The Problem of Random Flights

By: Pedro Barata de Tovar Sá

I am currently a 3rd year undergraduate Math student at University of Coimbra. During this year, I had the privilege of getting a grant from the Gulbenkian-foundation’s program “New Talents in Mathematics“. In the scope of this program, I proposed to investigate how chimney smoke disperses in the air, with the guidance of my supervisor Alexander Kovacec.

This in turn led us to study an important problem on probability theory. Let me explain. In 1905 Karl Pearson, the well known statistician, asked this question:

A man starts from a point O and walks l yards in a straight line, he then turns through any angle whatever and walks another l yards in a second straight line. He repeats the process n times. I require the probability that after these n steps he is a distance between r and r+dr from his origin O.

This problem which is basically concerned with the probability density of a sum of two dimensional random vectors was soon found to have numerous applications. Lord Rayleigh, Nobel laureate for physics considered a composition of n vibrations each of unit amplitude and arbitrary phase and sought the distribution of the resulting intensity. Using de Moivre’s formula the relation between cossines, sines and complex exponentials and interpreting the latter as unit vectors in the complex plane we see that the question is entirely equivalent to Pearson’s question.

As another application, Major Donald Ross, Nobel laureate for medicine in 1902 for work on malaria, asked: Suppose a box containing a million gnats were to be opened in the center of a large plain, and that the insects were allowed to wander freely in all directions, how many of them would be found after death at a given distance from the place where the box was opened?

The problem of Pearson was solved by the Belgian mathematician J. C. Kluyver in 1906 by means of a representation of the probability Prob(R_n ≤ r) via an integral of an expression involving Bessel functions J0 and J1. Here R_n is the random variable giving the distance of a particle from the start after it has done n random jumps of unit length in the plane. No elementary solution (closed expression) is known to this day and much efforts have been expended to integrate Kluyver’s solution efficiently.

The problem can be asked in higher dimensions as well. It is then interesting to see that in three dimensions an elementary solution exists. In fact such was found by L. R. G. Treloar 1947 for the British rubber industry: The title of his paper is: The statistical length of long chain molecules.

But, when I started to work on the topic, we didn’t know anything of this. Since we did not find the answers we wanted in the books on probability theory or random walks but thought the problem should have been studied before we put the question to the site math.stackexchange.com It was from there that a pseudonymous answer led us to a survey article by Dutka from 1985 which led us to know that the problem is known as the problem of Random flights. At that time we had already found Treloar’s solution completely independently and very different from his. In particular we discovered that the elementaricity of the three dimensional problem has to do with a theorem discovered by Archimedes: the area of the region of a sphere contained between two parallel planes intersecting it depends only on the distance between the planes.

Our findings can be read in the manuscript “On the probability to be after n random jumps of unit length in space within a distance r from the start: the problem of random flights“. In its final sections one also finds a report about the history of the problem.

I have now also a conjecture for a piecewise polynomial solution in odd dimensional space which I hope to be able to prove in the next months together with my supervisor.