Illustrated Physics
physics, drawn
Concept

Renormalisation — where it appears

The procedure of absorbing a theory's divergent or unknowable short-distance contributions into a few measured quantities, such as a particle's mass and charge. Once that is done the predictions for everything else are finite, and how the absorbed quantities change with scale becomes physics in its own right.

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 Where a moving charge keeps its field's energy and its momentum. Left: the density of field energy and of field momentum around a charged shell moving slowly to the right, by direction, each drawn as the mass per unit volume it contributes — energy density over c², momentum density over velocity. The energy density is the same in every direction; the momentum density goes as sin²θ, zero ahead and behind and largest broadside, because it is ε₀E × B and the magnetic field of a moving charge vanishes along its line of motion. Broadside it is twice the energy's; averaged over all directions sin²θ is 0.6667, and twice that is the 4/3. Right: the fraction of the field's energy, and of its momentum, lying within a distance r of the centre of a shell of radius a. Both are 1 − a/r, because both densities fall as the fourth power of distance: half of each lies within 2a and nine-tenths within 10a.

The mass a charge's field gets wrong by a third

Give a charged sphere's field its energy U and divide by c², and the field has a mass. Move the sphere and ask the field's momentum what the mass is, and it says four-thirds of that. The discrepancy took from 1881 to the 1960s to understand, and the answer is not a better calculation of the field. It is that a charge cannot hold itself together, and whatever does has energy too.

electromagnetism · Field energy

Named alongside it

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

AttractorBifurcationChaosClassical electron radiusElectromagnetic massField momentumFour-momentumLyapunov exponentMaxwell stressNonlinearityPeriod doublingPoincare stress

All concepts