Multistage slow relaxation in a Hamiltonian system: The Fermi-Pasta-Ulam model

The relaxation process toward equipartition of energy among normal modes in a Hamiltonian system with many degrees of freedom, the Fermi-Pasta-Ulam (FPU) model is investigated numerically. We introduce a general indicator of relaxation $\sigma$ which denotes the distance from equipartition state. In the time evolution of $\sigma$, some long-time interferences with relaxation, named “plateaus,” are observed. In order to examine the details of the plateaus, relaxation time of $\sigma$ and excitation time for each normal mode are measured as a function of the energy density ${\epsilon }_0=E_0/N$. As a result, multistage relaxation is detected in the finite-size system. Moreover, by an analysis of the Lyapunov spectrum, the spectrum of mode energy occupancy, and the power spectrum of mode energy, we characterize the multistage slow relaxation, and some dynamical phases are extracted: quasiperiodic motion, stagnant motion (escaping from quasiperiodic motion), local chaos, and stronger chaos with nonthermal noise. We emphasize that the plateaus are robust against the arranging microscopic state. In other words, we can often observe plateaus and multistage slow relaxation in the FPU phase space. Slow relaxation is expected to remain or vanish in the thermodynamic limit depending on indicators.

Figure 1

The time evolution of $\overline{\sigma }$ [Eq. (8)] is presented for $N=32$. Different curves correspond to the different values of ${\epsilon }_0, {\epsilon }_0=2.344\times {10}^{-3},\cdots ,3.695\times {10}^{-1}$, lower to higher ${\epsilon }_0$ from top to bottom in the figure. Relaxations are interfered with for a long time at lower ${\epsilon }_0$ with plateaus. The fluctuation on the plateaus for lower ${\epsilon }_0$ indicates recurrence phenomena, as reported in the original FPU paper [1].

Figure 2

Multistage relaxation in the time evolution of $\overline{\sigma }$ [Eq. (8)]. The condition of system size and energy density is $N=32({\epsilon }_0=4.141\times {10}^{-2},5.703\times {10}^{-2},6.484\times {10}^{-2},7.266\times {10}^{-2})$ and $N=16({\epsilon }_0=2.648\times {10}^{-1},2.055\times {10}^{-1},2.211\times {10}^{-1},2.289\times {10}^{-1},2.602\times {10}^{-1})$ with some different initial phase $\gamma$.

Figure 3

The way to measure the relaxation time ${\tau }_{{\sigma }^{\circ }}\left({\epsilon }_0\right)$ [Eq. (19)] for a high decision level ${\sigma }^{\circ }$ is described. The decision level ${\sigma }^{\circ }$ leads a data series $({\tau }_{{\sigma }^{\circ }},{\epsilon }_0)$. We put a lot of ${\sigma }^{\circ }$ (for simplification, only five decision levels are described in this figure) so the figure $t$ vs $\sigma$ [Eq. (8)] is filled by the lines of $\sigma ={\sigma }^{\circ }\left(\text{const}\right)$ with equal interval from top to bottom for observing the whole dynamics. The higher ${\sigma }^{\circ }$ grasps the early phase of dynamics (preplateaus), whereas the lower one provides the feature of later phase of dynamics (postplateaus).

Figure 4

The dependence of the relaxation time ${\tau }_{{\sigma }^{\circ }}$ [Eq. (19)] on energy density ${\epsilon }_0$ is shown for Fig. 3. The symbol is in common with Fig. 3. It should be noted that there is a large gap between the filled triangle and the open circle at ${\epsilon }_0={\epsilon }_{01}$. The large gap indicates the existence of the plateau with a long lifetime.

Figure 5

The way to measure the excitation time ${\tau }_k\left({\epsilon }_0\right)$ [Eq. (20)] is described. The decision level is fixed at ${\overline{\rho }}_k=0.25{\rho }^{*}$ throughout the present paper. ${\rho }^{*}=1/N$ is the equipartition value of the mode energy occupancy.

Figure 6

The dependence of relaxation time ${\tau }_{{\sigma }^{\circ }}$ [Eq. (19)] on energy density ${\epsilon }_0$ is presented for $N=32$, averaged over 10 trajectories with different initial phase $\gamma$. Different plot symbols correspond to different value of ${\sigma }^{\circ }$. There are two well-separated regions, the power-law range and much slower relaxation. A power law is obtained by large ${\sigma }^{\circ }$, namely in the early phase (preplateau). A separating data series is observed for small ${\sigma }^{\circ }$ (postplateau). The size of the gap between them corresponds to the effective lifetime of plateaus.

Figure 7

The dependence of relaxation time ${\tau }_{{\sigma }^{\circ }}$ [Eq. (19)] on the decision level ${\sigma }^{\circ }$ is presented for $N=32$, averaged over 10 trajectories with different initial phase $\gamma$. Different plot symbols correspond to different values of energy density ${\epsilon }_0$. The upper symbol and bottom symbols denote respectively lower and higher ${\epsilon }_0$. For lower ${\epsilon }_0$, large gaps which denote plateaus with long lifetimes are formed. Furthermore, multistepped structures can be observed.

