Illustrated Physics
physics, drawn
Concept

Mean square displacement — where it appears

The average of the squared distance a diffusing particle has moved, which grows in proportion to time for an ordinary random walk. It is the standard observable and it is blind to dimension: it is the same function in one, two and three dimensions, where the probability of ever returning home is 1, 1 and 0.3405.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Seven walks from one point. 7 random walks of 400 steps each, all starting at the same place. None of them goes anywhere in particular and none of them stays put; the typical distance reached after n steps is the square root of n, so quadrupling the time doubles the spread. The tracks are seeded, so this is a property of the figure rather than of any one run.

The jiggle that proved atoms

A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

thermodynamics · Diffusion The fraction of walks that have come home, against how long they have walked. The proportion of 1,600 lattice random walks that have returned to their starting point at least once, against the number of steps taken, on a logarithmic horizontal axis, in one, two and three dimensions. In one dimension almost every walk is home almost at once and the fraction climbs toward one. In two it climbs more slowly — the return is still certain, but only logarithmically, so a two-dimensional walk that has not come back after a thousand steps is unremarkable. In three the curve flattens: it reaches 0.3481 and stops, against the exact value 0.3405, drawn as a line. That number is Watson's integral, and it is the probability that a three-dimensional walk ever comes home at all. The difference between the cases is not one of degree. In one and two dimensions the expected number of returns is infinite and a diffusing particle visits every site eventually; in three it is finite, and a molecule released in a room will, with probability two-thirds, never pass through its starting point again. The same statement runs the other way round: a reaction that needs two diffusing partners to meet is a very different problem on a membrane from what it is in a cell.

The walk that comes home

A particle wandering at random on a line returns to where it started, with certainty. On a plane it returns, with certainty. In three dimensions the probability is 0.3405 — so two out of three molecules released in a room never pass through their starting point again, and the difference between the cases is not a matter of degree.

thermodynamics · Diffusion 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

Named alongside it

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

Random walkBrownian motionDiffusionAgeingAnomalous diffusionAvogadro numberDiffusion coefficientDimensionalityEquipartitionErgodicityFirst passageFluctuation dissipation

All concepts