Hydrodynamic Lyapunov modes and strong stochasticity threshold in the dynamic $XY$ model: An alternative scenario

Crossover from weak to strong chaos in high-dimensional Hamiltonian systems at the strong stochasticity threshold (SST) was anticipated to indicate a global transition in the geometric structure of phase space. Our recent study of Fermi-Pasta-Ulam models showed that corresponding to this transition the energy density dependence of all Lyapunov exponents is identical apart from a scaling factor. The current investigation of the dynamic $XY$ model discovers an alternative scenario for the energy dependence of the system dynamics at SSTs. Though similar in tendency, the Lyapunov exponents now show individually different energy dependencies except in the near-harmonic regime. Such a finding restricts the use of indices such as the largest Lyapunov exponent and the Ricci curvatures to characterize the global transition in the dynamics of high-dimensional Hamiltonian systems. These observations are consistent with our conjecture that the quasi-isotropy assumption works well only when parametric resonances are the dominant sources of dynamical instabilities. Moreover, numerical simulations demonstrate the existence of hydrodynamical Lyapunov modes (HLMs) in the dynamic $XY$ model and show that corresponding to the crossover in the Lyapunov exponents there is also a smooth transition in the energy density dependence of significance measures of HLMs. In particular, our numerical results confirm that strong chaos is essential for the appearance of HLMs.

Figure 1

(Color online) Time evolution of the coordinate ${\theta }_l$ for two randomly selected elements for four cases with energy density (a) $ϵ=0.04$, (b) 0.4, (c) 4, and (d) 40, respectively. Notice the change in the system dynamics with increasing energy density.

Figure 2

(Color online) (a) Time evolution of the quantity $⟨{\theta }_l^2⟩$ for the corresponding cases shown in Fig. 1. (b) The variation of the exponent $\gamma$ with the energy density $ϵ$. Notice the existence of three regimes with $\gamma \approx 0$, $\gamma \approx 1$, and $\gamma \approx 2$, respectively. The two threshold values are ${ϵ}_{c1}\approx 0.14$ and ${ϵ}_{c2}\approx 10$.

Figure 3

(Color online) Energy density dependence of (a) the temperature $T$ and (b) the potential energy density $v$. Both quantities change their behavior in a regime around ${ϵ}_M\approx 1$.

Figure 4

Lyapunov spectrum of a case with $ϵ=4$. Due to the Hamiltonian nature of the system dynamics the Lyapunov spectrum has the symmetry ${\lambda }^{\left(\alpha \right)}=-{\lambda }^{(2L-1-\alpha )}$. As can be seen from the inset, the system has four zero-value Lyapunov exponents. The system size used here is $L=64$.

Figure 5

(Color online) Largest Lyapunov exponent ${\lambda }^{\left(0\right)}$ vs energy density $ϵ$ for the dynamic $XY$ model. The $ϵ$ dependence of ${\lambda }^{\left(0\right)}$ is fitted to a power law with the exponent 2.0 in the low-energy regime $ϵ<{ϵ}_{c1}\approx 0.14$.

Figure 6

(Color online) $ϵ$ dependence of typical Lyapunov exponents in the positive branch of the Lyapunov spectrum for the dynamic $XY$ model. Here, $\alpha$ takes the value from 0 to 60 with the increment 10 (from top to bottom). The density of the Kolmogorov-Sinai entropy ${\rho }_h\equiv h_{\mathrm{KS}}∕N$ is plotted as a dashed line. Obviously, except in the low-energy regime $ϵ<{ϵ}_{c1}$ the profiles of change of all Lyapunov exponents are different from each other.

Figure 7

(Color online) Normalized Lyapunov exponents ${\lambda }^{\left(\alpha \right)}\left(ϵ\right)∕{\lambda }^{\left(\alpha \right)}\left({ϵ}_0\right)$ vs energy density $ϵ$. Here the more or less arbitrary value ${ϵ}_0=0.1$ was chosen. Note that except in the near-harmonic regime $ϵ<{ϵ}_{c1}$ data for different Lyapunov exponents scatter greatly instead of collapsing on a master curve.

Figure 8

(Color online) Lyapunov exponents normalized by the entropy density ${\rho }_h$ vs energy density $ϵ$ for the dynamic $XY$ model.

Figure 9

(Color online) Standard deviations of the finite-time Lyapunov exponents $\sigma \left({\lambda }_{\tau }^{\left(\alpha \right)}\right)$ vs energy density $ϵ$ for the dynamic $XY$ model. Dashed lines indicate the transition values ${ϵ}_{c1}=0.14$ and ${ϵ}_M=1$.

Figure 10

(a) Profile of a Lyapunov vector with $\alpha =46$ for a randomly selected value of $t$ and (b) the corresponding static LV structure factor $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ for the dynamic $XY$ model with $ϵ=4$. The Lyapunov spectrum for $ϵ=4$ is shown in Fig. 4. Here, $k_{\max }$ is defined as the wave number of the highest peak in the spectrum $S_u^{\left(\alpha \alpha \right)}\left(k\right)$.

Figure 11

(Color online) (a) Contour plot of static LV structure factors for the dynamic $XY$ model with $ϵ=4$. (b) Variation of $k_{\max }$ with the Lyapunov index $\alpha$. The Lyapunov vectors with $\alpha \approx 64$ are dominated by components with wave numbers comparable to $2\pi ∕L$, which is the smallest nontrivial wave number permitted by the periodic boundary conditions used. These facts together imply the existence of hydrodynamic Lyapunov modes in this system.

Figure 12

(a) $S\left(k_{\max }\right)$ and (b) spectral entropy $H_S\equiv -\sum S\left(k\right)\ln S\left(k\right)$ vs $\alpha$ for the dynamic $XY$ model with $ϵ=4$. The quantities $S_{\max }\left(ϵ\right)$, ${\alpha }_S\left(ϵ\right)$, $H_{\min }\left(ϵ\right)$, and ${\alpha }_H\left(ϵ\right)$ will be used to measure the significance of HLMs.

Figure 13

(a) $S_{\max }\left(ϵ\right)$, (b) $H_{\min }$, and (c) the normalized index ${\alpha }_H∕L$ vs $ϵ$ for the dynamic $XY$ model. Here, the threshold values of the energy density for transitions in the Lyapunov spectrum are shown as dashed lines. Notice the change in the behavior of these quantities at these threshold values.

Figure 14

(Color online) Energy density dependence of (a) Lyapunov exponents ${\lambda }^{\left(\alpha \right)}\left(ϵ\right)$ sampled from the positive branch of the Lyapunov spectrum, (b) normalized Lyapunov exponents ${\lambda }^{\left(\alpha \right)}\left(ϵ\right)∕{\lambda }^{\left(\alpha \right)}\left({ϵ}_0\right)$ with ${ϵ}_0=0.4$, and (c) Lyapunov exponents normalized by the Kolmogorov-Sinai entropy density ${\lambda }^{\left(\alpha \right)}∕{\rho }_h$ for the lattice ${\varphi }^4$ model.