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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 3 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

The first rung's shares were a limit of distributions — where the walk probably is after many steps. This one 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

Named alongside it

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

InvariantArcsine distributionBilliardsChaosConjugacyCurvatureDeterminismExpectationInvariant densityIterationLimitLyapunov exponent

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