scholarly journals Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length

2020 ◽  
Vol 493 (3) ◽  
pp. 3932-3937 ◽  
Author(s):  
T C N Boekholt ◽  
S F Portegies Zwart ◽  
M Valtonen
Keyword(s):  
Dynamical Systems ◽  
Free Fall ◽  
Quantitative Relation ◽  
Double Precision ◽  
Planck Length ◽  
Rounding Errors ◽  
Massive Black Holes ◽  
System A ◽  
N Body Problem ◽  
Three Body

ABSTRACT Chaos is present in most stellar dynamical systems and manifests itself through the exponential growth of small perturbations. Exponential divergence drives time irreversibility and increases the entropy in the system. A numerical consequence is that integrations of the N-body problem unavoidably magnify truncation and rounding errors to macroscopic scales. Hitherto, a quantitative relation between chaos in stellar dynamical systems and the level of irreversibility remained undetermined. In this work, we study chaotic three-body systems in free fall initially using the accurate and precise N-body code Brutus, which goes beyond standard double-precision arithmetic. We demonstrate that the fraction of irreversible solutions decreases as a power law with numerical accuracy. This can be derived from the distribution of amplification factors of small initial perturbations. Applying this result to systems consisting of three massive black holes with zero total angular momentum, we conclude that up to 5 per cent of such triples would require an accuracy of smaller than the Planck length in order to produce a time-reversible solution, thus rendering them fundamentally unpredictable.

scholarly journals Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

1973 ◽  
Vol 28 (2) ◽  
pp. 206-215
Author(s):  
Hanns Ruder
Keyword(s):  
Coordinate System ◽  
Perturbation Expansion ◽  
Rotational Energy ◽  
The Body ◽  
System A ◽  
N Body Problem ◽  
Definition Of ◽  
Fixed System ◽  
Body Fixed Coordinate System ◽  
Groundstate Energy

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.

scholarly journals Event-Response Ellipses: A Method to Quantify and Compare the Role of Dynamic Storage at the Catchment Scale in Snowmelt-Dominated Systems

Water ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 1824 ◽  
Author(s):  
Jessica Driscoll ◽  
Thomas Meixner ◽  
Noah Molotch ◽  
Ty Ferre ◽  
Mark Williams ◽  
...  
Keyword(s):  
Water Balance ◽  
Quantitative Relation ◽  
Catchment Scale ◽  
System A ◽  
Time Period ◽  
Ellipse Method ◽  
Streamflow Response ◽  
Dynamic Storage ◽  
Event Response

A method for quantifying the role of dynamic storage as a physical buffer between snowmelt and streamflow at the catchment scale is introduced in this paper. The method describes a quantitative relation between hydrologic events (e.g., snowmelt) and responses (e.g., streamflow) by generating event-response ellipses that can be used to (a) characterize and compare catchment-scale dynamic storage processes, and (b) assess the closure of the water balance. Event-response ellipses allow for the role of dynamic, short-term storage to be quantified and compared between seasons and between catchments. This method is presented as an idealization of the system: a time series of a snowmelt event as a portion of a sinusoidal wave function. The event function is then related to a response function, which is the original event function modified mathematically through phase and magnitude shifts to represent the streamflow response. The direct relation of these two functions creates an event-response ellipse with measurable characteristics (e.g., eccentricity, angle). The ellipse characteristics integrate the timing and magnitude difference between the hydrologic event and response to quantify physical buffering through dynamic storage. Next, method is applied to eleven snowmelt seasons in two well-instrumented headwater snowmelt-dominated catchments with known differences in storage capacities. Results show the time-period average daily values produce different event-response ellipse characteristics for the two catchments. Event-response ellipses were also generated for individual snowmelt seasons; however, these annual applications of the method show more scatter relative to the time period averaged values. The event-response ellipse method provides a method to compare and evaluate the connectivity between snowmelt and streamflow as well as assumptions of water balance.

scholarly journals Regular and Chaotic Motion in a Restricted Three–Body Problem of Astrophysical Interest

2000 ◽  
Vol 174 ◽  
pp. 281-285 ◽  
Author(s):  
J. C. Muzzio ◽  
F. C. Wachlin ◽  
D. D. Carpintero
Keyword(s):  
External Field ◽  
Body Problem ◽  
Stellar System ◽  
Three Dimensional ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Stellar Orbits ◽  
System A ◽  
Regular And Chaotic Motion ◽  
Three Body

AbstractWe have studied the motion of massless particles (stars) bound to a stellar system (a galactic satellite) that moves on a circular orbit in an external field (a galaxy). A large percentage of the stellar orbits turned out to be chaotic, contrary to what happens in the usual restricted three–body problem of celestial mechanics where most of the orbits are regular. The discrepancy is probably due to three facts: 1) Our study is not limited to orbits on the main planes of symmetry, but considers three–dimensional motion; 2) The force exerted by the satellite goes to zero (rather than to infinity) at the center of the satellite; 3) The potential of the satellite is triaxial, rather than spherical.

