Melting density and nature of the fluid-solid transition in the Bowen-Lyons-Radin-Winkler zipper-molecule hard-core model

The zipper-molecule model of Bowen, Lyons, Radin and Winkler (Phys. Rev. Lett. 96, 025701 (2006)) is the first continuum one-component hard-core system with a mathematically proven fluid-solid transition. Its order parameter $\mathbf{P}(d)$ , the probability in the infinite-volume canonical ensemble at density $d$ that the origin lies in an infinite fully-linked cluster of molecules, is proven to vanish on a low-density interval $(0,d_4)$ and to be positive on a high-density interval $(d_1,1)$ . The intermediate-density regime is unanalyzed: the melting density, any phase-coexistence interval and critical densities, and the order of the transition are all undetermined, and the authors' expectation that positivity extends below $d_1$ is unproven. The problem posed here is to characterize the positivity set of $\mathbf{P}$ and the nature of the melting transition, defined as in the source through the non-analyticity of $\mathbf{P}(d)$ in $d$ , throughout the intermediate regime.

Background

Whether a classical system of particles interacting only through hard-core (impenetrability) constraints can undergo a fluid-solid (freezing/melting) transition is one of the oldest open problems of statistical mechanics. In a hard-core system the interaction energy is purely kinetic and can be integrated out of the partition function, so any transition must be driven purely by entropy, a competition between disordered and ordered spatial arrangements. Simulations strongly indicate such a transition for the canonical hard-disk and hard-sphere models, but no analytic proof exists. Earlier rigorous phase transitions for particles moving in the continuum either broke a symmetry between particle species rather than a spatial symmetry, or used long-range attractions; neither mechanism is available in a one-component hard-core system.

Bowen, Lyons, Radin and Winkler constructed the first continuum one-component hard-core model with a proven fluid-solid transition. Their particles are 'zipper' molecules: solid unit hexagons of area 1 (so that number density coincides with packing density) whose edges carry a fringe of projections and holes (the shape is controlled by two parameters: the inradius $r$ , the radius of the largest disk inscribed in the molecule, and the fringe height $2\rho$ , the depth of each projection and hole). Two neighboring molecules can interlock, or 'link', in two modes: tightly linked when a projection is inserted more than halfway into the partner's hole, and loosely linked when it is inserted less than halfway. A molecule is fully linked when all its projections and holes engage neighbors, free when none do, and partially linked otherwise. The nonconvex shape constrains the relative orientation of neighboring molecules, which is what makes long-range order analytically detectable in this model.

The model is analyzed in the infinite-volume canonical ensemble, defined as the limit of uniform distributions on arrangements of molecules at fixed density $d$ in expanding periodic boxes, with the defining convention that the finite-box distribution is supported on arrangements with the largest number of degrees of freedom. The order parameter is $\mathbf{P}(d)$ , the probability, in that ensemble at density $d$ , that the origin lies in an infinite fully-linked cluster. The paper introduces constants $1 > d_1 > d_2 > d_3 > 0$ such that a large region of fully linked molecules with all links loose has density in $[d_2,d_1]$ (with $d_1$ the smallest possible such upper limit) and any large collection of free molecules has density at most $d_3$ , and proves the theorem that there exists $d_4 > 0$ with $\mathbf{P}(d) = 0$ for $d \in (0,d_4)$ and $\mathbf{P}(d) > 0$ for $d \in (d_1,1)$ . Non-analyticity of $\mathbf{P}(d)$ in $d$ is taken as the hallmark of the transition. The high-density half of the proof works with invariant measures, that is translation-invariant probability measures on infinite configurations: $\lambda_1$ , the unique invariant measure on the densest packings without tight links, and $\lambda_0$ , the unique invariant measure on tilings by the molecules; the extremal (variational) properties of $\lambda_1$ do the heavy lifting.

What the paper leaves open is the intermediate-density regime $(d_4, d_1)$ : nothing is proved there, so the precise melting density, meaning the threshold of the positivity set of $\mathbf{P}$ , is unknown. By analogy with simulations of hard spheres and hard disks the authors expect the positivity set to extend below $d_1$ , that is that melting occurs at a density below $d_1$ , 'but we do not know how to prove this'. Whether the ensemble exhibits phase coexistence at intermediate densities, and the resulting critical densities and the order of the transition, are likewise undetermined. The extension of the method to the round-core hard-disk and hard-sphere models is a distinct and still-famous open problem, discussed below only as context.

Problem Statement

Determine the fluid-solid phase behavior of the Bowen-Lyons-Radin-Winkler zipper-molecule model throughout the intermediate-density range, completing the phase diagram begun by the source theorem. Working within the model and its infinite-volume canonical ensemble exactly as defined in the cited paper (zipper molecules of unit area with projections and holes; tight and loose links; the maximal-degrees-of-freedom convention for the finite-box ensemble), answer the following.

Characterize the positivity set $\{d \in (0,1) : \mathbf{P}(d) > 0\}$ in the intermediate range $(d_4, d_1)$ : decide whether it extends strictly below $d_1$ , as the authors expected by analogy with hard-sphere and hard-disk simulations, or whether $\mathbf{P}(d) = 0$ on all of $(0, d_1)$ ; and determine the melting density $d_m = \inf\{d : \mathbf{P}(d) > 0\}$ , establishing in particular whether the positivity set is an interval so that a single melting density is well defined (monotonicity of $\mathbf{P}$ in $d$ is not known).
Determine the nature of the transition in the source's sense, which takes non-analyticity of $\mathbf{P}(d)$ in $d$ as the hallmark of the transition: establish where in the intermediate range $(d_4, d_1)$ the order parameter $\mathbf{P}(d)$ is non-analytic in $d$ .

A complete resolution determines the full density-dependence of the order parameter and the location and nature of the melting transition. Either sign of the answer to item 1, extension of positivity below $d_1$ or proof that no such extension occurs, is an equally valid resolution. The adaptation of the method to classical hard disks or hard spheres is a separate, still-open problem and is not part of this question.

References

Lewis Bowen, Russell Lyons, Charles Radin, Peter Winkler, Fluid-Solid Transition in a Hard-Core System, Physical Review Letters 96, 025701 (2006), DOI: 10.1103/PhysRevLett.96.025701, https://doi.org/10.1103/PhysRevLett.96.025701 (arXiv:cond-mat/0508405, https://arxiv.org/abs/cond-mat/0508405)
Lewis Bowen, Russell Lyons, Charles Radin, Peter Winkler, A Solidification Phenomenon in Random Packings, SIAM Journal on Mathematical Analysis 38(4), 1075-1089 (2006), DOI: 10.1137/050647785, https://doi.org/10.1137/050647785 (arXiv:cond-mat/0512686, https://arxiv.org/abs/cond-mat/0512686)
David Aristoff, Charles Radin, Rigidity in Solids, Journal of Statistical Physics 144, 1247-1255 (2011), DOI: 10.1007/s10955-011-0304-x, https://doi.org/10.1007/s10955-011-0304-x (arXiv:1105.4500, https://arxiv.org/abs/1105.4500)
Charles Radin, Wulff shape for equilibrium phases, arXiv:1610.08564 (2016), https://arxiv.org/abs/1610.08564; published in revised form under the title The Open Mathematics of Crystallization, Notices of the American Mathematical Society 64(6), 551-556 (2017), DOI: 10.1090/noti1534, https://doi.org/10.1090/noti1534
Andras Suto, Momentum coupling of classical atoms (2025), arXiv:2501.07369, https://arxiv.org/abs/2501.07369