Chapter 3
Broken Symmetry
3.1 Crystals
Symmetry breaking, though exterminating unlucky creatures in its cruel ways, often operates in a more refined and benevolent manner by creating structures. The
laws of the Universe are built in a way allowing matter to self-organize, enhancing
its complexity at the price of increasing entropy elsewhere. Atoms gather to form
molecules, or they assemble in crystals; molecules combine in polymeric chains and
colloidal particles, polymers evolve to self-replicating circuits, and colloidal particles evolve to cells, and these keep self-organizing to form animals, plants, ecological landscapes, and societies. On all scales, symmetry breaking turns uniform
spaces into patterns.
Crystals are periodic patterns on the atomic scale. Atoms or molecules assemble in a suitable order to optimize their interactions. The order is counteracted by
entropy, defined by the formula chiseled on the grave of Ludwig Boltzmann, its creator: S = k lnW , where the Boltzmann constant k multiplies the logarithm of W , the
number of possible realizations of a certain state. The perfect order is unique, the
logarithm of unity is zero, and so is its entropy. On the contrary, ways to disturb the
order are plentiful. There are many ways to remove an atom from its due position in
a perfectly ordered crystal, and this number is huge in a crystal visible to the naked
eye. There are still more ways to place a stray atom in the interstices of the crystal grid, disturbing its neighbors. Perfect crystals require strong interatomic bonds
and special conditions for their creation; therefore diamonds are not only (almost)
forever, but also very expensive. Thermodynamics teaches us that equilibrium structures strive to minimize their free energy F = E − T S, where E is the internal energy
depending on the strength of the interaction bonds and T is the temperature, which,
multiplying entropy, favors disorder. As a consequence of this simple formula, crystals melt when the temperature rises. This was known long before Boltzmann, and
the very term crystal stems from kryos (κρυoς ), the Greek word for frost.
Perfect order may come in different varieties. The order is rational, and was already precisely understood in the 19th century. The crystal structure is determined
23
© Springer Nature Switzerland AG 2020
L. Pismen, Morphogenesis Deconstructed, The Frontiers Collection,
https://doi.org/10.1007/978-3-030-36814-2_3
The original version of this chapter was revised: Figure 3.5 has been replaced. The correction to
this chapter is available at https://doi.org/10.1007/978-3-030-36814-2_9
Broken Symmetry
3.1 Crystals
Symmetry breaking, though exterminating unlucky creatures in its cruel ways, often operates in a more refined and benevolent manner by creating structures. The
laws of the Universe are built in a way allowing matter to self-organize, enhancing
its complexity at the price of increasing entropy elsewhere. Atoms gather to form
molecules, or they assemble in crystals; molecules combine in polymeric chains and
colloidal particles, polymers evolve to self-replicating circuits, and colloidal particles evolve to cells, and these keep self-organizing to form animals, plants, ecological landscapes, and societies. On all scales, symmetry breaking turns uniform
spaces into patterns.
Crystals are periodic patterns on the atomic scale. Atoms or molecules assemble in a suitable order to optimize their interactions. The order is counteracted by
entropy, defined by the formula chiseled on the grave of Ludwig Boltzmann, its creator: S = k lnW , where the Boltzmann constant k multiplies the logarithm of W , the
number of possible realizations of a certain state. The perfect order is unique, the
logarithm of unity is zero, and so is its entropy. On the contrary, ways to disturb the
order are plentiful. There are many ways to remove an atom from its due position in
a perfectly ordered crystal, and this number is huge in a crystal visible to the naked
eye. There are still more ways to place a stray atom in the interstices of the crystal grid, disturbing its neighbors. Perfect crystals require strong interatomic bonds
and special conditions for their creation; therefore diamonds are not only (almost)
forever, but also very expensive. Thermodynamics teaches us that equilibrium structures strive to minimize their free energy F = E − T S, where E is the internal energy
depending on the strength of the interaction bonds and T is the temperature, which,
multiplying entropy, favors disorder. As a consequence of this simple formula, crystals melt when the temperature rises. This was known long before Boltzmann, and
the very term crystal stems from kryos (κρυoς ), the Greek word for frost.
Perfect order may come in different varieties. The order is rational, and was already precisely understood in the 19th century. The crystal structure is determined
23
© Springer Nature Switzerland AG 2020
L. Pismen, Morphogenesis Deconstructed, The Frontiers Collection,
https://doi.org/10.1007/978-3-030-36814-2_3
The original version of this chapter was revised: Figure 3.5 has been replaced. The correction to
this chapter is available at https://doi.org/10.1007/978-3-030-36814-2_9