Chaos and thermalization in a classical chain of dipoles

We explore the connection between chaos, thermalization, and ergodicity in a linear chain of $N$ interacting dipoles. Starting from the ground state, and considering chains of different numbers of dipoles, we introduce single site excitations with excess energy $\mathrm{\Delta }K$. The time evolution of the chaoticity of the system and the energy localization along the chain is analyzed by computing, up to a very long time, the statistical average of the finite-time Lyapunov exponent $\lambda \left(t\right)$ and the participation ratio $\mathrm{\Pi }\left(t\right)$. For small $\mathrm{\Delta }K$, the evolution of $\lambda \left(t\right)$ and $\mathrm{\Pi }\left(t\right)$ indicates that the system becomes chaotic at approximately the same time as $\mathrm{\Pi }\left(t\right)$ reaches a steady state. For the largest considered values of $\mathrm{\Delta }K$ the system becomes chaotic at an extremely early stage in comparison with the energy relaxation times. We find that this fact is due to the presence of chaotic breathers that keep the system far from equipartition and ergodicity. Finally, we show numerically and analytically that the asymptotic values attained by the participation ratio $\mathrm{\Pi }\left(t\right)$ fairly correspond to thermal equilibrium.

Figure 1

Schematic representation of the dipole chain.

Figure 2

(a)–(c) Computed average $\widehat{\lambda }\left(t\right)$ for eight excess energies along with some error bars (standard deviation) for chains with $N=100, N=200,$ and $N=400$ dipoles, respectively. The red dashed line guides the eye for the $-1$ slope, which marks the expected $\lambda \left(t\right)\sim t^{-1}$ behavior for regular orbits. (d) Time evolution of the finite-time Lyapunov exponent of each of the 20 initial conditions (gray lines), the corresponding averaged finite-time Lyapunov exponent $\widehat{\lambda }\left(t\right)$ (color lines) and some error bars for a chain of $N=100$ dipoles and for the excess energies $\mathrm{\Delta }K=0.05{\mathrm{\Delta }}_s,0.25{\mathrm{\Delta }}_s,0.75{\mathrm{\Delta }}_s$, and $1.25{\mathrm{\Delta }}_s$.

Figure 3

Maximal Lyapunov exponent ${\lambda }_1$ as a function of the energy density $\epsilon =\mathrm{\Delta }K/N$ for dipole chains with $N=100$, 150, 200, 300, and 400. Note that a double logarithmic scale is used. The values of $a$ are the exponents that, for each $N$, are obtained from the fittings of the pairs $(\epsilon ,{\lambda }_1)$ to the power law of Eq. (8).

Figure 4

Time averaged participation ratio $\widehat{\mathrm{\Pi }}\left(t\right)$ for eight excess energies. The horizontal dashed lines mark the equilibrium values $\langle \mathrm{\Pi }\rangle$ assuming that it follows a Boltzmann distribution for the local energies. Chains with (a) $N=100$, (b) $N=200,$ and (c) $N=400$ dipoles are considered.

Figure 5

Color maps showing the time evolution of the local energies $E_k\left(t\right)$ [see Eq. (6)] for a chain with $N=200$ and $\mathrm{\Delta }K=1.1{\mathrm{\Delta }}_s$ but with different initial conditions ${\theta }_{100}\approx 0.6614$ [map (a)] and ${\theta }_{100}\approx 0.8267$ [map (b)], and with the corresponding momenta $p_{100}$ according to Eq. (5). Panel (c) shows the participation ratio $\mathrm{\Pi }\left(t\right)$ corresponding to the trajectories of the color maps (a) (red color) and (b) (blue color). The horizontal dashed line in panel (c) marks the equilibrium value $\langle \mathrm{\Pi }\rangle$ assuming that it follows a Boltzmann distribution for the local energies.