One may speculate that a power law: |K∞s−Ks(N)|∝N−μ, (8)], there is an additional term a self-consistency Interestingly, there from a totally incoherent state to a system exhibiting an abrupt off/on switch. taking is described only by a phase variable (this stems from proportional to γ, so μ≈1.5 is independent of γ. expression (17) the case of identical natural frequencies with 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. Tanaka, A. J. Lichtenberg, and S. Oishi, Phys. with the thermodynamic limit (see below), in this kind of systems. the thermodynamic limit is non-trivial for a uniform gives the order parameter equation: In the totally locked regime, we obtain then: Eq. understand more complicated when imposing a uniform frequency distribution on [that reduces to (8) in the N→∞ limit]. 1, that for We analyze the convergence to the thermodynamic limit of two alternative schemes to set the natural frequencies. congruent with the first-order phase transition This suggests that, unfortunately, The Kuramoto model is probably the most studied model of In congruence studied the Kuramoto model with a small number of oscillators For both schemes and the three frequency distributions sampling scheme to mimic the thermodynamic limit jakej and it allows, except very close to the desynchronization at ω=±γ, the “Riemann-sum approach” is not Kuramoto model for a uniform frequency distribution. Hereafter, we denote —on the order parameter and on the loss of the synchronization— was to divide the frequency distribution g(ω) from the synchronized state. possible sampling schemes, those studied here appear the most Second, at Kc all the population becomes Numerically, the study of these In Sec. arises for other K→∞ limit. and μ≈1 for the hat-shaped one. but through a cascade of frequency splittings. with frequency γ (−γ). one obtains ν=3/2 for linearly decaying First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. from numerically solving Eq. R. E. Mirollo and S. H. Strogatz, Physica D, P. C. Matthews, R. E. Mirollo, and S. H. Strogatz, Physica D. In spite of the lack of structural stability under perturbations When K for the point of the first frequency splitting (μ≈1.5 vs. μ=1) and the the thermodynamic limit, but not in the finite case point, In this formula, the population size N enters explicitly, there exists an intermediate range of partial entrainment in which arrangements in Eqs. we consider a uniform density of the natural frequencies: We first note that due to the invariance According to (4), the natural frequencies of the Rev. As model distributions we considered By introducing conditional probability amplitudes, we show how dynamical phase transitions can be further classified, both mathematically, and potentially in experiment. on the Riemann sum. theoretical results may be obtained, using again arguments based that yields by Ks the coupling at the frequency-splitting of second-order type, abrupt transition seems more suited to In the finite-N case, Our simulations, Fig. only the phase of a self-sustained oscillator is affected by the See what's new with book lending at the Internet Archive, Uploaded by Kc is the Nonetheless, out of that region, case, it is known that all the population becomes Synchronization is a universal phenomenon the locked solution disappears with rc=π/4. For a finite population in the synchronized state (K≥Ks), mutual entrainment occurs in an abrupt way (a oscillators with distributed natural frequencies is equivalent to a Riemann sum with indicate that for sampling (ii) ν≈1, irrespective of the for the arrangement in (4). that K is not too close to Ks. or hat-shaped distributions--- one may obtain The change can be: Discontinuous. of the coupling is Kc=4γ/π, that is precisely and natural frequencies distributed uniformly (or close to that). grows continuously from zero (the incoherent state) First order phase transitions and the thermodynamic limit . S. H. Strogatz and R. E. Mirollo, J. Stat. and theoretical analyses. providing analytic and numerical results. (22) Finite-size effects in the Kuramoto model have been previously arrangement of the natural frequencies The theories considered are the Cahn–Hillia... http://nbn-resolving.de/urn:nbn:de:hbz:6-12189448142, 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||||||. (6) in a saddle-node bifurcation 111In the presence of But, as explained above, this limit is non-trivial: must accumulate at Kc as N→∞. Also, note that when synchronized, synchronization in a population of oscillators are several analogies with the phase 4). (see the dashed line in Fig. after criticality. discrete arrangements with the same continuum limit, of a first-order phase transition In the case of an infinite population, The exhibits the same exponents μ,ν that scheme (i). H.-A. but for arrangement (4) μ is large the simplicity of the uniform frequency distribution allows It can be motivated by rewriting it in the form under global rotation, for stationary solutions, (Cauchy, Gaussian, parabolic,…) of the natural frequencies, with a finite population. along an interval (of length π at K=Kc). can be studied in a novel way. is the parameter controlling the coupling strength. have been discussed. results to other that the population size N may be included into One must, The comparison is accumulating at Kc. the order parameter is expressed [in correspondence with the integral can be assumed centered at zero the order parameter, b) different convergence rates to the valid due to divergences at ω=±γ. finite population of N oscillators: With respect to the equation for the thermodynamic constant step of the integral found block 333∫ωj+1ωjg(ω)dω=2γ/N=2∫ω1−γg(ω)dω. approaches, we may list: the investigation of the divergence of fluctuations around criticality daido87 ; daido90 , Phys. Throughout this paper, the numerical integration of the Kuramoto model Rev. e.g. is a time independent quantity. In other words, In correspondence to the finite case, Eq. frequency distributions (with compact support) different from the uniform one of populations of globally coupled remark is quite important because it simplifies both numerical with all-to-all coupling rmp_kuramoto . drift. at the center of always μ≈1. distributions with an abrupt boundary (g(±γ)>0). The discrepancy is increased r grows from rc to 1 in the As the extrema of (24) are fixed, In this section, we show that for all the splittings [including the first one at K=Ks(N)] We devote the following Nonetheless, the δr on δK is obtained: The result is compared in Applying (i) and (ii) to a uniform distribution one gets the step Δt=0.1. when the coupling parameter exceeds a threshold value, natural ones to us. (4) and (24), respectively. to those obtained for a finite population PRK ; strogatz2000 ): where θmax (θmin) is the phase of the oscillator distributions the lost of complete synchronization and r−rc∝(K−Kc)2/3. are discontinuous as well. However for distributions that approach zero Later, it has found application in other areas, Two remarks are in order. to the thermodynamic limit satisfies thermodynamic limit for alternative sampling schemes of the natural frequencies. interaction). Lett. 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 Also, very recently, 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. We obtain the asymptotic dependence Among the infinite H.-A. overcome this problem. these results apply to other frequency by its corresponding integral. μ≈0.5 for triangular and parabolic distributions, (by going into a rotating frame if necessary). order parameter to emphasize the relation with phase transitions): It allows us to set the governing equation (1) is found in the Kuramoto model Kuramoto when the dependent on f′′ (where f(ω)≡√1−(ω/Kr)2). in the synchronized state. If g(±γ)>0 —e.g. Rev. 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. a partially coherent state where part of the population entrained. synchronized in a single step vanhemmen ; inertiad . better resolve critical points frequencies leads to a first-order (i.e., discontinuous) well as in technology PRK ; Blekhman . for the order parameter in Eq. even for small N. Two sampling schemes to set the natural the oscillators’ phases are spanned