Fluctuation scaling in complex systems: Taylor's law and beyond1

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form ‘fluctuations ≈ constant × average^α’, where the exponent α is predominantly in the range [1/2, 1]. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names Taylor's law or fluctuation scaling. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling.     

Direct numerical simulation study of the interaction between the polymer effect and velocity gradient tensor in decaying homogeneous isotropic turbulence

Direct numerical simulation of decaying homogeneous isotropic turbulence (DHIT) of a polymer solution is performed. In order to understand the polymer effect on turbulence or additive-turbulence interaction, we directly investigate the influence of polymers on velocity gradient tensor including vorticity and strain. By visualizing vortex tubes and sheets, we observe a remarkable inhibition of vortex structures in an intermediate-scale field and a small-scale field but not for a large scale field in DHIT with polymers. The geometric study indicates a strong relevance among the vorticity vector, rate-of-strain tensor, and polymer conformation tensor. Joint probability density functions show that the polymer effect can increase "strain generation resistance" and "vorticity generation resistance", i.e., inhibit the generation of vortex sheets and tubes, ultimately leading to turbulence inhibition effects.     

Investigation of insect morphology by MRI: assessment of spatial and temporal resolution

Classically, the investigation of the internal morphology of insects relies on histologic methods, e.g., the preparation of thin tissue sections. However, the preparation of serial sections is time consuming and means the irreversible loss of the animal. In the present investigation, we have analyzed the potential of NMR imaging as a tool for the morphologic classification of insects with sufficient spatial resolution. With a 512 matrix, 15 mm FOV, 200 microm slice thickness, images with an in-plane spatial resolution of 30 microm are obtained with a signal-to-noise ratio of 70. These conditions require only seven averages, resulting in an experimental time of only 50 min. Such image quality already permits the differentiation of fine structural and morphologic details such as e.g., intestinal tracts and copulation organ in a beetle. Also, wing controlling dorsal muscle groups as well as leg structures and joints are clearly distinguishable. We conclude that the spatial resolution and contrast condition of MR imaging are quite promising for the new approach of zoological insect classification using NMR imaging. Further principally available technical enhancement of sensitivity and spatial resolution will provide an attractive alternative to invasive techniques for the classification of, sometimes, rare and precious insect specimen.     

Auditory temporal processing: two-tone flutter fusion and a model of temporal integration

Auditory temporal processing was examined using a flutter-fusion paradigm in which two tones were separated by a silent interval. The listener's task was to judge when the two tones, presented in a background noise, fused perceptually. The fusion point was studied in a series of six experiments. In the first five experiments, the duration of the first stimulus (T1) was the dependent variable. In the last experiment, the duration of the second stimulus (T2) was the dependent variable. An inverse relationship was found between T1 duration and the interstimulus interval (ISI) such that, when ISI was decreased, T1 duration had to be increased to maintain fusion. When ISI was plotted as a function of T1 duration, the data were represented by a negative exponential equation. Increasing the level of the tones, increasing the bandwidth of the background noise, or presenting the stimuli dichotically lowered the duration of T1 necessary for fusion. Changing the frequency of the tones had no effect on fusion. Decreasing the duration of T2 and holding T1 constant also resulted in fusion. A neurophysiological model implicating ON and OFF neural response interactions is postulated to account for the data.     

An extended car-following model considering the appearing probability of truck and driver's characteristics

In this paper, the appearing probability of truck is introduced and an extended car-following model is presented to analyze the traffic flow based on the consideration of driver's characteristics, under honk environment. The stability condition of this proposed model is obtained through linear stability analysis. In order to study the evolution properties of traffic wave near the critical point, the mKdV equation is derived by the reductive perturbation method. The results show that the traffic flow will become more disorder for the larger appearing probability of truck. Besides, the appearance of leading truck affects not only the stability of traffic flow, but also the effect of other aspects on traffic flow, such as: driver's reaction and honk effect. The effects of them on traffic flow are closely correlated with the appearing probability of truck. Finally, the numerical simulations under the periodic boundary condition are carried out to verify the proposed model. And they are consistent with the theoretical findings.     

