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1. Can exotic disordered “stealthy” particle configurations tolerate arbitrarily large holes?

   https://doi.org/10.1039/C7SM01028A  

   Zhang, G.; Stillinger, F. H.; Torquato, S. (January 2017, Soft Matter)

   The probability of finding a spherical cavity or “hole” of arbitrarily large size in typical disordered many-particle systems in the infinite-system-size limit ( e.g. , equilibrium liquid states) is non-zero. Such “hole” statistics are intimately linked to the thermodynamic and nonequilibrium physical properties of the system. Disordered “stealthy” many-particle configurations in d -dimensional Euclidean space  d are exotic amorphous states of matter that lie between a liquid and crystal that prohibit single-scattering events for a range of wave vectors and possess no Bragg peaks [Torquato et al. , Phys. Rev. X , 2015, 5 , 021020]. In this paper, we provide strong numerical evidence that disordered stealthy configurations across the first three space dimensions cannot tolerate arbitrarily large holes in the infinite-system-size limit, i.e. , the hole probability has compact support. This structural “rigidity” property apparently endows disordered stealthy systems with novel thermodynamic and physical properties, including desirable band-gap, optical and transport characteristics. We also determine the maximum hole size that any stealthy system can possess across the first three space dimensions. 

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2. On the universality of local dissipation scales in turbulent channel flow

   https://doi.org/10.1017/jfm.2015.664  

   Bailey, S. C.; Witte, B. M. (January 2016, Journal of Fluid Mechanics)

   Well-resolved measurements of the small-scale dissipation statistics within turbulent channel flow are reported for a range of Reynolds numbers from $$Re_{{\it\tau}}\approx 500$$ to 4000. In this flow, the local large-scale Reynolds number based on the longitudinal integral length scale is found to poorly describe the Reynolds number dependence of the small-scale statistics. When a length scale based on Townsend’s attached-eddy hypothesis is used to define the local large-scale Reynolds number, the Reynolds number scaling behaviour was found to be more consistent with that observed in homogeneous, isotropic turbulence. The Reynolds number scaling of the dissipation moments up to the sixth moment was examined and the results were found to be in good agreement with predicted scaling behaviour (Schumacher et al. , Proc. Natl Acad. Sci. USA , vol. 111, 2014, pp. 10961–10965). The probability density functions of the local dissipation scales (Yakhot, Physica D, vol. 215 (2), 2006, pp. 166–174) were also determined and, when the revised local large-scale Reynolds number is used for normalization, provide support for the existence of a universal distribution which scales differently for inner and outer regions. 

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3. Designer pair statistics of disordered many-particle systems with novel properties

   https://doi.org/10.1063/5.0189769  

   Wang, Haina; Torquato, Salvatore (January 2024, The Journal of Chemical Physics)

   The knowledge of exact analytical functional forms for the pair correlation function g2(r) and its corresponding structure factor S(k) of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing database of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform, and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperatures that precisely match targeted forms for both g2(r) and S(k). Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve many-particle systems with a wide range of translational order and self-diffusion coefficients D, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics. 

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4. Multiscale regression on unknown manifolds

   https://doi.org/10.3934/mine.2022028  

   Liao, Wenjing; Maggioni, Mauro; Vigogna, Stefano (January 2022, Mathematics in Engineering)

   null (Ed.)

   We consider the regression problem of estimating functions on $$ \mathbb{R}^D $$ but supported on a $ d $-dimensional manifold $$ \mathcal{M} ~~\subset \mathbb{R}^D $$ with $$ d \ll D $$. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $$ \mathcal{M} $$ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $$ \mathbb{R}^D $$. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees. 

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5. Random Close Packing as a Dynamical Phase Transition

   https://doi.org/10.1103/PhysRevLett.127.038002  

   Wilken, Sam; Guerra, Rodrigo E.; Levine, Dov; Chaikin, Paul M. (July 2021, Physical review letters)

   Kamien, Randall; Thomas, Jessica; Chaté, Hugues; Garisto, Robert; Agarwal, Abhishek; Dalena, Serena; Mitra, Samindranath (Ed.)

   Sphere packing is an ancient problem. The densest packing is known to be a face-centered cubic (FCC) crystal, with space-filling fraction ϕFCC = pi/sqrt(18) ≈ 0.74. The densest “random packing,” random close packing (RCP), is yet ill defined, although many experiments and simulations agree on a value ϕRCP ≈ 0.64. We introduce a simple absorbing-state model, biased random organization (BRO), which exhibits a Manna class dynamical phase transition between absorbing and active states that has as its densest critical point ϕcmax ≈ 0.64 ≈ ϕRCP and, like other Manna class models, is hyperuniform at criticality. The configurations we obtain from BRO appear to be structurally identical to RCP configurations from other protocols. This leads us to conjecture that the highest-density absorbing state for an isotropic biased random organization model produces an ensemble of configurations that characterizes the state conventionally known as RCP. 

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