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

## TL;DR

This paper explores the complex phase diagram of the Kuramoto model with non-local interactions and thermal noise, revealing both equilibrium and non-equilibrium phase transitions through exact analytical methods.

## Contribution

It provides the first exact analytical results for the Kuramoto model with non-local interactions in both noise-free and identical-frequency limits.

## Key findings

- Non-local interactions can both promote and hinder synchronization.
- Exact solutions are obtained using Ott-Antonsen ansatz and transfer matrix methods.
- The phase diagram includes rich equilibrium and non-equilibrium transitions.

## 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 nearest-neighbor 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.