Illustrated Physics
physics, drawn
Concept

Poisson process — where it appears

A sequence of events arriving independently at a constant average rate, so that the count in any interval is Poisson distributed. The waits between events are exponentially distributed and memoryless: the chance of the next event in the coming second does not depend on how long it has been.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The price of precision never falls below two. For a clock ticking forwards and backwards with a free-energy drop a per step, the product of its squared relative uncertainty and the entropy it has produced, in units of k: exactly a·coth(a/2), whatever the time elapsed. It approaches the bound of 2 only as the driving vanishes and grows as a once the driving is strong; the three clocks compared elsewhere sit at 2.04 (0.5 kT), 2.63 (2.0 kT), 8.01 (8.0 kT). No steady-state process of any design lies below the line at 2 — the thermodynamic uncertainty relation.

The precision a clock pays for in heat

Anything that counts — a molecular motor stepping along a filament, a chemical oscillator keeping time in a bacterium, a clock ticking on a wall — has to be driven, because a count that is not driven runs backwards as often as forwards. The heat it gives off buys it regularity, and there is an exact floor under the price: the squared relative uncertainty of the count, times the entropy produced, is never less than two in units of Boltzmann's constant.

thermodynamics · Entropy 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.

CloggingCritical pointDetailed balanceEntropy productionExponential distributionFluctuationsForce chainsGranular flowJammingMemorylessnessMolecular motorNonequilibrium steady state

All concepts