N-body Systems Research Articles

The Three-Body and n-Body Problems

This project explores the multi-body problem according to Newton's Laws of Gravitation in two dimensions. First, I will derive and numerically solve the differential equations which govern the motion of the three-body problem. Then, I will explore the chaotic nature of the system through a variety of demonstrations, plots, animations, and Explore windows. After, I will look at a few special periodic solutions to the three-body problem. Next, I will apply this system to astronomy, exploring the gravitational orbits that comprise our solar system. Finally, I will to generalize the three-body problem into a higher order n-body system (specifically, five bodies).

A Proposal for a Metaphysics of Self-Subsisting Structures. II. Quantum Physics

The paper presents an extension of the metaphysics of self-subsisting structures set out in a companion paper to the realm of non-relativistic quantum physics. The discussion is centered around a Pure Shape Dynamics model representing a relational implementation of a de Broglie-Bohm N-body system. An interpretation of this model in terms of self-subsisting structures is proposed and assessed against the background of the debate on the metaphysics of quantum physics, with a particular emphasis on the nature of the wave function. The analysis shows that elaborating an appropriate Leibnizian/Machian metaphysics of the quantum world requires a substantial revision of the notion of world-building relation.

Dynamical large deviations for long-range interacting inhomogeneous systems without collective effects.

We consider the long-term evolution of a spatially inhomogeneous long-range interacting N-body system. Placing ourselves in the dynamically hot limit, i.e., assuming that the system only weakly amplifies perturbations, we derive a large deviation principle for the system's empirical angle-averaged distribution function. This result extends the classical ensemble-averaged kinetic theory given by the so-called inhomogeneous Landau equation, as it specifies the probability of typical and large dynamical fluctuations. We detail the main properties of the associated large deviation Hamiltonian, particularly focusing on how it complies with the system's conservation laws and possesses a gradient structure.

A de Broglie–Bohm Model of Pure Shape Dynamics: N-Body system

We provide the construction of a de Broglie–Bohm model of the N-body system within the framework of Pure Shape Dynamics. The equation of state of the curve in shape space is worked out, with the instantaneous shape being guided by a wave function. In order to get a better understanding of the dynamical system, we also give some numerical analysis of the 3-body case. Remarkably enough, our simulations typically show the attractor-driven behaviour of complexity, well known in the classical case, thereby providing further evidence for the claim that the arrow of complexity is the ultimate cause of the experienced arrow of time.

Fewer Measurements from Shadow Tomography with N-Representability Conditions.

Classical shadow tomography provides a randomized scheme for approximating the quantum state and its properties at reduced computational cost with applications in quantum computing. In this Letter we present an algorithm for realizing fewer measurements in the shadow tomography of many-body systems. Accelerated tomography of the two-body reduced density matrix (2-RDM) is achieved by combining classical shadows with necessary constraints for the 2-RDM to represent an N-body system, known as N-representability conditions. We compute the ground-state energies and 2-RDMs of hydrogen chains and the N_{2} dissociation curve. The results demonstrate a significant reduction in the number of measurements with important applications to quantum many-body simulations on near-term quantum devices.

FARGOCPT: 2D Multiphysics code for simulating disk interactions with stars, planets, and particles

Context. Planet-disk interactions play a crucial role in the understanding of planet formation and disk evolution. There are multiple numerical tools available to simulate these interactions, including the commonly used FARGO code and its variants. Many of the codes have been extended over time to include additional physical processes, with a focus on their accurate modeling. Aims. We introduce FARGOCPT, an updated version of FARGO that incorporates other previous enhancements to the code, to provide a simulation environment tailored to studies of the interactions between stars, planets, and disks. It is meant to ensure an accurate representation of planet systems, hydrodynamics, and dust dynamics, with a focus on usability. Methods. The radiation-hydrodynamics part of FARGOCPT uses a second-order upwind scheme in 2D polar coordinates, supporting multiple equations of state, radiation transport, heating and cooling, and self-gravity. Shocks are considered using artificial viscosity. The integration of the N-body system is achieved by leveraging the REBOUND code. The dust module utilizes massless tracer particles, adapted to drag laws for the Stokes and Epstein regimes. Moreover, FARGOCPT provides mechanisms to simulate accretion onto stars and planets. Results. The code has been tested in practice in the context of multiple studies. Additionally, it comes with an automated test suite for checking the physics modules. It is available online. Conclusions. FARGOCPT offers a unique set of simulation capabilities within the current landscape of publicly available planet-disk interaction simulation tools. Its structured interface and underlying technical updates are intended to assist researchers in ongoing explorations of planet formation.

