Next Article in Journal
On Linear Coding over Finite Rings and Applications to Computing
Next Article in Special Issue
Information-Theoretic Bound on the Entropy Production to Maintain a Classical Nonequilibrium Distribution Using Ancillary Control
Previous Article in Journal
Can a Robot Have Free Will?
Previous Article in Special Issue
Stochastic Stirling Engine Operating in Contact with Active Baths
Article Menu
Issue 5 (May) cover image

Export Article

Open AccessArticle
Entropy 2017, 19(5), 238;

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

Department of Physics and Astronomy, Texas Christian University, Fort Worth, TX 76129, USA
Author to whom correspondence should be addressed.
Academic Editors: Andrea Puglisi, Alessandro Sarracino and Angelo Vulpiani
Received: 2 April 2017 / Revised: 15 May 2017 / Accepted: 15 May 2017 / Published: 20 May 2017
(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)
Full-Text   |   PDF [545 KB, uploaded 20 May 2017]   |  


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. View Full-Text
Keywords: Kolmogorov–Sinai entropy, Lyapunov exponents; periodic boundary conditions; chaotic dynamics; N-body simulation; stochastic thermodynamics Kolmogorov–Sinai entropy, Lyapunov exponents; periodic boundary conditions; chaotic dynamics; N-body simulation; stochastic thermodynamics

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

Share & Cite This Article

MDPI and ACS Style

Kumar, P.; Miller, B.N. Lyapunov Spectra of Coulombic and Gravitational Periodic Systems. Entropy 2017, 19, 238.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top