Animals in nature have an uncanny ability to operate as a single unit with collective movement when they group together. Flocks of birds have been found to fit statistical models, people walking together naturally match each others step, bacteria can form groups that then move as one, with each individual responding to the same chemical stimuli.
Back in 1855 a mathematical physicist turned doctor, called Adolf Eugen Fick, created what is now known as Fick’s laws of diffusion. The first is simply that the the diffusive flux is proportional to the concentration gradient and the second is that the rate of concentration change is proportional to the derivative of the concentration gradient. Later the same conclusion was reached when applying the method of random walks to particles in suspension and the final result was reached where the mean squared displacement of a particle is proportional to the time since it stared to move. This is considered the standard description of the diffusion process:
〈 χ2 〉 ∝ t
In chemistry Hund’s rule is one that may not always be remembered associated with his name, but is nonetheless always recalled. It is the fact that electrons will always attempt to avoid pairing with another electron in an orbital and so all orbitals in a shell will gain one electron before they start doubling up. This can be simply explained as this arrangement allows the electrons to exist nearer the nucleus and so makes the overall atom more stable.
Quite a while ago I wrote a post on materials that are called auxetic. This is where the Poisson ratio, the ratio of how much thinner a material gets when stretched a certain amount, is negative. This means that if you had a rod of auxetic material and you tried stretching it at the ends, it would get thicker in the middle. To the same degree compressing actually makes it get thinner, the opposite of most materials we encounter in every day life. The actual production of auxetic materials, a type of mechanical metamaterial, is actually quite easy with multiple methods being suggested such as moulding, 3D printing and the use of precise incisions into certain regular materials.
Complex numbers have a tendency to spring up in more advanced physics concepts such as electrical engineering, more sophisticated versions of refraction and of course in the waveform of particles in quantum mechanics. It is simply very useful to have a whole mathematical dimension off the real number line to work with when dealing with things of varying phase relation. The mathematical explanation of why we can’t just use a basic sinusoidal function for an electron’s wave is because such a result would give absolute solutions when we know that there has to be uncertainty in the answer to the wave equation.
Physics has a lot to do with patterns. We observed that an object’s acceleration was related to its mass; the heavier the object the more force was required to make it accelerate at the same rate. This led to arguably the simplest physics equation: F = m x a which is simply the above observation in mathematical form. In physics many events are correlated, which means that event A leads on to and determines event B. But in the example above both the acceleration and force were occurring in the same object; what should be done if A and B occur in different locations and at different times? It is not hard to believe that the massive break off of ice in the Arctic sea in 2007 (A) led to the uncharacteristically frozen rain in southern China in 2008 (B). This is a considerable problem for physicists to handle as the models of systems must have selected a set time (normally t=0) and a set space (normally the origin) for the event to begin at and have values of time and space in relation to these points.
The quality of being self similar is when a pattern manages to contain a smaller, and possibly distorted, version of the entire pattern. Mathematicians can generate these shapes to their heart’s content but in physics these patterns only appear sparingly. Below is a picture of the Hofstadter-Butterfly, a geometric pattern representing the theorised energy levels for electrons in a magnetic field.