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Statistics and Optimisation of Random Pan Stacking
The stacking of objects is a familiar operation. For example, a common conundrum of daily life is how to arrange a collection of pots and pans so that the stack will fit in a kitchen cupboard. Because of the varying form and convex nature of the pans, the height will vary depending on the order in which they are stacked. Other examples in the macroscopic realm, which typically involve identical objects, include the stacking of bowls, cups and chairs. At the microscopic level, stacking is associated with self-assembly. Columnar formation has been observed in bowlics, a kind of liquid crystal, and colloidal nanodisks. Under some conditions red blood cells may stack to form rouleaux. Stacking also features in processes involving granular materials.
In this talk, I will present a minimal model for the one-dimensional, random stacking of distinct objects with varying degrees of inter-penetrability. For a given set of pans we examine how the height of the stack depends on the order of the pans and we will determine the configurations with the minimum and maximum height. While it is easy to identify the former, the configuration that maximizes the height is not trivial. We examine the nature of the degeneracy when a given height may correspond to several different orderings of the pans. We also seek the distributions of pan angles that optimize the mean stack height.