Lyapunov instabilities of Lennard-Jones fluids

Recent work on many-particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive Lyapunov exponents (LEs), called hydrodynamic Lyapunov modes. Until now, however, these modes have been found only for hard-core systems. Here we report results on Lyapunov spectra and Lyapunov vectors (LVs) for Lennard-Jones fluids. By considering the Fourier transform of the coordinate fluctuation density $u^{\left(\alpha \right)}(x,t)$, it is found that the LVs with $\lambda \approx 0$ are highly dominated by a few components with low wave numbers. These numerical results provide strong evidence that hydrodynamic Lyapunov modes do exist in soft-potential systems, although the collective Lyapunov modes are more vague than in hard-core systems. In studying the density and temperature dependence of these modes, it is found that, when the value of the Lyapunov exponent ${\lambda }^{\left(\alpha \right)}$ is plotted as function of the dominant wave number $k_{\mathit{max}}$ of the corresponding LV, all data from simulations with different densities and temperatures collapse onto a single curve. This shows that the dispersion relation ${\lambda }^{\left(\alpha \right)}$ vs $k_{\mathit{max}}$ for hydrodynamical Lyapunov modes appears to be universal for the low-density cases studied here. Despite the wavelike character of the LVs, no steplike structure exists in the Lyapunov spectrum of the systems studied here, in contrast to the hard-core case. Further numerical simulations show that the finite-time LEs fluctuate strongly. We have also investigated localization features of LVs and propose a length scale to characterize the Hamiltonian spatiotemporal chaotic states.

Figure 1

Time evolution of temperature $T\equiv ⟨m{v}^2⟩$ and total energy. The inset shows that the total energy $E$ has small short-term fluctuations but no long-term drift. Detailed calculation shows that the standard deviation of the total energy $\sigma \left(E\right)=4\times {10}^{-7}$. The nearly constant state variables show that the system has already reached a stationary state. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 2

(a) Snapshot of the particle positions $x_i$ vs index of particles $i$ and (b) pair distribution function $G\left(r\right)$ obtained from the distances between all particles (upper panel) and from nearest neighbors only (lower panel) for the stationary state shown in Fig. 1 [see [25] for the definition of $G\left(r\right)$]. The sharp peaks in $G\left(r\right)$ imply that the state is a broken-chain state.

Figure 3

Lyapunov spectrum for the state shown in Fig. 2. Enlargement of the part in the regime ${\lambda }^{\left(\alpha \right)}\approx 0$ is shown in the inset. It is the result of an average over 58 samples with different initial conditions and for each sample the averaging period is $4\times {10}^{6}h$. Here no stepwise structure exists, in contrast to the case of hard-core systems. This is a typical result for our soft-potential system.

Figure 4

Distribution of the finite-time Lyapunov exponent ${\lambda }_{\tau }^{\left(\alpha \right)}$ where $\tau$ is equal to the period of reorthonormalization and the index of LEs $\alpha$ is equal to 90, 94, and 98, respectively. The strong fluctuations of ${\lambda }_{\tau }^{\left(\alpha \right)}$ are one of the possible reasons for the disappearance of the stepwise structures in the Lyapunov spectrum. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 5

Normalized Lyapunov exponents ${\lambda }^{\left(\alpha \right)}∕{\lambda }^{\left(1\right)}$ with $L=200$, 400, 1000, 2000, and 4000, respectively. The Lyapunov spectrum becomes more and more bent as the particle density $\rho =N∕L$ is decreased. This implies the separation of two time scales. Here $N=100$ and $T=0.2$.

Figure 6

$u^{\left(\alpha \right)}(x,t)$ for LVs with index $\alpha =1$, 10, 90, and 95, respectively. $u^{\left(\alpha \right)}(x,t)$ is visualized by plotting vertically the fluctuations $\delta x_i^{\left(\alpha \right)}$ at the corresponding positions $x_i$ along the $x$ axis. Notice that the LVs with $\alpha =1$ and 10 are more localized while those with $\alpha =90$ and 95 are more distributed. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 7

Time evolution of $u^{\left(95\right)}(x,t)$ for parameters as in Fig. 6. No clear wave structure can be detected.

