Diffusion — where it appears
Named by 32 essays across 9 fields — each of them below, with the objects they name alongside it.
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.
Momentum going sideways
Viscosity is usually described as friction between layers of fluid, which gets the effect right and the mechanism wrong. What is actually happening is that momentum is being conducted across the flow — by the same equation, with the same solutions, as a drop of ink spreading.
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.
The correction that took a century
Newton derived the speed of sound in 1687 and got 290 metres a second against a measured 340. The arithmetic was right; the assumption was not. Heat cannot cross a wavelength in a period, so the compressions are adiabatic — and the factor that repairs the answer turns out to be a count of the ways a molecule can move.
The surface that pulls toward the stronger side
Surface tension is usually treated as a constant of a liquid, and it is not — it depends on temperature and on what is dissolved, and a difference in it along a surface is a force along that surface — which drags the liquid underneath and needs no pressure difference at all.
The viscosity that does not care how much gas there is
Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been.
The ring the drop leaves behind
A drop of coffee dries into a ring rather than a disc, and nothing about coffee is responsible. The pattern is produced by a boundary condition — an edge that cannot move — and it survives replacing the coffee with anything else that will stay suspended.
The pressure that comes from counting
Dissolve a teaspoon of anything in a litre of water, put a membrane between it and pure water, and the solution will hold up a column of water two and a half metres tall. Nothing is pulling. The pressure does not depend on what was dissolved, only on how many particles it made — which is the ideal gas law, with the solute in place of the gas.
The last curve to go
Chaos does not arrive all at once. Order is destroyed a resonance at a time, and there is a coupling — 0.971635, known to six figures — at which the final barrier separating one part of the phase space from another gives way. Below it a chaotic orbit is still trapped; above it nothing stops it.
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.
How far a field gets into metal
Every static argument about conductors says the field inside one is zero, and every one assumes the electrons have had time to move. Give them less time and the field gets in — 9.2 millimetres into copper at mains frequency, 2.1 microns at a gigahertz — and the metal box that silences a radio does almost nothing about the cable running past it.
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.
The heat that arrives as a wave
Two fluids with two velocities give two wave equations, not one. In the first the components move together and the density oscillates, which is ordinary sound. In the second they move oppositely, the density stays put, and what oscillates is the temperature — so a heat pulse in liquid helium has a speed, a front and a reflection.
The gradient that drives the other thing
A concentration gradient drives a flow of matter and a temperature gradient drives a flow of heat. Each also drives the other, by coefficients that are equal — a relation nobody could have guessed and which follows from the fact that the underlying motion runs the same forwards and backwards in time.
The shear that only reaches so far
Viscosity carries momentum sideways without limit when the driving is steady. Reverse the driving and it stops: the motion fills a depth set by the viscosity and the frequency, arrives there late, and beyond that depth the fluid does not know the wall exists.
The membrane that almost holds
Van 't Hoff's law gives the osmotic pressure a perfectly selective membrane would develop, and no membrane is. What a real one develops is a fraction of it — a number between zero and one that belongs to the membrane and the solute together, and that decides whether a solution is isotonic in effect or only on paper.
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.
The gas that flows towards more of itself
Fick's law says a substance diffuses from where there is more of it to where there is less. In a mixture of three gases, nitrogen can do the opposite for hours — flowing into the bulb that already holds more nitrogen, and then stopping while a difference remains — and in a welded bar of steel, carbon crosses into the side that is already richer. Nothing is wrong with the second law. Diffusion flattens chemical potential, and with more than two components, or a second element changing it, that is not the same as flattening concentration.
The drift a sound leaves behind
A sound wave moves fluid back and forth and puts it back where it started. Over many cycles it does not: a steady circulation appears, four cells to a wavelength, driven entirely from inside a boundary layer seventy micrometres thick. Its speed contains the sound speed and the amplitude, and it contains no viscosity at all — so making the fluid thinner does not make the drift weaker.
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.
The salt that sinks through a stable sea
Warm salty water lying on cold fresh water is heavier at the bottom at every depth, and by every test of static stability it should stay where it is. It does not. Heat leaks sideways out of a thin tongue of sinking water a hundred times faster than salt does, so the tongue arrives cooled and still salty — heavier than its new surroundings — and keeps going. A column can be stable by density and unstable by diffusion, and most of the subtropical ocean is.
The speed limit that does not care how big the molecules are
Two molecules in water cannot react faster than they can find each other, and how fast they find each other is set by diffusion. A bigger molecule is an easier target, so it seems a reaction between big molecules should be faster. It is not: a molecule twice as big is twice as easy to hit and wanders half as fast, and the two cancel exactly. Every reaction that happens at the first touch runs at the same rate, about seven billion per molar per second in water, whether the partners are ions or proteins — a speed limit written by the viscosity of water and the temperature, and nothing else.
