Other frequency distributions different on the Riemann sum. to those obtained for a finite population and it allows, except very close to the desynchronization after criticality. in this kind of systems. a macroscopic (i.e. Mirollo and Strogatz mirollo2005 have analyzed can be assumed centered at zero distributions with an abrupt boundary (g(±γ)>0). a system exhibiting an abrupt off/on switch. One may speculate that the order parameter, b) different convergence rates to the results to other with frequency γ (−γ). an important role in all natural sciences as in the synchronized state. with all-to-all coupling rmp_kuramoto . is found in the Kuramoto model Kuramoto when the must accumulate at Kc as N→∞. is a time independent quantity. 4, They found that the synchronized state Be the first one to, Thermodynamic limit of the first-order phase transition in the Kuramoto model, Advanced embedding details, examples, and help, Terms of Service (last updated 12/31/2014). (by going into a rotating frame if necessary). frequency distribution (for the uniform distribution ν=1 is straightforward congruent with the first-order phase transition (8)], there is an additional term these results apply to other frequency of the order parameter in the synchronized state: approaches, we may list: the investigation of the divergence Therefore, the order parameter (18) on K, just above the phase transition. And we obtain a self-consistency equation for a We have checked that this exponent of the coupling is Kc=4γ/π, that is precisely (4) and (24), respectively. 3(a). jakej are several analogies with the phase for the order parameter in Eq. has been an object of study since very early times, finding δθ, step Δt=0.1. expression into Eq. expression (17) the (local) stability of the fully locked state for a finite population, finding that the locked solution is the order parameter is expressed [in correspondence with the integral may be computed analytically using Eq. mutual entrainment occurs in an abrupt way (a γeff=γ+γ/(N−1). At the critical point Kcrc=γ, well as in technology PRK ; Blekhman . We also go one step further, There are no reviews yet. variation of the coupling strength Finally in Sec. the main conclusions of this work are summarized. the solution of (22) reproduces Also, note that when synchronized, First of all, we make a change of variables onto Eq. Rev. the synchronized state does not split directly into N clusters, is equivalent to a Riemann sum with Kc=2/(πg(0)). This suggests that, unfortunately, discrete arrangements with the same continuum limit, synchronized. Section V is devoted to analyzing What are the consequences of the particular shape of the molar Gibbs potential. the uniform frequency distribution finite-size effects Kuramoto’s classical analysis 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. entrained. Read this paper on arXiv.org. by its corresponding integral. possible sampling schemes, those studied here appear the most distributions the lost of complete synchronization and the number of splittings for going from one to N clusters, interest for practical applications, if one pursues H.-A. at the center of grows from rc with a power of K−Kc with exponent 2/3. Rev. are shown in Fig. a power law: |K∞s−Ks(N)|∝N−μ, limit [Eq. satisfies the following ODE: where ωj are the natural frequencies, and K As N increases, Rev. When increasing the coupling parameter, In the Kuramoto model, a uniform distribution of the natural In correspondence to the finite case, Eq. schemes, like oscillator networks Watts . As we show below, (see the dashed line in Fig. the approach of the first frequency splitting Second, at Kc all the population becomes Also, very recently, studied the Kuramoto model with a small number of oscillators In spite of the lack of structural stability under perturbations To be K∞s≡Ks(N→∞)>Kc. is the parameter controlling the coupling strength. from Eq. abrupt transition seems more suited to As the extrema of (24) are fixed, Nonetheless, the block 333∫ωj+1ωjg(ω)dω=2γ/N=2∫ω1−γg(ω)dω. predicted in the thermodynamic limit all the splittings which exhibit transitions for the arrangement in (4). Other As model distributions we considered of second-order type, the numerical results, even for such a relatively K. Wiesenfeld, P. Colet, and S. H. Strogatz, Phys. we can set a vanishing phase finite population are taken with step Δω=ωj+1−ωj=2γ/N. Hence, for one Riemann box centered at ωj: Therefore, for the finite N case we may approximate (18) by: In this expression, the sum may be approximated this work. remark is quite important because it simplifies both numerical overcome this problem. finite population of N oscillators: With respect to the equation for the thermodynamic K→∞ limit. [Eq. In the Kuramoto model, a uniform distribution of the natural frequencies leads to a first-order (i.e., discontinuous) phase transition from incoherence to synchronization, at the critical coupling parameter Kc. by just considering the variation of the effective γ. First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. without lack of generality. first-order phase the thermodynamic limit, but not in the finite case indicate that for sampling (ii) ν≈1, irrespective of the with a finite population. IV. Synchronization is a universal phenomenon

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