Kuramoto model of synchronization:equilibrium and non equilibrium aspects

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1 Kuramoto model of synchronization:equilibrium and non equilibrium aspects Stefano Ruffo Dipartimento di Fisica e Astronomia and CSDC, Università degli Studi di Firenze and INFN, Italy 113th Statistical Mechanics Conference, Rutgers, may , honouring Brezin, Jona-Lasinio and Parisi University of Florence Center for the Study of Complex Dynamics Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.1/31

2 Plan Synchronization Kuramoto and Sakaguchi models, the role of noise Inertia Equilibrium vs. non equilibrium Complete phase diagram: First and second order phase transition Hysteresis and bistability Linear stability analysis Steady state perturbative expression α-kuramoto Zero-mode dominance Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.2/31

3 References S. Gupta, M. Potters and S. Ruffo, One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes, Phys. Rev. E, 85, 6621 (212). S. Gupta, A. Campa and S. Ruffo, Overdamped dynamics of long-range systems on a one-dimensional lattice: Dominance of the mean-field mode and phase transition, Phys. Rev. E, 86, 6113 (212). S. Gupta, A. Campa and S. Ruffo, Nonequilibrium first-order transition in coupled oscillator systems with inertia and noise, Phys. Rev. E, 89, (214) S. Gupta, A. Campa and S. Ruffo, Kuramoto model of synchronization: equilibrium and nonequilibrium aspects, J. Stat. Mech.: Theory and Exp., R81 (214) A. Campa, S. Gupta and S. Ruffo, Nonequilibrium inhomogeneous steady state distribution in disordered, mean-field rotator systems, arxiv: , to be published J. Stat. Mech.:Theory and Exp. (215) Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.3/31

4 FPU-I Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.4/31

5 FPU-II Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.5/31

6 Huygens It is quite worth noting that when we suspended two clocks... from two hooks imbedded in the same wooden beam, the motions of each pendulum in opposite swings were so much in agreement that they never receded the least bit from each other and the sound of each was always heard simultaneously. Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.6/31

7 Synchronization Spontaneous synchronization: Coordination of events to operate a system in unison, in the absence of any ordering field Flashing fireflies Synchronized firing of cardiac pacemaker cells Phase synchronization in electrical power distribution networks Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.7/31

8 The Kuramoto model A framework to study spontaneous synchronization: N globally coupled oscillators with distributed natural frequencies dθ i dt = ω i + K N NX sin(θ j θ i ) j=1 K coupling constant, ω i quenched random variables with distribution g(ω) Order parameter, fraction of phase-locked oscillators: r = 1/N P j exp(iθ j) High K: Synchronized phase, r > Low K: Incoherent phase, r =. For unimodal g(ω) continuous transition on tuning K. θ i i-th oscillator 1 r K c K Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.8/31

9 The Kuramoto-Sakaguchi model r = r K c K Stochastic fluctuations of the ω i in time dθ i dt = ω i + K N NX sin(θ j θ i ) + η i (t) j=1 < η i (t) >=, < η i (t)η j (t ) >= 2Dδ ij δ(t t ) D K c (D) r = r K Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.9/31

10 2nd order dynamics Two dynamical variables: θ i (Phase); v i (Angular velocity) dθ i dt = v i m dv i dt = γv i + ω i + K N NX sin(θ j θ i ) + η i (t) j=1 where m is the inertia and γ the friction constant. Motivation: An adaptive frequency can explain the slower approach to synchronization observed in a particular firefly (the Pteropyx mallacae) Ermentrout (1991) Phase dynamics in electric power distribution networks in the mean-field limit Filatrella, Nielsen and Pedersen (28), Rohden, Sorge, Timme and Witthaut (212), Olmi and Torcini (214) Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.1/31