The heat that arrives before it could
Release a pulse of heat at one point, and the diffusion equation — the law that has described heat flow since Fourier — says the temperature rises everywhere at once, however far away. A thermometer a kilometre off registers a change the instant the heat is released. That is an infinite signal speed, and relativity forbids it. The flaw is harmless for a kettle and fatal for a relativistic fluid. The repair is a relaxation time: heat flow takes a moment to respond to a temperature gradient. With it the equation carries heat at a finite speed, sends a sharp front ahead of the spreading warmth, and turns short-wavelength heat into a damped wave.
The ice that grows more slowly the thicker it gets
A lake under air ten degrees below freezing grows eleven centimetres of ice on the first day and only another eighteen in the following six. The ice forms at its underside, and every gram of it releases latent heat that must be carried up through the ice already there before more can form: the ice is its own insulation. The thickness therefore grows as the square root of time, and the rule, Stefan's, depends on temperature and time only through their product — the freezing degree-days that lake-ice forecasters count. A few centimetres of snow on top slows it more than any amount of cold speeds it up.
The puddle that could dry in a second
A liquid in equilibrium with its vapour loses molecules exactly as fast as the vapour delivers them, and kinetic theory says how fast that is. For water at room temperature it is two and a half kilograms per square metre every second — a puddle two and a half millimetres deep gone in a second. Puddles take hours because almost every molecule that leaves is knocked back by the air within a fraction of a micrometre. The ceiling binds only where the vapour can escape in a single free flight: in vacuum, where water freezes itself, in the smallest droplets, and on the filament of a lamp.
The bubble that grows by counting its sides
A froth of soap bubbles squeezed between two sheets of glass coarsens over a few days: small cells vanish and large ones swell. The obvious guess is that size decides which cells win, as it does for two bubbles joined by a tube, where the small one always empties into the large one. In a froth it is wrong. John von Neumann showed in 1952 that a cell's area changes at a rate fixed entirely by how many sides it has — shrinking if fewer than six, growing if more — and not at all by how big it is. The proof is a single sum of angles.
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.
The metal that feels colder than the wood
A steel railing and a wooden bench in the same shade are at the same temperature, and the railing feels much colder. The usual explanation, that metal conducts heat better, is half the answer, and it gets water wrong. When two bodies touch, the surface between them jumps at once to a temperature that then stays fixed, a weighted average of the two in which each body's weight is the square root of its conductivity times its heat capacity per volume. Skin at 33 °C touching steel at 20 °C meets it at 21; touching wood, at 30. The nerves report the meeting temperature, not the object's.
The pond that is hottest at the bottom
Water heated from below rises, which is why a pond in the sun is never much warmer than the air: the heat is carried to the surface and lost there. Put a metre of salt gradient in the way and the hot water at the bottom is also the heaviest, cannot rise, and keeps its heat, behind an insulating layer that sunlight passes straight through. Ponds built this way reach eighty degrees under an ordinary sky. The salt does not collect any extra light. What it buys is temperature.
The rings a reaction lays down in a gel
Let one salt soak into a tube of jelly that holds another, and where they meet a solid should form, spreading smoothly behind the advancing front. It does not. It forms in sharp, separate bands, close together near the entrance and further and further apart, each one a fixed multiple of the distance of the last. Raphael Liesegang saw the rings in 1896 and they have been found since in rocks, in agate, in gallstones and in microchips. Nothing in the jelly is patterned to begin with. The pattern comes from a solid that will not start until the solution is overloaded, and that then empties the solution around it.
The heating a quantum rotor refuses
Kick a freely turning rotor hard, at regular intervals, and classical mechanics says its motion becomes chaotic and its energy grows without limit, the momentum random-walking upward with every kick. Do the same to a quantum rotor and it follows the classical heating for a while, then stops — not because anything damps it, but because the waves of its momentum states interfere so as to cancel further spreading. The correspondence principle, which promises classical behaviour when quantum numbers are large, is broken here in the most direct way: by a quantum system that refuses to get hot.
The markers that move because atoms are unequal
In 1947 Ernest Kirkendall and Alice Smigelskas wound fine molybdenum wires around a bar of brass, plated it thickly with copper, and heated it for fifty-six days. The wires, which do not dissolve in either metal, ended closer together: the brass inside them had shrunk. Zinc had diffused out of the brass faster than copper diffused in, and the lattice planes of the brass had been taken away along with the zinc. If atoms in a metal moved by swapping places with each other, the two flows would have had to be equal. They were not, and the only way the books balance is if atoms move by jumping into vacancies — empty lattice sites — that flow the other way. The wires were the first direct evidence that diffusion in metals runs on holes.
Named alongside it
The objects these essays reach for when they reach for this one.
Random walkMean free pathViscosityEquilibriumTransportBrownian motionEvaporationIrreversibilitySurface tensionAnderson localisationBoundary layerBuoyancy