Ordering According to Size of Disks in a Narrow Channel
Dan Liu, Michael Karbach, Gerhard Müller
TL;DR
The paper investigates whether purely steric interactions can drive size-based ordering in polydisperse granular disks confined to a narrow channel. Using an exact configurational-statistics framework built on 16-tile jammed patterns and 17 species of statistically interacting quasiparticles, the authors derive how jammed macrostates are counted and how energies from a tunable jamming protocol bias the outcomes in a granular grand canonical ensemble. A single parameter $\Delta\mathcal{V}$ controls whether segregation or alternation is favored, with a border case $\Delta\mathcal{V}=0$ allowing high mixing entropy. The work demonstrates that symmetry-breaking size ordering can arise from steric effects alone and provides exact, protocol-sensitive predictions for entropy and excess volume, offering guidance for potential experimental realizations with appropriate stochastic modeling of jamming dynamics.
Abstract
A long and narrow channel confines disks of two sizes. The disks are randomly agitated in a widened channel under moderate pressure, then jammed according to a tunable protocol. We present exact results that characterize jammed macrostates (volume, entropy, jamming patterns). The analysis divides jammed disk sequences into overlapping tiles out of which statistically interacting quasiparticles are constructed. The fractions of small and large disks are controlled by a chemical potential adapted to configurational statistics of granular matter. The results show regimes for the energy parameters (determined by the jamming protocol) that either enhance or suppress the mixing of disk sizes. Size segregation or size alternation driven by steric forces alone are manifestations of a broken symmetry.