11 Previous studies No noise: Simulations for a Lorentzian g(ω) show a first-order synchronization transition Tanaka, Lichtenberg and Oishi (1997) Analysis in the continuum limit, based on a suitable Fokker-Planck equation in the limit N for a Lorentzian g(ω): either larger inertia or larger ω spread makes the system harder to synchronize Acebron and Spigler (1998); Acebron, Bonilla and Spigler (2) HOWEVER, THE COMPLETE PHASE DIAGRAM HAS NOT BEEN ADDRESSED Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.11/31

12 Rescaling One can analyze the model in the reduced parameter space (T, σ, m) dθ i dt = v i dv i dt = 1 m v i + σω i + 1 N NX sin(θ j θ i ) + η i (t) j=1 where now: Two steps g(ω) has zero average and unit width < η i (t)η j (t ) >= 2T m δ ij δ(t t ) Subtracting average motion: θ i θ i + < ω > t,v i v i + < ω > t, ω i ω i + < ω > Rescaling: t = t p K/m,v i = v ip m/k,1/ m = γ/ Km, σ = γσ/k, T = T/K. Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.12/31

13 Equilibrium and nonequilibrium The role of σ σ =, no external drive detailed balance equilibrium stationary state σ > non equilibrium stationary state (Gawedzki and Nardini using Jona-Lasinio et al. MFT) Continuous transition lines m = T =, σ >, Kuramoto m =, T >, σ >, Sakaguchi, continuous transition, critical line σ = Hamiltonian system + heat-bath (Brownian Mean Field model), continuous transition, critical line Chavanis (213) m σ Kuramoto model critical point r = Critical line r r r = BMF critical line T Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.13/31

14 Phase diagram-i Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.14/31

15 Phase diagram-ii (b).5 σ coh σ inc σ.25 T m Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.15/31

16 Hysteresis Order parameter T=.25,N=5 m=1 m= Adiabatically tuned σ σ inc σ coh Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.16/31

17 Bistability For m = 2, T =.25, N = 1, and a Gaussian g(ω) with zero mean and unit width, (left) shows, at σ =.195, r vs. time in the stationary state, while (right) shows the distribution P(r) at several σ s around.195. Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.17/31

18 N continuum limit Single-particle distibution f(θ, v, ω, t): Fraction of oscillators at time t and for each ω which have phase θ and angular velocity v (Periodic in θ and normalized). Evolution by Kramers equation f t f = v θ + v σω r sin(ψ θ) f + T 2 f v m m v 2, with self-consistent order parameter r exp(iψ) = ZZZ dθdvdωg(ω) exp(iθ)f(θ, v, ω, t) Homogeneous (r = ) solution f inc = 1 2π 1 exp (v σω m) 2 «2πT 2T Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.18/31

19 Linear stability results Stability analysis gives σ inc : f(θ, v, ω, t) = f inc (θ, v, ω) + e λt δf(θ, v, ω) 2T e mt = X p= ( mt) p (1 + p mt ) p! + Z g(ω)dω 1 + p mt + i σω T + λ T m. Acebron, Bonilla and Spigler (2) The equation for λ has at most one solution with a positive real part and, when the solution exists, it is necessarily real. Neutral stability λ = gives the stability surface σ inc (m, T). Similarly, one can define σ coh (m, T). The two surfaces enclose the first-order transition surface σ c (m, T). Taking proper limits, the surface σ inc (m, T) meets the critical lines on the (T, σ) and (m, T) planes. The intersection of the surface with the (m, σ) plane gives an implicit formula for σnoiseless inc (m, σ). Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.19/31

20 Landau free energy σ < σ inc σ = σ inc σ inc < σ < σ c (i) (ii) (iii) F(r) σ = σ c 1 r σ c < σ < σ coh (iv) σ = σ coh σ > σ coh (v) (vi) (vii) Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.2/31

21 Mean-field metastability Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.21/31

22 Synchronized steady state = v f θ + v σω r sin(ψ θ) f + T 2 f v m m v 2 Z re iψ = dθdvdω g(ω)e iθ f(θ, v, ω) f sync steady (θ, v, ω) = Φ v P 2T n= b nφ n b n (θ) = P k= ( m) k c n,k (θ) asymptotic series Recursion relations for c p,k : v 2T Hermite basis c, c,2 c,4 c,6 c 1,1 c 1,3 c 1,5 c 2,2 c 2,4 c 2,6 c 3,3 c 3,5 c 4,4 c 4,6 c 5,5 c 6,6 Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.22/31

