousands, or even millions of times.

For example, to estimate the likelihood of throwing exactly three sixes from five dice, you could construct a model that generates five random integers chosen with a uniform distribution from one to six, then run this (very simple) simulation a thousand times and count how many times three sixes occur. If you thought one or more of the dice were biased then you could adjust the distributions in your model accordingly. Of course, in this example it is not difficult to find the mathematically exact answer - but Monte Carlo simulations are usually used in more complicated scenarios where an exact answer is not known or is very difficult to compute.

The main drawback of Monte Carlo simulations is their dependence on a set of known or assumed probability distributions for each type of event. If you assume and apply an incorrect probability distribution for one event type, this can have a big effect on the results of the simulation (for example, if we assumed that dice were fair in the dice simulation above, and the actual dice were not fair, then our simulation results would be worthless).