Chaos in the Hamiltonian mean-field model

We study the dynamical properties of the canonical ordered phase of the Hamiltonian mean-field (HMF) model, in which $N$ particles, globally coupled via pairwise attractive interactions, form a rotating cluster. Using a combination of numerical and analytical arguments, we first show that the largest Lyapunov exponent remains strictly positive in the infinite-size limit, converging to its asymptotic value with $1/\ln N$ corrections. We then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered, clustered phase and the disordered phase present at high-energy densities. We also show that the full spectrum of Lyapunov exponents consists of a bulk component converging to the (zero) value taken by a test oscillator forced by the mean field, plus subextensive bands of $O\left(\ln N\right)$ exponents taking finite values. We finally investigate the robustness of these results by studying a “2D” extension of the HMF model where each particle is endowed with 4 degrees of freedom, thus allowing the emergence of chaos at the level of a single particle. Altogether, these results illustrate the subtle effects of global (or long-range) coupling and the importance of the order in which the infinite-time and infinite-size limits are taken: For an infinite-size HMF system represented by the Vlasov equation, no chaos is present, while chaos exists and subsists for any finite system size.

Figure 1

(a) Equilibrium energy distribution Eq. (9) for three different values of the internal energy: $U=0.1$, $U=0.2$, and $U=0.5$ (from bottom to top). The value of $T$ in Eq. (9) is computed by Eq. (8). Recall the shifted energy axis, Eq. (4), for $h$. (b) Power spectrum $S\left(f\right)$, multiplied by the size $N$, for $U=0.5$ and three different system sizes, $N=250$ (solid black line), 500 (dotted-dashed red line), and 1000 (dashed green line).

Figure 2

(a) Mean-square displacement of the global magnetization phase $\Delta {\phi }^2=\langle {[\phi \left(t\right)-\phi \left(0\right)]}^2\rangle$ versus time. Mean-square displacements have been computed by averaging over time span of order $1\times {10}^7$ with sliding windows of duration 160 000–320 000 for three different system sizes: from top to bottom $N=1000$ (black dots), $N=4000$ (red squares), and $N=10000$ (green diamonds). (b) Same quantities as in (a) but rescaled to better highlight the crossover from ballistic to diffusive behavior. Time has been rescaled by system size, $t^{\prime }=t/N$, and mean-square displacements by the rescaled time, $\Delta {\phi }^2\to \Delta {\phi }^2/t^{\prime }$. The dashed lines mark the linear ballistic growth, and the vertical dot-dashed (blue) line highlights the beginning of deviations from linear growth at the rescaled crossover time ${\tau }_{\mathrm{diff}}/N$.

Figure 3

Critical scaling of the magnetization $M$. (a) Size dependence of the magnetization at the critical point, $M(U_{\mathrm{c}},N)$. (b) Magnetization $M(U,N)$ as a function of the energy difference $\Delta U=U-U_{\mathrm{c}}$ for $N=100,400,1000$, and 4000 (from top to bottom). (c) Same data as panel (b) with rescaled axes.

Figure 4

Largest LE ${\lambda }_1$ of the HMF model. (a) ${\lambda }_1$ versus the system size $N$ for $U=0.5$ (black circles) and $U=0.7$ (red squares). The dashed lines show the quadratic extrapolations to the asymptotic values. (b), (c) Time series of the finite-time exponent ${\lambda }^{\tau }$ for $N=100,400,1000$ with $U=0.25$ [panel (b), from top to bottom] and for $U=0.20,0.25,0.30$ with $N=400$ [panel (c), from top to bottom]. The laminar and burst states have different values of the time-averaged LE as shown in Fig. 6c below. (d) Time fraction in the burst state $F_{\mathrm{burst}}$ as a function of the size $N$ for $U=0.25$ (black circles), $0.30$ (red squares), and $0.35$ (green diamonds).

