8 Synchronization of Coupled Oscillators—Phase Transitions …
141
λ j =
α
n
− 1
−
α
n
exp
2πi
j
n
−
α
n
exp
−4πi
j
n
+ (
α
n
+ 1) exp
−2πi
j
n
, j = 0, 1, . . . , n − 1.
(8.13)
Considering only the real parts, we have:
Re[λ j ] =
α
n
1 − cos
4π
j
n
− 1 + cos
2π
j
n
.
(8.14)
Note that λ 0 = 0. This zero eigenvalue is associated with the zero determinant of
the above matrix, Eq. (8.11), and ultimately with the normalized total population,
n
j=1 p j = 1. It is obvious from Eq. (8.14) that all eigenvalues for j ≥ 1 have negative real parts for sufficiently small α. Real parts are zero for α = (n/4) sec
2
(π j/n).
The smallest values of sec
2
(x), and hence the first eigenvalues to become positive, occur for arguments furthest away from the asymptote at x = π/2. This happens simultaneously for j = 1 and j = n − 1 and leads to a critical point point at
α c = (n/4) sec
2
(π/n). For large values of n, the secant squared will approach unity.
We thus have lim n→∞ α c = n/4. The heuristic approach of the previous section led
to α c ≈ n/2. Even though the proportionality factors differ by a factor of two, the
heuristic treatment and the analytical result of this section agree in that they both
have α c increase in directly proportionality to n.
Figure 8.4 shows the results of numerical simulations of Eq. (8.10); for different
values of the number of states, n, the order parameter, r , is plotted as a function
of the coupling parameter α. It appears that the critical exponent follows the γ =
1/2 prediction of the heuristic approach to very good accuracy. A critical exponent
γ = 1/2 was already established for n = 4 in Ref. [5]. We also obtained the plots
for n = 6 and n = 8. For these values we likewise found a critical exponent of 1/2.
For values n 10 the numerical simulations become inaccurate, especially in the
vicinity of the phase transition. There are therefore insufficient data to verify the
3/(2
√
n) prefactor in Eq. (8.9).
8.2.3 The Temperature Dependence
If transitions as in Eq. (8.5) are thermally activated, then the transition rate k generally
follows an Arrhenius dependence and features the temperature, T , in the denominator
of the exponent, i.e. k ∝ exp [−b/T ], where b is positive [19]. This means that
the coupling parameter α in Eq. (8.5) should be replaced by α/T if we wish to
include temperature dependence. We can now see how the system behaves in the
experimentally more likely scenario where the temperature is varied while coupling
constants are fixed and constant. Equation (8.7) has the form r ∝
√ α − α c , where α c
denotes the critical value of α. For a constant α = α 0 and a varying temperature T ,
this formula takes the form r ∝
√ α 0 /T − α 0 /T c ∝
√
1/T − 1/T c . When T is close
to T c , the latter expression is well approximated by r ∝
√
T c − T . This is the same
141
λ j =
α
n
− 1
−
α
n
exp
2πi
j
n
−
α
n
exp
−4πi
j
n
+ (
α
n
+ 1) exp
−2πi
j
n
, j = 0, 1, . . . , n − 1.
(8.13)
Considering only the real parts, we have:
Re[λ j ] =
α
n
1 − cos
4π
j
n
− 1 + cos
2π
j
n
.
(8.14)
Note that λ 0 = 0. This zero eigenvalue is associated with the zero determinant of
the above matrix, Eq. (8.11), and ultimately with the normalized total population,
n
j=1 p j = 1. It is obvious from Eq. (8.14) that all eigenvalues for j ≥ 1 have negative real parts for sufficiently small α. Real parts are zero for α = (n/4) sec
2
(π j/n).
The smallest values of sec
2
(x), and hence the first eigenvalues to become positive, occur for arguments furthest away from the asymptote at x = π/2. This happens simultaneously for j = 1 and j = n − 1 and leads to a critical point point at
α c = (n/4) sec
2
(π/n). For large values of n, the secant squared will approach unity.
We thus have lim n→∞ α c = n/4. The heuristic approach of the previous section led
to α c ≈ n/2. Even though the proportionality factors differ by a factor of two, the
heuristic treatment and the analytical result of this section agree in that they both
have α c increase in directly proportionality to n.
Figure 8.4 shows the results of numerical simulations of Eq. (8.10); for different
values of the number of states, n, the order parameter, r , is plotted as a function
of the coupling parameter α. It appears that the critical exponent follows the γ =
1/2 prediction of the heuristic approach to very good accuracy. A critical exponent
γ = 1/2 was already established for n = 4 in Ref. [5]. We also obtained the plots
for n = 6 and n = 8. For these values we likewise found a critical exponent of 1/2.
For values n 10 the numerical simulations become inaccurate, especially in the
vicinity of the phase transition. There are therefore insufficient data to verify the
3/(2
√
n) prefactor in Eq. (8.9).
8.2.3 The Temperature Dependence
If transitions as in Eq. (8.5) are thermally activated, then the transition rate k generally
follows an Arrhenius dependence and features the temperature, T , in the denominator
of the exponent, i.e. k ∝ exp [−b/T ], where b is positive [19]. This means that
the coupling parameter α in Eq. (8.5) should be replaced by α/T if we wish to
include temperature dependence. We can now see how the system behaves in the
experimentally more likely scenario where the temperature is varied while coupling
constants are fixed and constant. Equation (8.7) has the form r ∝
√ α − α c , where α c
denotes the critical value of α. For a constant α = α 0 and a varying temperature T ,
this formula takes the form r ∝
√ α 0 /T − α 0 /T c ∝
√
1/T − 1/T c . When T is close
to T c , the latter expression is well approximated by r ∝
√
T c − T . This is the same