Illustrated Physics
physics, drawn
Concept

Heavy tailed distribution — where it appears

A probability distribution whose tail falls off as a power law rather than exponentially, so rare huge values dominate its averages. Lévy flights, earthquake sizes and some laser-cooled atoms' waiting times follow one, and their variance or even mean can be infinite.

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

What one trajectory says, and what the crowd says. For the walk with α = 0.5: the time-averaged mean-square displacement of each of 12 single trajectories, each watched for 10000 time units — the average of the squared displacement over every window of a given lag along that one path — against the lag, on logarithmic axes (thin lines), and the mean-square displacement of an ensemble of 800 walkers at the same times (thick). The ensemble spreads as t^0.5. Every single trajectory's time average grows nearly in proportion to the lag — mean slope 0.82, where the ensemble's is 0.5 — as if it were ordinary diffusion, and the trajectories disagree with one another about the diffusion coefficient by more than a factor of ten. A time average along one path and an average over many paths, which for ordinary diffusion are the same thing, here give different laws. That is weak ergodicity breaking, and it is what a single-molecule tracking experiment has to be read against.

The diffusion that slows the longer it is watched

A random walker whose steps come at irregular times still diffuses normally, provided the waits have a finite average. Let the waits have no average — mostly short, occasionally enormous, as they are for a molecule that keeps getting stuck — and three things change at once. The spread grows more slowly than time. A diffusion coefficient measured by following one particle depends on how long it was followed. And following one particle for a long time no longer tells what many particles do: averages along a path and averages over a crowd give different laws. Tracking single molecules inside living cells runs straight into all three.

thermodynamics · Diffusion A peak that keeps growing while the light stays on. The momentum distribution of 20,000 atoms in a one-dimensional dark-state scheme after 100, 1,000 and 10,000 scattering times, as a density per unit of ħk, in bins a hundredth of ħk wide. There is no steady state. The peak at zero grows taller and narrower for as long as the light is on — its height reaches 29 per ħk by the last time — and the share of atoms within a tenth of a photon's momentum climbs from 55 to 87 to 96 per cent. The atoms are not pushed into the peak; they fall into it at random and the light stops touching them.

The atoms the light stops touching

Every photon an atom scatters kicks it by the same small amount in a random direction, so a gas that keeps scattering light cannot be colder than one kick. The way below is not a stronger friction but a weaker touch: arrange for the light to stop scattering off atoms that are nearly at rest. Then the coldest atoms are not cooled, only left alone, for times that have no average — and the gas never settles into a temperature at all, but piles into a peak that narrows for as long as the light stays on.

astrophysics · Radiation pressure

Named alongside it

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

ErgodicityRandom walkAgeingAnomalous diffusionDark stateDiffusion coefficientLaser coolingMean square displacementOptical pumpingRaman coolingRecoil limitSingle particle tracking

All concepts