Temperature fluctuations in mesoscopic systems.

Temperature is a fundamental concept in thermodynamics. In macroscopic thermodynamics, systems possess their own intrinsic temperature which equals the reservoir temperature when they equilibrate. In stochastic thermodynamics for simple systems at the microscopic level, thermodynamic quantities other than temperature (a deterministic parameter of the reservoir) are stochastic. To bridge the disparity in the perspectives about temperature between the micro- and macroregimes, we assign a generic mesoscopic N-body system an intrinsic fluctuating temperature T in this work. We simplify the complicated dynamics of numerous particles to one stochastic differential equationwith respect to T, where the noise term accounts for finite-size effects arising from random energy transfer between the system and the reservoir. Our analysis reveals that these fluctuations make the extensive quantities (in the thermodynamic limit) deviate from being extensive. Moreover, we derive finite-size corrections, characterized by heat capacity of the system, to the Jarzynski equality. A possible violation of the principle of maximum work that scales with N^{-1} is also discussed. Additionally, we examine the impact of temperature fluctuations in a finite-size Carnot engine. We show that irreversible entropy production resulting from the temperature fluctuations of the working substance diminishes the average efficiency of the cycle as η_{C}-〈η〉∼N^{-1}, highlighting the unattainability of the Carnot efficiency η_{C} for mesoscopic heat engines even under the quasistatic limit. Our general framework paves the way for further exploration of nonequilibrium thermodynamics and the corresponding finite-size effects in a mesoscopic regime.

Integrability of Φ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Phi ^4$$\\end{document} matrix model as N-body harmonic oscillator system

We study a Hermitian matrix model with a kinetic term given by Tr(HΦ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ Tr (H \\Phi ^2 )$$\\end{document}, where H is a positive definite Hermitian matrix, similar as in the Kontsevich Matrix model, but with its potential Φ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi ^3$$\\end{document} replaced by Φ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi ^4$$\\end{document}. We show that its partition function solves an integrable Schrödinger-type equation for a non-interacting N-body Harmonic oscillator system.

Machine Learning Implementation on TTV Analysis using TTVFast

Exoplanet research has undergone rapid development lately due to the increasing number of space telescopes and satellite surveys launched in recent years. Two missions with the most exoplanet discoveries using transit methods are Kepler and TESS missions. Both missions focused on photometric surveys to detect planets by observing the periodic dimming of a stars brightness as a planet passes in front of it. There is another method that could take advantage of transit observations called transit timing variation method. By observing the deviation of the transit time of transiting planet we could infer the existence of another body in the system. Performing exoplanet parameter estimation using TTV data is an extensive process. Estimating exoplanet parameters using TTV, we must fit the TTV data using a wide range of parameters using n-body simulation. N-body simulation in this scale is computationally costly. To estimate planet parameters, we need to run thousands of n-body simulations. With the increasing trend of using machine learning methods in the research process, we try to implement the machine learning method to make TTV analysis more effective and efficient. We implement the application of machine learning to n-body simulation using REBOUND and TTVFAST and compare the results. REBOUND is a Python-based library designed for simulating the dynamics of n-body systems, particularly celestial bodies like planets, stars, and other objects that interact gravitationally. While TTVFAST is a modified n-body simulation code that is specifically designed to calculate TTV on transiting planetary systems. We found that TTVFAST is much faster than REBOUND when generating samples for training and testing while still maintaining similar accuracy. Also, the machine learning model generated from both data samples is performed similarly.

On particular integrability in classical mechanics

In this study the notion of particular integrability in Classical Mechanics, introduced in Turbiner (2013 J. Phys. A: Math. Theor. 46 025203), is revisited within the formalism of symplectic geometry. A particular integral I is a function not necessarily conserved in the whole phase space T∗Q but when restricted to a certain invariant subspace W⊆T∗Q it becomes a Liouville first integral. For natural Hamiltonian systems, it is demonstrated that such a function I allows us to construct a lower dimensional Hamiltonian in W . This symmetry reduction is intimately related with a phenomenon beyond separation of variables and it is based on an adaptive application of the classical results due to Lie and Liouville on integrability. Three physically relevant systems are used to illustrate the underlying key aspects of the symplectic theory approach to particular integrability: (I) the integrable central-force problem, (II) the chaotic two-body Coulomb system in a constant magnetic field as well as (III) the N-body system.