Magnetic resonance elastography of the brain in a mouse model of Alzheimer's disease: initial results

The increasing prevalence of Alzheimer's disease (AD) has provided motivation for developing novel methods for assessing the disease and the effects of potential treatments. Magnetic resonance elastography (MRE) is an MRI-based method for quantitatively imaging the shear tissue stiffness in vivo. The objective of this research was to determine whether this new imaging biomarker has potential for characterizing neurodegenerative disease. Methods were developed and tested for applying MRE to evaluate the mouse brain, using a conventional large bore 3.0T MRI system. The technique was then applied to study APP-PS1 mice, a well-characterized model of AD. Five APP-PS1 mice and 8 age-matched wild-type mice were imaged immediately following sacrifice. Brain shear stiffness measurements in APP-PS1 mice averaged 22.5% lower than those for wild-type mice (P = .0031). The results indicate that mouse brain MRE is feasible at 3.0T, and brain shear stiffness has merit for further investigation as a potential new biomarker for Alzheimer's disease.     

A look at the abandoned contributions to cosmology of Dirac,Sciama, and Dicke

The separate contributions to cosmology of the above researchers are revisited and a cosmology encompassing their basic ideas is proposed. We study Dirac's article (1938) on the large number hypothesis, Sciama's proposal (1953) of realizing Mach's principle, and Dicke's considerations (1957) on a flat‐space representation of general relativity with a variable speed of light (VSL). Dicke's tentative theory can be formulated in a way which is compatible with Sciama's hypothesis on the gravitational constant G. Additionally, such a cosmological model is shown to satisfy Dirac's second ‘large number’ hypothesis on the total number of particles in the universe being proportional to the square of the epoch. In the same context, Dirac's first hypothesis on an epoch‐dependent G – contrary to his prediction – does not necessarily produce a visible time dependence of G. While Dicke's proposal reproduces the classical tests of GR in first approximation, the cosmological redshift is described by a shortening of measuring rods rather than an expansion of space. Since the temporal evolution of the horizon R is governed by , the flatness and horizon problems do not arise in the common form.     

Subgrid-scale dynamics and model test in a turbulent stratified jet with coexistence of stable and unstable stratification

The subgrid-scale dynamics of stratified flows is studied in a horizontally introduced turbulent jet with coexistence of stable and unstable stratification of a low Richardson number case and a high Richardson number case. The positive production of subgrid-scale kinetic energy and the production of scalar variance suggest the forward energy cascade. The subgrid-scale buoyant destruction plays a role as a sink of subgrid-scale kinetic energy in the stable stratification while holds a role of turbulent generation in the unstable stratification. The role-switch of buoyant destruction in the stable stratification of high-Ri case implies the occurrence of a destabilising process triggered by the coupled instability mechanisms. The energy balance assumption related to the production of and the dissipation of subgrid-scale kinetic energy as well as the subgrid-scale buoyant destruction may fail. The a-priori test suggests the scale-invariant dynamic and standard Smagorinsky models not to work properly here, while the scale-dependent dynamic model gives a decent performance but with restrictions of the ratio between two testing filter scales.     

Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field B_ij (x,x′,t) and isometries of the correlation space K³ equipped with a (pseudo-) Riemannian metrics ds²(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds²(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (M^t, ds²(t)), M^t ? K³. Here the metric ds²(t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds²(t) in K³ and present symmetries (or isometric motions in K³) of the metric ds²(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds²(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λ_g is a second-order differential invariant and show how λ_g can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.     

Stochastic analysis of a chemical reaction with spatial and temporal structures

A stochastic analysis of the spatial and temporal structures in the Prigogine-Lefever-Nicolis model (the Brusselator) is presented. The analysis is carried out through a Langevin equation derived from a multivariate master equation using the Poisson representation method, which is used to calculate the spatial correlation functions and the fluctuation spectra in the Gaussian approximation. The case of an infinite three-dimensional system is considered in detail. The calculations for the spatial correlation functions and the fluctuation spectra for a finite system subject to different kinds of boundary conditions are also given.     

