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 10-12 2015, honouring Brezin,
Jona-Lasinio and Parisi
Center for the Study of Complex Dynamics
University of Florence
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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
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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,
066201 (2012).
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,
061130 (2012).
S. Gupta, A. Campa and S. Ruffo, Nonequilibrium first-order transition in coupled oscillator
systems with inertia and noise, Phys. Rev. E, 89, 022123 (2014)
S. Gupta, A. Campa and S. Ruffo, Kuramoto model of synchronization: equilibrium and
nonequilibrium aspects, J. Stat. Mech.: Theory and Exp., R08001 (2014)
A. Campa, S. Gupta and S. Ruffo, Nonequilibrium inhomogeneous steady state distribution in
disordered, mean-field rotator systems, arXiv:1502.05559, to be published J. Stat.
Mech.:Theory and Exp. (2015)
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FPU-I
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FPU-II
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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.
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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
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The Kuramoto model
A framework to study spontaneous synchronization:
N globally coupled oscillators with distributed natural frequencies
N
dθi
K̃ X
= ωi +
sin(θj − θi )
dt
N j=1
K̃ coupling constant, ωi quenched random variables with distribution g(ω)
P
Order parameter, fraction of phase-locked oscillators: r = |1/N j exp(iθj )|
High K̃: Synchronized phase , r > 0
Low K̃: Incoherent phase, r = 0.
For unimodal g(ω) continuous transition on tuning K̃.
r
i-th
oscillator
1
θi
0
ec
K
e
K
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The Kuramoto-Sakaguchi model
r=0
r 6= 0
0
Ke
Kec
Stochastic fluctuations of the ωi in time
N
dθi
K̃ X
= ωi +
sin(θj − θi ) + ηi (t)
dt
N j=1
< ηi (t) >= 0 , < ηi (t)ηj (t′ ) >= 2Dδij δ(t − t′ )
D
Kec(D)
r=0
0
r 6= 0
Ke
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2nd order dynamics
Two dynamical variables: θi (Phase); vi (Angular velocity)
dθi
= vi
dt
N
dvi
K X
m
= −γvi + ωi +
sin(θj − θi ) + ηi (t)
dt
N 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 (2008), Rohden, Sorge, Timme
and Witthaut (2012), Olmi and Torcini (2014)
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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 (2000)
HOWEVER, THE COMPLETE PHASE DIAGRAM HAS NOT BEEN ADDRESSED
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Rescaling
One can analyze the model in the reduced parameter space (T, σ, m)
dθi
= vi
dt
N
1
1 X
dvi
= − √ vi + σωi +
sin(θj − θi ) + ηi (t)
dt
m
N j=1
where now:
g(ω) has zero average and unit width
< ηi (t)ηj (t′ ) >=
2T
√
δ δ(t
m ij
− t′ )
Two steps
Subtracting average motion: θi ⇒ θi + < ω > t,vi ⇒ vi + < ω > t, ωi ⇒ ωi + < ω >
p
p
√
√
′
′
′
Rescaling: t = t K/m,vi = vi m/K,1/ m = γ/ Km, σ ′ = γσ/K, T ′ = T /K.
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Equilibrium and nonequilibrium
The role of σ
σ = 0, ⇒ no external drive ⇒ detailed balance ⇒ equilibrium stationary state
σ > 0 ⇒ non equilibrium stationary state (Gawedzki and Nardini using
Jona-Lasinio et al. MFT)
Continuous transition lines
m = T = 0, σ > 0, Kuramoto
m = 0, T > 0, σ > 0, Sakaguchi, continuous transition, critical line
σ = 0 Hamiltonian system + heat-bath (Brownian Mean Field model), continuous
transition, critical line Chavanis (2013)
σ
Kuramoto model
critical point
r=0
Critical line
r 6= 0
0
BM
Fc
r 6= 0
riti
cal
line
T
m
r=0
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Phase diagram-I
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Phase diagram-II
(b)
coh
0.5 σ inc
σ
σ 0.25
0
T 0.2
0.4
50
25
m
0
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Hysteresis
Order parameter
T=0.25,N=500
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
m=1
m=100
0
σ inc
0.25
Adiabatically tuned σ
0.5
σ coh
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Bistability
For m = 20, T = 0.25, N = 100, and a Gaussian g(ω) with zero mean and unit width, (left)
shows, at σ = 0.195, r vs. time in the stationary state, while (right) shows the distribution
P (r) at several σ’s around 0.195.
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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
”
T ∂2f
∂f
∂f
∂ “ v
= −v
+
,
√ − σω − r sin(ψ − θ) f + √
∂t
∂θ
∂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 = 0) solution
f inc
√
„
«
(v − σω m)2
1
1
√
exp −
=
2π 2πT
2T
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Linear stability results
Stability analysis gives σ inc : f (θ, v, ω, t) = f inc (θ, v, ω) + eλt δf (θ, v, ω)
∞
X
(−mT )p (1 +
2T
=
emT
p=0
p
mT
)
p!
+∞
Z
−∞
g(ω)dω
1+
p
mT
+ i σω
+
T
λ
√
T m
.
Acebron, Bonilla and Spigler (2000)
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 = 0 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 σc (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
inc
(m, σ).
