Ordering According to Size of Disks in a Narrow Channel

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.

💡 Research Summary

The paper investigates the statistical mechanics of jammed configurations of binary‑size disks confined in a long, narrow channel. The authors consider disks of two diameters, σL (large) and σS (small), placed in a channel of width H such that each disk contacts either a neighboring disk or a wall at three points, guaranteeing mechanical stability. Under the geometric constraint 1 ≤ σL/σS < H/σS < 1 + √(3/4), any jammed microstate can be built from a set of 16 overlapping “tiles”, each tile representing a pair of adjacent disks with one disk overlapping the next tile. The tiles are grouped into six distinct volume classes (Va–Vf) and are listed in Table 1 of the paper.

A reference state (pseudo‑vacuum) consisting solely of large disks arranged as …121212… is introduced. From this reference, 17 species of statistically interacting quasiparticles are defined (Table 3). Species 1 and 2 are “compact” particles that preserve the reference state, species 3–14 are “hosts” that replace one or two large disks by small disks, and species 15–17 are “tags” that modify any host. Each species m carries an excess volume ΔVm relative to the reference segment and a quantum number sm that counts how many small disks the particle introduces (sm = 0, 1, 2).

The combinatorial structure follows a generalized Pauli principle: the number of available slots dm for a particle of species m is dm = Am − ∑n gmn(Nn − δmn), where Am are capacity constants and gmn are interaction coefficients previously determined for a similar system. The statistical weight of a configuration with occupation numbers {Nm} is W({Nm}) = ∏m (dm + Nm − 1)!/(Nm! (dm − 1)!).

Two energetic contributions are considered before jamming. First, a “granular temperature” Tk (β = 1/Tk) quantifies the intensity of random agitation, analogous to thermal energy in colloids. Second, work against the pistons at the channel ends contributes a term pm ΔVm, where pm is taken as the external pressure P (set to unity) multiplied by the excess volume of the particle. The activation energy for species m is therefore εm = pm − µ sm, where µ plays the role of a chemical potential controlling the average fraction NS of small disks. The grand partition function is Z̄ = ∏m (1 + wm)^(−Am) wm! with wm satisfying the nonlinear equations e^{β εm} = (1 + wm) ∏n (1 + wn)^{−gnm}.

Solving the linear system X G N̄ = A (with Gmn = gmn + wm δmn) yields the average particle densities N̄m as functions of β and µ. From these densities the excess volume per particle V̄ = ∑m N̄m ΔVm and the configurational entropy per particle S̄ = ∑m