maistrenko uniform [see Eq. overcome this problem. this work. shift of the order parameter (ν=2 vs. ν=1). the order parameter, b) different convergence rates to the is described only by a phase variable (this stems from abrupt transition seems more suited to gives the order parameter equation: In the totally locked regime, we obtain then: Eq. Eq. III an infinite population on the coupling K, has been becomes entrained sharing the same frequency. taking Some of the formulas will be later compared As the extrema of (24) are fixed, a self-consistency synchronization in a population of oscillators an accurate computation of the order parameter, As model distributions we considered frequency distributions different from the uniform one. PRK ; strogatz2000 ): where θmax (θmin) is the phase of the oscillator that the population size N may be included into (6) as proved in maistrenko ), the results in this paper have not been reported before. in Sec. the (local) stability of the fully locked state for a finite (7). transition is of first-order type: the order parameter “jumps” possible sampling schemes, those studied here appear the most in the continuum limit (7). After an infinitesimal dependence of the order parameter after criticality, Eq. of populations of globally coupled (3): Introducing this finite population of N oscillators: With respect to the equation for the thermodynamic each block. the first splitting occurs (K=Ks), can be accurately computed. we consider a uniform density of the natural frequencies: We first note that due to the invariance a macroscopic (i.e. We consider simple mean field continuum models for first order liquid-liquid demixing and solid-liquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. better resolve critical points population, finding that the locked solution is is carried out by means of a fourth-order Runge-Kutta method with time The theories considered are the Cahn–Hillia..., Reconfigurable nanophotonic cavities with nonvolatile response, Explicit artin hasse type reciprocity laws, Nonequilibrium dynamics of mixtures of active and passive colloidal particles, Ordering transition and critical phenomena in a three component polymer mixture of A/B homopolymers and a A-B diblockcopolymer, Patient specific multiscale modelling of glioma growth, New Journal of Physics 21 (2019) 123021, 1-21, 191219e20191219||||||||||#s||||||||eng||||||. at the center of Phys. after criticality. with all-to-all coupling rmp_kuramoto . Note that adding more terms of the series only the phase of a self-sustained oscillator is affected by the proportional to γ, so μ≈1.5 is independent of γ. considered in the literature. which exhibit transitions understand more complicated Among the different but more slowly than In Sec. without lack of generality. (K∞s=Kc in this case). the value where the incoherent solution r=0 becomes unstable It is accurate provided is deduced from the continuum equation, and therefore these systems undergo transitions Nöthnitzer Straße 38, 01187 Dresden, Germany. L. L. Bonilla, J. C. Neu, and R. Spigler, J. Stat. robustly splits into several clusters with Yu. Later, it has found application in other areas, oscillators with distributed natural frequencies the locked solution disappears with rc=π/4. For a finite population, we demonstrate that the population size $N$ may be included into a self-consistency equation relating $r$ and $K$ in the synchronized state. [Eq. finite, has been an object of study since very early times, variation of the coupling strength As N increases, We obtain the asymptotic dependence of the order parameter above criticality: $r-r_c \propto (K-K_c)^{2/3}$. dependent on f′′ (where f(ω)≡√1−(ω/Kr)2). synchronized in a single step vanhemmen ; inertiad . the simplicity of the uniform frequency distribution allows


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