One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes

Gupta, Shamik; Potters, Max; Ruffo, Stefano

doi:10.1103/physreve.85.066201

Cited by 28 publications

(34 citation statements)

References 21 publications

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“…While we provided strong numerical evidence for the existence of both the synchronized and the incoherent phase, only the latter could be treated analytically to obtain the corresponding linear stability threshold that bounds the firstorder transition point from below. It would be interesting to consider possible extension of our studies to systems with non-mean-field couplings, taking hints from similar previous studies in specific limits of the dynamics [34][35][36][37].…”

Section: Discussionmentioning

confidence: 99%

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

2014

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We study the dynamics of a system of coupled oscillators of distributed natural frequencies, by including the features of both thermal noise, parametrized by a temperature, and inertial terms, parametrized by a moment of inertia. For a general unimodal frequency distribution, we report here the complete phase diagram of the model in the space of dimensionless moment of inertia, temperature, and width of the frequency distribution. We demonstrate that the system undergoes a nonequilibrium first-order phase transition from a synchronized phase at low parameter values to an incoherent phase at high values. We provide strong numerical evidence for the existence of both the synchronized and the incoherent phase, treating the latter analytically to obtain the corresponding linear stability threshold that bounds the first-order transition point from below. In the limit of zero noise and inertia, when the dynamics reduces to the one of the Kuramoto model, we recover the associated known continuous transition. At finite noise and inertia but in the absence of natural frequencies, the dynamics becomes that of a well-studied model of long-range interactions, the Hamiltonian mean-field model. Close to the first-order phase transition, we show that the escape time out of metastable states scales exponentially with the number of oscillators, which we explain to be stemming from the long-range nature of the interaction between the oscillators.

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Section: Discussionmentioning

confidence: 99%

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

2014

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“…With a proper choice of the origin, the coordinate of the ith site is x i = ia. On each site resides an oscillator, with the dynamics of the ith oscillator governed by the evolution equation [48,49] …”

Section: The Kuramoto Model With a Power-law Coupling Between Oscillamentioning

confidence: 99%

Kuramoto model of synchronization: equilibrium and nonequilibrium aspects

Gupta¹,

Campa²,

Ruffo³

2014

J. Stat. Mech.

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102

136

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Abstract. The phenomenon of spontaneous synchronization, particularly within the framework of the Kuramoto model, has been a subject of intense research over the years. The model comprises oscillators with distributed natural frequencies interacting through a mean-field coupling, and serves as a paradigm to study synchronization. In this review, we put forward a general framework in which we discuss in a unified way known results with more recent developments obtained for a generalized Kuramoto model that includes inertial effects and noise. We describe the model from a different perspective, highlighting the longrange nature of the interaction between the oscillators, and emphasizing the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view. In this review, we first introduce the model and discuss both for the noiseless and noisy dynamics and for unimodal frequency distributions the synchronization transition that occurs in the stationary state. We then introduce the generalized model, and analyze its dynamics using tools from statistical mechanics. In particular, we discuss its synchronization phase diagram for unimodal frequency distributions. Next, we describe deviations from the mean-field setting of the Kuramoto model. To this end, we consider the generalized Kuramoto dynamics on a one-dimensional periodic lattice on the sites of which the oscillators reside and interact with one another with a coupling that decays as an inverse power-law of their separation along the lattice. For two specific cases, namely, in the absence of noise and inertia, and in the case when the natural frequencies are the same for all the oscillators, we discuss how the long-time transition to synchrony is governed by the dynamics of the mean-field mode (zero Fourier mode) of the spatial distribution of the oscillator phases.

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“…Recent results within such a setup and with a focus similar to the present review may be found in Refs. [57,58].…”

Section: Discussionmentioning

confidence: 99%

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

Gherardini

Gupta

Ruffo

2018

Contemporary Physics

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Spontaneous synchronization is a remarkable collective effect observed in nature, whereby a population of oscillating units, which have diverse natural frequencies and are in weak interaction with one another, evolves to spontaneously exhibit collective oscillations at a common frequency. The Kuramoto model provides the basic analytical framework to study spontaneous synchronization. The model comprises limit-cycle oscillators with distributed natural frequencies interacting through a mean-field coupling. Although more than forty years have passed since its introduction, the model continues to occupy the centre-stage of research in the field of non-linear dynamics, and is also widely applied to model diverse physical situations. In this brief review, starting with a derivation of the Kuramoto model and the synchronization phenomenon it exhibits, we summarize recent results on the study of a generalized Kuramoto model that includes inertial effects and stochastic noise. We describe the dynamics of the generalized model from a different yet a rather useful perspective, namely, that of long-range interacting systems driven out of equilibrium by quenched disordered external torques. A system is said to be long-range interacting if the inter-particle potential decays slowly as a function of distance. Using tools of statistical physics, we highlight the equilibrium and nonequilibrium aspects of the dynamics of the generalized Kuramoto model, and uncover a rather rich and complex phase diagram that it exhibits, which underlines the basic theme of intriguing emergent phenomena that are exhibited by many-body complex systems.

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One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes

Cited by 28 publications

References 21 publications

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

Kuramoto model of synchronization: equilibrium and nonequilibrium aspects

Spontaneous synchronisation and nonequilibrium statistical mechanics of coupled phase oscillators

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