First-order like phase transition induced by quenched coupling disorder

TL;DR

This paper studies how quenched coupling disorder induces a first-order like phase transition in a population of oscillators, with a detailed analysis of the transition thresholds at zero and finite temperature.

Contribution

It introduces a mean-field theory for the stochastic Kuramoto model with quenched disorder, revealing a first-order like transition at zero temperature and a continuous transition at finite temperature.

Findings

Discontinuous transition at T=0 with a critical threshold matching previous models.

Transition becomes continuous at T>0 with a higher critical threshold.

Derived an exact formula for synchronization threshold at T>0.

Abstract

We investigate the collective dynamics of a population of XY model-type oscillators, globally coupled via non-separable interactions that are randomly chosen from a positive or negative value, and subject to thermal noise controlled by temperature $T$ . For a finite ratio of positive versus negative coupling, we find that the system at $T = 0$ exhibits a discontinuous, first-order like phase transition from the incoherent to the fully coherent state. We determine the critical threshold for this synchronization transition using a linear stability analysis for the fully coherent state and a heuristic stability argument for the incoherent state. Our theoretical results are supported by extensive numerical simulations which clearly display a first order like transition. Remarkably, the synchronization threshold induced by the type of random coupling considered here is identical to the one found in studies which consider uniform input or output strengths for each oscillator node [Hong and Strogatz, Phys. Rev. E 84, 046202 (2011); H. Hong and S. H. Strogatz, Phys. Rev. Lett. 106, 054102 (2011)]. When thermal noise is present, $T > 0$, the transition from incoherence to the partial coherence is continuous and the critical threshold is now larger compared to the deterministic case, T = 0. We formulate an exact mean-field theory applicable to heterogeneous network structures, based on recent work for the stochastic Kuramoto model using graphon objects representing graphs in the mean-field limit. Applying stability results for the incoherent solution, we derive an exact formula for the synchronization threshold for $T > 0$. In the limit $T \rightarrow 0$ we retrieve the result for the deterministic case with $T = 0$.