Physicists Use New Absorbent State Model to Study Random Closed Packaging

Sphere wrapping, a mathematical problem in which non-overlapping spheres are arranged in a given space, has been studied extensively in the past. The densest possible packing has been proven to be a face-centered cubic crystal (FCC) with a space-filling fraction of ϕFCC = π / √18≈0.74.

The most dense random packaging possible, called Closed Random Packaging (RCP), on the other hand, is still poorly defined. Previous studies and simulations, however, predicted that its volume fraction would be ϕRCP≈0.64.

Researchers at New York University and the Technion-Israel Institute of Technology recently conducted a study to delve deeper into the characteristics of CPR, using a new absorbent state model they developed. Their article, published in Physical examination letters, confirmed the original predictions of RCP value, while also representing RCP as a dynamic phase transition.

The work was inspired by a series of experiments conducted by David Pine and Jerry Gollub on the reversibility of particulate suspensions in periodic shear flow. One of the physicists on the team, Paul M. Chaikin, recently invented a model called random organization (RO), which explained the findings Pine and Gollub collected in terms of dynamic phase transition between resting and active states. .

“Using the RO model and other models of similar absorbent states, Dov Levine and Daniel Hexner showed that at a critical point these models are hyperuniform, a quality that is often associated with fluctuations in density of disappearance at large. scale, ”said Sam Wilken, one of the researchers who conducted the study, told Phys.org. “This was confirmed in my thesis and in a subsequent article. In my thesis, I extended the RO model to include repulsive interactions and renamed it Bias Random Organization (BRO) to obtain a quantitative fit for my experiments on sheared suspensions. “

Absorbent state models are derived from toy models that describe the spread or containment of viruses or disease. These toy models show that in high density regions (i.e. heavily populated areas) particles (i.e. people) overlap and are considered active (i.e. i.e. infected).

The active particles then receive random displacements and spread out in a given space, to reduce their density and their activity so that they can eventually become inactive or disappear. Alternatively, they could infect neighboring, inactive, absorbent regions with which there was no previous overlap in activity.

“The competition between infection and dilution determines the fate of a system, which either finds a configuration where no particles overlap (an absorbing state) or continuously evolves (an active equilibrium state),” explained Wilken. “These dynamically disparate states are separated by a critical point (here a critical density) characteristic of a second order phase transition.”

RO, the model developed by Chaikin, is one of the earliest models of continuous absorbing states (i.e. reaching a continuum of space), as opposed to network models (i.e. physical models specifically defined on a network). The BRO model, introduced by Wilken in his thesis, mixes random and repulsively directed displacements of active particles and thus increases the critical density of the system.

The BRO model was originally developed for the purpose of studying the structures of dilute suspensions. Nevertheless, Wilken and his colleagues felt that it was imperative to study the densest possible critical states of the model, because dense packagings of particles are a particularly ancient and fundamental problem in physics.

“Surprisingly, our model does not crystallize in the dense critical state limit, where there are small displacements, and instead approximates what has been called Random Close-Packing (RCP),” said Wilken. “In this work, we demonstrate that the BRO model belongs to a well-studied class of absorbing state models called the Manna class, sharing universal dynamic exponents like the scaling of the side-overlapping particle fraction. active of the transition, as well as as the divergence of the power law of time to arrive at the state of equilibrium near the critical point. “

In their study, Wilken and colleagues found that critical states at small displacement sizes not only approach fractional-volume CPR, but also exhibit structural behaviors that had not previously been associated with RCP. These behaviors included the divergence of the correlation function of nearest neighbor pairs, as well as isostatic coordination (Z = 6, on average each particle has six neighbors that touch each other).

“In addition, we show that the long-range density fluctuations (in S (q)) of the critical states go to zero in the large size limit as a power law (S (q) ~ q ^ alpha), where alpha is a universal exponent of the Manna class, “said Wilken.” We believe that the association of RCP with a dynamic phase transition of the Manna class allows a clearer path to the mathematical study of RCP, in particular because that previously studied simulation models, such as Lubachevsky-Stillinger and soft sphere relaxation, produce structurally identical density correlations. “

The researchers found that past simulations and theoretical models converge at RCP, suggesting that it is a special condition, as physicist JD Bernal first hypothesized in 1960. Fact Interestingly, in the BRO model used by Wilken and colleagues, the RCP emerged as the highest critical density point. Other existing approaches describing RCP impose constraints such as isostaticity, interference, and hyperuniformity, all of which are emerging properties in the researchers’ BRO model.

In the future, the work may inspire further studies focusing on RCP and the applications of their model to the problem of packaging spheres. So far, the team has mainly explored the structural and dynamic features of the BRO model in 2D bi-dispersed and 3D monodisperse systems, but they would soon like to use the model to examine other systems as well.

“In preliminary studies, we found that in 1D and 2D BRO leads to tightly packed crystalline phases, while in 3D and 4D this leads to disordered packaging,” said Wilken. “The introduction of shear in 3D BRO simulations leads to crystallization and it shows the interesting role that dimensionality and isotropy play in the geometry and frustration of packaging spheres. In the future, we plan to investigate these roles as well as the implications on the configurational entropy of tight random states. ”

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