A role for a brachistochrone in the broken ergodicity scenario

We show that the time evolution of ergodic decomposition is outlined by a variational principle for a rather universal class of complex systems with a quasicontinuous spectrum of intrinsic relaxation timescales. In turn, this variational approach enables us to single out a logarithmic scaling law for the barriers on the hierarchical level. Such a logarithmic dependence has already been shown to hold valid for certain complex systems. Moreover, it represents a key factor to explain the experimentally ubiquitous Debye–Kohlrausch relaxation law and plays a central role on different theoretical model descriptions within this field.