Correlation functions and their universal connection during an extremely slow equilibration process

We study the equilibration process of a one-dimensional lattice with transverse motions and external magnetic field. Starting from certain initial states, the system commonly reaches a metastable transient state shortly and then stays there for an extremely long time before it finally arrives in the ergodic equilibrium state. The relaxation time $T_{\mathrm{eq}}$ diverges even much more rapidly than exponential, which, compared with the widely reported power-law or even exponential divergence in many other systems, implies much higher stability of the transient state. Two correlation functions, the spatiotemporal correlation of the local energy and the autocorrelation of global heat current, are studied in both the metastable transient and the final equilibrium states. It is revealed that the correlations behave entirely differently in the two different states. In the former case they suggest normal heat diffusion and normal heat conduction; whereas in the latter one they indicate super heat diffusion and anomalous heat conduction. More importantly, we confirm that a general relation which connects the two correlations keeps valid not only in the equilibrium state but in the metastable transient state as well. The universality of the connection is, thus, extended.

Figure 1

(a) The instantaneous ensemble average kinetic energy $\langle E_k\left(t\right)\rangle$ versus time $t$ with $N=512$ and various $B$. The curves are basically $N$ independent. The number of independent runs for each curve ranges from several hundred to several thousand. (b) $\langle E_k\left(t\right)\rangle$ versus time $t$ for the systems in which stochastic momentum exchange with various $\eta$ is applied. Each average is taken over 256 independent runs.

Figure 2

(a) The average potential energy $\langle E_p^{\mathrm{meta}}\rangle$ at the metastable state versus the magnetic field $B$. $E$ denotes the initial per-particle energy. Each average is taken over about 3000 independent runs. The numerical results (symbols) agree with the analytical expectations (curves) quite well, in particular, in the large $B$ cases. In the large $B$ limit, it decays as $B^{-2}$ and $B^{-4}$ for linear and PQ dynamics, respectively. (b) The relaxation time $T_{\mathrm{eq}}$ versus the magnetic field $B$ for the PQ dynamics with $N=512$. It diverges with $B$ very rapidly.

Figure 3

(a) The snapshot of ${\rho }_{\mathrm{E}}(i,t)$ at $t=2000$ for the PQ system without (solid black curve) and with (other curves) ME. The inset: ${\rho }_{\mathrm{E}}(i,t)$ at $t=0$. (b) The MSD of the spatiotemporal correlation $r^2\left(t\right)$ for the PQ system without (black stars) and with (other symbols) ME. The corresponding expectations $t^1$ and $t^{1.15}$ are plotted as black dotted and red dashed lines, respectively, for reference.

Figure 4

(a) The autocorrelation of global heat flux $C_{JJ}\left(t\right)$ for the PQ system in metastable transient and equilibrium states. $N=2048$. Data binning over contiguous $t$ regimes has been performed in order to reduce statistical fluctuations. $C_{JJ}\left(t\right)$ decays asymptotically as $t^{-1.5}$ and $t^{-0.85}$, respectively. (b) Power spectra $P_J\left(\omega \right)$ of the global heat flux $J$ in the same cases.

Figure 5

Power-spectra $P\left(\omega \right)$ in the metastable transient and the equilibrium states. $B=4.5$ and $N=512$. The high peak locates at $\omega =4.5$, which agrees with Eq. (A2) exactly. The inset: the magnification of the low-frequency regime where a new peak emerges. Two straight lines with slopes 0.5 and 1.5 are plotted for reference.