Illustrated Physics
physics, drawn
Concept

Random walk — where it appears

A path built from independent steps, whose net displacement grows as the square root of their number rather than in proportion to it. The square root is why diffusion is fast over small distances and hopeless over large ones, and it is the same arithmetic wherever independent errors accumulate.

Named by 10 essays across 6 fields — each of them below, with the objects they name alongside it.

A spike, spreading. The solution of the diffusion equation at three times, with a seeded random walk histogrammed behind it. The area under every curve is the same because nothing is lost; only the width changes, and it grows as the square root of the time.

The equation that only runs forwards, and the walk underneath it

A drop of ink spreads and never gathers. The equation describing it is one of the few in physics that is not reversible — and underneath it is nothing but a coin being tossed.

thermodynamics · Diffusion Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that.

A nucleus with no clock

A half-life is a precise number and no individual nucleus has one. Each has the same chance of decaying in the next second as it had on the day it formed, and the exponential curve is a property of the population rather than of any member of it.

quantum · Decay 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 Nine hundred steps and hardly anywhere. On the left, a walk of 900 steps of unit length in uniformly random directions, which is what a photon does inside a star: it goes a mean free path, scatters, and starts again in a direction that has forgotten the last one. After 900 steps it is 7.5 lengths from where it began, against 900 if it had gone straight. On the right, the root-mean-square distance over 240 independent walks against the number of steps, both logarithmic: a straight line of slope 0.4920 against an exact one half. The square root is the whole of the result and it is brutal. Escaping a body of radius R takes not R/λ steps but (R/λ)² of them, so a mean free path a thousand times smaller costs a million times as long. That is the difference between a photon leaving the Sun's core and a neutrino doing it: one takes a hundred thousand years and the other takes two and a third seconds, through the same material, and the only thing that differs is λ. What the picture cannot show is the sense in which the escaping energy is not the photon that started: it is absorbed and re-emitted countless times, at falling temperature, so what leaves is a gamma ray's worth of energy arriving as a great many visible photons.

The light that takes a hundred thousand years to leave

A neutrino made in the Sun's core is at the surface in two and a third seconds. A photon made beside it takes something like a hundred thousand years, through the same material, over the same seven hundred thousand kilometres — and the whole of the difference is one length, entering the answer squared.

astrophysics · Kinetic theory A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device.

The delay that is a random variable

A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

optics · Dispersion The potential at a point is where its walkers end up. A square divided into a 24-step grid, with its top edge held at a potential of 1 and the other three edges at 0. From the probe point (0.29, 0.71), 4000 random walkers each step to one of their four neighbours with equal chance until they touch an edge; six of them are drawn, each ending with a dot on the edge it reached. The fraction that end on the held edge is 0.405 ± 0.008, one standard error, after an average of 122 steps. Solving Laplace's equation on the same grid by repeatedly replacing every value with the average of its four neighbours gives 0.408, and the series solution for the continuous square gives 0.408. The walkers were never told the equation: a value that is the average of its neighbours and a probability of ending somewhere are the same arithmetic.

The potential is where the wanderers stop

Start a random walker at a point between charged conductors and let it wander until it touches one of them. The average potential of the surfaces the walkers touch is the potential at the starting point — exactly, with no equation solved — and the charge a conductor keeps at each place on its surface is the chance that a walker arriving from far away touches it there first.

electromagnetism · Potential Light that walks through a cloud. Seven photons entering a slab 8 scattering mean free paths thick from above, straight down, and followed until they leave, with every scattering equally likely to send them in any direction; their paths are projected onto the page. Of 20000 photons followed the same way, 82.5 per cent come back out of the top and 17.5 per cent out of the bottom. Diffusion theory predicts 18.2 per cent through the bottom. Only 10 photons cross without scattering at all, where e⁻⁸ predicts 6.7 — within the counting error of so few: nearly all of what is transmitted has walked, and the direction it arrived from is forgotten on the way.

The cloud light has to walk through

A beam crossing thirty scattering lengths of anything should keep e⁻³⁰ of itself — a ten-millionth of a millionth. A cloud thirty scattering lengths thick lets through nearly a third of the sunlight falling on it. The light has not crossed; it has walked, one scattering at a time, and a walk through a slab obeys a law with the shape of Ohm's rather than of an exponential.

optics · Scattering A wall pulled away fast leaves the energy behind. The energy of a ball bouncing in a box whose wall is moved, against the length of the box, for 3 wall speeds — each a fraction of the ball's own starting speed — with the adiabatic prediction drawn dashed. Every collision is solved for exactly rather than stepped, so nothing here assumes the wall is slow. Moved slowly, the wall takes energy from the ball at the adiabatic rate, and the energy falls as the inverse square of the length. Moved as fast as the ball is moving, it takes almost nothing: the ball cannot catch a wall retreating faster than it travels, so the collisions stop and the energy stops falling. That is the difference between a gas pushing a piston and a gas expanding into a vacuum, and it is drawn here for one particle. At the end of the range the slowest wall leaves the energy at 0.058 of its starting value and the fastest at 1.000, against an adiabatic 0.065.

The wall that moves while the ball is in flight

Energy is conserved because the rules do not depend on the time. Move the walls of a box and the rules do depend on the time, so what is inside gains or loses without limit — and how much depends entirely on how fast. Moved slowly, a wall takes energy at exactly the rate the adiabatic law says; moved faster than the ball travels, it takes none at all, and the same box is a piston or a vacuum according to a speed.

mechanics · Energy With interference kept, transmission falls exponentially; without it, only as one over the thickness. Light through stacks of randomly thick transparent layers, alternating indices 1 and 2.6, with thicknesses scattered by ±50 per cent about a quarter wave, on a logarithmic scale against the number of layers. The falling line is the transmission with the waves' interference kept, computed exactly by multiplying transfer matrices and averaged as the logarithm over 40 random stacks and five wavelengths. It falls in a straight line: the transmission drops by a factor of e every 17 layers, however thick the stack, which is exponential decay — localisation. The upper curve is the same stacks with every surface's reflection and transmission added as intensities, so that no interference survives. It falls only as one over the thickness, checked to grow in exact proportion, which is the diffusion of light through a cloud: at 400 layers it still transmits 1.0 per cent where the coherent stack typically transmits 4.5·10⁻¹¹.

The walk that interference can stop

Light scattered many times walks through a cloud, and a walk always gets through eventually — a slab twice as thick lets through half as much. Keep the waves' interference instead of adding intensities, and in one dimension the same disorder does something a walk cannot: it stops the light exponentially, traps it in modes with nothing special about where they sit, and turns transmission from a number into a spread over powers of ten. Whether the same can happen to light in three dimensions has been claimed, retracted and argued for thirty years.

optics · Scattering

Named alongside it

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

DiffusionIrreversibilityMean free pathBrownian motionCoherent backscatteringMean square displacementMultiple scatteringOptical depthRadiative transferActionAdiabatic invariantAlbedo

All concepts