2.5 Phase Transitions
19
most precise, and it is from there that the prevailing exponent of the power law is
extracted.
In other applications, where the variable x is a physical property rather than a
rank, the log-log plot is drawn in another way, with larger sizes or stronger events
in the tail and frequent ones at the high end, but the cause of deviations is the same:
there are very few large planets, strong earthquakes, and world wars, while on the
other hand small cosmic debris, hardly felt tremors, or minor skirmishes may escape
attention. The reader can say at this point that objects obeying a power law have no
place in a book on morphogenesis. They are here for the sake of contrast: “universal”
laws indifferent to content, compiled statistically, and offering no sure predictions
is all we are left with in a world without form. Another reason is that we should pay
respect to Dragon Kings defying forecasts. Disasters may drive progress. We would
very likely not be here now if dinosaurs had not been put down by a Dragon King.
2.5 Phase Transitions
Liquid droplets aggregate still more readily than solid particles. Why have I not yet
mentioned liquids? Both liquids and gases are fluids, and both are without form, differing only by density, and smoothly merging one into the other at a critical point.
Although we clearly distinguish between air and ocean here on the Earth, the distinction would not be so clear-cut elsewhere. There is a lot of water vapor in the
atmosphere. What would happen if the oceans vaporized, as they probably did on
Venus? Is the interior of stars gaseous or liquid or neither? It is a plasma, of course,
since it is ionized, and this property trumps distinctions in density.
The separation between a liquid and its vapor is just one example of a phase transition, encountered in many other settings, from magnetic spins to separated neighborhoods. The equilibrium state of a physical system is determined by the minimum
of the free energy, F = E − T S. Molecules of the same kind attract each other, and
the energy level E is lowered when they come close together. But the entropy S is at
its highest level when they occupy all the available space, which they would always
do if they do not interact, as in an ideal gas. As the temperature T grows, entropy
prevails, and the phases will not separate above some critical temperature. This kind
of a transition between two disordered phases differs qualitatively from solidification, which is always abrupt, as it establishes a certain order, even if imperfect.
We have to go down to a milder climate, closer to what we are able to endure, to see liquid and vapor as distinct coexisting phases. They are separated by
a sharp interface carrying extra energy, and therefore tending to minimize its area.
Molecules within the bulk of a liquid interact on all sides with molecules of the same
kind, while molecules near the surface have agreeable neighbors only on one side
(Fig. 2.6, left). If this is the boundary with the vapor phase, there are few molecules
of the same kind there. If this is the boundary with another fluid phase, interactions
with molecules on the other side are less favorable – this is the origin of surface
tension.
19
most precise, and it is from there that the prevailing exponent of the power law is
extracted.
In other applications, where the variable x is a physical property rather than a
rank, the log-log plot is drawn in another way, with larger sizes or stronger events
in the tail and frequent ones at the high end, but the cause of deviations is the same:
there are very few large planets, strong earthquakes, and world wars, while on the
other hand small cosmic debris, hardly felt tremors, or minor skirmishes may escape
attention. The reader can say at this point that objects obeying a power law have no
place in a book on morphogenesis. They are here for the sake of contrast: “universal”
laws indifferent to content, compiled statistically, and offering no sure predictions
is all we are left with in a world without form. Another reason is that we should pay
respect to Dragon Kings defying forecasts. Disasters may drive progress. We would
very likely not be here now if dinosaurs had not been put down by a Dragon King.
2.5 Phase Transitions
Liquid droplets aggregate still more readily than solid particles. Why have I not yet
mentioned liquids? Both liquids and gases are fluids, and both are without form, differing only by density, and smoothly merging one into the other at a critical point.
Although we clearly distinguish between air and ocean here on the Earth, the distinction would not be so clear-cut elsewhere. There is a lot of water vapor in the
atmosphere. What would happen if the oceans vaporized, as they probably did on
Venus? Is the interior of stars gaseous or liquid or neither? It is a plasma, of course,
since it is ionized, and this property trumps distinctions in density.
The separation between a liquid and its vapor is just one example of a phase transition, encountered in many other settings, from magnetic spins to separated neighborhoods. The equilibrium state of a physical system is determined by the minimum
of the free energy, F = E − T S. Molecules of the same kind attract each other, and
the energy level E is lowered when they come close together. But the entropy S is at
its highest level when they occupy all the available space, which they would always
do if they do not interact, as in an ideal gas. As the temperature T grows, entropy
prevails, and the phases will not separate above some critical temperature. This kind
of a transition between two disordered phases differs qualitatively from solidification, which is always abrupt, as it establishes a certain order, even if imperfect.
We have to go down to a milder climate, closer to what we are able to endure, to see liquid and vapor as distinct coexisting phases. They are separated by
a sharp interface carrying extra energy, and therefore tending to minimize its area.
Molecules within the bulk of a liquid interact on all sides with molecules of the same
kind, while molecules near the surface have agreeable neighbors only on one side
(Fig. 2.6, left). If this is the boundary with the vapor phase, there are few molecules
of the same kind there. If this is the boundary with another fluid phase, interactions
with molecules on the other side are less favorable – this is the origin of surface
tension.