Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

• Published in: Physica A Vol. 390; no. 20; pp. 3290 - 3307
• Main Authors: Antonopoulos, Ch, Bountis, T., Basios, V.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2011
• Subjects: [formula omitted]-Gaussian distributions; Approximation; Chaos theory; Edge of chaos; Mathematical analysis; Mathematical models; Microplasmas; Multi-dimensional Hamiltonian systems; Nonextensive statistical mechanics; Orbits; Quasi-stationary states; Statistical distributions; Sums; Weak chaos; Weak chaos; Multi-dimensional Hamiltonian systems; Nonextensive statistical mechanics; q-Gaussian distributions; Quasi-stationary states; Edge of chaos
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2011.05.026

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈106) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos. ► We discovered pdfs well approximated by q-Gaussian functions for long times. ► These pdfs represent quasi-stationary states and tend to Gaussians for longer times. ► We observed q>1 and tend to Gaussians as the energy begins to be shared by all dofs. ► For the microplasma, our pdfs allowed us to find two energy regimes of “weak” chaos.