8 Synchronization of Coupled Oscillators—Phase Transitions …
145
Ising models) is zero in the absence of an external field. However, as the temperature
T goes to zero, a singularity is approached. Complete alignment, i.e. r = 1, occurs
at T = 0 [22]. Lev Landau gave a good intuitive explanation of this behavior [24].
We will give an explanation in the same vein for our coupled oscillators, but before
doing so it is instructive to reiterate Landau’s account.
Let −J and +J be the energies for neighboring spins in the parallel and antiparallel alignment, respectively. Next imagine a cyclic 1D array of N spins all in parallel
alignment. We randomly pick two non-neighboring spins on this cycle. There are
W = N (N − 3) ways to make this choice. The chosen spins divide the cycle into
two segments. After flipping all the magnets in one of the segments, we have two
magnetic domains. The flipping involves two interfaces where the energy increases
from −J to +J and thus requires 4J of free energy. However, it is also associated
with an entropy increase of S = k B ln W ≈ 2k B ln N . For finite T and N → ∞,
we always have T S 4J . This means that no finite coupling energy is sufficient
to overcome the thermal noise and give full alignment of all spins. Only at T = 0 is
it possible to achieve r > 0.
Going back to the system depicted in Fig. 8.3a, a similar line of reasoning is
found to apply. Assume we add one oscillator to the system with N tot oscillators.
The randomization over n states is associated with an entropy of S = k B ln n. This
quantity obviously increases with n. As was mentioned before, the energy that is
driving the transitions features in the exponents of the transition rates k i (cf. Eq. (8.5)).
If an oscillator is added to state i, there is a change in the quantities α( p i+2 − p i ) and
α( p i − p i−2 ) that are in the exponents of k i+1 and k i−1 , respectively. For an increasing
number of states, one oscillator is coupled to an ever smaller fraction of the entire
population of oscillators. The changes in the coupling energy will therefore become
ever more insignificant upon increase of n when compared to k B T ln n, i.e. the free
energy change due to entropy effects. This explains why the value of the coupling
constant α at which the phase transition occurs increases with n. Equivalently, the
value of T at which the phase transition occurs decreases with n and we have T c → 0
as n → ∞.
The resiliency of the critical exponent value of 1/2 would be an interesting venue
for future research. We have shown that in a nonequilibrium setting as in Fig. 8.3a,
the value of 1/2 persists for any number of states. But what would happen if, for
instance, we let the rates k i , where i = 1...n, depend on time, i.e., k i goes up or down
as an oscillator spends more time in state i? How much can we modify the system
in Fig. 8.3a before we find another critical behavior?
8.3.4 Implications for Coupled Oscillators in the Wet Lab
We have seen and come to understand that the 2nd order phase transition with a
critical exponent of 1/2 is very robust. The mean-field value of the critical exponent
arises because the mass action (Eq. (8.10)) implies that each oscillator in the system
is coupled to infinitely many other oscillators. Coupling an oscillator to a smaller
145
Ising models) is zero in the absence of an external field. However, as the temperature
T goes to zero, a singularity is approached. Complete alignment, i.e. r = 1, occurs
at T = 0 [22]. Lev Landau gave a good intuitive explanation of this behavior [24].
We will give an explanation in the same vein for our coupled oscillators, but before
doing so it is instructive to reiterate Landau’s account.
Let −J and +J be the energies for neighboring spins in the parallel and antiparallel alignment, respectively. Next imagine a cyclic 1D array of N spins all in parallel
alignment. We randomly pick two non-neighboring spins on this cycle. There are
W = N (N − 3) ways to make this choice. The chosen spins divide the cycle into
two segments. After flipping all the magnets in one of the segments, we have two
magnetic domains. The flipping involves two interfaces where the energy increases
from −J to +J and thus requires 4J of free energy. However, it is also associated
with an entropy increase of S = k B ln W ≈ 2k B ln N . For finite T and N → ∞,
we always have T S 4J . This means that no finite coupling energy is sufficient
to overcome the thermal noise and give full alignment of all spins. Only at T = 0 is
it possible to achieve r > 0.
Going back to the system depicted in Fig. 8.3a, a similar line of reasoning is
found to apply. Assume we add one oscillator to the system with N tot oscillators.
The randomization over n states is associated with an entropy of S = k B ln n. This
quantity obviously increases with n. As was mentioned before, the energy that is
driving the transitions features in the exponents of the transition rates k i (cf. Eq. (8.5)).
If an oscillator is added to state i, there is a change in the quantities α( p i+2 − p i ) and
α( p i − p i−2 ) that are in the exponents of k i+1 and k i−1 , respectively. For an increasing
number of states, one oscillator is coupled to an ever smaller fraction of the entire
population of oscillators. The changes in the coupling energy will therefore become
ever more insignificant upon increase of n when compared to k B T ln n, i.e. the free
energy change due to entropy effects. This explains why the value of the coupling
constant α at which the phase transition occurs increases with n. Equivalently, the
value of T at which the phase transition occurs decreases with n and we have T c → 0
as n → ∞.
The resiliency of the critical exponent value of 1/2 would be an interesting venue
for future research. We have shown that in a nonequilibrium setting as in Fig. 8.3a,
the value of 1/2 persists for any number of states. But what would happen if, for
instance, we let the rates k i , where i = 1...n, depend on time, i.e., k i goes up or down
as an oscillator spends more time in state i? How much can we modify the system
in Fig. 8.3a before we find another critical behavior?
8.3.4 Implications for Coupled Oscillators in the Wet Lab
We have seen and come to understand that the 2nd order phase transition with a
critical exponent of 1/2 is very robust. The mean-field value of the critical exponent
arises because the mass action (Eq. (8.10)) implies that each oscillator in the system
is coupled to infinitely many other oscillators. Coupling an oscillator to a smaller