Download or read book Chaotic Dynamics and Thermodynamics of Periodic Systems with Long-range Forces written by Pankaj Kumar and published by . This book was released on 2016 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gravitational and electromagnetic interactions form the backbone of our theoretical understanding of the universe. While, in general, such interactions are analytically inexpressible for three-dimensional infinite systems, one-dimensional modeling allows one to treat the long-range forces exactly. Not only are one-dimensional systems of profound intrinsic interest, physicists often rely on one-dimensional models as a starting point in the analysis of their more complicated higher-dimensional counterparts. In the analysis of large systems considered in cosmology and plasma physics, periodic boundary conditions are a natural choice and have been utilized in the study of one dimensional Coulombic and gravitational systems. Such studies often employ numerical simulations to validate the theoretical predictions, and in cases where theoretical relations have not been mathematically formulated, numerical simulations serve as a powerful method in characterizing the system's physical properties. In this dissertation, analytic techniques are formulated to express the exact phase-space dynamics of spatially-periodic one-dimensional Coulombic and gravitational systems. Closed-form versions of the Hamiltonian and the electric field are derived for single-component and two-component Coulombic systems, placing the two on the same footing as the gravitational counterpart. Furthermore, it is demonstrated that a three-body variant of the spatially-periodic Coulombic or gravitational system may be reduced isomorphically to a periodic system of a single particle in a two-dimensional rhombic potential. The analytic results are utilized for developing and implementing efficient computational tools to study the dynamical and the thermodynamic properties of the systems without resorting to numerical approximations. Event-driven algorithms are devised to obtain Lyapunov spectra, radial distribution function, pressure, caloric curve, and Poincare surface of section through an N-body molecular-dynamics approach. The simulation results for the three-body systems show that the motion exhibits chaotic, quasiperiodic, and periodic behaviors in segmented regions of the phase space. The results for the large versions of the single-component and two-component Coulombic systems show no clear-cut indication of a phase transition. However, as predicted by the theoretical treatment, the simulated temperature dependencies of energy, pressure as well as Lyapunov exponent for the gravitational system indicate a phase transition and the critical temperature obtained in simulation agrees well with that from the theory.