2017/06/30 by Shamik Gupta
Computer Science · Physics and Astronomy · #Ansatz #Context (archaeology) #Ising model #Kuramoto model #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Phase (matter) #Phase diagram #Synchronization (alternating current) #Theoretical and Computational Physics #Thermal equilibrium #cond-mat.stat-mech #nlin.AO #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1751-8121/aa88d7
published as J. Phys. A 50, 424001 (2017) · Invited contribution to the J. Phys. A Special Issue "Emerging Talents" http://iopscience.iop.org/journal/1751-8121/page/Emerging-Talents; v2: minor revision, close to the published version
openalex created_date 2017/06/30 · openalex publication_date 2017/08/29 · arxiv created 2017/09/18 · arxiv updated 2017/09/20 · openalex updated_date 2026/08/05
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.