8 Synchronization of Coupled Oscillators—Phase Transitions …
139
ε, we find from Eq. (8.6): α/n ≈ 1/(4ε) log [(1 + ε)/(1 − ε)]. Expanding the righthand-side for small ε up to second order, we next obtain α/n ≈ (1/2) + (1/6)ε
2 .
From the latter expression, we solve for ε and thus derive an approximation for the
amplitude, ε, as a function of the coupling parameter α:
ε ≈
6
n
α −
n
2
.
(8.7)
In order to quantify to what extent a distribution on a cycle as in Fig. 8.3a is
homogeneous, an order parameter, r , is commonly defined as:
re
iψ
=
n
k=1
p k exp
2iπk
n
,
(8.8)
where the “i” in the numerator of both exponents denotes
√
−1. This definition is due
to Lord Rayleigh [2, 26]. It is obvious that we get r = 0 if there is no synchronization
and all the p k ’s are identical. We have r = 1 if there is maximal synchronization and
all molecules are in the same state j, i.e. p k = 1 if k = j and p k = 0 if k = j.
We consider our model (Fig. 8.3) for a large value of n and we again take a simple
approach to come to an upper bound for the value of the order parameter. Imagine that
p k = (1 + ε)/n for 1 ≤ k ≤ n/2 and p k = (1 − ε)/n for n/2 < k ≤ n. The sum on
the right-hand-side of Eq. (8.8) now reduces to (2ε)
n/2
k=1 (1/n) exp [2iπk/n]. For
large n we can approximate the summation with an integral over a half of a period L:
(1/L)
L/2
x=0 exp [2iπx/L] dx = i/π. With this result and with Eq. (8.7), we derive
for the order parameter as a function of α:
r = 0 if α < n/2 and r ≈
3
2
1
n
α − n/2 if α > n/2,
(8.9)
where we took 2
√
6/π ≈ 3/2. Equation (8.9) makes strong statements about the
phase transition. As the number of states in the cycle, n, increases, the phase transition
occurs for ever larger values of the coupling parameter as α c ≈ n/2, where α c stands
for the critical value of α. However, the square root over the (α − n/2)-term means
that the critical exponent maintains its mean-field value of 1/2 (cf. Eq. (8.2)) for all
values of n. In the next section we will verify some of these predictions with both
analytical and numerical work. In the Conclusions and Discussion section of this
article we will put these results in the larger phase-transition context.
8.2.2 The System of ODEs
The system of coupled ordinary differential equations associated with the setup
shown in Fig. 8.3a is:
139
ε, we find from Eq. (8.6): α/n ≈ 1/(4ε) log [(1 + ε)/(1 − ε)]. Expanding the righthand-side for small ε up to second order, we next obtain α/n ≈ (1/2) + (1/6)ε
2 .
From the latter expression, we solve for ε and thus derive an approximation for the
amplitude, ε, as a function of the coupling parameter α:
ε ≈
6
n
α −
n
2
.
(8.7)
In order to quantify to what extent a distribution on a cycle as in Fig. 8.3a is
homogeneous, an order parameter, r , is commonly defined as:
re
iψ
=
n
k=1
p k exp
2iπk
n
,
(8.8)
where the “i” in the numerator of both exponents denotes
√
−1. This definition is due
to Lord Rayleigh [2, 26]. It is obvious that we get r = 0 if there is no synchronization
and all the p k ’s are identical. We have r = 1 if there is maximal synchronization and
all molecules are in the same state j, i.e. p k = 1 if k = j and p k = 0 if k = j.
We consider our model (Fig. 8.3) for a large value of n and we again take a simple
approach to come to an upper bound for the value of the order parameter. Imagine that
p k = (1 + ε)/n for 1 ≤ k ≤ n/2 and p k = (1 − ε)/n for n/2 < k ≤ n. The sum on
the right-hand-side of Eq. (8.8) now reduces to (2ε)
n/2
k=1 (1/n) exp [2iπk/n]. For
large n we can approximate the summation with an integral over a half of a period L:
(1/L)
L/2
x=0 exp [2iπx/L] dx = i/π. With this result and with Eq. (8.7), we derive
for the order parameter as a function of α:
r = 0 if α < n/2 and r ≈
3
2
1
n
α − n/2 if α > n/2,
(8.9)
where we took 2
√
6/π ≈ 3/2. Equation (8.9) makes strong statements about the
phase transition. As the number of states in the cycle, n, increases, the phase transition
occurs for ever larger values of the coupling parameter as α c ≈ n/2, where α c stands
for the critical value of α. However, the square root over the (α − n/2)-term means
that the critical exponent maintains its mean-field value of 1/2 (cf. Eq. (8.2)) for all
values of n. In the next section we will verify some of these predictions with both
analytical and numerical work. In the Conclusions and Discussion section of this
article we will put these results in the larger phase-transition context.
8.2.2 The System of ODEs
The system of coupled ordinary differential equations associated with the setup
shown in Fig. 8.3a is: