Microscopic chaos from Brownian motion in a one-dimensional anharmonic oscillator chain

Abstract

The problem of relating microscopic chaos to macroscopic behavior in a many-degrees-of-freedom system is numerically investigated by analyzing statistical properties associated to the position and momentum of a heavy impurity embedded in a chain of nearest-neighbor anharmonic Fermi-Pasta-Ulam oscillators. For this model we have found that the behavior of the relaxation time of the momentum autocorrelation function of the impurity is different depending on the dynamical regime (either regular or chaotic) of the lattice.

Macroscopic evidence of microscopic dynamics in the Fermi-Pasta-Ulam oscillator chain from nonlinear time-series analysis

“…This asymptotic behavior is also displayed in Fig. 2 for each ǫ value considered using the numerically computed values of the diffusion coefficient in each dynamical regime [8]. As in Refs.…”

“…Then the 2(N + 1) equations of motion were numerically integrated to obtain the time evolution of the system, whose state is represented by the variable Γ(t) = ({x i (t)}, {p i (t)}) ∈ ℜ 2(N +1) . After thermal equilibrium between the impurity and the FPU chain with N = 300 000 unit mass oscillators is attained, the behavior of the heavy impurity is almost identical to a Brownian motion for all ǫ values studied [10], as can be inferred from the values of the diffusion coefficient and the exponential fit to the MACF [8]. So, we can consider the heavy impurity as a genuine BP.…”

“…This behavior, for both low an high ǫ values, is unaltered by the inclusion of a heavy impurity [8]. The transition between these two dynamical regimes was first detected by means of a change in the scaling behavior of the largest Lyapunov exponent (LLE), which measures the exponential rate of divergence of two originally close trajectories in phase space, as ǫ is varied from a low to a high value for the FPU model [7] and by a change in the scaling of the relaxation time of the MACF as a function of ǫ for the FPU chain coupled to a heavy impurity [8].…”

“…As can be readily seen, the shape of h (m) (r, τ ) is very similar for the weakly chaotic case ǫ = 0.01 and the strongly chaotic case ǫ = 10. This result can be readily explained by the fact that the BP, in both dynamical regimes, performs Brownian motion [8]. Now, a Brownian signal is a stationary Gaussian process characterized by a power spectrum S(ω) ∝ ω −2 , which allows to obtain an explicit form for the (r, τ ) entropy in the limit…”

Macroscopic detection of the strong stochasticity threshold in Fermi-Pasta-Ulam chains of oscillators

“…Afterwards the heavy impurity performs Brownian motion for all ⑀ values studied [12]. The momentum autocorrelation function (MACF) 0 ͑t͒ϵ͗P͑t͒P͑0͒͘ t / ͗P 2 ͑0͒͘ t of the heavy impurity was obtained by computing the time averages ͗¯͘ t over a time interval of t =2ϫ 10 5 .…”