Non-Maxwellian behavior and quasistationary regimes near the modal solutions of the Fermi-Pasta-Ulamβsystem

Abstract

In a recent paper [M. Leo, R. A. Leo, and P. Tempesta, J. Stat. Mech. (2011) P03003], it has been shown that the π/2-mode exact nonlinear solution of the Fermi-Pasta-Ulam β system, with periodic boundary conditions, admits two energy density thresholds. For values of the energy density ε below or above these thresholds, the solution is stable. Between them, the behavior of the solution is unstable, first recurrent and then chaotic. In this paper, we study the chaotic behavior between the two thresholds from a statistical point of view, by analyzing the distribution function of a dynamical variable that is zero when the solution is stable and fluctuates around zero when it is unstable. For mesoscopic systems clear numerical evidence emerges that near the second threshold, in a large range of the energy density, the numerical distribution is fitted accurately with a q-Gaussian distribution for very large integration times, suggesting the existence of a quasistationary state possessing a weakly chaotic behavior. A normal distribution is recovered in the thermodynamic limit.

Dynamics and statistics of the Fermi–Pasta–Ulamβ-model with different ranges of particle interactions

“…However, if the orbit is trapped for a long time near islands of regular motion, the MLE does not quickly converge and when it does, one cannot tell from its value whether the dynamics can be described as weakly or strongly chaotic. Now, long-range systems are known to possess long-living quasi-stationary states (QSS) [13][14][15][16][17][18][19][20][21][22][23][24][25][26], whose statistical properties are very dierent from what is expected within the framework of classical BG thermostatistics [27]. More specifically, when one studies such QSS in the spirit of the central limit theorem, one finds that the pdfs of sums of their variables are well approximated by q-Gaussian functions (with 1 < q < 3) or q-statistics [28][29][30][31][32].…”

Beyond Boltzmann–Gibbs–Shannon in Physics and Elsewhere

“…Let us end this section by mentioning other selected entropic applications beyond BG in physics: long-range interacting many-body classical Hamiltonian systems (XY model [ 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 ], Heisenberg model [ 98 , 99 , 100 ], Fermi–Pasta–Ulam (FPU) model [ 101 , 102 , 103 , 104 ]) (see [ 105 , 106 ] for earlier related approaches of the original FPU model and also [ 107 ], where the existence of non-Maxwellian compact-support momenta distributions are detected for special initial conditions); quantum-entangled low-dimensional Hamiltonian systems [ 108 , 109 , 110 ]; plasma physics [ 111 , 112 , 113 , 114 , 115 ]; turbulence [ 87 , 116 ]; astrophysics, cosmology, and black holes [ 89 , 117 , 118 , 119 , 120 , 121 , 122 ]; nonlinear dynamical systems [ 123 , 124 , 125 , 126 , 127 , 128 ]; nonlinear quantum mechanics [ 129 , 130 , 131 , 132 ]; anomalous diffusion, type II superconductors, and repulsive short-range interacting systems with overdamping […”

On the robustness of the q-Gaussian family

“…Like Gaussian distributions, q-Gaussians also are ubiquitous in Nature. Indeed, analytical, experimental and numerical investigations in biology [35], economics [4,30], high energy physics [37], anomalous diffusion processes [1,31], dynamics of many-body classical Hamiltonian systems [2,5,14,18], cold atoms [6,15,16], dissipative and conservative low dimensional maps [25,26], turbulence [3] among others 1 , have shown that q-Gaussian distributions appear in the probabilistic analysis of many systems in which long-range interactions are present, or ergodicity lacks. These evidences strongly support the existence of a generalized CLT involving q-Gaussians.…”