Relaxation in a perfect funnel

We have exactly solved the relaxational dynamics of a model protein which possesses a kinetically perfect funnel-like energy landscape. We find that the dependence of the relaxation time, $\tau$, on the density of states (DOS) and the energy level spacing distributions of the model displays several main types of behavior depending on the temperature $T$. This allows us to identify possible generic features of the relaxation. For some ranges of $T$, $\tau$ is insensitive to the density of states; for intermediate values of $T$ it depends on the energy level spacing distribution rather than on the DOS directly, and it becomes gradually more dependent on DOS with increasing temperature; finally, the relaxation can also be determined exclusively by the presence of a deep gap in the energy spectrum rather than by the detailed features of the density of states. We found that the behavior of $\tau$ crucially depends on the degeneracy of the energy spectrum. For the special case of exponentially increasing degeneracy, we were able to identify a characteristic temperature which roughly separates the relaxational regimes controlled by energetics and by entropy, respectively. Finally, the validity of our theory is discussed when roughness of energy landscape is added.