8 Synchronization of Coupled Oscillators—Phase Transitions …
133
v for a sufficiently small value of P. The area inside the dotted curve in Fig. 8.1a
is where the multivaluedness occurs. The T = T c isotherm is the border curve that
divides the P-v diagram into the two regions. The part of the T < T c curve where
d P/dv > 0 is not realistic. It was James Clerk Maxwell who realized that, between
the points A and B on the dotted curve (cf. Fig. 8.1a), the isotherm has to be replaced
by a horizontal segment (dashed in Fig. 8.1a). It is along this line that liquid and gas
coexist and that the phase transition takes place: as volume is increased, pressure
stays the same and the system responds by evaporating more liquid.
On the T = T c curve there is a point where d
2 P/dv
2
= d P/dv = 0. This is the
so-called critical point. It is a unique point (P c , v c , T c ) in P, v, T -space where the
liquid phase and the gas phase are not distinguishable. In the vicinity of the critical
point, the different physical quantities follow power laws. From the Van der Waals
Equation it can be derived that at leading order
v gas − v liquid
∝ (T c − T )
1/2 or, in
terms of the density ρ = 1/v:
ρ liquid − ρ gas ∝ (T c − T )
1/2
.
(8.2)
In other words, as the temperature is brought from T c to a small T below the critical
temperature, the liquid and gas densities start to differ proportionally to
√ T . This
is a symmetry breaking and Fig. 8.1b illustrates how it occurs. The liquid phase
occupies a smaller volume and therefore a smaller phase-space volume than the
gas phase. A smaller phase-space volume means fewer microstates and therefore,
following the traditional Boltzmann definition [19], a lower entropy. The quantity
(ρ liquid − ρ gas ), cf. Eq. (8.2), can therefore be taken as an order parameter.
The critical behavior described by Eq. (8.2) does not depend on a or b. The
exponent 1/2 is therefore expected to apply universally. Experiment does show universality. However, it appears that the actual exponent is not 1/2, but close to 1/3
[10, 13].
8.1.2 Magnetization
Magnetization is less omnipresent in daily life than evaporation. However, as a phase
transition it is easier to model than the transition from liquid to gas. Figure 8.2a shows
a 2D Ising model. On each lattice point there is an atom whose spin, s, can be pointed
either upward (s = 1) or downward (s = −1). Parallel spins (↑↑) have less energy
than spins with opposite orientation (↑↓). Assuming that an individual spin only
interacts with its four nearest neighbors and that there is no external magnetic field,
we have for the magnetic energy of the entire system
H = −J
i, j
s i s j ,
(8.3)
133
v for a sufficiently small value of P. The area inside the dotted curve in Fig. 8.1a
is where the multivaluedness occurs. The T = T c isotherm is the border curve that
divides the P-v diagram into the two regions. The part of the T < T c curve where
d P/dv > 0 is not realistic. It was James Clerk Maxwell who realized that, between
the points A and B on the dotted curve (cf. Fig. 8.1a), the isotherm has to be replaced
by a horizontal segment (dashed in Fig. 8.1a). It is along this line that liquid and gas
coexist and that the phase transition takes place: as volume is increased, pressure
stays the same and the system responds by evaporating more liquid.
On the T = T c curve there is a point where d
2 P/dv
2
= d P/dv = 0. This is the
so-called critical point. It is a unique point (P c , v c , T c ) in P, v, T -space where the
liquid phase and the gas phase are not distinguishable. In the vicinity of the critical
point, the different physical quantities follow power laws. From the Van der Waals
Equation it can be derived that at leading order
v gas − v liquid
∝ (T c − T )
1/2 or, in
terms of the density ρ = 1/v:
ρ liquid − ρ gas ∝ (T c − T )
1/2
.
(8.2)
In other words, as the temperature is brought from T c to a small T below the critical
temperature, the liquid and gas densities start to differ proportionally to
√ T . This
is a symmetry breaking and Fig. 8.1b illustrates how it occurs. The liquid phase
occupies a smaller volume and therefore a smaller phase-space volume than the
gas phase. A smaller phase-space volume means fewer microstates and therefore,
following the traditional Boltzmann definition [19], a lower entropy. The quantity
(ρ liquid − ρ gas ), cf. Eq. (8.2), can therefore be taken as an order parameter.
The critical behavior described by Eq. (8.2) does not depend on a or b. The
exponent 1/2 is therefore expected to apply universally. Experiment does show universality. However, it appears that the actual exponent is not 1/2, but close to 1/3
[10, 13].
8.1.2 Magnetization
Magnetization is less omnipresent in daily life than evaporation. However, as a phase
transition it is easier to model than the transition from liquid to gas. Figure 8.2a shows
a 2D Ising model. On each lattice point there is an atom whose spin, s, can be pointed
either upward (s = 1) or downward (s = −1). Parallel spins (↑↑) have less energy
than spins with opposite orientation (↑↓). Assuming that an individual spin only
interacts with its four nearest neighbors and that there is no external magnetic field,
we have for the magnetic energy of the entire system
H = −J
i, j
s i s j ,
(8.3)