Chaos and relaxation to equilibrium in systems with long-range interactions

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become trapped in nonequilibrium stationary states. For a finite number of particles a nonequilibrium state has a finite lifetime, so that eventually a system will relax to thermodynamic equilibrium. The time that a system remains trapped in a quasistationary state (QSS) scales with the number of particles as $N^{\delta }$, with $\delta >0$, and diverges in the thermodynamic limit. In this paper we will explore the role of chaotic dynamics on the time that a system remains trapped in a QSS. We discover that chaos, measured by the Lyapunov exponents, favors faster relaxation to equilibrium. Surprisingly, weak chaos favors faster relaxation than strong chaos.

Figure 1

Results of molecular dynamics simulations showing the short-time oscillations of the magnetization $M_{\theta }$ for two initial WB distributions. The top curve (solid line) is for the distribution that does not satisfy the GVC. The bottom curve (solid line) is for the initial distribution that satisfies the GVC. The horizontal dashed lines show the value of the initial magnetization for each case. Notice that for the distribution that satisfies the GVC, strong oscillations are suppressed and the final magnetization is close to the initial one. In both cases, $\epsilon =0.1$ and ${\varepsilon }_{\theta }={\varepsilon }_{\varphi }=0.6$, and the corresponding virial magnetization is $\sim 0.5$. Due to symmetry, we show only $M_{\theta }$ and not $M_{\varphi }$ ($M_{\varphi }\approx -M_{\theta }$).

Figure 2

Evolution of the largest Lyapunov exponent (LLE), ${\lambda }_1$, to its stationary value, for the HMF model. The convergence to the same value, ${\lambda }_1\simeq 0.45{\tau }_D^{-1}$, was found for different numbers of particles. This result is in agreement with the previous studies where the same behavior was observed and attributed to the instability of particles near a separatrix [40].

Figure 3

Lyapunov exponent distribution (top) and Poincaré sections (bottom) of the test-particle model for $\epsilon =0.0$ (left column), $\epsilon =0.5$ (middle column), and $\epsilon =1.0$ (right column). The Poincaré sections corresponding to $\epsilon =0.0, \epsilon =0.5$, and $\epsilon =1.0$ are of 50 test particles with energy $2.5, 2.8$, and $3.2$, respectively, taken when $p_{\varphi }=0$. We see a correlation between Poincaré sections with nonchaotic regular orbits and LED dominated by low exponents.

Figure 4

Average value of the LED, $\langle \lambda \rangle$, vs. the coupling parameter $\epsilon$. As expected, stronger coupling is related with presence of more chaos in the test-particle dynamics.

Figure 5

Evolution of the kurtosis for the HMF model ($\epsilon =0.0$) with different numbers of particles, with time rescaled by ${10}^4{\tau }_D$ (left) and $N^{\delta }$ (right), $\delta =1.0$. The initial particle distribution satisfied the GVC, with magnetization $M_0=0.431852$ and mean energy ${\varepsilon }_{\theta }={\varepsilon }_{\varphi }=0.6$. The collapse suggests $\delta$ equal to $1.0$, as predicted by the kinetic equation analysis developed in Ref. [41] for the HMF model.

Figure 6

Evolution of the kurtosis for the HMF-ladder model with different numbers of particles, with time rescaled by ${10}^4{\tau }_D$ (left) and $N^{\delta }$ (right), $\delta =0.75$. The initial condition satisfied the GVC, with magnetization $M_0=0.440772$, subsystem energy ${\varepsilon }_{\theta }={\varepsilon }_{\varphi }=0.6$, and a coupling of $\epsilon =0.01$.

Figure 7

Exponent $\delta$ (from the relaxation time ${\tau }_{\times }\sim N^{\delta }$) vs. average value of the LED, $\langle \lambda \rangle$. Each pair $(\delta ,\langle \lambda \rangle )$ corresponds to a different value of the coupling parameter $\epsilon$ (see Fig. 4). We see that weak chaos favors relaxation to equilibrium more than strong chaos. Error bars restrict the range of reasonable data collapse.