The 4-body problem in a (1+1)-dimensional self-gravitating system

Lauritzen, A.; Gustainis, Peter; Mann, Robert B.

doi:10.1063/1.4815834

Cited by 5 publications

(5 citation statements)

References 14 publications

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“…Furthermore, analytic reduction of the systems' equations of motion in rhombic coordinates may be utilized to extend the tangent-space approach [19] for calculating the spectrum of Lyapunov exponents for these spatially periodic systems in simulation. Our work also paves the way to study the dynamics of 4-body periodic versions, the results of which could be compared with those already known for the 4-body free-boundary gravitational case [6].…”

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confidence: 81%

“…Not only can they be mapped to systems subject to experimental study [1], a one-dimensional gravitation-like interaction has actually been observed in the laboratory [2]. Moreover, onedimensional systems have been of great intrinsic interest to theoretical and computational physicists (see [3][4][5][6][7] and references therein). Small versions of the onedimensional systems (N ≥ 3) are of particular interest as they are relatively less convoluted to analyze yet may exhibit interesting chaotic dynamics.…”

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confidence: 99%

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Dynamics of Coulombic and gravitational periodic systems

Kumar

Miller

2016

Phys. Rev. E

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We study the dynamics and the phase-space structures of Coulombic and self-gravitating versions of the classical one-dimensional 3-body system with periodic boundary conditions. We demonstrate that such a 3-body system may be reduced isomorphically to a spatially periodic system of a single particle experiencing a two-dimensional potential on a rhombic plane. For the case of both Coulombic and gravitational versions, exact expressions of the Hamiltonian have been derived in rhombic coordinates. We simulate the phase-space evolution through an event-driven algorithm that utilizes analytic solutions to the equations of motion. The simulation results show that the motion exhibits chaotic, quasiperiodic, and periodic behaviors in segmented regions of the phase space. While there is no evidence of global chaos in either the Coulombic or the gravitational system, the former exhibits a transition from a completely non-chaotic phase space at low energies to a mixed behavior. Gradual yet striking transitions from mild to intense chaos are indicated with changing energy, a behavior that differentiates the spatially periodic systems studied in this paper from the well-understood free-boundary versions of the 3-body problem. Our treatment of the 3-body systems is the first one of its kind and opens avenues for analysis of the dynamical properties exhibited by spatially periodic versions of various classes of systems studied in plasma and gravitational physics as well as in cosmology.

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confidence: 81%

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confidence: 99%

Dynamics of Coulombic and gravitational periodic systems

Kumar

Miller

2016

Phys. Rev. E

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“…As expected, the sums were found to be close to zero. It should be noted that versions of the systems with N < 5 may exhibit largely segmented phase-space distributions with a coexistence of chaotic and stable regions for a given energy value (N = 2 being purely trivial with completely integrable phase space) [11,12,25,42]. Hence, there is a considerable probability that a randomly chosen initial condition ends up in one of the stable regions whose trajectories exhibit no stochasticity and which results in values of LCEs close to zero in simulation.…”

Section: Resultsmentioning

confidence: 99%

“…One dimensional systems are of great interest to physicists in terms of their intrinsic properties and as a starting point in the analysis of their more-complicated higher-dimensional counterparts (see [8][9][10][11][12] and references therein). Not only can one-dimensional systems map the behaviors of experimental results [13], one-dimensional interactions have, in fact, been recently emulated in the laboratory [14].…”

Section: Introductionmentioning

confidence: 99%

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

Kumar

Miller

2017

Entropy

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An open question in nonlinear dynamics is the relation between the Kolmogorov entropy and the largest Lyapunov exponent of a given orbit. Both have been shown to have diagnostic capability for phase transitions in thermodynamic systems. For systems with long-range interactions, the choice of boundary plays a critical role and appropriate boundary conditions must be invoked. In this work, we compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact expressions for time evolution of the tangent-space vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov entropy for each system for a given system size. Our approach forms an effective and approximation-free instrument for studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in small as well as large versions of the spatially periodic systems. When a phase transition exists, we find that the largest Lyapunov exponent serves as a precursor of the transition that becomes more pronounced as the system size increases.

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“…One dimensional systems are of great interest to physicists in terms of their intrinsic properties and as a starting point in the analysis of their more-complicated higher-dimensional counterparts (see [1][2][3][4][5] and references therein). In the analysis of large systems considered in plasma and gravitational physics, periodic boundary conditions are preferred [6][7][8][9] and have been utilized in the study of one-dimensional plasma and gravitational systems [5,[10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Lyapunov spectra of Coulombic and gravitational periodic systems

Kumar¹,

Miller²

2016

Preprint

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We compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact time evolution of tangentspace vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov-entropy density for each system at different degrees of freedom. Our approach forms an effective and approximation-free tool toward studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in large versions of the spatially periodic systems.

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The 4-body problem in a (1+1)-dimensional self-gravitating system

Cited by 5 publications

References 14 publications

Dynamics of Coulombic and gravitational periodic systems

Dynamics of Coulombic and gravitational periodic systems

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

Lyapunov spectra of Coulombic and gravitational periodic systems

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