Figure 8

Intermittent behavior of the peak wave number $k_{*}$ and spectral entropy $H_s\left(t\right)$ for the spatial Fourier spectrum of $u^{\left(95\right)}(x,t)$. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 9

(a) Variation of the peak wave number $k_{*}$ with time. (b), (c) Two typical snapshots of ${\mathrm{LV}}_{95}$, off and on states at $t=44$ and 176, respectively. (d),(e) Their spatial Fourier transforms. The spectrum for the off state has a sharp peak at small $k_{*}$ while that for the on state has no dominant peak. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 10

Time-averaged spectral entropy $⟨H_s\left(t\right)⟩$ vs index of LVs. The gradual decrease of $⟨H_s\left(t\right)⟩$ from $\alpha \approx 0$ means that LVs corresponding to smaller positive LEs are more localized in Fourier space, i.e., they have more wavelike character, than those corresponding to larger LEs. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 11

(Color online) Contour plot of the spectrum $S_{uu}^{\left(\alpha \right)}\left(k\right)$ for $L=1000$ and 2000. There is a sharp peak at $k\approx 0$ and $\lambda \approx 0$. To guide eyes, a dashed line is plotted to show how the peak wave number $k_{\mathit{max}}$ changes with the variation of $\lambda$. The sudden jump in $k_{\mathit{max}}$ is marked with an arrow. In total 58 samples for the case of $L=1000$ (ten for $L=2000$) are used for averaging for each period of $4\times {10}^{6}h$. Here $T=0.2$ and $N=100$.

Figure 12

The Lyapunov exponent ${\lambda }^{\left(\alpha \right)}$ is plotted as function of the wave number $k_{\mathit{max}}$ of the highest peak in the time-averaged spatial Fourier spectrum of LVs. The dashed line is of the form ${\lambda }^{\left(\alpha \right)}\sim k_{\mathit{max}}^{1.2}$. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 13

(Top panel) The wave number $k_{\mathit{max}}$ of the highest peak in the time-averaged spatial Fourier spectrum of LVs. (Middle panel) The height $S_{uu}^{\left(\alpha \right)}\left(k_{\mathit{max}}\right)$ of the highest peak in the time-averaged spectrum. (Bottom panel) The spectral entropy $H_S$ for the averaged spectrum $S_{uu}^{\left(\alpha \right)}\left(k\right)$. The sudden jump in $k_{\mathit{max}}$ implies the separation of time scales. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 14

Enlargement of Fig. 13 for the part in the regime $\alpha \simeq N$.

Figure 15

The Lyapunov spectrum ${\lambda }^{\left(\alpha \right)}$ (top panel), $d^2\lambda ∕d{\alpha }^2$ (middle panel), and $k_{\mathit{max}}$ (bottom panel) vs the index $\alpha$. The parameter settings used here are $N=100$, $L=1000$, and $T=0.2$.

Figure 16

Same as top panel of Fig. 8, but with different density $\rho =\left(\mathrm{a}\right)$ $1∕2$, (b) $1∕4$, and (c) $1∕8$, respectively. Here $T=0.2$ and the particle number $N=100$.

Figure 17

(Color online) (Top panel) the wave number $k_{\mathit{max}}$ of the highest peak in the time-averaged spatial Fourier spectrum of LVs as a function of LV index $\alpha$. (Bottom Panel) the Lyapunov exponent ${\lambda }^{\left(\alpha \right)}$ as function of $k_{\mathit{max}}$. Note that in the bottom panel, all data from simulations with different densities and temperatures collapse to a single curve. Fitting of the low-wave-number part to a power-law function ${\lambda }^{\alpha }\sim k_{\mathit{max}}^{\gamma }$ gives $\gamma \approx 1.2\pm 0.1$. Here $N=100$ and $T=0.2$.

