Large scale limit of interface fluctuation models

The contour lines of the saturated surface of the (2 + 1)-dimensional restricted solid-on-solid (RSOS) growth model are investigated by numerical method. It is shown that the calculated contour lines are conformal invariant curves with fractal dimension df = 1.34, and they belong to the universality class at large-scale limit, called the Schramm–Loewner evolution with diffusivity κ = 4. This is identical to the value obtained from the inverse cascade of surface quasigeostrophic (SQG) turbulence [Phys. Rev. Lett.98 (2007) 024501]. We also found that the measured fractal dimensions of contours on the (2 + 1)-dimensional RSOS saturated surfaces do not coincide well with that of SLE4 df = 1 + κ/8.

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The Maximum Entropy of a Metric Space

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haab003 ◽

2021 ◽

Author(s):

Tom Leinster ◽

Emily Roff

Keyword(s):

Probability Measure ◽

Maximum Entropy ◽

Metric Space ◽

Large Scale ◽

Hausdorff Space ◽

Geometric Information ◽

Finite Case ◽

Maximum Value ◽

Maximizing Measure ◽

Scale Limit

Abstract We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These generalize the Shannon and Rényi entropies of information theory. We prove that on any space X, there is a single probability measure maximizing all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows, and its asymptotics determine geometric information about X, including the volume and dimension. And the large-scale limit of the maximizing measure itself provides an answer to the question: what is the canonical measure on a metric space? Primarily, we work not with entropy itself but its exponential, which in its finite form is already in use as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.

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From Kinetic Theory to Navier–Stokes Hydrodynamics

10.1093/oso/9780199592357.003.0005 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Kinetic Theory ◽

Knudsen Number ◽

Asymptotic Expansions ◽

Large Scale ◽

Local Equilibrium ◽

Navier Stokes ◽

Fluid Equations ◽

Scale Limit

This Chapter illustrates the derivation of the macroscopic fluid equations, starting from Boltzmann’s kinetic theory. Two routes are presented, the heuristic derivation based on the enslaving of fast modes to slow ones, and the Hilbert–Chapman–Enskog procedure, based on low-Knudsen number asymptotic expansions. The former is handier but mathematically less rigorous than the latter. Either ways, the assumption of weak departure from local equilibrium proves crucial in recovering hydrodynamics as a large-scale limit of kinetic theory.

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Large-scale limit of dynamic-torsion theory

Soviet Physics Journal ◽

10.1007/bf00900088 ◽

1987 ◽

Vol 30 (5) ◽

pp. 392-396

Author(s):

M. A. Katanaev

Keyword(s):

Large Scale ◽

Torsion Theory ◽

Dynamic Torsion ◽

Scale Limit

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Testing tidal alignment models for anisotropic correlations of halo ellipticities with N-body simulations

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa718 ◽

2020 ◽

Vol 494 (1) ◽

pp. 694-702 ◽

Cited By ~ 1

Author(s):

Teppei Okumura ◽

Atsushi Taruya ◽

Takahiro Nishimichi

Keyword(s):

Dark Matter ◽

Large Volume ◽

Correlation Functions ◽

Clustering Analysis ◽

Large Scale ◽

Acoustic Oscillations ◽

Model Predictions ◽

Alignment Model ◽

Linear Alignment ◽

Scale Limit

ABSTRACT There is a growing interest of using the intrinsic alignment (IA) of galaxy images as a tool to extract cosmological information complimentary to galaxy clustering analysis. Recently, Okumura & Taruya derived useful formulas for the intrinsic ellipticity–ellipticity correlation, the gravitational shear–intrinsic ellipticity correlation, and the velocity–intrinsic ellipticity correlation functions based on the linear alignment (LA) model. In this paper, using large-volume N-body simulations, we measure these alignment statistics for dark-matter haloes in real and redshift space and compare them to the LA and non-linear alignment model predictions. We find that anisotropic features of baryon acoustic oscillations in the IA statistics can be accurately predicted by our models. The anisotropy due to redshift-space distortions (RSDs) is also well described in the large-scale limit. Our results indicate that one can extract the cosmological information encoded in the IA through the Alcock–Paczynski and RSD effects.

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Quantum coherence of cosmological perturbations

Modern Physics Letters A ◽

10.1142/s0217732317501917 ◽

2017 ◽

Vol 32 (35) ◽

pp. 1750191 ◽

Cited By ~ 6

Author(s):

Massimo Giovannini

Keyword(s):

Quantum Coherence ◽

Large Scale ◽

Optical Cavity ◽

Single Mode ◽

Second Order ◽

Cosmological Perturbations ◽

Single Mode Approximation ◽

Optical Limit ◽

Zero Time ◽

Scale Limit

In this paper, the degrees of quantum coherence of cosmological perturbations of different spins are computed in the large-scale limit and compared with the standard results holding for a single mode of the electromagnetic field in an optical cavity. The degree of second-order coherence of curvature inhomogeneities (and, more generally, of the scalar modes of the geometry) reproduces faithfully the optical limit. For the vector and tensor fluctuations, the numerical values of the normalized degrees of second-order coherence in the zero time-delay limit are always larger than unity (which is the Poisson benchmark value) but differ from the corresponding expressions obtainable in the framework of the single-mode approximation. General lessons are drawn on the quantum coherence of large-scale cosmological fluctuations.

