Illustrated Physics
physics, drawn
Concept

Memorylessness — where it appears

The property of a process whose future does not depend on how long it has already lasted, which forces an exponential distribution of waiting times. A nucleus that has waited a billion years is in the state it started in, which is what makes a half-life a constant and radiometric dating possible.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

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 How far apart the two paths can be. Fringe visibility against the difference between the two path lengths, for light at 550 nm with a bandwidth of 100 nm. Each curve is the modulus of the Fourier transform of its own line shape, summed over the spectrum here rather than taken from a standard result, and the three shapes have the same width at half height. The conventional coherence length λ²/Δλ is 3.02 µm for this light, and what the curves show is that the convention is a rounding of three genuinely different behaviours: a flat band halves at 1.83 µm, a Gaussian line halves at 1.33 µm, a Lorentzian line halves at 0.68 µm. The flat band comes back — a rectangle's transform rings — and the Lorentzian's tails keep a little visibility very much further out than its width suggests. Nothing here is about the apparatus: the fade is the source forgetting its own phase.

How far a wave can remember

Split a beam, delay one half, and put them back together. The fringes are bright while the delay is short and fade as it grows, and the distance at which they die is fixed by nothing but the width of the source's spectral line. Watching them fade is reading the line shape.

optics · Coherence Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

The exponential that is only true in the middle

A decay law is not an assumption about nuclei; it is the Fourier transform of an energy distribution. Do that transform honestly and the exponential fails at both ends — flat at the start, because the state is normalisable, and an inverse square at the end, because no system has states of arbitrarily negative energy. Neither departure is a correction that could be made small.

quantum · Decay A hopper that runs in bursts. Grains discharged against time, on a logarithmic scale, from a silo of 1 mm glass beads through outlets 3.5, 4.0, 4.4 grain diameters wide; each time the outlet jams it is cleared after 2 s (a tap on the silo) and the flow restarts. Avalanche sizes are drawn at random from an exponential distribution with each outlet's mean. At 3.5 diameters the mean avalanche is 53 grains and the outlet jammed 57 times in two minutes, passing 2,537 grains; at 4.0 diameters the mean avalanche is 1,000 grains and the outlet jammed 37 times in two minutes, passing 33,568 grains; at 4.4 diameters the mean avalanche is 45,821 grains and the outlet jammed 3 times in two minutes, passing 116,983 grains. While it flows, each outlet runs at its full Beverloo rate; a narrower outlet does not flow more slowly, it stops more often, and the flat stretches are the waits.

The outlet that jams by chance

A hopper whose outlet is a few grains wide does not flow slowly. It flows at the full rate and then stops dead, when a handful of grains wedge into an arch across the opening, and it flows again only when something knocks the arch down. The number of grains that pass between one jam and the next is random and exponentially distributed, as if every grain leaving had the same small chance of being the last. Its average rises so steeply with the opening's size that it looks like a threshold, and whether there is one is still argued: two laws agree wherever an outlet can be watched, and they part only after millions of grains.

fluids · Granular matter

Named alongside it

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

BandwidthFourier transformHalf-lifeSpectrumBeatsCloggingCoherenceCritical pointDecayDecoherenceEnvelopeExponential decay

All concepts