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# Physics of Pattern Formation

## 1 Introduction

Nature is full of all kinds of patterns - from the stripes of a zebra or the surface ripples on sand dunes, to the scales of a lizard or even the shapes of various types of galaxies. Some of these patterns are created by genetic programming that is a result of evolution. For example, a female panther may be able to judge the fitness of a male from the appearance of the spot pattern on the male's fur, which helps to choose a mating partner that will maximize the probability of healthy offspring. Some other patterns, such as aurora borealis, are a result of simple physical processes that obviously don't have any "life" in them and can't be seen as working for any particular cause. To be precise, evolution isn't *consciously* working for any cause either, but it is undoubtedly effective in developing organisms with better survival possibilities.

Natural patterns and the human sensory system have the curious property that two patterns can appear "similar", as judged by a human observer, without being anywhere near identical. If you take two photos that have pictures of two different zebras' stripe patterns in them, you can immediately see that the patterns are similar to each other, and totally different from some other black and white patterns like that on a Dalmatian dog's fur. However, if you use some simple algorithm to calculate the *correlation* between the bitmaps of two zebra patterns, it can be very small. It's very hard to define what we mean with similarity of patterns exactly enough to be able to tell a computer how to recognize the similarity.

This article gives a short introduction to pattern forming phenomena and their relation to physical concepts of instabilities and chaos. Here these are described in words, without any complicated mathematics. In another Hub, meant for the readers who have studied physics or math in the university, I will give examples of how to prove that the solutions of certain field equations can be unstable and pattern-forming.

## 2 Field systems

Physical science is a study of systems, states and dynamics. A system can be many kinds of things, like a single oxygen molecule, the water contained in an aquarium, the earth's atmosphere or even our Solar system. When studying the physics of a system, we first want to know how to completely define the temporary state of the system, or at least the properties of the state that we are interested in.

If an astronomer wants to describe the state at which our solar system is right now, they will probably give coordinates and velocities of all planets relative to the position and velocity of the sun, and possible also the orientations of the planets to inlude information about planetary rotation. If a physical chemist wants to describe the state of an oxygen molecule, they will give an electronic wavefunction, but are not interested in the quantum state of the nuclei. If someone studies acoustics and wants to describe the state of the air contained inside a room, they are interested in the air pressure as function of position, because that is the medium in which sound waves exist.

Sometimes the state is described by giving values to some set of variables, as in the description of the solar system, where the coordinates and velocities of the planets were specified. The coordinates are called "*degrees of freedom*". In other cases, the system is something we call a field, and in that case the state is a function of position coordinates. A field is a system with infinitely many degrees of freedom, because to fully specify a function of x, y and z, we have to know its value at every single point of the region of space it is defined in. Some examples of fields are an air pressure field, electromagnetic field, temperature field and a gravitational field.

When one says that some field has patterns in it, it is meant that the field function is not a constant (have same value at every point of space), but also that the variation in the field values is not just any kind of random noise either. The inhomogeneities of the field must have a certain regularity in them to qualify as patterns.

After we know how to give all necessary info about the state of the system, the next step is to seek out the equations of motion for the system. With such dynamical equations, we are able to predict the future development of the system's state, given that we now its temporary state now. For a classical mechanical system of bodies or particles, the equation of motion is Newton's second law F=ma, where F is a force we need to know, m is the mass of a body and a is the acceleration (second time derivative of position). For example, in classical mechanics of the solar system, the force F is given by Newton's law of gravitation.

The dynamical equations that describe the behavior of a field are *partial differential equations*, containing derivatives of the field function with respect to both the position and time coordinates. Commonly encountered field equations include heat conduction equation, wave equation and Maxwell's equations. These can be divided to different classes, based on the order of the derivatives they contain, and on whether they are *linear or nonlinear* equations. Pattern formation exclusively happerns in systems that obey nonlinear dynamical equations.

