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_{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.