Emergent Hierarchical Structures in Complex-System Dynamics

TLDR

The study relates its observations to theoretical models and experimental findings on relaxation in complex systems. The authors introduce a method for studying thermal relaxation in multiminima energy landscapes. The method identifies all configurations linked to a minimum by paths below an energy lid, defines a pocket of phase‑space states, and solves the master equation for relaxation, illustrated on instances of the Travelling‑Salesman Problem. The analysis shows that the number of accessible configurations and the density of states grow exponentially with the energy lid, low‑temperature dynamics involve successive local equilibrations in expanding phase‑space regions, and the propagator decays algebraically with a temperature‑dependent exponent.

Abstract

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.