A variational approach to relaxation in ultrametric spaces

We investigate the relaxation behavior of complex systems endowed with a rugged free energy landscape from a variational perspective. We focus first on the dynamics in generic ultrametric spaces and then we specialize our generic results to a coarse-grained description of RNA folding, where ultrametricity holds as a limit description of conformation space. Using variational calculus in the generic context, we obtain the brachistochrone or fastest relaxation pathway for the ultrametric distance as a function of the barrier size and conclude that the brachistochrone may only be realized by those systems that follow the exponential or Debye law. Our approach not only reproduces the phenomenological generic relaxation law in the ultrametric limit, but is also justified a posteriori in specialized contexts: It reproduces meaningful folding pathways for the search in conformational space performed by renaturing RNA molecules which are targets of natural selection, reflecting a maximization in the efficiency of the folding process.