Figure 18

Time-averaged participation number $p^{\left(\alpha \right)}$ vs index of LVs (upper panel) and ${\lambda }^{\left(\alpha \right)}$ (lower panel). The parameter settings used here are $N=100$, $L=110$, and $T=1.6$.

Figure 19

Time-averaged participation number $p^{\left(1\right)}$ and $p^{(N-2)}$ vs particle number $N$. The parameter settings used here are $L∕N=1.1$ and $T=1.6$.

Figure 20

Time-averaged participation number $p^{\left(1\right)}$ and $p^{(N-2)}$ vs temperature $T$. The parameter setting used here is: $N=200$ and $L=220$.

Figure 21

The standard deviation of the total energy $\sigma \left(E\right)\equiv \sqrt{\mathbf{⟨}{(E-⟨E⟩)}^2\mathbf{⟩}}$ versus the integration step size $h$, where ⟨⋯⟩ denotes the time average. The parameter settings used here are $N=5$, $L=20$, and $T=0.1$.

Figure 22

Lyapunov exponent ${\lambda }^{\left(\alpha \right)}$ (top panel) and the standard deviation of the finite-time Lyapunov exponents $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\left({\tau }_{GS}\right)\mathbf{\right)}\equiv \sqrt{\mathbf{⟨}{[\lambda \left({\tau }_{GS}\right)-⟨\lambda \left({\tau }_{GS}\right)⟩]}^2\mathbf{⟩}}$ (bottom panel) versus the index of the Lyapunov exponents $\alpha$. The data of $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\left({\tau }_{GS}\right)\mathbf{\right)}$ from simulations using the fourth Runge-Kutta integrator are also presented for comparison. ${\tau }_{GS}$ is the period between reorthogonalizations of the offset vectors and here ${\tau }_{GS}=400h$. The integration step used is $h=0.002$. The length of the trajectory used for time averaging is ${\tau }_0=1.28\times {10}^{10}h$.

Figure 23

The values of the exponents ${\lambda }^{\left(3\right)}$ and ${\lambda }^{\left(4\right)}$ versus the integration step size $h$. Here the reorthogonalization period is fixed at ${\tau }_{GS}=0.8$ and sufficient long trajectories are used for averaging in order to achieve a good convergence of the data gotten. Other parameters used here are the same as in Fig. 21.

Figure 24

Fluctuation strength $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\left({\tau }_{GS}\right)\mathbf{\right)}$ of the short-time Lyapunov exponent ${\lambda }^{\left(4\right)}\left({\tau }_{GS}\right)$ versus the integration step size $h$. Other parameters used here are the same as in Fig. 22.

Figure 25

${\lambda }^{\left(\alpha \right)}$ and $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\left({\tau }_{GS}\right)\mathbf{\right)}$ for a system with $N=40$, $L=160$, and $T=0.8$. The interaction potential is of the form stated in Eq. (4). The system is integrated with the fourth-order Runge-Kutta algorithm.

Figure 26

Lyapunov spectrum for a system with the Stoddard-Ford Lennard-Jones potential in Eq. (3). Apart from the choice of the potential cutoff all the parameters are the same as in Fig. 3.

Figure 27

The part-time-averaged value of ${\lambda }^{\left(99\right)}$ versus the averaging time.

Figure 28

The time evolution of the peak wave number $k_{*}$ (upper panel) and the instantaneous spectral entropy $H_s\left(t\right)$ (lower panel) for the Lyapunov vector no. 95 of the system used in Fig. 26. The similarity to Fig. 8 is obvious.

Figure 29

(Color online) The contour plot of the spectrum $S_{uu}^{\left(\alpha \right)}$ (upper panel) and the dispersion relation ${\lambda }^{\left(\alpha \right)}$ versus $k_{\mathit{max}}$ (lower panel) for the system shown in Fig. 26. See Figs. 11, 12 for similar results of the model with the interaction potential stated in Eq. (2).