Illustrated Physics
physics, drawn
Concept

Self-similarity — where it appears

The property of looking statistically the same at different magnifications, so that a part of an object is a reduced copy of the whole. Fractals grown by diffusion are self-similar over a range of scales, which is why their mass follows a power of their size.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Every value the orbit of x → r x(1 − x) settles on. The values a long orbit of x → r x(1 − x) visits, one column of the picture for each of 320 settings of r between 2.8 and 4. A single point means the orbit settles to one value, two means it alternates, and each branching doubles the count with the gaps shrinking by a constant factor. The superstable settings marked run 3.23607, 3.49856, 3.55464, located by bisection on the map itself. They accumulate at r = 3.569946, and past it the orbit visits a band of values rather than a list of them. The bands are not noise: the map has no random number in it, and the same initial value gives the same orbit every time.

The map a dripping tap turns out to be

A universal result about one-dimensional maps is worth nothing to a physicist unless a real system is one, and a tap, a convection cell and a driven circuit are continuous systems with no map in sight. What makes them maps is dissipation — and the systems that have none take a different route entirely.

mechanics · Chaos A cluster grown by random walkers. 3000 particles, each released far away to wander at random on a square grid until it first touches the cluster and sticks there, starting from one seed at the centre. Colour marks the order of arrival: the first third (violet), the second (blue), the last (red). The cluster is 220 grid steps across, branched and open, and almost every late arrival is at the outer tips: walkers wander into the outer branches long before they can find their way into the gaps between them. The gaps never fill.

The cluster that grows where the walkers arrive

Let particles wander in one at a time from far away, each sticking where it first touches a growing clump. The rule has nothing in it about shape, and the clump that results is a lacy, branched thing whose mass grows more slowly than its area, so it gets emptier the bigger it gets. The reason is that a random walker reaches the outermost tips long before it can find its way into a gap between branches. The same arithmetic — the probability that a walker arrives is a solution of Laplace's equation — shapes electrodeposits, lightning-like discharges in plastic, fingers of water pushed into oil, and colonies of bacteria short of food.

thermodynamics · Diffusion

Named alongside it

The objects these essays reach for when they reach for this one.

AttractorBifurcationChaosDiffusionFractal dimensionHarmonic measureInstabilityLaplace equationLyapunov exponentNonlinearityPeriod doublingRandom walk

All concepts