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, ea ch 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 suboptima l tour grows exponentially with the energy lid, ii) the density of sta tes within the pocket also shows exponential growth, iii) the low-temp erature dynamical behaviour is characterized by a sequence of local eq uilibrations 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 an d experimental findings on relaxation in complex systems.