8 Synchronization of Coupled Oscillators—Phase Transitions …
143
differential equations actually readily reduces to second order. This is because of the
fixed and normalized total population, i.e. p 1 + p 2 + p 3 = 1.
For n = 4 each state is coupled to two of the four others states. In Ref. [5] the
system of ODEs is numerically solved and, in addition, stochastic simulations are
performed and a heuristic derivation is presented. The critical exponent comes out
to be 1/2. For n = 4, we still face a highly structured dynamical system and it is
remarkable that the statistical approach and the associated mean-field prediction
apparently already apply.
In the system of Fig. 8.3a, each state “interacts” with its two neighbor states. So
for general n, each oscillator effectively connects with a fraction of about 2/n of the
entire population. It appears that for n = 4, the fraction of 1/2 is sufficiently high
to warrant a mean-field approach. As n is increased, we expect the legitimacy of
the mean-field approach to break down. In Ref. [5] the continuum limit, n → ∞, of
the system in Fig. 8.3a is investigated. In that limit the flow of probability density
around the cycle in Fig. 8.3a is described by a PDE that, after some manipulation,
appears equivalent to a Burgers’ Equation [15]. It turns out that for n → ∞, a phase
transition no longer occurs as α is varied. The results presented in Sect. 8.2 of this
article are consistent with this observation: in the n → ∞ limit, the phase transition
is pushed to the α → ∞ limit. Going to n = 10 with our numerical simulations,
we did not observe a change of the mean-field exponent of 1/2. All in all, both the
simple heuristic approach and the full simulation of the ODEs show that the critical
exponent of the phase transition keeps the mean-field value of 1/2, but that the phase
transition occurs for ever higher values of the coupling parameter, α, as the number
of states, n, is increased. We will come back to this in the penultimate subsection.
8.3.2 Entropy and Dissipative Structures
As was mentioned before, our system as depicted in Fig. 8.3 has irreversible transitions. Unlike the systems discussed in Sects. 8.1.1 and 8.1.2, it is not at equilibrium.
Our system produces entropy and in this subsection we will come to a quantitative
assessment of the involved entropies.
One oscillator with a probability p k of being in state k, comes with an associated
entropy of S = −
n
k=1 p k log p k . For the case of a homogeneous distribution, i.e.
p k = 1/n ∀k, it is readily found that S = log n. If we have p k = (1 + ε)/n for half
of the n states and p k = (1 − ε)/n for the other half, then we find S ≈ log n −
ε
2
/2 after we use the approximation log(1 ± ε) ≈ ±ε − ε
2
/2. In other words, the
nonhomogeneous distribution of oscillators over states leads to an entropy decrease.
Clustering decreases entropy.
Consider again the “bump” in Fig. 8.3b. If the probability in each state were
1/n, then the flux from each state to the next would be J i→(i+1) = p i k i = 1/n.
However, when there is a jump as in Fig. 8.3b, the two fluxes adjacent to the jump
carry nonzero exponents (cf. Eq. (8.5)). From state i to i + 1, the probability goes
from (1 − ε)/n to (1 + ε)/n. This leads to J i→(i+1) = (1/n)(1 − ε) exp [2αε/n]
143
differential equations actually readily reduces to second order. This is because of the
fixed and normalized total population, i.e. p 1 + p 2 + p 3 = 1.
For n = 4 each state is coupled to two of the four others states. In Ref. [5] the
system of ODEs is numerically solved and, in addition, stochastic simulations are
performed and a heuristic derivation is presented. The critical exponent comes out
to be 1/2. For n = 4, we still face a highly structured dynamical system and it is
remarkable that the statistical approach and the associated mean-field prediction
apparently already apply.
In the system of Fig. 8.3a, each state “interacts” with its two neighbor states. So
for general n, each oscillator effectively connects with a fraction of about 2/n of the
entire population. It appears that for n = 4, the fraction of 1/2 is sufficiently high
to warrant a mean-field approach. As n is increased, we expect the legitimacy of
the mean-field approach to break down. In Ref. [5] the continuum limit, n → ∞, of
the system in Fig. 8.3a is investigated. In that limit the flow of probability density
around the cycle in Fig. 8.3a is described by a PDE that, after some manipulation,
appears equivalent to a Burgers’ Equation [15]. It turns out that for n → ∞, a phase
transition no longer occurs as α is varied. The results presented in Sect. 8.2 of this
article are consistent with this observation: in the n → ∞ limit, the phase transition
is pushed to the α → ∞ limit. Going to n = 10 with our numerical simulations,
we did not observe a change of the mean-field exponent of 1/2. All in all, both the
simple heuristic approach and the full simulation of the ODEs show that the critical
exponent of the phase transition keeps the mean-field value of 1/2, but that the phase
transition occurs for ever higher values of the coupling parameter, α, as the number
of states, n, is increased. We will come back to this in the penultimate subsection.
8.3.2 Entropy and Dissipative Structures
As was mentioned before, our system as depicted in Fig. 8.3 has irreversible transitions. Unlike the systems discussed in Sects. 8.1.1 and 8.1.2, it is not at equilibrium.
Our system produces entropy and in this subsection we will come to a quantitative
assessment of the involved entropies.
One oscillator with a probability p k of being in state k, comes with an associated
entropy of S = −
n
k=1 p k log p k . For the case of a homogeneous distribution, i.e.
p k = 1/n ∀k, it is readily found that S = log n. If we have p k = (1 + ε)/n for half
of the n states and p k = (1 − ε)/n for the other half, then we find S ≈ log n −
ε
2
/2 after we use the approximation log(1 ± ε) ≈ ±ε − ε
2
/2. In other words, the
nonhomogeneous distribution of oscillators over states leads to an entropy decrease.
Clustering decreases entropy.
Consider again the “bump” in Fig. 8.3b. If the probability in each state were
1/n, then the flux from each state to the next would be J i→(i+1) = p i k i = 1/n.
However, when there is a jump as in Fig. 8.3b, the two fluxes adjacent to the jump
carry nonzero exponents (cf. Eq. (8.5)). From state i to i + 1, the probability goes
from (1 − ε)/n to (1 + ε)/n. This leads to J i→(i+1) = (1/n)(1 − ε) exp [2αε/n]