SAMPLING DYNAMICAL SYSTEMS

1999 ◽  
Vol 3 (4) ◽  
pp. 602-609 ◽  
Author(s):  
Melvin J. Hinich
Keyword(s):  
Dynamical Systems ◽  
Sampling Rate ◽  
Identification Problem ◽  
Frequency Component ◽  
Linear Dynamical Systems ◽  
System A ◽  
Fixed Sample ◽  
Single Input ◽  
Exogenous Time Series

Linear dynamical systems are widely used in many different fields from engineering to economics. One simple but important class of such systems is called the single-input transfer function model. Suppose that all variables of the system are sampled for a period using a fixed sample rate. The central issue of this paper is the determination of the smallest sampling rate that will yield a sample that will allow the investigator to identify the discrete-time representation of the system. A critical sampling rate exists that will identify the model. This rate, called the Nyquist rate, is twice the highest frequency component of the system. Sampling at a lower rate will result in an identification problem that is serious. The standard assumptions made about the model and the unobserved innovation errors in the model protect the investigators from the identification problem and resulting biases of undersampling. The critical assumption that is needed to identify an undersampled system is that at least one of the exogenous time series be white noise.

Periodic orbits in the free-fall three-body problem

2016 ◽  
Vol 60 (12) ◽  
pp. 1083-1089 ◽  
Author(s):  
V. V. Orlov ◽  
V. A. Titov ◽  
L. A. Shombina
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Free Fall ◽  
Three Body Problem ◽  
Three Body

scholarly journals Improving Weather Forecast Skill through Reduced-Precision Data Assimilation

2017 ◽  
Vol 146 (1) ◽  
pp. 49-62 ◽  
Author(s):  
Sam Hatfield ◽  
Aneesh Subramanian ◽  
Tim Palmer ◽  
Peter Düben
Keyword(s):  
Data Assimilation ◽  
Observation Error ◽  
Double Precision ◽  
Ensemble Size ◽  
Error Statistics ◽  
Rounding Errors ◽  
Precision Data ◽  
Ensemble Data Assimilation ◽  
Ensemble Data ◽  
Computational Resources

Abstract A new approach for improving the accuracy of data assimilation, by trading numerical precision for ensemble size, is introduced. Data assimilation is inherently uncertain because of the use of noisy observations and imperfect models. Thus, the larger rounding errors incurred from reducing precision may be within the tolerance of the system. Lower-precision arithmetic is cheaper, and so by reducing precision in ensemble data assimilation, computational resources can be redistributed toward, for example, a larger ensemble size. Because larger ensembles provide a better estimate of the underlying distribution and are less reliant on covariance inflation and localization, lowering precision could actually permit an improvement in the accuracy of weather forecasts. Here, this idea is tested on an ensemble data assimilation system comprising the Lorenz ’96 toy atmospheric model and the ensemble square root filter. The system is run at double-, single-, and half-precision (the latter using an emulation tool), and the performance of each precision is measured through mean error statistics and rank histograms. The sensitivity of these results to the observation error and the length of the observation window are addressed. Then, by reinvesting the saved computational resources from reducing precision into the ensemble size, assimilation error can be reduced for (hypothetically) no extra cost. This results in increased forecasting skill, with respect to double-precision assimilation.

scholarly journals TESTING κ-POINCARÉ WITH NEUTRAL KAONS

2000 ◽  
Vol 15 (34) ◽  
pp. 2119-2127 ◽  
Author(s):  
GIOVANNI AMELINO-CAMELIA ◽  
FRANCO BUCCELLA
Keyword(s):  
Quantum Gravity ◽  
Gamma Rays ◽  
Experimental Tests ◽  
Neutral Kaon ◽  
Planck Length ◽  
Cpt Symmetry ◽  
Simple Estimate ◽  
System A ◽  
Neutral Kaons ◽  
Deformation Length

In recent work on experimental tests of quantum-gravity-motivated phenomenological models, a significant role has been played by the so-called "κ" deformations of Poincaré symmetries. Sensitivity to values of the relevant deformation length λ as small as 5×10-33 m has been achieved in recent analyses comparing the structure of κ-Poincaré symmetries with data on the gamma rays we detect from distant astrophysical sources. We investigate violations of CPT symmetry which may be associated with κ-Poincaré in the physics of the neutral-kaon system. A simple estimate indicates that experiments on the neutral kaons may actually be more λ-sensitive than the corresponding astrophysical experiments, and may already allow to probe values of λ of order the Planck length.

AN INTRODUCTION TO QUANTITATIVE POINCARÉ RECURRENCE IN DYNAMICAL SYSTEMS

2009 ◽  
Vol 21 (08) ◽  
pp. 949-979 ◽  
Author(s):  
BENOIT SAUSSOL
Keyword(s):  
Dynamical Systems ◽  
Thermodynamic Formalism ◽  
Dimension Theory ◽  
Recurrence Rates ◽  
One Dimensional ◽  
Expanding Map ◽  
System A ◽  
Higher Dimensional ◽  
Technical Details ◽  
Hyperbolic Behavior

We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

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