The uncertainty of propagation that is inherent to non-linear dynamical systems is among the fundamental problems that lead to their unpredictability, or even their indeterminism if our space-time was a discrete one. To clarify this point, despite the numerous research input, no numerical simulation has been published with the purpose of enabling compute explicitly the uncertainty propagation during time, probably due to the complexity of the n-body problem even with point-like masses. This problem is in a general way apprehended by utilizing arithmetical approaches such as Gaussian Mixture Models and Polynomial Chaos Expansions. Therefore, there is a genuine need to develop numerical models that can enhance the computation of uncertainty propagation of the n-body problem by considering the non-point-like case of interactions into a billiard.
To this end, a team of researchers led by Dr. Philippe Guillemant from the Aix-Marseille University in France investigated the possibility to quantify the truly multiple evolution of trajectories in an isolated dynamical system, that is occurring after the decay of its physical information (position and momentum phase states) below a critical value, due to a finite density of information. Their work is currently published in the research journal, Annals of Physics.
The research team commenced their work by computing conservation laws in such a way that the bounded density of information in a discrete space-time was taken into account. They selected an idealized 2D billiard to study the influence of the key parameters on the loss of information, since it was seen to be among the simplest toy models using basic physical laws. Eventually, an evaluation of the growing law of the number of multiple histories was undertaken, enabling estimation of the number of extra dimensions that were necessary to maintain determinism after the critical step.
The authors observed that based on the computations undertaken, the amount of deterministic information that is calculable using physical laws was seen to be of the same order as the amount of information that was originally contained in the initial conditions. Furthermore, they noted that after certain duration, all billiard states became possible final states, independent of initial conditions.
The Philippe Guillemant and colleagues study has successfully presented a numerical model capable of computing the uncertainty propagation of the n-body problem by considering the non-point-like case of interactions into a billiard. It has been seen that, due to the possible incompleteness of governing laws at discrete level, a 3D discrete space–time would need 3 additional dimensions to specify final conditions, and even 3 other ones to also describe alternative present paths, like in many-worlds theory. Therefore, if our space–time is really a discrete one, one would need to introduce 6 extra dimensions in order to provide out of time supplementary constraints that specify which history of our multiverse should be played.
Philippe Guillemant, Marc Medale, Cherifa Abid. A discrete classical space–time could require 6 extra-dimensions. Annals of Physics volume 388 (2018) pages 428–442.
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