σnoiseless
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Landau free energy
σ < σ inc
σ = σ inc
(i)
σ inc < σ < σc
(iii)
(ii)
F (r)
σ = σc
(iv)
0
r
1
σc < σ < σ
(v)
11111
00000
00000
11111
coh
σ > σ coh
σ = σ coh
(vi)
(vii)
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Mean-field metastability
∆
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Synchronized steady state
”
∂ “ v
T ∂2f
∂f
+
0 = −v
√ − σω − r sin(ψ − θ) f + √
∂θ
∂v
m
m ∂v 2
Z
reiψ =
dθdvdω g(ω)eiθ f (θ, v, ω)
sync
(θ, v, ω)
fsteady
“
√v
2T
”P
∞
n=0 bn Φn
“
√v
2T
= Φ0
P
√ k
bn (θ) = ∞
(
m) cn,k (θ) asymptotic series
k=0
”
Hermite basis
Recursion relations for cp,k :
c0,0
0
c0,2
0
c0,4
0
c0,6
c1,1
0
c1,3
0
c1,5
0
c2,2
0
c2,4
0
c2,6
c3,3
0
c3,5
0
c4,4
0
c4,6
c5,5
0
c6,6
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Some details of the derivation
Inserting the expansion in Hermite into the stationary Kramers equation, one gets the
Brinkman hierarchy
√
n
∂bn−1 (θ, ω) p
∂bn+1 (θ, ω)
+ (n + 1)T
+ √ bn (θ, ω)+
nT
∂θ
∂θ
m
Expanding the bn ’s in powers of
cn,k
√
r
n
bn−1 (θ, ω)[r sin(θ)−σω] = 0
T
m, one obtains recursion equations for the coefficients
√
∂cn−1,k (θ, ω) p
∂cn+1,k (θ, ω) √
nT
+ (n + 1)T
+ nT a(θ, ω)cn−1,k (θ, ω)+ncn,k+1 (θ, ω) = 0
∂θ
∂θ
for n, k = 0, 1, 2, . . . (with c−1,k (θ, ω) ≡ 0), 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.
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Borel transform-I
Brezin, Parisi, Zinn-Justin, .
.
.
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Borel transform-II
Z ∞
∞
X
a
k
tk AB (x) =
dt exp(−t)BA(tx)
A(x) =
ak xk , BA(t) ≡
k!
0
k=0
k=0
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
n(θ)
n(θ)
∞
X
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
1
2
3
θ
4
5
6
0
1
2
3
4
5
6
θ
m = 0.25, T = 0.25, σ = 0.295. Left panel: Borel resumed with ktrunc = 38, Right panel:
direct summation with ktrunc = 22.
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Simulations
0.45
0.12
0.4
0.1
0.35
0.08
0.25
p(θ)
n(θ)
0.3
0.2
0.15
0.06
0.04
0.1
0.02
0.05
0
0
0
1
2
3
θ
4
5
6
0
1
2
3
4
5
6
θ
(left panel): particle density n(θ), m = 0.25,T = 0.25,σ = 0.295,ktrunc = 12, N = 106
(right panel): pressure p(θ), same parameter values
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Local temperature
0.45
0.3
0.4
0.35
n(θ)
0.3
0.25
0.28
0.2
T(θ)
0.29
0.27
0.15
n(θ)
T(θ)
0.1
0.26
0.05
0
0.25
0
1
2
3
4
5
6
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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. 6= exp(−β(K + U )), not a product measure
Dynamics matters: Phase transitions different for underdamped and overdamped
dynamics
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α-Kuramoto
K
dθi
= ωi +
e
dt
N
N
X
j=1,j6=i
sin(θj − θi )
|xj − xi |α
ωi 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
∂ “ ∂θ ”
=−
ρ
∂t
∂θ
∂t
∂θ(ω, x, t)
= ω + κ(α)K
∂t
Z
′
dθ dω
′
′
′ sin(θ − θ)
′
′
′
′
dx
ρ(θ
;
ω
,
x
,
t)g(ω
)
|x′ − x|α
Linear stability analysis of the homogeneous state
ρ(θ; ω, x, t) =
1
+ δρ(θ; ω, x, t)
2π
Dispersion relation
ck (α)K
1−
2
Z
∞
−∞
dω
g(ω)
=0
(λk ± iω)
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Stability of the incoherent state
If g(ω) is symmetric around the mean and non increasing then λk is either real positive or
zero.
The dispersion relation rewrites
1 − ck (α)K
Z
∞
dω
0
λk
g(ω) = 0
2
2
λk + ω
The limit λk → 0 gives the critical couplings
(k)
Kc
(0)
Kc
=
2
ck (α)πg(0)
(1)
< Kc
(2)
< Kc
< ....
and the growth rates λk
2K
(k)
πg(0)Kc
Z
0
∞
hλ i
K λ2 /2
λk
k
k
e
Erfc
g(ω)
=
dω 2
= 1.
√
(k)
λk + ω 2
2
Kc
for Gaussian g(ω).
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Zero-mode dominance
1
r0(t)
r1(t)
r2(t)
r3(t)
0.1
0.01
0.001
0.0001
1e-05
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time t
(0)
α = 0.5 , K = 15 , Kc
(1)
≈ 1.59577 , Kc
(2)
≈ 4.26696 , Kc
(3)
≈ 6.53664 , Kc
≈ 7.71516, . . . ,
so that the Fourier modes 0, 1, 2, 3 are all linearly unstable.
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