Concept

Drift — where it appears

The average step of a random walk or process: how far it moves per unit of time once its fluctuations are averaged out. When the step depends on where the walk stands, the drift is an average over the places it visits, weighted by how often it visits them.

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

Downhill on average, and never on purpose. The logarithms of 4 Collatz orbits plotted against step number, each wandering upward and downward and each ending at one. A separate sample of four thousand starts gives an average fall of -0.15 per step.

The heuristic that cannot be a proof

There is a two-line argument that the Collatz conjecture is true, it is convincing, and everybody who works on the problem believes it. It also cannot be turned into a proof, and understanding exactly where it fails is more instructive than the argument itself.

dynamics · Collatz

Two losing games that win together

Game A is a coin that wins 49.5% of the time. Game B tosses a bad coin when the capital is a multiple of three and a good one otherwise, and it loses too, because the capital spends more than a third of its time on multiples of three. Choose between the two games at random and the walk drifts upwards by 0.0157 a round; play A, B, B over and over and it gains 0.0574. Nothing is wrong with the arithmetic. The losing coin A wins by knocking the capital off the bad remainder.

probability · Random walk

Named alongside it

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

Random walk Collatz Expectation Expected value Heuristic Iteration Logarithm Markov chain Paradox Periodicity Proof Stationary distribution

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