23 Some details of the derivation Inserting the expansion in Hermite into the stationary Kramers equation, one gets the Brinkman hierarchy nt b n 1 (θ, ω) θ + p (n + 1)T b n+1(θ, ω) + n r n b n (θ, ω)+ θ m T b n 1(θ, ω)[r sin(θ) σω] = Expanding the b n s in powers of m, one obtains recursion equations for the coefficients c n,k nt c n 1,k (θ, ω) θ + p (n + 1)T c n+1,k(θ, ω) + nta(θ, ω)c n 1,k (θ, ω)+nc n,k+1 (θ, ω) = θ for n, k =,1, 2,... (with c 1,k (θ, ω) ), where a(θ, ω) [r sin(θ) σω]/t. This system of equations can be solved recursively using the scheme illustrated by the arrows in the sparse matrix of coefficients. Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.23/31

24 Borel transform-i Brezin, Parisi, Zinn-Justin,... Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.24/31

25 Borel transform-ii A(x) = X a k x k, BA(t) k= X k= a k k! tk A B (x) = Z dt exp( t)ba(tx) n(θ).25.2 n(θ) θ θ m =.25, T =.25, σ =.295. Left panel: Borel resumed with k trunc = 38, Right panel: direct summation with k trunc = 22. Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.25/31

26 Simulations n(θ).25.2 p(θ) θ θ (left panel): particle density n(θ), m =.25,T =.25,σ =.295,k trunc = 12, N = 1 6 (right panel): pressure p(θ), same parameter values Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.26/31

27 Local temperature n(θ) n(θ) T(θ) T(θ) Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.27/31

28 A non equilibrium perspective Kuramoto model as an overdamped limit of a long-range interacting system. General dynamics: (1) External drive, (2) Quenched vs. annealed randomness. No quenched randomness Equilibrium Prob. distr. exp( β(k + U)), product measure Phase transition given by U same for underdamped and overdamped dynamics Quenched randomness Non equilibrium stationary state Prob. distr. exp( β(k + U)), not a product measure Dynamics matters: Phase transitions different for underdamped and overdamped dynamics Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.28/31

29 α-kuramoto dθ i dt = ω i + K e N NX j=1,j i sin(θ j θ i ) x j x i α ω i is a quenched random variable with distribution g(ω) In the continuum limit, the local density ρ(θ; ω, x, t) satisfies the continuity equation ρ t = θ ρ θ t θ(ω, x, t) t = ω + κ(α)k Z dθ dω dx sin(θ θ) x x α ρ(θ ; ω, x, t)g(ω ) Linear stability analysis of the homogeneous state ρ(θ; ω, x, t) = 1 2π + δρ(θ; ω, x, t) Dispersion relation 1 c k(α)k 2 Z dω g(ω) (λ k ± iω) = Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.29/31

30 Stability of the incoherent state If g(ω) is symmetric around the mean and non increasing then λ k is either real positive or zero. The dispersion relation rewrites 1 c k (α)k Z dω λ 2 k λ k + ω2 g(ω) = The limit λ k gives the critical couplings K (k) c = 2 c k (α)πg() K () c < K (1) c < K (2) c <.... and the growth rates λ k 2K πg()k (k) c Z dω λ 2 k λ k + ω2 g(ω) = K K (k) c h e λ2 k /2 λk i Erfc = 1. 2 for Gaussian g(ω). Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.3/31

31 Zero-mode dominance 1.1 r (t) r 1 (t) r 2 (t) r 3 (t) e Time t α =.5, K = 15, K () c , K (1) c , K (2) c , K (3) c ,..., so that the Fourier modes, 1, 2, 3 are all linearly unstable. Kuramoto model of synchronization:equilibrium and non equilibrium aspects p.31/31