The one-dimensional Boltzmann gas: The ergodic hypothesis and the phase portrait of small systems

The concept of ergodicity and its application to microcanonical systems composed of few particles of different mases are clarified. The distribution functions in position and velocity are theoretically derived and numerically verified. Moreover, we deal with a one-dimensional Boltzmann gas where the order relation (connected to the one dimensionality) brings constraints depending on the two classes of boundary conditions enforced (reflecting, periodic). The numerical simulations on a one-dimensional Boltzmann gas act as real experiments and allow us to play on the constraints to which the system is subjected.     

RANS predictions of turbulent diffusion flames: comparison of a reactor and a flamelet combustion model to the well stirred approach

The flame stabilisation process in turbulent non-premixed flames is not fully understood and several models have been developed to describe the turbulence–chemistry interaction. This work compares the performance of the multiple Representative Interactive Flamelet (mRIF) model, the Volume Reactor Fraction Model (VRFM), and the Well-Stirred reactor (WS) model in describing such flames. The predicted ignition delay and flame lift-off length of n-heptane sprays are compared to experimental results published within the Engine Combustion Network (ECN). All of the models predict the trend of ignition delay reasonably well. At a low gas pressure (42 bar) the ignition delay is overpredicted compared to the experimental data, but the difference between the models is not significant. However, the predicted lift-off lengths differ. At high pressure (87 bar) the difference between the models is small. All models slightly underpredict the lift-off length compared to the experimental data. At low gas pressure (42 bar) the mRIF model gives the best results. The VRFM and WS models predict excessively short lift-off lengths, but the VRFM model gives better results than the WS model. The flame structures of the models are also compared. The WS model and the VRFM model yield a well defined flame stabilisation point whereas the mRIF model does not. The flame of the mRIF model is more diffuse and the model is not able to predict flame propagation. All models were able to predict the experimental trends in lift-off and ignition delay, but certain differences between them are demonstrated.     

Mean magnetic field and noise cross-correlation in magnetohydrodynamic turbulence: results from a one-dimensional model

We show that a recently proposed [J. Fleischer, P.H. Diamond, Phys. Rev. E 58, R2709 (1998)] one-dimensional Burgers-like model for magnetohydrodynamics (MHD) is in effect identical to existing models for drifting lines and sedimenting lattices. We use the model to demonstrate, contrary to claims in the literature, that the energy spectrum of MHD turbulence should be independent of mean magnetic field and that cross-correlations between the noise sources for the velocity and magnetic fields cannot change the structure of the equations under renormalisation. We comment on the scaling and the multiscaling properties of the stochastically forced version of the model. Received 29 October 1998 and Received in final form 8 December 1998     

使用基于动态程序规划的时间延迟法分析直线磁化等离子体漂移波湍流角向传播速度和带状流结构

对等离子体湍流速度场的有效探测,有助于更加深入了解磁约束等离子体湍流以及实现对某些理论预言现象和结构(如带状流)的充分辨识.本文将基于动态程序规划的时间延迟估算技术成功应用于直线磁化等离子体装置中热阴极放电条件下的漂移波湍流角向速度涨落的实验分析,并且其结果清晰再现了漂移波湍流中通过非线性能量耦合自发产生的带状流结构.通过对采用不同频段等离子体湍流涨落通过基于动态程序规划的时间延迟估算分析所再现的带状流结构特征进行比较,进一步就该算法对载波信号中非相干噪声相对水平的抗干扰能力进行了定性评估.这些工作的成功开展,对于通过采用基于动态程序规划的时间延迟估算分析技术更为深入有效探索磁约束等离子体湍流行为特征,尤其是速度涨落场的演化提供了重要的借鉴和参考价值.     

