Abstract
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.
Highlights
Our understanding of the collective behavior of macroscopic systems is encompassed in the laws of thermodynamics
While phase transitions are rigorously defined in the thermodynamic limit, it is remarkable that their manifestations can sometimes be observed in very small systems
Our study provides insight into the chaotic dynamics of the two versions of the periodic system under conditions of varying energy and degrees of freedom
Summary
Our understanding of the collective behavior of macroscopic systems is encompassed in the laws of thermodynamics. Position and velocity can be followed for a given system, the largest LCE may be calculated by finding the rate of divergence between a reference trajectory and a nearby test trajectory obtained by pertubing the former [26] This numerical approach was extended to the case of systems with periodic boundary conditions by Kumar and Miller and was applied to find the largest LCE for a spatially-periodic one-dimensional Coulombic system [12]. We extend the numerical approach presented in [25] to compute the complete Lyapunov spectra of spatially-periodic one-dimensional Coulombic and gravitational systems and show that the energy dependence of the largest LCE emulates that of the sum of all the positive LCEs for both versions of the system.
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