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1. Local order metrics for many-particle systems across length scales

   https://doi.org/10.1103/PhysRevResearch.6.033262  

   Maher, Charles Emmett; Torquato, Salvatore (September 2024, Physical Review Research)

   Formulating order metrics that sensitively quantify the degree of order/disorder in many-particle systems in $d$-dimensional Euclidean space ${\mathbb{R}}^d$across length scales is an outstanding challenge in physics, chemistry, and materials science. Since an infinite set of $n$-particle correlation functions is required to fully characterize a system, one must settle for a reduced set of structural information, in practice. We initiate a program to use the local number variance ${\sigma }_N^2\left(R\right)$associated with a spherical sampling window of radius $R$(which encodes pair correlations) and an integral measure derived from it ${\mathrm{\Sigma }}_N(R_i,R_j)$that depends on two specified radial distances $R_i$and $R_j$. Across the first three space dimensions ( $d=1,2,3$), we find these metrics can sensitively describe and categorize the degree of order/disorder of 41 different models of antihyperuniform, nonhyperuniform, disordered hyperuniform, and ordered hyperuniform many-particle systems at a specified length scale $R$. Using our local variance metrics, we demonstrate the importance of assessing order/disorder with respect to a specific value of $R$. These local order metrics could also aid in the inverse design of structures with prescribed length-scale-specific degrees of order/disorder that yield desired physical properties. In future work, it would be fruitful to explore the use of higher-order moments of the number of points within a spherical window of radius $R$[S. Torquato , ] to devise even more sensitive order metrics. Published by the American Physical Society2024 

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2. Stochastic theory and direct numerical simulations of the relative motion of high-inertia particle pairs in isotropic turbulence

   https://doi.org/10.1017/jfm.2016.859  

   Dhariwal, Rohit; Rani, Sarma L.; Koch, Donald L. (February 2017, Journal of Fluid Mechanics)

   The relative velocities and positions of monodisperse high-inertia particle pairs in isotropic turbulence are studied using direct numerical simulations (DNS), as well as Langevin simulations (LS) based on a probability density function (PDF) kinetic model for pair relative motion. In a prior study (Rani et al. , J. Fluid Mech. , vol. 756, 2014, pp. 870–902), the authors developed a stochastic theory that involved deriving closures in the limit of high Stokes number for the diffusivity tensor in the PDF equation for monodisperse particle pairs. The diffusivity contained the time integral of the Eulerian two-time correlation of fluid relative velocities seen by pairs that are nearly stationary. The two-time correlation was analytically resolved through the approximation that the temporal change in the fluid relative velocities seen by a pair occurs principally due to the advection of smaller eddies past the pair by large-scale eddies. Accordingly, two diffusivity expressions were obtained based on whether the pair centre of mass remained fixed during flow time scales, or moved in response to integral-scale eddies. In the current study, a quantitative analysis of the (Rani et al. 2014) stochastic theory is performed through a comparison of the pair statistics obtained using LS with those from DNS. LS consist of evolving the Langevin equations for pair separation and relative velocity, which is statistically equivalent to solving the classical Fokker–Planck form of the pair PDF equation. Langevin simulations of particle-pair dispersion were performed using three closure forms of the diffusivity – i.e. the one containing the time integral of the Eulerian two-time correlation of the seen fluid relative velocities and the two analytical diffusivity expressions. In the first closure form, the two-time correlation was computed using DNS of forced isotropic turbulence laden with stationary particles. The two analytical closure forms have the advantage that they can be evaluated using a model for the turbulence energy spectrum that closely matched the DNS spectrum. The three diffusivities are analysed to quantify the effects of the approximations made in deriving them. Pair relative-motion statistics obtained from the three sets of Langevin simulations are compared with the results from the DNS of (moving) particle-laden forced isotropic turbulence for $$St_{\unicode[STIX]{x1D702}}=10,20,40,80$$ and $$Re_{\unicode[STIX]{x1D706}}=76,131$$ . Here, $$St_{\unicode[STIX]{x1D702}}$$ is the particle Stokes number based on the Kolmogorov time scale and $$Re_{\unicode[STIX]{x1D706}}$$  is the Taylor micro-scale Reynolds number. Statistics such as the radial distribution function (RDF), the variance and kurtosis of particle-pair relative velocities and the particle collision kernel were computed using both Langevin and DNS runs, and compared. The RDFs from the stochastic runs were in good agreement with those from the DNS. Also computed were the PDFs $$\unicode[STIX]{x1D6FA}(U|r)$$ and $$\unicode[STIX]{x1D6FA}(U_{r}|r)$$ of relative velocity $$U$$ and of the radial component of relative velocity $$U_{r}$$ respectively, both PDFs conditioned on separation $$r$$ . The first closure form, involving the Eulerian two-time correlation of fluid relative velocities, showed the best agreement with the DNS results for the PDFs. 

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3. On the Impulse Response of Singular Discrete LTI Systems and Three Fourier Transform Pairs

   https://doi.org/10.3390/signals5030023  

   Zhou, Qihou (September 2024, Signals)

   A basic tenet of linear invariant systems is that they are sufficiently described by either the impulse response function or the frequency transfer function. This implies that we can always obtain one from the other. However, when the transfer function contains uncanceled poles, the impulse function cannot be obtained by the standard inverse Fourier transform method. Specifically, when the input consists of a uniform train of pulses and the output sequence has a finite duration, the transfer function contains multiple poles on the unit cycle. We show how the impulse function can be obtained from the frequency transfer function for such marginally stable systems. We discuss three interesting discrete Fourier transform pairs that are used in demonstrating the equivalence of the impulse response and transfer functions for such systems. The Fourier transform pairs can be used to yield various trigonometric sums involving sin⁡πk/Nsin⁡πLk/N, where k is the integer summing variable and N is a multiple of integer L. 

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4. Binary mixtures of charged colloids: a potential route to synthesize disordered hyperuniform materials

   https://doi.org/10.1039/C8CP02616E  

   Chen, Duyu; Lomba, Enrique; Torquato, Salvatore (January 2018, Physical Chemistry Chemical Physics)

   Disordered hyperuniform materials are a new, exotic class of amorphous matter that exhibits crystal-like behavior, in the sense that volume-fraction fluctuations are suppressed at large length scales, and yet they are isotropic and do not display diffraction Bragg peaks. These materials are endowed with novel photonic, phononic, transport and mechanical properties, which are useful for a wide range of applications. Motivated by the need to fabricate large samples of disordered hyperuniform systems at the nanoscale, we study the small-wavenumber behavior of the spectral density of binary mixtures of charged colloids in suspension. The interaction between the colloids is approximated by a repulsive hard-core Yukawa potential. We find that at dimensionless temperatures below 0.05 and dimensionless inverse screening lengths below 1.0, which are experimentally accessible, the disordered systems become effectively hyperuniform. Moreover, as the temperature and inverse screening length decrease, the level of hyperuniformity increases, as quantified by the “hyperuniformity index”. Our results suggest an alternative approach to synthesize large samples of effectively disordered hyperuniform materials at the nanoscale under standard laboratory conditions. In contrast with the usual route to synthesize disordered hyperuniform materials by jamming particles, this approach is free from the burden of applying high pressure to compress the systems. 

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5. A reduction principle for Fourier coefficients of automorphic forms

   https://doi.org/10.1007/s00209-021-02784-w  

   Gourevitch, Dmitry; Gustafsson, Henrik P.; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha (March 2022, Mathematische Zeitschrift)

   Abstract We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients. 

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