Physical Cyclic Animations

We address the problem of synthesizing physical animations that can loop seamlessly. We formulate a variational approach by deriving a physical law in a periodic time domain. The trajectory of the animation is represented as a parametric closed curve, and the physical law corresponds to minimizing the bending energy of the curve. Compared to traditional keyframe animation approaches, our formulation is constraint-free, which allows us to apply a standard Gauss--Newton solver. We further propose a fast projection method to efficiently generate an initial guess close to the desired animation. Our method can handle a variety of physical cyclic animations, including clothes, soft bodies with collisions, and N-body systems.

Novel model for particle emission in small collision systems

Collider experiments provide an opportunity to produce particles at close distances and momenta. The measured correlation functions between particles can provide information on both the effective emission source and the interaction potential. In recent years, experiments at the LHC have shown that precision studies of the strong interaction are possible using correlation techniques, provided a good handle on the source function. The current work presents a new numerical framework called Common Emission in CATS (CECA), capable of simulating the effective emission source of an N-body system based on the properties of the single particles. The framework differentiates between primordial particle emission and particle production through resonances, allowing to verify the hypothesis proposed by the ALICE Collaboration that a common baryon–baryon emission source is present in small collision systems. The new framework is used to analyze ALICE data on text{ pp } and text{ p }Lambda correlations and compare the results to previous studies based on the common emission source scenario. It is demonstrated that the best fit to the text{ p }Lambda correlation data is obtained using a scattering length of 1.15pm 0.07 fm in the S = 1 channel. The new CECA framework provides an essential tool for precision studies in two-body systems and a consistent description of the source function in many-body systems.

Existence of Hyperbolic Motions to a Class of Hamiltonians and Generalized N-Body System via a Geometric Approach

Existence of Hyperbolic Motions to a Class of Hamiltonians and Generalized N-Body System via a Geometric Approach

Bounds on energy and angular momentum loss in the full n-body problem

A new bound on the amended potential in an n-body system is derived and applied to the partitioning of energy and angular momentum in a disrupting gravitational aggregate. This result provides analytic insight into how energy and angular momentum can be lost or partitioned between different collections of bodies as they escape from each other. To better understand the possible outcomes in such a situation, some specific numerical tests are also performed for systems of N = 3, 4, 5, 6. The results confirm that disrupting systems always lose energy, with a characteristic stochastic distribution of the lost energy. We also find that the system loses most of its angular momentum, although individual escaping components can retain significant nonzero angular momentum vectors that can be uniformly distributed in space as the number of bodies in the system increases.

Dynamics of intermediate mass black holes in globular clusters

Context. We recently introduced a new method for simulating collisional gravitational N-body systems with approximately linear time scaling with N. Our method is based on the multi-particle collision (MPC) scheme, previously applied in fluid dynamics and plasma physics. We were able to simulate globular clusters with a realistic number of stellar particles (at least up to several times 106) on a standard workstation. Aims. We simulated clusters hosting an intermediate mass black hole (IMBH), probing a broad range of BH-cluster and BH–average-star mass ratios, unrestricted by the computational constraints that affect direct N-body codes. Methods. We set up a grid of hybrid particle-in-cell-MPC N-body simulations using our implementation of the MPC method, MPCDSS. We used either single mass models or models with a Salpeter mass function (a single power law with an exponent of −2.35), with the IMBH initially sitting at the centre. The force exerted by and on the IMBH was evaluated with a direct sum scheme with or without softening. For all simulations we measured the evolution of the Lagrangian radii and core density and velocity dispersion over time. In addition, we also measured the evolution of the velocity anisotropy profiles. Results. We find that models with an IMBH undergo core collapse at earlier times, the larger the IMBH mass the shallower they are, with an approximately constant central density at core collapse. The presence of an IMBH tends to lower the central velocity dispersion. These results hold independently of the mass function of the model. For the models with Salpeter MF, we observed that equipartition of kinetic energies is never achieved, even long after core collapse. Orbital anisotropy at large radii appears to be driven by energetic escapers on radial orbits, triggered by strong collisions with the IMBH in the core. We measured the wander radius, that is the distance of the IMBH from the centre of mass of the parent system over time, finding that its distribution has positive kurtosis. Conclusions. Among the results we obtained, which mostly confirm or extend previously known trends that had been established over the range of parameters accessible to direct N-body simulations, we underline that the leptokurtic nature of the IMBH wander radius distribution might lead to IMBHs presenting as off-centre more frequently than expected, with implications on observational IMBH detection.

