Simple examples with features of renormalization for turbulent transport

Two simple exactly solvable models for turbulent transport are introduced and discussed here with complete mathematical rigour. These models illustrate several different facets of super-diffusion and renormalization for turbulent transport. The first model involves time dependent velocity fields with suitable long-range correlations and the complete renormalization theory is developed here in detail. In addition rigorous examples are developed by using variants of this model where the effective equation for the ensemble average at large scales and long times is diffusive despite the fact that each realization exhibits catastrophic large-scale instability. The second model introduced previously by the authors involves transport-diffusion in simple shear layers with turbulent velocity statistics. The theories of renormalized eddy diffusivity and higher-order statistics are surveyed here. An extreme limiting case of the theory involving turbulent velocity statistics with long-range spatial correlations but gaussian white noise in time is discussed in detail. Both the renormalized theory of eddy diffusivity and exact explicit equations for second-order correlations related to the pair distance function are developed in complete detail here in this instructive limiting case.

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ANALYSIS AND SIMULATION OF LONG-RANGE CORRELATIONS IN CURVED SPACE

International Journal of Modern Physics C ◽

10.1142/s0129183109014308 ◽

2009 ◽

Vol 20 (08) ◽

pp. 1211-1232 ◽

Cited By ~ 4

Author(s):

ALI REZA MEHRABI ◽

MUHAMMAD SAHIMI

Keyword(s):

Long Range ◽

Large Scale ◽

Dimensional Space ◽

Three Dimensional ◽

Amorphous Semiconductors ◽

Curved Space ◽

Long Range Correlations ◽

Planar Surfaces ◽

Positively Curved ◽

Industrial Problem

Numerical simulation and analysis of long-range correlations in curved space are studied. The study is motivated by the problem of constructing accurate models of large-scale porous media which usually contain long-range correlations in their various properties (such as their permeability, porosity, and elastic moduli) within and between their strata that are typically curved layers. The problem is, however, relevant to many other important models and phenomena in which extended correlations in curved space play a prominent role. Examples include the nonlinear σ-model in a curved space, models for describing the long-range structural correlations of amorphous semiconductors that consist of polytopes (tilings of positively-curved three-dimensional space), long-range correlations in the extrapolar total zone, and models in which the Universe is created by bubble nucleations and contain long-range correlations in the fluctuations in the curved spacetime. The study is also relevant to the important industrial problem of designing highly curved objects, such as cars and ships, which use composite materials that contain extended correlations in their property values. We study such correlations along two- and three-dimensional curves, as well as curved surfaces. We show that such correlations are well-defined only on developable surfaces, i.e. those that can be flattened to form planar surfaces without any stretching or distortion, and preserve the distance between two points on such surfaces after the stretching. If a given curved surface is not developable, but can be approximated as piecewise developable, one may still define and analyze extended correlations on it. Representative examples are presented and analyzed.

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Quantifying Evolution of Short and Long-Range Correlations in Chinese Narrative Texts across 2000 Years

Complexity ◽

10.1155/2018/9362468 ◽

2018 ◽

Vol 2018 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Heng Chen ◽

Haitao Liu

Keyword(s):

Long Range ◽

Word Length ◽

Large Scale ◽

Sentence Length ◽

Small Scale ◽

Evolutionary Trend ◽

Narrative Texts ◽

Long Range Correlations ◽

Cultural Aspects ◽

The Social

We investigate how short and long-range word length correlations evolve in Chinese narrative texts. The results show that, for short-range word length correlations, no significant linear evolutionary trend was found. But for long-range correlations, there are two opposite tendencies for two different regimes: the Hurst exponent of small-scale (box size n ranges from 10 to 100) word length correlations decreases over time, and the exponent of large-scale (box size n ranges from 101 to 1000) shows an increasing tendency. The increase of word length is corroborated as an essential regularity of word evolution in written Chinese. Further analyses show that a significant correlation coefficient is obtained between Hurst exponents from the small-scale correlations and mean word length across time. These indicate that word length correlation evolution possesses different self-adaptive mechanisms in terms of different scales of distances between words. We speculate that the increase of word length and sentence length in written Chinese may account for this phenomenon, in terms of both the social-cultural aspects and the self-adapting properties of language structures.

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Scaling, multifractality, and long-range correlations in well log data of large-scale porous media

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2011.01.010 ◽

2011 ◽

Vol 390 (11) ◽

pp. 2096-2111 ◽

Cited By ~ 29

Author(s):

Hassan Dashtian ◽

G. Reza Jafari ◽

Muhammad Sahimi ◽

Mohsen Masihi

Keyword(s):

Porous Media ◽

Long Range ◽

Large Scale ◽

Well Log ◽

Log Data ◽

Long Range Correlations

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Long-Range Correlations in the Fossil Record and the Fractal Nature of Macroevolution

Advances in Complex Systems ◽

10.1142/s021952599800017x ◽

1998 ◽

Vol 01 (02n03) ◽

pp. 255-266 ◽

Cited By ~ 10

Author(s):

Richard V. Solé ◽

Susanna C. Manrubia ◽

Juan Pérez-Mercader ◽

Michael Benton ◽

Per Bak

Keyword(s):

