Randomized form factor calculation

 

The randomized approach is based on the recognition that the formula defining the form factors can be taken to represent the probability of a quite simple event if the underlying probability distributions are defined properly.

An appropriate such event would be a surface j being hit by a particle leaving surface i. Let us denote the event that a particle leaves surface by . Expressing the probability of the ``hit'' of surface j by the total probability theorem we get:

The hitting of surface j can be broken down into the separate events of hitting the various differential elements composing . Since hitting of and hitting of are exclusive events if :

 

Now the probability distributions involved in the equations are defined:

1. Assume the origin of the particle to be selected randomly by uniform distribution:

   

2. Let the direction in which the particle leaves the surface be selected by a distribution proportional to the cosine of the angle between the direction and the surface normal:

   

The denominator guarantees that the integration of the probability over the whole hemisphere yields 1, hence it deserves the name of probability density function. Since the solid angle of from is where r is the distance between and , and is the angle of the surface normal of and the direction of , the probability of equation 1.15 is:

where is the indicator function of the event `` is visible from ''.

Substituting these into the original probability formula:

This is exactly the same as the formula for form factor . This probability, however, can be estimated by random simulation. Let us generate n particles randomly using uniform distribution on the surface i to select the origin, and a cosine density function to determine the direction. The origin and the direction define a ray which may intersect other surfaces. That surface will be hit whose intersection point is the closest to the surface from which the particle comes. If shooting n rays randomly surface j has been hit times, then the probability or the form factor can be estimated by the relative frequency:

Two problems have been left unsolved:

• How can we select n to minimize the calculations but to sustain a given level of accuracy?
• How can we generate uniform distribution on a surface and cosine density function in the direction?

Addressing the problem of the determination of the necessary number of attempts, we can use the laws of large numbers.

The inequality of Bernstein and Chebyshev [Rén81] states that if the absolute value of the difference of the event frequency and the probability is expected not to exceed with probability , then the minimum number of attempts (n) is:

The generation of random distributions can rely on random numbers of uniform distribution in [0..1] produced by the pseudo-random algorithm of programming language libraries. Let the probability distribution function of the desired distribution be P(x). A random variable x which has P(x) probability distribution can be generated by transforming the random variable r that is uniformly distributed in [0..1] applying the following transformation: