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Abstract
Packing spheres efficiently in large dimension dd is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on the packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be efficiently constructed by this method, up to a packing fraction close to 7 \, d \, 2^{-d}7d2−d. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space of the interaction potentials. Our method crucially exploits a recent exact formulation of the thermodynamics and the dynamics of simple liquids in infinite dimension.
Highlights
Hard-sphere systems have been used since long to describe the behaviour of gases, most notably by van der Waals in 1873 [40], whose equation of state provided the first example of a thermodynamic phase transition, by Metropolis et al in 1953 [41], who introduced the famous “Metropolis algorithm" to study the liquid equation of state in d = 2, and by Alder and Wainwright in 1957 [42] who showed the existence of a first order phase transition between the liquid and the crystal states in d = 3
The liquid is a collection of metastable glassy phases, or in other words, the allowed hard sphere configurations are organised in disjoint clusters in phase space, precisely as it happens in satisfiability problems [14]
As discussed in the Introduction, the Gibbs-Boltzmann uniform measure over hard sphere configurations can be biased by adding an arbitrary interaction potential to the hard core
Summary
The sphere packing problem consists in finding the densest arrangements of equal-sized spheres in the d-dimensional Euclidean space. Starting from an underconstrained situation and increasing the density of constraints, one generically encounters a “dynamic" phase transition where the solution space breaks apart into a very large number of disjoint clusters. At this dynamic transition, uniform sampling of solutions becomes exponentially hard in the number of variables [15]. We shall follow this idea in the context of particle systems, and look for the soft part of the interparticle potential that, in the mean-field infinite-dimensional analytical formulation, yields the highest possible packing fraction of the dynamic transition.
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