Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions

Abstract

In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distributed natural frequencies, which serves as a paradigm to investigate spontaneous collective synchronization in many-body interacting systems, we report on a very rich phase diagram in presence of thermal noise and an additional non-local interaction on a one-dimensional periodic lattice. Remarkably, the phase diagram involves both equilibrium and non-equilibrium phase transitions. In two contrasting limits of the dynamics, we obtain exact analytical results for the phase transitions. These two limits correspond to (i) the absence of thermal noise, when the dynamics reduces to that of a non-linear dynamical system, and (ii) the oscillators having the same natural frequency, when the dynamics becomes that of a statistical system in contact with a heat bath and relaxing to a statistical equilibrium state. In the former case, our exact analysis is based on the use of the so-called Ott-Antonsen ansatz to derive a reduced set of nonlinear partial differential equations for the macroscopic evolution of the system. Our results for the case of statistical equilibrium are on the other hand obtained by extending the well-known transfer matrix approach for nearestneighbor Ising model to consider non-local interactions. The work offers a case study of exact analysis in many-body interacting systems. The results obtained underline the crucial role of additional non-local interactions in either destroying or enhancing the possibility of observing synchrony in mean-field systems exhibiting spontaneous synchronization. 0 K T ∆ K c (T = 0, ∆ = 0) = −1 K c (T, ∆ = 0) = 2T − 1 ∆ c (K = 0, T = 0) = 1/2 T c (∆ = 0, K = 0) = 1/2 ∆ c (K = 0, T ) K c (T = 0, ∆) = 2∆ − 1 CONTENTS 11 dimensionless quantities:

Equilibration in the Nosé–Hoover Isokinetic Ensemble: Effect of Inter-Particle Interactions

“…Here, we discuss the Nosé-Hoover equilibrium properties for model (18). The initial condition corresponds to the θ j 's independently and uniformly distributed in [0, 2π) and the p j 's independently sampled from a Gaussian distribution with zero mean and width equal to 0.5.…”

“…Although ensemble equivalence is not guaranteed for LRI systems, it has been argued that inequivalence arises when one has a first-order phase transition in the canonical ensemble, and not when one has a second-order transition [17]. Consequently, we may regard the phase diagram of model (18) to be equivalent within microcanonical and canonical ensembles. For an explicit demonstration of ensemble equivalence for the model (18), one may refer to [14].…”

“…where the time evolution of the variable ζ is given by Equation (12). In order to derive the desired caloric curve of model (18) within the canonical ensemble, we start with the canonical partition function of the system at temperature T given by…”

“…with I n (x) = (1/(2π)) 2π 0 dθ exp(x cos θ) cos(nθ) being the modified Bessel function of first kind and of order n. It may be shown by following the arguments given in [18] that z s is nothing but the stationary magnetization m eq . Equation (28) has a trivial solution m eq = 0 valid at all temperatures, while a non-zero solution exists for β ≥ β c = 2/J [10].…”

Synchronization behavior in a ternary phase model

“…, N. Inspired by Ref. (17), we consider a spatial continuum of oscillators which can be described by an explicitly position-dependent probability density.…”