Figure 5

Localization of the Lyapunov vector. (a) ${\alpha }_i$ versus $i$ for $N=4000$ (black solid line), $N=10000$ (red dotted-dashed line), and $N=20000$ (green dashed line) at $U=0.5$. The dotted black line marks a decay as $1/i$. (b) Average amplitude of the vector components as a function of the corresponding single-oscillator energy, $\overline{\alpha }\left(h\right)$, in a system of $N={10}^5$ oscillators at $U=0.5$. The vertical dashed red line marks the single-oscillator separatrix energy $e_{\mathrm{s}}=2M$.

Figure 6

Intermittency for $U=0.25$ and $N=400$. (a) Time series of the finite-time LE ${\lambda }^{\tau }$ (time window $\tau =2$), the instantaneous value $Y_2$ of the inverse participation ratio, and the energy difference $h_{\mathrm{M}}-e_{\mathrm{s}}$ between the dominating oscillator (the one with the largest amplitude $A_i$ of the Lyapunov vector component) and the separatrix. (b) Density plot with respect to $h_{\mathrm{M}}-e_{\mathrm{s}}$ and $Y_2$. The color code shows the frequency in the logarithmic scale. The black indicates null density. Two peaks corresponding to the burst and the nonburst states are clearly visible. (c) Lyapunov exponent ${\lambda }_1$ (black circles) versus $N$ for $U=0.25$. The conditioned Lyapunov exponents (see text) are also shown for the burst state (red triangles) and for the nonburst state (green diamonds).

Figure 7

Single-oscillator LE versus the energy of the single particle, forced by $N$ globally coupled oscillators at $U=0.5$. In panel (a) the solid curves correspond to $N=100$, 1000, 4000, and $10000$ (from top to bottom); the three rescaled curves in panel (b) correspond to $N=1000$, 4000, and $10000$. The curves overlap except in the peak region, or near the separatrix energy. The vertical dashed lines indicate the energy $e_{\mathrm{s}}$ of the separatrix. Panel (c) shows a close-up of the peak at $h=e_{\mathrm{s}}$, the LE being scaled by the maximum value ${\lambda }_{\mathrm{M}}$ of each curve and plotted against rescaled single-particle energies (crosses, triangles, diamonds, and circles correspond to $N=100$, 1000, 4000, and $10000$, respectively).

Figure 8

Comparison between the maximum LE ${\lambda }_{\mathrm{M}}$ of a single forced oscillator (black squares) at separatrix energy $e_{\mathrm{s}}$ and the first LE ${\lambda }_1$ of the full HMF model for $U=0.5$ (red circles). The dashed lines highlight the quadratic behavior ${\lambda }_1,{\lambda }_{\mathrm{M}}\sim b_0+b_1/\ln N+b_2/{\left(\ln N\right)}^2$ with $b_0=0$ for ${\lambda }_{\mathrm{M}}$. (a) The LEs versus $1/\ln N$. (b) The LEs are multiplied by $\ln N$. The absence of the constant term $b_0$ is confirmed by the linearly arranged symbols for ${\lambda }_{\mathrm{M}}$.

Figure 9

Schematic representation of the toy model for the coupling pressure (see text). Pointlike particles sit at the logarithmic coordinates $x_i=\ln \sqrt{A_i}$ and are either diffusive (red circles with overarrows) or nondiffusive (blue). Diffusive particles turn nondiffusive with a rate ${\alpha }_1$ and nondiffusive ones start to diffuse with a rate ${\alpha }_2\sim {\alpha }_1/\sqrt{N}$, as the corresponding oscillators approach and leave the separatrix, respectively. The coupling is zero as long as particles are no farther than $\ln N$ from the rightmost particle, but otherwise acts as a barrier, preventing particles from being farther than $\ln N$ from the rightmost one. The net effect is a drift to the right.

