Monte Carlo Methods

The Geant4 toolkit uses a combination of the composition and rejection Monte Carlo methods. Only the basic formalism of these methods is outlined here. For a complete account of the Monte Carlo methods, the interested user is referred to the publications of Butcher and Messel, Messel and Crawford, or Ford and Nelson [1,2,3].

Suppose we wish to sample $x$ in the interval $[x_1,\ x_2]$ from the distribution $f(x)$ and the normalised probability density function can be written as :

$$f(x) = \sum_{i=1}^{n} N_{i} f_{i} (x) g_{i} (x)$$ (2.1)

where $N_i>0$, $f_i(x)$ are normalised density functions on $[x_1,\ x_2]$ , and $0 \leq g_i (x) \leq 1$.

According to this method, $x$ can sampled in the following way:

1. select a random integer $i \in \{1,2,\cdots n\}$ with probability proportional to $N_i$
2. select a value $x_0$ from the distribution $f_i(x)$
3. calculate $g_i (x_0)$ and accept $x = x_0$ with probability $g_i (x_0)$;
4. if $x_0$ is rejected restart from step 1.

It can be shown that this scheme is correct and the mean number of tries to accept a value is $\sum_{i} N_i$.

In practice, a good method of sampling from the distribution $f(x)$ has the following properties:

• all the subdistributions $f_i(x)$ can be sampled easily;
• the rejection functions $g_i(x)$ can be evaluated easily/quickly;
• the mean number of tries is not too large.

Thus the different possible decompositions of the distribution $f(x)$ are not equivalent from the practical point of view (e.g. they can be very different in computational speed) and it can be useful to optimise the decomposition.

A remark of practical importance : if our distribution is not normalised

$$\int_{x_1}^{x_2} f(x)dx = C > 0$$

the method can be used in the same manner; the mean number of tries in this case is $\sum_ {i} N_i/C$.