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

Abstract

The largest Lyapunov exponent of a system composed by a heavy impurity embedded in a chain of anharmonic nearest-neighbor Fermi-Pasta-Ulam oscillators is numerically computed for various values of the impurity mass M. A crossover between weak and strong chaos is obtained at the same value epsilon(T) of the energy density epsilon (energy per degree of freedom) for all the considered values of the impurity mass M. The threshold epsilon(T) coincides with the value of the energy density epsilon at which a change of scaling of the relaxation time of the momentum autocorrelation function of the impurity occurs and that was obtained in a previous work [Phys. Rev. E 65, 036228 (2002)]]. The complete Lyapunov spectrum does not depend significantly on the impurity mass M. These results suggest that the impurity does not contribute significantly to the dynamical instability (chaos) of the chain and can be considered as a probe for the dynamics of the system to which the impurity is coupled. Finally, it is shown that the Kolmogorov-Sinai entropy of the chain has a crossover from weak to strong chaos at the same value of the energy density as the crossover value epsilon(T) of largest Lyapunov exponent. Implications of this result are discussed.

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

“…The most striking feature is that the (r, τ ) entropy has a well defined saturation value h (m) (r, τ ) ≈ const., which is reached asymptotically as the embedding dimension m increases, for a small resolution r. Furthermore, as can be readily seen, there is no need to take large delay time values in order to obtain the plateau value displayed. The finite value of the (r, τ ) entropy is consistent with the fact that the LLE, and thus the h KS entropy, are well defined quantities in the thermodynamic limit, both for the FPU chain [14] and for the MFPU model [9]. It is also clear that there is no way to obtain the aforementioned plateau value using the position time series alone, as shown in Fig.…”

“…4, is a confirmation that the momentum of the BP is a better probe of the microscopic dynamics of the oscillator chain than its position. However, the saturation value h (m) (r, τ ) ≈ 0.086 obtained for r < 1 is higher than the true microscopic value h KS ≈ 0.12 × 10 −4 [9]. In Fig.…”

“…We can say that the difference in the behavior of the (r, τ ) entropy for low and high ǫ values means that some distinction between different dynamical regimes has indeed been observed by nonlinear time-series analysis. Now, it is important to recall that, in the particular case of ǫ = 0.01, the h KS entropy has a finite, albeit very small, value [9]. The apparent dynamical randomness, i.e.…”

“…Our starting point comes from the observation that the Kolmogorov-Sinai entropy h KS , which has been computed from the complete microscopic dynamics of this system as a function of ǫ, displays the aforementioned transition in the scaling behavior of h KS [9]. Since in the description of the microscopic dynamics the momentum of all the oscillators of the system is considered and the dynamics of the chain is not affected by the presence of the heavy impurity, we can ask whether some evidence of the above mentioned transition between dynamical regimes can be detected by applying the methods of nonlinear time series analysis to the momentum time series of the heavy impurity.…”

Probing Hamiltonian dynamics by means of the 0–1 test for chaos

“…As can be readily appreciated, these results seem to indicate a periodic, non ergodic dynamics underlying both time series. For the artificial time series this result seems at odds with the information available from the LLE, λ 1 = 0.12 for ǫ = 10, which clearly indicates that the system is strongly chaotic [21,24]. In the case of the experimental data, the power spectrum P (ω) ∼ ω −2 indicates that the motion is of Brownian (stochastic) character [13].…”