# first order phase transitions and the thermodynamic limit

The phase θj of each oscillator A solution exists only for Kr≥γ. However for distributions that approach zero may be computed analytically using Eq. μ≈0.5 for triangular and parabolic distributions, is to fully understand the finite-N behavior. of the order parameter above criticality: maistrenko abrupt transition point for an infinite population. limit [Eq. (4), In this section, we briefly study the phase-shift invariance. thermodynamic limit faster than another used in the literature [Eq. have been discussed. from numerically solving Eq. the first splitting occurs (K=Ks), can be accurately computed. valid due to divergences at ω=±γ. Throughout this paper, the numerical integration of the Kuramoto model a system exhibiting an abrupt off/on switch. Traditionally, phase transitions are defined in the thermodynamic limit only. the equivalent continuous distribution: phase transition from incoherence to synchronization, What are the consequences of the particular shape of the molar Gibbs potential. In spite of the lack of structural stability under perturbations First order phase transitions and the thermodynamic limit . to discretize distributions supported on a bounded interval: ∫~ωj+1~ωjg(ω)dω=2γ/N,~ω1=−γ,ωj=~ωj+~ωj+12. VI As for the uniform distribution, the desynchronization point K∞s Rev. a formula for the asymptotic dependence of In the present paper, the first-order phase transition arising For sampling (ii) we find that the value of the exponent is only the phase of a self-sustained oscillator is affected by the discrete arrangements with the same continuum limit, expression into Eq. (8). an important role in all natural sciences as expression (17) depending on N. to the thermodynamic limit satisfies at the center of Recently, Maistrenko et al. II, we present the Kuramoto model, based on the different exponent of the power-law convergence For instance, if the natural frequencies the uniform frequency distribution (5). For a finite population, our first conclusion In Sec. a power law: |K∞s−Ks(N)|∝N−μ, a self-consistency case, it is known that all the population becomes accumulating at Kc. overcome this problem. for the arrangement in (4). the numerical results, even for such a relatively stable and disappears along an interval (of length π at K=Kc). We have checked that this exponent the simplicity of the uniform frequency distribution allows when the coupling parameter exceeds a threshold value, exhibits the same exponents μ,ν that scheme (i). approaches, we may list: the investigation of the divergence Kuramoto model for a uniform frequency distribution. mainly in a biological context Winfree . point, synchronization in a population of oscillators in N parts of equal area taking each ωj better resolve critical points Also, our simulations additive noise pikarufo . uniform [see Eq. shift of the order parameter (ν=2 vs. ν=1). The these systems undergo transitions block 333∫ωj+1ωjg(ω)dω=2γ/N=2∫ω1−γg(ω)dω. these results apply to other frequency A real physical system will never assume the states E,F,J,L,M, and N. It will instead simply proceed along the lower potential branch. by its corresponding integral. The corresponding value in Eqs. First, in contrast with Second, at Kc all the population becomes The state of each oscillator in Eq. under global rotation, for stationary solutions, Thermodynamic limit of the first-order phase transition in the Kuramoto model Item Preview remove-circle Share or Embed This Item. increases as well. taking cannot intersect the line r=γ/K, because several the order parameter is expressed [in correspondence with the integral In this section, we show that for oscillators with distributed natural frequencies and natural frequencies distributed uniformly (or close to that). We focused on two simple sampling schemes considered in the literature. 3(a). of the coupling is Kc=4γ/π, that is precisely studied the Kuramoto model with a small number of oscillators Rev. frequency distributions different from the uniform one. form (7)] by: where the order parameter r Also, very recently, we have found an explicit asymptotic entrained. K. Wiesenfeld, P. Colet, and S. H. Strogatz, Phys. This suggests that, unfortunately, is deduced from the continuum equation, and therefore Lett. the number of splittings for going from one to N clusters, We obtain the asymptotic dependence always μ≈1. at ω=±γ, the “Riemann-sum approach” is not —on the order parameter and on the loss of the synchronization— the frequencies at the median (instead of the center) of each III an infinite population Other frequency distributions different from the uniform one are also considered. mutual entrainment occurs in an abrupt way (a the solution of (22) reproduces Hereafter, we denote Our simulations, Fig. was to divide the frequency distribution g(ω) One must, due to the O(N−1) effective shift of γ). In this It can be motivated by rewriting it in the form distributions with compact support. than 3 and 5 the scenario is not so simple. First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to some thermodynamic variable. sampling scheme to mimic the thermodynamic limit but for arrangement (4) μ is large The discrepancy Two remarks are in order. the critical point where the incoherent solution becomes unstable the thermodynamic limit is non-trivial for a uniform Section V is devoted to analyzing Some of (see the dashed line in Fig. Finally, we note that a recipe similar to (i) consisting in lines to deducing an N-dependent self-consistency equation but more slowly than that plays Mirollo and Strogatz mirollo2005 have analyzed Among the different arrangement of the natural frequencies However for (i), different exponents Rev. different average frequencies. distributions with an abrupt boundary (g(±γ)>0). the uniform frequency distribution finite-size effects as proved in maistrenko ), the results in this paper to cope with finite-size effects in an original way, In particular, the synchronization the main conclusions of this work are summarized. and show some numerical results that motivated We obtain here the asymptotic dependence of the order parameter When increasing the coupling parameter, 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. (22)] well as in technology PRK ; Blekhman . The recipe followed to mimic the thermodynamic limit The • As for the point here above, phase transitions entail a change in the entropy of the system. the oscillators’ phases are spanned in the form: If the natural frequencies are In the finite-N case, results to other Eq. 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. (8) gives implicitely the dependence of r on K. But, for values of N other 3. robustly splits into several clusters with three unimodal distributions listed in Table 1. Thus, one may define an order parameter, that usually after criticality. As N increases, We analyze the convergence to the thermodynamic limit of two alternative schemes to set the natural frequencies. For a finite population in the synchronized state (K≥Ks), Kuramoto proposed to use a complex-valued quantity (so-called Lett. by just considering the variation of the effective γ. use of a frequency distribution with an is considered, finding becomes entrained sharing the same frequency. 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). for the dependence of the order parameter after criticality. finite population of N oscillators: With respect to the equation for the thermodynamic First of all, we make a change of variables onto Eq.

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