3.5.5  The Monte Carlo Method

The Monte Carlo technique is very well used in experimental high energy physics. Fermi introduced it in the decade of 1940.

The technique uses the random generation of numbers to calculate and to simulate. For example, to calculate integrals that analytically could be very complicate, to calculate the acceptance of complicate spectrometers, etc. For instance, to simulate the production of particles, the pass of particles through material medium, the decay of a particle into two or more particles, the polarization of a particle, etc.

Practically any phenomenon in the nature can be simulated in the computer, using the Monte Carlo technique. The limitations that some decades ago the technique had, as reduced memory in the computers, slowness in the computer operative systems, generation of truly random numbers, reduced storage capacity of the computers, etc. have been superated by the new generation of computers. Therefore any phenomenon can be simulated in the computers using the Monte Carlo technique.

We will illustrate with an example the Monte Carlo technique.

Example:

The decay of a particle of mass $M$ into two particles of mass $m_1$ and $m_2$ respectively. See Figure 11.

The momenta of the daughter particles are equal in module and of opposite directions, in the $M$ center of mass rest frame. The situation is illustrated by the Figure 11. These magnitudes are fixed when the masses $M$, $m_1$, and $m_2$ are fixed.

The difference from event to event is the angle $(\theta)$, that the momenta of the particles made with the y axis, and the azimuthal angle $(\phi)$.

The numbers $cos\theta$ and $cos\phi$ are generated randomly; both are between $(-1, +1)$. Using the computer, it is generated a random number $\epsilon$. This has a value between $(0, +1)$. The function $(2\epsilon-1)$ is the distribution for $cos\theta$ and for $cos\phi$. The function goes between $(-1, +1)$.

The distributions of $cos\theta$ and of $cos\phi$ are flat. The student can verify these propositions for himself or by herself. He or she must write a computer program to generate the histograms of $cos\theta$ and $cos\phi$.

The student can generate non uniform distributions directly -using a weight function-, or weighing the uniform distributions in the way he or she chooses.

The base of the Monte Carlo method is the generation of random numbers. And every computational technique where random numbers are involved is called Monte Carlo method.

The Monte Carlo method is very helpful in high energy physics, both theoretically and experimentally. In the last one to calculate the detector efficiencies, for the detectors are very complicate; and in the first one to calculate extremely complicate integrals.