Brachistochrone approach to the relaxation dynamics of complex hierarchical systems

We investigate the structure and the relaxation dynamics of complex hierarchical systems from a variational point of view. First, an additional argument for the use of the ultrametric caricature to describe disordered systems is provided. Focusing on ultrametric models, we show that two relevant dynamical limit behaviors of such models, the limit of convergence of the dynamics and the transition from compact to noncompact exploration, are in fact realizations of brachistochrone relaxation pathways. In turn, by making use of a rugged model whose conformation space topology deviates from ultrametricity under selective controllable conditions, we show that while the exponent of the resulting relaxation law behaves as ruggedness-dependent, its functional form is robust with respect to the introduction of ruggedness. Finally, within this rugged context, the relaxation dynamics of the two above-mentioned limit behaviors are shown to correspond to characteristic relaxation laws: Debye–Kohlrausch and power decay, respectively.