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Inertio-gravity waves under the non-traditional -plane approximation: singularity in the large-scale limit

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.764 ◽

2016 ◽

Vol 810 ◽

pp. 475-488 ◽

Cited By ~ 2

Author(s):

Jun-Ichi Yano

Keyword(s):

Gravity Waves ◽

Taylor Expansion ◽

Large Scale ◽

Rotation Axis ◽

Vertical Direction ◽

Wave Frequency ◽

Coriolis Parameter ◽

Higher Order Terms ◽

Solution Behaviour ◽

Scale Limit

The low horizontal wavenumber limit of waves on a plane under a constant rotation rate (the so-called $f$-plane) is degenerate: all wave frequencies asymptotically approach the inertial frequency. This degeneracy has no serious consequence when the rotation axis is perpendicular to the plane (traditional $f$-plane approximation). However, when the rotation axis is tilted from the vertical direction (non-traditional $f$-plane approximation), we encounter a type of ‘singularity’ in the sense that each term of the Taylor expansion of the wave frequency in the horizontal Coriolis parameter diverges in the limit of low horizontal wavenumber. Such a drastic change of the solution behaviour by adding the horizontal Coriolis parameter in the low horizontal wavenumber limit is rather counter-intuitive, because the conventional scale analysis suggests that the horizontal Coriolis effect is negligible in this limit. However, the degeneracy of the system makes this effect critical with a need for considering higher-order terms. Two possible asymptotic limits are proposed for resolving this degeneracy. One of them, as it turns out, amounts to a representation of the wave frequency by a Taylor expansion in the horizontal wavenumber.

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On the Asymptotic Insensitivity of the Supermarket Model in Processor Sharing Systems

Proceedings of the ACM on Measurement and Analysis of Computing Systems ◽

10.1145/3460089 ◽

2021 ◽

Vol 5 (2) ◽

pp. 1-28

Author(s):

Grzegorz Kielanski ◽

Benny Van Houdt

Keyword(s):

Fixed Point ◽

Load Balancing ◽

Large Scale ◽

Unique Fixed Point ◽

Length Distribution ◽

Processor Sharing ◽

Global Attraction ◽

Queue Length Distribution ◽

Supermarket Model ◽

Scale Limit

The supermarket model is a popular load balancing model where each incoming job is assigned to a server with the least number of jobs among d randomly selected servers. Several authors have shown that the large scale limit in case of processor sharing servers has a unique insensitive fixed point, which naturally leads to the belief that the queue length distribution in such a system is insensitive to the job size distribution as the number of servers tends to infinity. Simulation results that support this belief have also been reported. However, global attraction of the unique fixed point of the large scale limit was not proven except for exponential job sizes, which is needed to formally prove asymptotic insensitivity. The difficulty lies in the fact that with processor sharing servers, the limiting system is in general not monotone. In this paper we focus on the class of hyperexponential distributions of order 2 and demonstrate that for this class of distributions global attraction of the unique fixed point can still be established using monotonicity by picking a suitable state space and partial order. This allows us to formally show that we have asymptotic insensitivity within this class of job size distributions. We further demonstrate that our result can be leveraged to prove asymptotic insensitivity within this class of distributions for other load balancing systems.

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Large-scale heterogeneous service systems with general packing constraints

Advances in Applied Probability ◽

10.1017/apr.2016.79 ◽

2017 ◽

Vol 49 (1) ◽

pp. 61-83 ◽

Cited By ~ 7

Author(s):

A. L. Stolyar

Keyword(s):

Steady State ◽

Large Scale ◽

Blocking Probability ◽

Asymptotic Optimality ◽

Natural Extension ◽

System Equilibrium ◽

The Mean ◽

Heterogeneous Service ◽

Subcritical System ◽

Scale Limit

Abstract A service system with multiple types of customers, arriving according to Poisson processes, is considered. The system is heterogeneous in that the servers can also be of multiple types. Each customer has an independent, exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to `packing' constraints, which depend on the server type. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered such that the customer arrival rates grow to ∞. We consider two variants of the model. For the infinite-server model, we prove asymptotic optimality of the greedy random (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steady state. (This version of GRAND generalizes that introduced by Stolyar and Zhong (2015) for homogeneous systems, with all servers of the same type.) We then introduce a natural extension of the GRAND algorithm for finite-server systems with blocking. Assuming subcritical system load, we prove existence, uniqueness, and local stability of the large-scale system equilibrium point such that no blocking occurs. This result strongly suggests a conjecture that the steady-state blocking probability under the algorithm vanishes in the large-scale limit.

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