Relaxation in a perfect funnel

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Abstract
We have exactly solved the relaxational dynamics of a model protein that possesses a kinetically perfect funnel-like energy landscape. We find that the dependence of the relaxation time τ 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, τ 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 τ 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 that 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.
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