Emergent Hierarchical Structures in Complex-System Dynamics

A method is introduced for studying thermal relaxation in multiminima energy landscapes. All the configurations connected to a given energy minimum by paths never exceeding a chosen "energy lid" are found, each equipped with a set of pointers to its neighbours. This information defines a phase space pocket around the minimum, in which the master equation for the relaxation process is directly solved. As an example we analyse some instances of the Travelling-Salesman Problem. We find that i) the number of configurations accessible from a given suboptimal tour grows exponentially with the energy lid, ii) the density of states within the pocket also shows exponential growth, iii) the low-temperature dynamical behaviour is characterized by a sequence of local equilibrations in increasingly larger regions of phase space and finally iv) the propagator decays algebraically with a temperature-dependent exponent. These observations are related to both theoretical models and experimental findings on relaxation in complex systems.