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Concept

Ergodicity

The property that a single trajectory spends time in each region in proportion to that region's size. It makes a long-run average along one orbit equal to an average over the whole space, which is the assumption that statistical mechanics rests on and which is hard to prove in any particular case.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Where a long orbit of the logistic map spends its time. A histogram of 60000 iterates in 32 bins, with the density the map preserves drawn over it as the exact share each bin should hold.

The histogram an orbit leaves

When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.

dynamics · Iteration One disc, and two paths that stop being near each other. Two nearly identical billiard paths drawn on an empty square and on a square with a circular obstacle, with the separation between them plotted against distance travelled.

The obstacle that makes a table chaotic

Put one round post in the middle of a square table and every trace of order goes. Two paths that start a hundred-thousandth of a degree apart end up on opposite sides of the table, and the reason is that a wall curving outwards multiplies a gap where a flat one only adds to it.

dynamics · Billiards The time a single walk spends in each state, against the share it should hold. Paired bars for each state, one the fraction of a long run's time spent there and one the computed stationary share, above a table of expected return times.

The time spent and the share held

Stationary shares are a limit of distributions — where the walk probably is after many steps. Here the question is about a single walk: the fraction of its time spent in each state is that state's share, and the expected wait between visits is exactly the reciprocal.

probability · Markov chains How long four orbits of the doubling map spend in each quarter. Grouped bars for four starting points of the doubling map, each group showing the fraction of time the orbit spends in each quarter of the unit interval, against a dashed line at one quarter.

Almost every orbit is fair

Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.

dynamics · Iteration Where x ↦ βx mod 1 spends its time, the golden ratio φ. A histogram of 2000000 orbit points of the β-map for the golden ratio φ in 50 bins, with Parry's staircase density drawn over it; they agree, and the density steps down at the orbit of 1.

Where a base-β orbit spends its time

Follow x ↦ βx mod 1 for a million steps and count how often the orbit visits each part of the interval. For the doubling map the answer is evenly; for the golden ratio it is a staircase with one step, spending 1.17 times the average near 0 and 0.72 times it near 1. The step sits exactly where the orbit of the number 1 lands, and in a base whose 1 never has a finite expansion the staircase has infinitely many steps.

dynamics · Symbolic dynamics

Named alongside it

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

InvariantInvariant measureArcsine distributionBilliardsBinary expansionChaosConjugacyCurvatureDensityDeterminismDoubling mapExpectation

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