Figure 8

The excitation time of each normal mode ${\tau }_k$ [Eq. (20)] is plotted as a function of energy density ${\epsilon }_0$. Different symbols correspond to different mode number $k$. The figure shows the existence of roughly two different mode groups, the fast and slow modes. The former have power law $a/{\epsilon }_0^b$; on the other hand, the latter separate from the former. Their dependence on ${\epsilon }_0$ is similar to the two-stage relaxation that is reported in Fig. 6 with respect to ${\tau }_{{\sigma }^{\circ }}$; moreover, if one takes a closer look at the figure, then one can see a more complex structure. The variety of time required for energy attainment of each normal mode is confirmed, which indicates the multistage property.

Figure 9

The dependence of relaxation time ${\tau }_{{\sigma }^{\circ }}$ [Eq. (19)] on energy density ${\epsilon }_0$ in the finite-size system $N=12,16$ is shown. As is the case in $N=32$, well-separated relaxation processes are shown. These figures show the existence of roughly two-stage relaxation. The multistage relaxation can be seen prominently for ${\tau }_{{\sigma }^{\circ }}$ in $N=16$. The multistage property can be observed more clearly in Figs. 10 and 11.

Figure 10

The dependence of relaxation time ${\tau }_{{\sigma }^{\circ }}$ [Eq. (19)] on the decision levels ${\sigma }^{\circ }$ is presented for $N=12,16$. Different plot symbols correspond to different values of energy density ${\epsilon }_0$. The upper symbol and bottom symbols denote respectively lower and higher ${\epsilon }_0$. For lower ${\epsilon }_0$, the separation of preplateau and postplateau dynamics can be seen with large gaps, which indicate the plateaus in lower ${\epsilon }_0$. Moreover, one can detect stepped structures in the relaxation which indicate multistage relaxation.

Figure 11

The dependence of excitation time ${\tau }_k$ [Eq. (20)] on excitation energy density ${\epsilon }_0$ in finite-size system $N=12,16$ is shown. One can confirm that there is variety of excitation times of each normal mode, so the existence of not only two but also more separated relaxation is suggested.

Figure 12

The time convergenc of the Kolmogorov-Sinai entropy (15) is shown for 48 different energy densities $({\epsilon }_0=2.344\times {10}^{-3},\cdots ,3.695\times {10}^{-1})$ in $N=12,16,32$. The difference of plot color denotes the difference of ${\epsilon }_0$, and the colors are the same as described in the caption to Fig. 1. The lower to higher ${\epsilon }_0$ are laid out from bottom to top. The black solid line is the $1/t$ decay, which can be seen for ideal periodic dynamics. At lower ${\epsilon }_0$, the trajectories almost behave as periodic solutions; on the other hand, for higher ${\epsilon }_0$, the system is detached from periodic motion and shows a chaotic feature.

Figure 13

The time evolution of $\overline{\sigma }$ (a) and the time convergence of the KS entropy (b) are shown for $N=32,E_0=1.825({\epsilon }_0=5.703\times 1{-}^{-2}),\gamma =0.5$. $\overline{N_{\mathrm{ex}}}$ in panel (a) is a subsidiary indicator which denotes the effective size of the excited modes defined by the long-time average of $logS\left(t\right)$. The left dashed-square (blue) and right dashed-square (red) regions in panel (a) denote the main plateau and slow development of $\overline{\sigma }\left(t\right)$. Both of them brake on the nondisabled growth of $\overline{\sigma }\left(t\right)$. According to the convergence of KS entropy, we can detect four dynamical phases. Phase I (green): The system behaves as a periodic solution with $1/t$ decay. Phase II (blue): We can see the power law with a more shallow slope than $1/t$, which means indicates the gradual separation from periodic motion, called stagnant motion. Phase III (red): KS entropy transiently converges which indicates the local chaos. Phase IV (purple): The remarkable growth of KS entropy is confirmed. This strong chaos leads to the attainment of equipartition.

Figure 14

The Lyapunov spectrum in dynamical phases I, II, III, and IV.

Figure 15

The spectrum of normal mode energy occupancy in dynamical phases I, II, III, and IV.

Figure 16

The averaged power spectrum of mode energy in dynamical phases I, II, III, and IV.

Figure 17

The robustness of the plateaus against arranging microscopic states is represented. The spectrum $\overline{{\rho }_k}$ of the plateau is extracted at $t=2\times {10}^4$ [asterisk in (a)] from the original trajectory, and coordinates and momenta are arranged with preservation of the spectrum $\overline{{\rho }_k}$, and the manipulated trajectories are driven adopting the arranged data as initial conditions. (a) The original dynamics; (b) the manipulated trajectories for some arranging pattern of initial conditions; and (c) the convergence of KS entropy. The manipulated trajectories present with long-time freezing plateaus similarly to original one.

Figure 18

The dependence of relaxation time ${\tau }_{{\sigma }^{\circ }}$ [Eq. (19)] on energy density ${\epsilon }_0$ is shown. Different symbols correspond to different decision levels ${\sigma }^{\circ }$. Toward the thermodynamic limit, we can see the “graceful-sticky overlap,” and the function of relaxation time is unified into power law $a/{\epsilon }_0^b$.

Figure 19

The dependence of excitation time ${\tau }_k$ [Eq. (20)] on energy density ${\epsilon }_0$ is shown. Different symbols correspond to different mode number $k$. The branching shape of ${\tau }_k$ seems to survives toward the thermodynamic limit.