38
3 Broken Symmetry
front, placed on such a level of the long-range inhibitor (marked by the dashed line
in Fig. 3.14) that the areas U and D are equal.
The FN system and models of this kind have been applied to a variety of patternforming processes, from the BZ reaction to nerve conduction to ecological interactions to coloration of animal fur, sometimes reflecting the actual mechanism in a
simplified way, sometimes producing apparently similar patterns in spite of having
no connection with reality. When the real world is recalcitrant, we build ourselves
toy models based on equations simple enough for us to solve. Sometimes a toy
model provides illuminating insights into behavior in the real world. More often, it
remains what its name implies, a plaything for mathematically inclined physicists
(Dyson, 2004).
The possibility of oscillations on a microscale, labelled “time crystals”, was proposed by Frank Wilczek (2012). This was followed by a string of papers proving
what should be quite clear: that no time-periodic state is possible without an external drive. Quantum effects notwithstanding, oscillations cannot take place at equilibrium. “Time crystals” were realized in a laboratory (Lukin et al, 2017), but only
when sustained by a periodic drive. The period of oscillations is an integer multiple
of the driving period, so that oscillations of such “time crystals” are not autonomous.
A “space-time crystal”, that would be seen when oscillations propagate as waves, is
a still more fancy term sounding as if it is connected to relativity theory, although it
does not touch upon it in any way.
3.6 Branching Patterns
Another common structure, besides stationary or dynamic patterns, seen in various
non-equilibrium systems, from phase transitions to living forms, is the dendritic
structure. This is typically generated by growth. If a crystal grows by accretion of
atoms diffusing from a solution, its flat surface is unstable. If there is slight protuberance on the surface, this will be better exposed to the solution, and will grow further.
On the other hand, a dimple will be less accessible and will be left further behind.
The farther a protuberance grows and the closer it comes to the source of material,
its sides will lose stability and it will start to branch out in its turn (Langer, 1980).
The result is a dendritic structure like the one in the left-hand panel of Fig. 3.15.
Some bacterial colonies are structured in the same way, as they grow in response to
the supply of a nutrient (Ben-Jacob et al, 1994).
This instability never saturates, the interphase boundary remains unstable, and
a dendrite keeps branching and growing. Of course, this goes for an ideal dendrite
in an infinite medium. In reality, any instability saturates at some level that is not
accounted for in the basic model. The growing crystal will slow down when its tip
becomes too sharp, close to the molecular scale or, in the case of a bacterial colony,
to the size of an individual microbe.
Branching patterns also can be caused by hydrodynamic or elastic instabilities. Viscous fingering develops through the Saffman–Taylor (1958) instability in
3 Broken Symmetry
front, placed on such a level of the long-range inhibitor (marked by the dashed line
in Fig. 3.14) that the areas U and D are equal.
The FN system and models of this kind have been applied to a variety of patternforming processes, from the BZ reaction to nerve conduction to ecological interactions to coloration of animal fur, sometimes reflecting the actual mechanism in a
simplified way, sometimes producing apparently similar patterns in spite of having
no connection with reality. When the real world is recalcitrant, we build ourselves
toy models based on equations simple enough for us to solve. Sometimes a toy
model provides illuminating insights into behavior in the real world. More often, it
remains what its name implies, a plaything for mathematically inclined physicists
(Dyson, 2004).
The possibility of oscillations on a microscale, labelled “time crystals”, was proposed by Frank Wilczek (2012). This was followed by a string of papers proving
what should be quite clear: that no time-periodic state is possible without an external drive. Quantum effects notwithstanding, oscillations cannot take place at equilibrium. “Time crystals” were realized in a laboratory (Lukin et al, 2017), but only
when sustained by a periodic drive. The period of oscillations is an integer multiple
of the driving period, so that oscillations of such “time crystals” are not autonomous.
A “space-time crystal”, that would be seen when oscillations propagate as waves, is
a still more fancy term sounding as if it is connected to relativity theory, although it
does not touch upon it in any way.
3.6 Branching Patterns
Another common structure, besides stationary or dynamic patterns, seen in various
non-equilibrium systems, from phase transitions to living forms, is the dendritic
structure. This is typically generated by growth. If a crystal grows by accretion of
atoms diffusing from a solution, its flat surface is unstable. If there is slight protuberance on the surface, this will be better exposed to the solution, and will grow further.
On the other hand, a dimple will be less accessible and will be left further behind.
The farther a protuberance grows and the closer it comes to the source of material,
its sides will lose stability and it will start to branch out in its turn (Langer, 1980).
The result is a dendritic structure like the one in the left-hand panel of Fig. 3.15.
Some bacterial colonies are structured in the same way, as they grow in response to
the supply of a nutrient (Ben-Jacob et al, 1994).
This instability never saturates, the interphase boundary remains unstable, and
a dendrite keeps branching and growing. Of course, this goes for an ideal dendrite
in an infinite medium. In reality, any instability saturates at some level that is not
accounted for in the basic model. The growing crystal will slow down when its tip
becomes too sharp, close to the molecular scale or, in the case of a bacterial colony,
to the size of an individual microbe.
Branching patterns also can be caused by hydrodynamic or elastic instabilities. Viscous fingering develops through the Saffman–Taylor (1958) instability in