Synchronization transitions in globally coupled rotors in the presence of noise and inertia: Exact results

We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied SakaguchiKuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and AcebrónBonilla-Spigler models. We derive an exact solution of the self-consistent equations for the order parameter in the stationary state, valid for arbitrary parameters in the dynamics, and demonstrate nontrivial phase transitions to synchrony that include reentrant synchronous regimes. Copyright c © EPLA, 2014 Introduction. – Synchronization in a large population of coupled oscillators of distributed natural frequencies is a remarkable example of a nonequilibrium phase transition. The paradigmatic minimal model to study synchronization is the one due to Kuramoto, introduced almost 40 years ago [1], based on an earlier work by Winfree [2]. Over the years, many details of the Kuramoto model [3,4], and applications to various physical [5], chemical [6], biological [7], engineering [8], and even social problems [9] have been addressed in the literature. The Kuramoto model comprises oscillators that are described by their phases, have natural frequencies given by a common distribution, and are subject to a global mean-field coupling. The phases follow a first-order dynamics in time. In the simplest setup of a purely sinusoidal coupling without a phase shift, and for a unimodal distribution of frequencies, the model exhibits a continuous (second-order) transition from an unsynchronized to a synchronized phase as the coupling constant exceeds a critical threshold. The phase transition appears as a Hopf bifurcation for the complex order parameter. The dynamics of the Kuramoto model is intrinsically dissipative. When all the oscillators have the same frequency, the analogue of the model in the realm of energy-conserving Hamiltonian dynamics is the so-called Hamiltonian mean-field model (HMF) [10,11]. In this case, the dynamical equations are the Hamilton equations: the oscillator phases follow a second-order dynamics in time, i.e., the system constituents are in fact not oscillators, but rotors. In order to include the effects of interaction with an external heat bath, it is natural to consider the HMF evolution in the presence of a Gaussian thermostat. In the resulting Brownian mean-field (BMF) model, the dynamical equations are damped and noise-driven [12,13]. Both the HMF and the BMF model have an equilibrium stationary state that exhibits a continuous phase transition between a synchronized phase at low values of energy/temperature and an unsynchronized phase at high values. On considering the BMF model with nonidentical oscillator frequencies, the dynamics violates the detailed balance leading to a nonequilibrium stationary state (NESS) [14]. In the overdamped limit, the dynamics reduces to that of the noisy Kuramoto model involving Kuramoto dynamics in the presence of Gaussian noise, which was introduced to model stochastic fluctuations of the natural frequencies in time [15]. The resulting phase diagram is complex, with both continuous and first-order transitions [14]. In this work, we study a generic model of globally coupled rotors, in which two types of deviations from equilibrium are included: i) a distribution of torques acting on the rotors, similar to the distribution of frequencies in the Kuramoto model, and ii) a phase shift in the coupling, that makes the latter non-Hamiltonian. We consider the rotors to have quite generally different