First post-Newtonian N-body problem in Einstein–Cartan theory with the Weyssenhoff fluid: Lagrangian and first integrals

The rotational dynamics of an N-body system at the first post-Newtonian order in Einstein–Cartan theory is derived. This result is achieved by performing the point-particle limit of the equations of motion of the Weyssenhoff fluid, which models the quantum spin effects residing inside the bodies. For the special case of binary systems, we determine the Lagrangian function and the resulting first integrals underlying the translational dynamics and the spin precession.

Dynamical state of the globular clusters Rup 106 and IC 4499

The dynamical evolution of globular clusters is theoretically described by a series of well-known events typical of N-body systems. Still, the identification of observational signatures able to empirically describe the stage of dynamical evolution of a stellar system with a density typical of a globular cluster represents a challenge. In this paper, we study the dynamical age of the globular clusters Rup 106 and IC 4499. To this aim, we study the radial distribution of the blue straggler stars (BSSs) via the A+ parameter and of the slope of the main sequence mass function. Both tracers show that Rup 106 and IC 4499 are dynamically young clusters where dynamical friction has just started to segregate massive stars towards their respective centres. Furthermore, we observe that the BSSs are more centrally concentrated in both clusters than the reference population. Similarly, we find that in both cases the slope of the mass function significantly decreases as a function of the cluster-centric distance. This result provides additional support for the use of the radial distribution of the BSSs as a powerful and observationally convenient indicator of the cluster dynamical age.

Active learning to overcome exponential-wall problem for effective structure prediction of chemical-disordered materials

Chemical-disordered materials have a wide range of applications whereas the determination of their structures or configurations is one of the most important and challenging problems. Traditional methods are extremely inefficient or intractable for large systems due to the notorious exponential-wall issue that the number of possible structures increase exponentially for N-body systems. Herein, we introduce an efficient approach to predict the thermodynamically stable structures of chemical-disordered materials via active-learning accompanied by first-principles calculations. Our method, named LAsou, can efficiently compress the sampling space and dramatically reduce the computational cost. Three distinct and typical finite-size systems are investigated, including the anion-disordered BaSc(OxF1−x)3 (x = 0.667), the cation-disordered Ca1−xMnxCO3 (x = 0.25) with larger size and the defect-disordered ε-FeCx (x = 0.5) with larger space. The commonly used enumeration method requires to explicitly calculate 2664, 1033, and 10496 configurations, respectively, while the LAsou method just needs to explicitly calculate about 15, 20, and 10 configurations, respectively. Besides the finite-size system, our LAsou method is ready for quasi-infinite size systems empowering materials design.

Reversible time-step adaptation for the integration of few-body systems

ABSTRACT The time-step criterion plays a crucial role in direct N-body codes. If not chosen carefully, it will cause a secular drift in the energy error. Shared, adaptive time-step criteria commonly adopt the minimum pairwise time-step, which suffers from discontinuities in the time evolution of the time-step. This has a large impact on the functioning of time-step symmetrization algorithms. We provide new demonstrations of previous findings that a smooth and weighted average over all pairwise time-steps in the N-body system, improves the level of energy conservation. Furthermore, we compare the performance of 27 different time-step criteria, by considering three methods for weighting time-steps and nine symmetrization methods. We present performance tests for strongly chaotic few-body systems, including unstable triples, giant planets in a resonant chain, and the current Solar System. We find that the harmonic symmetrization methods (methods A3 and B3 in our notation) are the most robust, in the sense that the symmetrized time-step remains close to the time-step function. Furthermore, based on our Solar System experiment, we find that our new weighting method based on direct pair-wise averaging (method W2 in our notation), is slightly preferred over the other methods.

Gradient descent algorithm to search for periodic [formula omitted]-body orbits

We describe a gradient-descent based algorithm to compute approximate periodic orbits of an N-body system. Given initial position and velocity vectors, the N-body system is numerically evolved using a Runge–Kutta solver which is end-to-end differentiable, such that loss gradients can be computed. The frame of the trajectories which is most similar to the starting conditions, approximately one orbital period after t=0, is computed, and the L1 norm between the initial conditions and the conditions of this frame is minimized. In this manner, the trajectories are optimized to exhibit periodic behavior. We apply this algorithm to the three-body problem to find quasi-stable orbits, but this algorithm can be applied to other chaotic systems.