Figure 10

(a) Average single-particle residence time $t_r$ near the separatrix versus $N$ for $U=0.5$. The residence time is estimated by measuring the crossing time needed for a particle to pass from an energy $e_{\mathrm{L}}$ to an energy $e_{\mathrm{R}}$ or vice versa, where $e_{\mathrm{L}}<e_{\mathrm{R}}$ are the two energies corresponding to the half maximum of the curve ${\lambda }_0\left(h\right)$ reported in Fig. 7. The average is taken over $5\times {10}^5$ to $2\times {10}^6$ events for each size $N$. The dashed red line indicates $N^{1/12}$. (b) Single-particle finite-time diffusion $D\left(t\right)$ versus time $t$ for $U=0.5$. Dashed (black), dot-dashed (red), and solid (blue) lines correspond to $N={10}^3$, $N={10}^4$, and $N={10}^5$, respectively. Inset: Effective diffusion coefficient $D_{\mathrm{s}}$ (see text) as a function of $1/\ln N$. Black circles and red squares correspond to $U=0.5$ and $U=0.7$, respectively. The dashed lines mark the linear extrapolation to the asymptotic value.

Figure 11

Critical scaling of the largest LE ${\lambda }_1(U,N)$. (a) ${\lambda }_1$ as a function of $1/\ln N$ for different energies $U=0.70,0.73$, and $0.745$ from top to bottom. (b) Extrapolated values of the asymptotic largest LE ${\lambda }_{\infty }$ as a function of the distance from the criticality $|U-U_{\mathrm{c}}|$. (c) Size dependence of ${\lambda }_1$ at the critical point $U=U_{\mathrm{c}}$.

Figure 12

Full Lyapunov spectrum at $U=0.7$. (a) Lyapunov spectrum ${\lambda }_i$ as a function of the rescaled index $r\equiv (i-0.5)/N$ for system sizes $N=32,64,...,1024$ from top right to bottom left, as indicated by the arrow. Inset: Close-up of the first part, with sizes $N=2048,4096,...,16384$ added. (b) ${\lambda }_i$ versus $N$ at fixed rescaled indices $r=0.2,0.4,0.6$, and $0.8$ (from top to bottom). The dashed lines indicate ${\lambda }_i\sim 1/\sqrt{N}$. (c) $a^{\left(N\right)}\left(r^{\prime }\right)$, as defined in Eq. (45), plotted as a function of the logarithmically rescaled index $r^{\prime }=(i-1)/\ln N$ for $N=1024,2048,...,16384$. They show reasonable behavior toward the convergence, implying the logarithmic size dependence (44) for these subextensive LEs (see text).

Figure 13

Phase diagram of the generalized HMF model. The shaded area denotes the region where microcanonical and canonical ensembles differ from one another (i.e., the coexistence region of two different thermodynamic phases) occurring in correspondence of discontinuous canonical transitions. Solid and dashed lines correspond to continuous and discontinuous transitions, respectively, within the canonical ensemble. The red dots indicate the three parameter values we have studied.

Figure 14

Lyapunov exponents in the generalized HMF model. (a) Largest LE of the full system, ${\lambda }_1$ versus $1/\ln N$ for $A=0.2,U=1.4$ (black circles); $A=1.0,U=1.5$ (green diamonds); and $A=6.0,U=5.0$ (red squares). The dashed lines indicate linear fits to the scaling regime. (b) Full spectrum ${\lambda }_i$ versus $r=(i-0.5)/N$ with $N=32,64,128,256$, and 512 (from top to bottom) for $A=1.0,U=1.5$. Inset: ${\lambda }_i$ versus $N$ for fixed rescaled indices $r=0.5,1.0,1.5$ from top to bottom. (c) Distribution of the single-oscillator energy $h$ for $A=1.0,U=1.5$ and systems of size $N=40,100,400,1000$, $10000$. The arrow indicates increasing system size. In the inset: Energy-dependent single-oscillator LE for the same oscillator numbers $N$. The vertical dashed line marks the position of the mean-field saddle energy $e_{\mathrm{S}}=2+A(P+1)$, with the numerical estimate $P=0.469\left(1\right)$. (d) The LE ${\overline{\lambda }}_0$ of a single oscillator forced by the full system of size $N$ for $A=1.0,U=1.5$. The red dashed and green solid lines show the results of a quadratic and a linear fitting, respectively, which result in finite asymptotic values of  ${\overline{\lambda }}_0$.