On leveraging the chaotic and combinatorial nature of deterministic n-body dynamics on the unit m-sphere in order to implement a pseudo-random number generator

The goal of this paper is to describe how to implement a pseudo-random number generator by using deterministic n-body dynamics on the unit m-sphere. Throughout this paper we identify several types of patterns in dynamics, along with ways to interrupt the formation of these patterns. 1 Nondeterministic and deterministic/chaotic-deterministic physical behaviour In terms of the laws of physics – as far as how we model them today, anyway – there are two types of ways that a system can evolve over time: nondeterministically and deterministically. An uncomplicated nondeterministic case is the evolution of an electron’s position over time. As per the rules of quantum mechanics, our knowledge of an electron’s position is given by a radially symmetric probability function that falls off with increasing radial distance in a nonlinear way. This means we have only a cloudy idea of where the electron is most likely to be found before we actually probe for its true position by using a photon. Upon probing for the electron’s position, we would observe that it is most likely fairly close to the centre of the probability cloud, but that the two angles – θ ,φ of the standard spherical coordinates – are truly and unpredictably random. If we were to repeat this probe experiment many times, we would find that there is absolutely no pattern in how the two angles that we measure are distributed over time. An uncomplicated deterministic case is that of the Newtonian gravitational relationship between two bodies of equal mass that are on opposite sides of a shared circular orbit path. This coplanar orbit relationship is anything but random, insomuch that it produces prolonged and stable repetitious cyclical motion, which in turn forms a pattern. If one of the two bodies is perturbed so that it is moved just slightly off of the circular orbit path, then we would find that the two bodies simply form new, elliptical orbit paths, and continue to undergo prolonged and stable pattern-forming repetitious cyclical motion. A complicated deterministic case is that of the Newtonian gravitational relationships between three bodies of equal mass that are equidistantly distributed along a shared circular orbit path. Similar to the uncomplicated case of two bodies, this coplanar orbit relationship is anything but random, insomuch that it also produces prolonged pattern-forming repetitious cyclical motion. Unlike the uncomplicated case however, this case is unstable. If one of the three bodies is perturbed so that the three bodies are no longer equidistant, then we would find that this slight asymmetry in the strength of the gravitational interaction amongst the three bodies naturally feeds upon itself over time, causing two of the bodies to speed toward each other at an accelerating rate. Ultimately, this dynamic asymmetry in the strength of the gravitational interaction amongst the three bodies generally interrupts any chance that they will ever again re-establish anything remotely similar to their original prolonged equidistant circular orbit relationship. This devolution from predictable, repetitious cyclical motion into pseudorandom, non-repetitious acyclical motion due to some slight asymmetry that feeds upon itself over time can be described as the transition into chaos. For more information on the chaotic nature of deterministic many-body dynamics, see [1, 2]. ∗Little Red River Park, SK Canada – email: shalayka@gmail.com

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