Numerical verification of the microscopic time reversibility of Newton's equations of motion: Fighting exponential divergence [IMA]

Numerical solutions to Newton’s equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible. This is due to the exponential growth of errors introduced by the integration scheme and the numerical round-off in the least significant figure. This secular growth of error is sometimes attributed to the increase in entropy of the system even though Newton’s equations of motion are strictly time reversible. We demonstrate that when numerical errors are reduced to below the physical perturbation and its exponential growth during integration the microscopic reversibility is retrieved. Time reversibility itself is not a guarantee for a definitive solution to the chaotic N-body problem. However, time reversible algorithms may be used to find initial conditions for which perturbed trajectories converge rather than diverge. The ability to calculate such a converging pair of solutions is a striking illustration which shows that it is possible to compute a definitive solution to a highly unstable problem. This works as follows: If you (i) use a code which is capable of producing a definitive solution (and which will therefore handle converging pairs of solutions correctly), (ii) use it to study the statistical result of some other problem, and then (iii) find that some other code produces a solution S with statistical properties which are indistinguishable from those of the definitive solution, then solution S may be deemed veracious.

S. Zwart and T. Boekholt
Tue, 6 Feb 18
28/62

Comments: Accepted for publication in Communications in Nonlinear Science and Numerical Simulation. Calculations are performed with Brutus as part of the AMUSE framework. Simultion data will become available online

|

Stability and self-organization of planetary systems [EPA]

We show that stability of planetary systems is intimately connected with their internal order. An arbitrary initial distribution of planets is susceptible to catastrophic events in which planets either collide or are ejected from the planetary system. These instabilities are a fundamental consequence of chaotic dynamics and of Arnold diffusion characteristic of many body gravitational interactions. To ensure stability over astronomical time scale of a realistic planetary system — in which planets have masses comparable or those of planets in the solar system — the motion must be quasi-periodic. A dynamical mechanism is proposed which naturally evolves a planetary system to a periodic state from an arbitrary initial condition. A planetary self-organization predicted by the theory is similar to the one found in our solar system.

R. Pakter and Y. Levin
Mon, 29 Jan 18
6/54

Turbulent superstructures in Rayleigh-Bénard convection [CL]

Turbulent Rayleigh-B\’enard convection displays a large-scale order in the form of rolls and cells on lengths larger than the layer height once the fluctuations of temperature and velocity are removed. These turbulent superstructures are reminiscent of the patterns close to the onset of convection. They are analyzed by numerical simulations of turbulent convection in fluids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to $10^7$. For each case, we identify characteristic scales and times that separate the fast, small-scale turbulent fluctuations from the gradually changing large-scale superstructures. The characteristic scales of the large-scale patterns, which change with Prandtl and Rayleigh number, are also found to be correlated with the boundary layer dynamics, and in particular the clustering of thermal plumes at the top and bottom plates. Our analysis suggests a scale separation and thus the existence of a simplified description of the turbulent superstructures in geo- and astrophysical settings.

A. Pandey, J. Scheel and J. Schumacher
Tue, 16 Jan 18
43/79

Comments: 16 pages (incl. Supplementary Material), 12 figures (all with downsized figure size)

Chaotic dynamics in the planar gravitational many-body problem with rigid body rotations [EPA]

The discovery of Pluto’s small moons in the last decade brought attention to the dynamics of the dwarf planet’s satellites. Recent work has considered resonant interactions in the orbits of Pluto’s small moons, with the Pluto-Charon system apparently inducing rotational chaos in non-spherical moons without the need of resonance. However, New Horizons observations suggest that despinning due to tidal dissipation has not taken place. Still, a tidally evolving Styx does appear to exhibit intermittent obliquity variations and episodes of tumbling, suggesting some form of chaos in the rotational dynamics. With these systems in mind, we study a planar $N$-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar $N$-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid ‘Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits non-spherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in $N$-body systems such as the Pluto system where some of the bodies are highly non-spherical.

J. Kwiecinski, A. Kovacs, A. Krause, et. al.
Tue, 9 Jan 18
94/94

Comments: 27 pages, 7 figures. arXiv admin note: text overlap with arXiv:1701.05594 by other authors

Dynamical analysis of bounded and unbounded orbits in a generalized Hénon-Heiles system [CL]

The H\’enon-Heiles potential was first proposed as a simplified version of the gravitational potential experimented by a star in the presence of a galactic center. Currently, this system is considered a paradigm in dynamical systems because despite its simplicity exhibits a very complex dynamical behavior. In the present paper, we perform a series expansion up to the fifth-order of a potential with axial and reflection symmetries, which after some transformations, leads to a generalized H\’enon-Heiles potential. Such new system is analyzed qualitatively in both regimes of bounded and unbounded motion via the Poincar\’e sections method and plotting the exit basins. On the other hand, the quantitative analysis is performed through the Lyapunov exponents and the basin entropy, respectively. We find that in both regimes the chaoticity of the system decreases as long as the test particle energy gets far from the critical energy. Additionally, we may conclude that despite the inclusion of higher order terms in the series expansion, the new system shows wider zones of regularity (islands) than the ones present in the H\’enon-Heiles system.

F. Dubeibe, A. Riano-Doncel and E. Zotos
Thu, 7 Dec 17
10/72

Quantum chaos of dark matter in the Solar System [CL]

We perform time-dependent analysis of quantum dynamics of dark matter particles in the Solar System. It is shown that this problem has similarities with a microwave ionization of Rydberg atoms studied previously experimentally and analytically. On this basis it is shown that the quantum effects for chaotic dark matter dynamics become significant for dark matter mass ratio to electron mass being smaller than $2 \times 10^{-15}$. Below this border multiphoton diffusion over Rydberg states of dark matter atom becomes exponentially localized in analogy with the Anderson localization in disordered solids. The life time of dark matter in the Solar System is determined in dependence on mass ratio in the localized phase and a few photon ionization regime. Various implications of these quantum results are discussed for the capture of dark matter from Galaxy and its steady-state density distribution.

D. Shepelyansky
Thu, 23 Nov 17
48/52

We perform time-dependent analysis of quantum dynamics of dark matter particles in the Solar System. It is shown that this problem has similarities with a microwave ionization of Rydberg atoms studied previously experimentally and analytically. On this basis it is shown that the quantum effects for chaotic dark matter dynamics become significant for dark matter mass ratio to electron mass being smaller than $2 \times 10^{-15}$. Below this border multiphoton diffusion over Rydberg states of dark matter atom becomes exponentially localized in analogy with the Anderson localization in disordered solids. The life time of dark matter in the Solar System is determined in dependence on mass ratio in the localized phase and a few photon ionization regime. Various implications of these quantum results are discussed for the capture of dark matter from Galaxy and its steady-state density distribution.