Long Range ◽

Fossil Record ◽

Large Scale ◽

Time Frame ◽

Data Series ◽

Dynamical Process ◽

Characteristic Time Scale ◽

Long Range Correlations ◽

Theoretical Approaches ◽

Change Of Scale

Recent studies on the fossil record time series has suggested that there is consistent evidence for self-similarity i.e. long-range correlations with power-law behavior. The existence of such fractal strucutes means that, when looking at a given time frame, some basic properties remain the same if a change of scale is performed. In other words, there is no characteristic time scale, as we could expect if some type of periodic or other low-dimensional dynamics were present. A possible explanation for such long-range order is a dynamical process operating at all scales, as it is the case for systems in the neighborhood of critical points. In this paper these results are further explored by extending previous data analysis and examining the relevance of recent theoretical approaches to the statistical features of the fossil record. The presence of long-range correlations is shown through Hurst analysis using non-interpolated data series from the Fossil Record 2 database. As shown in previous studies, such correlations span over hundreds of millions of years and are compared with a simple model of large-scale evolution displaying self-organized criticality.

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QUANTUM MACROSTATISTICAL THEORY OF NONEQUILIBRIUM STEADY STATES

Reviews in Mathematical Physics ◽

10.1142/s0129055x05002492 ◽

2005 ◽

Vol 17 (09) ◽

pp. 977-1020 ◽

Cited By ~ 8

Author(s):

GEOFFREY L. SEWELL

Keyword(s):

Long Range ◽

Large Scale ◽

Stochastic Systems ◽

Local Equilibrium ◽

Transport Coefficients ◽

Steady States ◽

Nonequilibrium Steady States ◽

Striking Difference ◽

Long Range Correlations ◽

Scale Properties

We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centered on the large scale properties of the locally conserved hydrodynamical observables, and our basic physical assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis, we obtain a picture wherein the fluctuations of the hydrodynamical variables about a nonequilibrium steady state execute a Gaussian Markov process of a generalized Onsager–Machlup type, which is completely determined by the position dependent transport coefficients and the equilibrium entropy function of the system. This picture reveals that the transport coefficients satisfy a generalized form of the Onsager reciprocity relations in the nonequilibrium situation and that the spatial correlations of the hydrodynamical observables are generically of long range. This last result constitutes a model-independent quantum mechanical generalization of that obtained for special classical stochastic systems and marks a striking difference between the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations.

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Explicit expressions for eddy-diffusivity fields and effective large-scale advection in turbulent transport

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.220 ◽

2016 ◽

Vol 795 ◽

pp. 524-548 ◽

Cited By ~ 5

Author(s):

S. Boi ◽

A. Mazzino ◽

G. Lacorata

Keyword(s):

Large Scale ◽

Spectral Gap ◽

Environmental Flows ◽

Turbulent Transport ◽

Eddy Diffusivity ◽

Small Scale ◽

Eddy Diffusivities ◽

Perturbative Methods ◽

Tracer Dispersion ◽

Explicit Expressions

Large-scale transport is investigated in terms of new explicit expressions for eddy diffusivities and effective advection obtained from asymptotic perturbative methods. The carrier flow is formed by a large-scale component plus a small-scale contribution mimicking a turbulent flow. The scalar dynamics is observed in its pre-asymptotic regimes (i.e. on scales comparable to those of the large-scale velocity). The resulting eddy diffusivity is thus a tensor field which explicitly depends on the large-scale velocity. Small-scale interactions also cause the emergence of an effective large-scale (compressible) advection field which, as a result of the present study however, turns out to be of negligible importance. Two issues are addressed by means of Lagrangian simulations: quantifying the possible deterioration of the eddy-diffusivity/effective advection description by reducing to zero the spectral gap separating the large-scale velocity component from the small-scale component; comparing the accuracy of our closure against other simple, reasonable, options. Answering these questions is important in view of possible applications of our closure to tracer dispersion in environmental flows.

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Green function of correlated genes in a minimal mechanical model of protein evolution

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1716215115 ◽

2018 ◽

Vol 115 (20) ◽

pp. E4559-E4568 ◽

Cited By ~ 15

Author(s):

Sandipan Dutta ◽

Jean-Pierre Eckmann ◽

Albert Libchaber ◽

Tsvi Tlusty

Keyword(s):

Amino Acids ◽

Green Function ◽

Long Range ◽

Protein Evolution ◽

Large Scale ◽

Strongly Coupled ◽

Long Range Correlations ◽

Cooperative Interactions ◽

Physical Interactions ◽

The Green Function

The function of proteins arises from cooperative interactions and rearrangements of their amino acids, which exhibit large-scale dynamical modes. Long-range correlations have also been revealed in protein sequences, and this has motivated the search for physical links between the observed genetic and dynamic cooperativity. We outline here a simplified theory of protein, which relates sequence correlations to physical interactions and to the emergence of mechanical function. Our protein is modeled as a strongly coupled amino acid network with interactions and motions that are captured by the mechanical propagator, the Green function. The propagator describes how the gene determines the connectivity of the amino acids and thereby, the transmission of forces. Mutations introduce localized perturbations to the propagator that scatter the force field. The emergence of function is manifested by a topological transition when a band of such perturbations divides the protein into subdomains. We find that epistasis—the interaction among mutations in the gene—is related to the nonlinearity of the Green function, which can be interpreted as a sum over multiple scattering paths. We apply this mechanical framework to simulations of protein evolution and observe long-range epistasis, which facilitates collective functional modes.

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