8 Synchronization of Coupled Oscillators—Phase Transitions …
135
8.1.3 Universality in Phase Transitions
For both the liquid-gas transition and the magnetization there is a “competition”
between the system’s tendency to settle in the lowest energy arrangement and the
thermal agitation due to k B T . A phase transition occurs when the system exhibits
a discontinuity upon variation of temperature. Though the physical details and the
basic equations are different for the magnetization and the liquid-gas transition, in
both cases we observed that simple theory predicts a value of 1/2 for an exponent
that actual experiment finds to be significantly lower.
The reason for the discrepancy is the same in both cases. The mean-field approach
is not justified in a 3D space where each particle only interacts with a limited number
of other particles. The mean-field approximation replaces the product of the two spins,
s i s j , in Eq. (8.3) with the product of the average spins, i.e. m
2 . We can actually rewrite
s i s j = (m + δ i )(m + δ j ), where δ can only take the values (1 − m) and (−1 − m).
What the mean field approximation essentially does is neglect all the δ i δ j products
in Eq. (8.3). This will lead to incorrect energies particularly if there are clusters with
parallel spins.
Our treatment of the liquid-gas phase transition was fully based on the Van der
Waals Equation. This equation does not acknowledge local density variations. The
mean-field approximation was therefore implicit when the critical exponent of 1/2
was derived.
Through analytical solutions and approximations, series expansions, and numerical simulations, the critical exponents for the Ising model in spaces of different
dimensionality have been obtained [10]. There is no phase transition at finite temperature in 1D. In 2D and 3D the critical exponents are 1/8 and 0.32, respectively. It
turns out, finally, that in case of more than three dimensions the number of neighbors
is sufficiently large for the mean-field approximation to apply and obtain 1/2 for the
critical exponent.
8.1.4 Phase Transitions for Coupled Oscillators
Already in the 17th century Christiaan Huygens noticed that two clocks, when hanging side-by-side on a wall, will synchronize their ticking over time. For a modern
scientist or engineer it is not hard to understand that such clocks are mechanically
coupled through little shockwaves that propagate through the wall. A contemporary
and animate version of Huygens’ clocks occurred when the London Millenium Footbridge across the Thames was opened in June of 2000. Pedestrians walking across the
bridge began to synchronize their stepping leading the bridge to sway with alarming
and unforeseen amplitude [12].
Setups with N coupled oscillators are commonly modelled with a system due to
Kuramoto [26]:
135
8.1.3 Universality in Phase Transitions
For both the liquid-gas transition and the magnetization there is a “competition”
between the system’s tendency to settle in the lowest energy arrangement and the
thermal agitation due to k B T . A phase transition occurs when the system exhibits
a discontinuity upon variation of temperature. Though the physical details and the
basic equations are different for the magnetization and the liquid-gas transition, in
both cases we observed that simple theory predicts a value of 1/2 for an exponent
that actual experiment finds to be significantly lower.
The reason for the discrepancy is the same in both cases. The mean-field approach
is not justified in a 3D space where each particle only interacts with a limited number
of other particles. The mean-field approximation replaces the product of the two spins,
s i s j , in Eq. (8.3) with the product of the average spins, i.e. m
2 . We can actually rewrite
s i s j = (m + δ i )(m + δ j ), where δ can only take the values (1 − m) and (−1 − m).
What the mean field approximation essentially does is neglect all the δ i δ j products
in Eq. (8.3). This will lead to incorrect energies particularly if there are clusters with
parallel spins.
Our treatment of the liquid-gas phase transition was fully based on the Van der
Waals Equation. This equation does not acknowledge local density variations. The
mean-field approximation was therefore implicit when the critical exponent of 1/2
was derived.
Through analytical solutions and approximations, series expansions, and numerical simulations, the critical exponents for the Ising model in spaces of different
dimensionality have been obtained [10]. There is no phase transition at finite temperature in 1D. In 2D and 3D the critical exponents are 1/8 and 0.32, respectively. It
turns out, finally, that in case of more than three dimensions the number of neighbors
is sufficiently large for the mean-field approximation to apply and obtain 1/2 for the
critical exponent.
8.1.4 Phase Transitions for Coupled Oscillators
Already in the 17th century Christiaan Huygens noticed that two clocks, when hanging side-by-side on a wall, will synchronize their ticking over time. For a modern
scientist or engineer it is not hard to understand that such clocks are mechanically
coupled through little shockwaves that propagate through the wall. A contemporary
and animate version of Huygens’ clocks occurred when the London Millenium Footbridge across the Thames was opened in June of 2000. Pedestrians walking across the
bridge began to synchronize their stepping leading the bridge to sway with alarming
and unforeseen amplitude [12].
Setups with N coupled oscillators are commonly modelled with a system due to
Kuramoto [26]: