Illustrated Physics
physics, drawn
Concept

Diffusion coefficient — where it appears

The constant that sets how fast random motion spreads particles or heat, so that the mean square displacement grows as twice it times the time in each direction. It is measured in square metres per second, and for a sphere in a liquid it is kT divided by the Stokes drag coefficient.

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 signal that dies, and comes back when the spins are turned over. The size of the signal from an ensemble of proton spins whose precession rates are spread by 20 Hz (root mean square) about the mean, with a true relaxation time T₂ of 80 ms, against time after they are tipped. The free signal falls to 1/e in about 10 ms and to 0.03 by 20 ms: not because any spin has relaxed but because they have fanned out. A π pulse at 30 ms (marked) turns every spin over, and at 60 ms the signal returns to 0.472 — exactly e^(−2τ/T₂), the dashed envelope, since every static difference in rate has cancelled. Without the pulse the signal at 60 ms is below one part in a million.

The echo that gives back everything but the wandering

Tip a sample of protons in a slightly uneven magnetic field and their signal dies within milliseconds. None of them has relaxed: they have fanned out, each precessing at the rate its own patch of field sets. Turn every spin over halfway through and the fan closes again, and the signal comes back as an echo, smaller only by the decay that was real. What the echo cannot return is a phase gained while a spin moved, and that one loss, which grows as the cube of the waiting time, is how a scanner finds a stroke within minutes.

quantum · Spin

Named alongside it

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

Random walkAgeingAnomalous diffusionDephasingErgodicityHeavy-tailed distributionLarmor precessionMagnetic field gradientMagnetic resonanceMean square displacementReversibilitySingle-particle tracking

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