Preface: Long-range interactions and synchronization

Abstract

Spontaneous synchronization is a general phenomenon in which a large population of coupled oscillators of diverse natural frequencies self-organize to operate in unison. The phenomenon occurs in physical and biological systems over a wide range of spatial and temporal scales, e.g., in electrochemical and electronic oscillators, Josephson junctions, laser arrays, animal flocking, pedestrians on footbridges, audience clapping, etc. Besides the obvious necessity of the synchronous firings of cardiac cells to keep the heart beating, synchrony is desired in many manmade systems such as parallel computing, electrical power-grids. On the contrary, synchrony could also be hazardous, e.g., in neurons, leading to impaired brain function in Parkinson's disease and epilepsy. Due to this wide range of applications, collective synchrony in networks of oscillators has attracted the attention of physicists, applied mathematicians and researchers from many other fields. An essential aspect of synchronizing systems is that long-range order naturally appear in these systems, which questions the fact whether long-range interactions may be particular suitable to synchronization. In this context, it is interesting to remind that long-range interacting system required several adaptations from statistical mechanics à la Gibbs Boltzmann, in order to deal with the peculiarities of these systems: negative specific heat, breaking of ergodicity or lack of extensivity. As for synchrony, it is still lacking a theoretical framework to use the tools from statistical mechanics. The present issue presents a collection of exciting recent theoretical developments in the field of synchronization and long-range interactions, in order to highlight the mutual progresses of these twin areas.

Critical exponent for the Lyapunov exponent and phase transitions—the generalized Hamiltonian mean-field model

“…The dynamics of classical many-body systems with long range interactions in a D-dimensional space, with potential decaying at large distances r as r −D [1], are described exactly by a Vlasov equation, where a Kac prescription is used in order to have a properly defined continuum limit [2] (for a survey of recent results on long-range systems and its implications in synchronization of non-linear systems see [3]). In this limit, particles interact only through their mean field [4][5][6][7][8], the system never reaches thermodynamic equilibrium, and usually settles into a non-Gaussian stationary state [5].…”