Patterns in space and/or time form in fluid dynamics (Rayleigh-Bénard convection and Taylor-Couette flow), laser physics, materials science (dendrites in the formation of solids from liquid melts), biology (morphogenesis), and chemistry (Belousov-Zhabotinsky reactions). External constraints, such as temperature gradients, drive most of these systems out of equilibrium.
Novelty.
The parts of the system can be viewed as the molecular constituents or small uniform parts of the system. In either case, the whole system has a property (a pattern) that the parts do not have.
Discontinuity.
When some parameter becomes larger than a critical value, the system transitions from a uniform state to a non-uniform state.
Universality.
Similar patterns, such as convection rolls in fluids, can be observed in diverse systems regardless of the microscopic details of the fluid. Often, there is a single parameter, such as the Reynolds number, which involves a combination of fluid properties, that determines the type of patterns that form. Cross and Hohenberg highlighted how the models and mechanisms of pattern formation across physics, chemistry, and biology have similarities. Turing’s model for pattern formation in biology associated it with concentration gradients of reacting and diffusing molecules. However, Gierer and Meinhardt showed that it is sufficient to have a network with competition between short-range positive feedback and long-range negative feedback. This could occur in a circuit of cellular signals.
Self-organisation.
The formation of a particular pattern occurs spontaneously, resulting from the interaction of the many components of the system.
Effective theories.
A crystal growing from a liquid melt can form shapes such as dendrites. This process involves instabilities of the shape of the crystal-liquid interface. The interface dynamics are completely described by a few partial differential equations that can be derived from macroscopic laws of thermodynamics and heat conduction. A helpful review is by Langer.
Diversity.
Diverse patterns are observed, particularly in biological systems. In toy models, such as the Turing model, with just a few parameters, a diverse range of patterns, both in time and space, can be produced by varying the parameters. Many repeated iterations can lead to a diversity of structures. This may result from a sensitive dependence on initial conditions and history. For example, every snowflake is different because, as it falls, it passes through a slightly different environment, with small variations in temperature and humidity, compared to others.
Toy models.
Turing proposed a model for morphogenesis in 1952 that involved two coupled reaction-diffusion equations. Homogeneous concentrations of the two chemicals become unstable when the difference between the two diffusion constants becomes sufficiently large. A two-dimensional version of the model can produce diverse patterns, many resembling those found in animals. However, after more than seventy years of extensive study, many developmental biologists remain sceptical of the relevance of the model, partly because it is not clear whether it has a microscopic basis. Kicheva et al., argue that “pattern formation is an emergent behaviour that results from the coordination of events occurring across molecular, cellular, and tissue scales.”
Other toy models include Diffusion Limited Aggregation, due to Witten and Sander, and Barnsley’s iterated function system for fractals that produces a pattern like a fern.
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