The stochastisation hypothesis and the spacing of planetary systems

Abstract

Il était générallement admis que les probabilités n'étaient que l'expression de notre ignorance.Pour les systèmes instables, il n'en est pas ainsi. Les probabilités acquièrent une signication dynamique intrinsèque.Peu à peu se dessine dès lors une nouvelle rationalité dans laquelle probabilité n'est pas ignorance et science ne se confond pas avec certitude. C'est à ce prix que la notion d'évolution et avec elle les notions d'événement et de créativité font leur entrée dans les lois fondamentales de la nature.Ilya Progogine, Les lois du Chaos, Ed. Flammarion, 1994 Abstract. We introduce the stochastisation hypothesis which aims to provide a framework to deal with physical systems in random environment. We apply the stochastisation hypothesis in two dierent cases : in the study of the dynamics of a protoplanetary nebula and in the Chaotic long term behaviour of a generic planetary system using previous works of S. Albeverio and al., L. Nottale and J. Laskar. These results give both a particular law for the distribution of planetary orbits.

Self-organizing systems in planetary physics: Harmonic resonances of planet and moon orbits

“…The harmonic orbit resonance model makes a very specific prediction about five harmonic orbit and distance ratios (rather than the logarithmic spacing assumed in the generalized Titius-Bode law), which can be tested with numerical simulations. Such a quantized geometric pattern is also distinctly different from random systems (Dole 1970;Lecar 1973;Dworak and Kopacz 1997;Dworak and Kopacz 1997;Hayes and Tremaine 1998;Lynch 2003;Neslusan 2004;Cresson 2011).…”

Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena

“…Inspired by the work of Feynman and Kolmogorov, J. C. Zambrini [107,51] developed a Euclidean approach to Schrödinger processes looked upon as Bernstein processes where a couple of forward and backward processes, suitably connected, are considered. C. Léonard [83], on one hand, and Cresson and Darses [49,50] managed in a way to connect Nelson's and Zambrini's work.…”

“…A stochastic model which can be brought in connection with laws of the Titius-Bode type has been introduced in [2], based on the observation that in a symmetric diffusion process with drift term of gradient type confinement phenomena can occur, due to the formation of different ergodic components, in correspondence with hypersurfaces where the density of the invariant measure vanishes, see also [92,91,6]. This and similar models have been further explored in various other contexts, see [97,49]. A. Truman and coworkers have obtained very interesting results on the dynamics of planetesimal formation in an original nebulous cloud and asymptotic convergence of their orbits to circular resp.…”

Semiclassical stochastic mechanics for the Coulomb potential with applications to modelling dark matter

“…From an alternative perspective, Nelson's stochastic mechanics can be viewed as a stochastic perturbation of classical Newtonian mechanics, which happens to agree with the predictions of standard quantum mechanics. In this sense the equations we consider here can be viewed as a stochastic perturbation of a classical two body problem, and several such perturbations and their application to problems in astronomy have been studied previously in the literature for instance 1,7,8,17,23,28 . Here we have to emphasise that the Bohr correspondence limit for our family of wave functions gives rise to an additional Bohm type potential proportional to the modulus squared of the logarithmic derivative of |ψ| 2 .…”