Diffusive anomalies in a long-range Hamiltonian system

We scrutinize the anomalies in diffusion observed in an extended long-range system of classical rotors, the HMF model. Under suitable preparation, the system falls into long-lived quasi-stationary states for which superdiffusion of rotor phases has been reported. In the present work, we investigate diffusive motion by monitoring the evolution of full distributions of unfolded phases. After a transient, numerical histograms can be fitted by the $q$-Gaussian form $P\left(x\right)\propto {\{1+(q-1){[x∕\beta ]}^2\}}^{1∕(1-q)}$, with parameter $q$ increasing with time before reaching a steady value $q\simeq 3∕2$ (squared Lorentzian). From the analysis of the second moment of numerical distributions, we also discuss the relaxation to equilibrium and show that diffusive motion in quasistationary trajectories depends strongly on system size.

Figure 1

Histograms of rotor phases at different instants of the dynamics (symbols). Simulations for $N=1000$ were performed starting from fully magnetized initial conditions at $\epsilon =0.69$ (conditions leading to QS states). Countings were accumulated over 1000 realizations, at times $t_k=102.4\times 2^k$, with $k=1,2,\dots ,9$, growing in the direction of the arrow up to $t=52428.8$. Solid lines correspond to $q$-Gaussian fittings. Histograms were shifted for visualization. Right-side panel: log-log representation of the fitted data.

Figure 2

Averaged time series of (a) parameter $q$ (symbols), (b) temperature $T$, (c) deviation $\sigma$, and (d) diffusion exponent $\gamma$, for $\epsilon =0.69$ and different values of $N$ ($N=500\times 2^k$, with $k=0,\dots ,9$). Bold lines correspond to $N=500$, as reference, and $N$ increases in the direction of the arrows up to $N=256000$. Averages were taken over $2.56\times {10}^5∕N$ realizations, starting from a fully magnetized configuration at $t=0$. Although small numbers of realizations are used for large sizes, curves are reproducible. In panel (a), the fitting error is approximately 0.03. Dotted lines are drawn as reference. In (b), they correspond to temperatures at equilibrium $(T_{\mathrm{EQ}}=0.476)$ and at QS states in the TL $(T_{\mathrm{QS}}=0.38)$. In (c), they correspond to ballistic motion $(\gamma =2)$ and to normal diffusion $(\gamma =1)$.

Figure 3

Averaged time series of (a) parameter $q$ (symbols), (b) temperature $T$, (c) deviation $\sigma$, and (d) local exponent $\gamma$, as a function of $t∕N^{1.7}$. Data are the same as presented in Fig. 2.

Figure 4

Histograms of rotor phases at different instants of the dynamics (symbols). Simulations were performed for $N=500$ and $\epsilon =0.69$, starting from an equilibrated initial condition. Countings were accumulated over 200 realizations, at times $t_k=0.1\times 4^k$, $k=3,4,\dots ,10$, growing in the direction of the arrow up to $t\simeq 1.05\times {10}^5$. The $q$-Gaussian function with $q=1.53$ was plotted for comparison (solid line). Histograms were shifted for visualization.

Figure 5

Time series of (a) temperature $T$, (b) deviation $\sigma$, and (c) diffusion exponent $\gamma$, for $N=500$ and $\epsilon =0.69$. At $t=0$, the system is in an equilibrium configuration. The outcome of a single run is exhibited.