## 3 Instabilities and patterns

The dynamics of some fields has the property that they strive to become constant fields. For example, if a field describing the temperature at different points of a solid object has variations in it, the temperature differences tend to even out because of *heat conduction*, until the temperature is the same at all points. Once the temperature field has become a constant, it will remain constant indefinitely unless somehow disturbed externally. There's no reason why there would be a spontaneous flow of heat from one part on an object to another part, unless this heat flow acts to even out temperature differences. This is one way to state an important physical law, the *Second Law of Thermodynamics*. We say that the constant temperature field is a *stationary solution* of the heat conduction equation.

Let's consider one special type of a field system, a thin layer of water on a horizontal plastic surface. The state of this system is specified by giving a function h(x,y), the local thickness of the water layer at coordinates x and y. Physicists have invented a field equation that gives the time development of this function h(x,y). It is simply called *thin film equation*. When the properties of this equation are studied, it immediately becomes apparent that a film of constant thickness is a stationary solution to it.

But now, this seems a bit counterintuitive. We know from experience that water on a plastic surface doesn't want to form a thin film that has a large contact area with the plastic. It wants to form spherical drops to *minimize the contact area* with the water-repelling (hydrophobic) plastic, just like the water on the leaves in the image above.

This apparent paradox can be resolved mathematically. The thin film equation is something we call a nonlinear differential equation. Equations of this type can have *unstable stationary solutions.* The flat water film on plastic is such an unstable solution. Such a film is so unstable to external disturbances, that a slightest amount of sound waves or air currents coming to contact with it immediate causes the film to break up and form drops of water like in the image.

This kind of instabilities explain many things, including the fact that our universe is not composed of a gas of contant density. Such a constant-density universe would be *gravitationally unstable*, and it would immediately start becoming more dense at some places than the others, eventually forming stars that are hugely more dense than the interstellar space between them.

This happens because the combined field equations of gravity and matter density are nonlinear partial differential equations, just like the thin film equation was. The fact that the force of gravity is strong enough to destabilise the constant density universe, is an example of an anthropic coincidence, an unlikely fortunate thing that made the creation of living nature possible.

One special type of a potentially unstable field system, related to patterns in living nature, is a chemical reaction-diffusion system. In that kind of systems, a combination of chemical reactions taking place in a liquid solution causes pattern formation in the concentration fields of the reactants. If the reacting substances are colorful, we can also see those patterns. The most famous system of this type is the *Belousov-Zhabotinsky reaction* (see *YouTube video*). The formation of biological patterns such as the zebra stripes, is suspected to be a result of reaction-diffusion phenomena. This *hypothesis* was originally presented by the mathematician *Alan Turing* (the same man who broke the Enigma code of the Nazi military). One could even speculate that the first primordial living cells, that formed on our planet almost 4 billion years ago, might have been a result of some obscure pattern forming reaction.

Pattern-forming instabilities often possess the property of having some characteristic length scale, which is the order of magnitude of the size of the patterns they create. On the other hand, some instabilities produce *fractal patterns*, which don't have a characteristic length.

## 4 Some words about chaos

When something is called "chaotic", it is usually something unpredictable. For example, if some country goes to chaos and there is a civil war, it's often difficult to predict the future government. In physics, we have all kinds of dynamical equations, that predict the future behavior of some physical system completely, given that we know the state of the system at some initial moment. This is called *determinism*, and may give the impression that there is no room for chaos in physics.

There are, however, situations where we can't fully predict something with the law of physics. For example, systems that must be described using *quantum mechanics*, such as a single atom or molecule, behave deterministically only as far as they aren't observed. A measurement that determines the state of the quantum system, will always also disturb the system in an unpredictable way, and only statistical predictions can be made for the result of the measurement.

Another form of practical indeterminism is called *deterministic chaos*. It is caused by the some process's sensitivity to initial conditions. A very small numerical inaccuracy in our knowledge of the degrees of freedom of the initial state can make our prediction of some later state of the system grossly inaccurate. One situation where this kind of chaos occurs, is turbulent flow in gases and liquids. The pattern formation processes described in the previous chapter are also chaotic - it is practically impossible to predict the exact patterns that form in them.

The concept of chaos also appears in disciplines other that physical science. For instance, the behavior of prices in the *stock market* is chaotic, as are many other things that depend on the behavior of some human population.

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