The development and investigation of a strongly non-equilibrium model of heat transfer in fluid with allowance for the spatial and temporal non-locality and energy dissipation

The differential equation of heat transfer with allowance for energy dissipation and spatial and temporal nonlocality has been derived by the relaxation of heat flux and temperature gradient in the Fourier law formula for the heat flux at the use of the heat balance equation. An investigation of the numerical solution of the heat-transfer problem at a laminar fluid flow in a plane duct has shown the impossibility of an instantaneous acceptance of the boundary condition of the first kind — the process of its settling at small values of relaxation coefficients takes a finite time interval the duration of which is determined by the thermophysical and relaxation properties of the fluid. At large values of relaxation coefficients, the use of the boundary condition of the first kind is possible only at Fo → ∞. The friction heat consideration leads to the alteration of temperature profiles, which is due to the rise of the intervals of elevated temperatures in the zone of the maximal velocity gradients. With increasing relaxation coefficients, the smoothing of temperature profiles occurs, and at their certain high values, the fluid cooling occurs at a gradientless temperature variation along the transverse spatial variable and, consequently, the temperature proves to be dependent only on time and on longitudinal coordinate.     

The clinical application of whole-body diffusion-weighted imaging in the early assessment of chemotherapeutic effects in lymphoma: the initial experience

At present, accurate assessment of therapeutic efficacy at the early stage of treatment is still a challenge for radiologists. As a new non-radiation whole body imaging technology, Whole body-diffusion weighted imaging (WB-DWI) had shown promising application prospects in therapeutic assessment, which confirmed by many premier animal studies. Here we report that in the chemotherapeutic assessment of malignant non-Hodgkin's lymphoma, WB-DWI can not only detect the morphological change of solid infiltrated lesion as the convention (such as CT, PET, etc.) but also provide information about the growth and decline process of tumor cells in the lesion combining with the dynamic changes of apparent diffusion coefficient (ADC) value, which is sooner than the morphological changes.     

Analysis of the spatial and temporal distributions between successive earthquakes: Nonextensive statistical mechanics viewpoint

The spatial and temporal distributions between successive earthquakes are treated in the framework of nonextensive statistical mechanics. We find temporal distributions exhibit the power law behavior; q-exponential with q>1. It means the earthquakes are strongly correlated in time. The spatial distributions obey the q-exponential form with q<1. We also examine the dependence of the q exponent on magnitude range, covering period, time interval and size of the region where data are gathered. The conjecture of Abe et al. [S. Abe, N. Suzuki, Physica A 350 (2005) 588] has been examined for different categories of data. The results show a strange relation between q values of the spatial and temporal distributions.     

Extension of a temporal model of frequency discrimination: intensity effects in normal and hearing-impaired listeners

The effects of intensity on the difference limen for frequency (DLF) in normal-hearing and in hearing-impaired listeners are incorporated into the temporal model of frequency discrimination proposed by Goldstein and Srulovicz [Psychophysics and Physiology of Hearing, edited by E. F. Evans and J.P. Wilson (Academic, New York, 1977)]. A simple extension of the temporal mode, which includes the dependence of phase locking on intensity, is sufficient to predict the effects of intensity on the DLF in normal-hearing listeners. To account for elevated DLFs in hearing-impaired listeners the impairment is modeled as a reduction in the synchrony of the discharge from VIIIth-nerve fibers that innervate the region of hearing loss. Constraints on the optimal processor and the validity of the temporal model at high frequencies are discussed.     

Temporal and spatial correlations in a viscoelastic model of heterogeneous faults

We study the temporal and spatial correlations in a one-dimensional model of a heterogeneous fault zone, in the presence of viscoelastic effects. As a function of dynamical weakening and of dissipation, the system exhibits three different “phases": one in which there are no time correlations between the events, a second, in which there are “Omori's law” type temporal correlations, and a third, runaway phase with quasiperiodic system size events. Received 30 May 2000 and Received in final form 7 September 2000