Grain Sorting in the One-dimensional Sand Pile Model
Jérôme Durand-Lose
Labri, ura cnrs 1 304,
Université Bordeaux I,
351, cours de la Libération,
F-33 405 Talence Cedex, France
Abstract
The evolution of a one-dimensional pile is studied, empty at first, it receives a grain in its first stack at each iteration. The final position of grains is singular: grains are sorted according to their parity. They are sorted on trapezoidal areas alternating on both sides of a diagonal line of slope 
. This is explained and proved by means of a local study. Each generated pile, encoded in height differences, is the concatenation of four patterns: 22, 1313, 0202, and 11. The relative length of the first two patterns and the last two patterns converges to . Asymptotic expansions are made and it is proved that all the lengths of the pile are increasing proportionally to the square root of the number of iterations.
