Fluctuation Exponent Research Articles

Fluctuation exponents of the KPZ equation on a large torus

AbstractWe study the one‐dimensional KPZ equation on a large torus, started at equilibrium. The main results are optimal variance bounds in the super‐relaxation regime and part of the relaxation regime.

KPZ-type fluctuation exponents for interacting diffusions in equilibrium

We consider systems of N diffusions in equilibrium interacting through a potential V. We study a “height function,” which, for the special choice V(x)=e−x, coincides with the partition function of a stationary semidiscrete polymer, also known as the (stationary) O’Connell–Yor polymer. For a general class of smooth convex potentials (generalizing the O’Connell–Yor case), we obtain the order of fluctuations of the height function by proving matching upper and lower bounds for the variance of order N2/3, the expected scaling for models lying in the KPZ universality class. The models we study are not expected to be integrable, and our methods are analytic and nonperturbative, making no use of explicit formulas or any results for the O’Connell–Yor polymer.

Long- and short-term fluctuations compared for several organ systems across sleep stages.

Some details of cardiovascular and cardio-respiratory regulation and their changes during different sleep stages remain still unknown. In this paper we compared the fluctuations of heart rate, pulse rate, respiration frequency, and pulse transit times as well as EEG alpha-band power on time scales from 6 to 200s during different sleep stages in order to better understand regulatory pathways. The five considered time series were derived from ECG, photoplethysmogram, nasal air flow, and central electrode EEG measurements from full-night polysomnography recordings of 246 subjects with suspected sleep disorders. We applied detrended fluctuation analysis, distinguishing between short-term (6-16s) and long-term (50-200s) correlations, i.e., scaling behavior characterized by the fluctuation exponents α 1 and α 2 related with parasympathetic and sympathetic control, respectively. While heart rate (and pulse rate) are characterized by sex and age-dependent short-term correlations, their long-term correlations exhibit the well-known sleep stage dependence: weak long-term correlations during non-REM sleep and pronounced long-term correlations during REM sleep and wakefulness. In contrast, pulse transit times, which are believed to be mainly affected by blood pressure and arterial stiffness, do not show differences between short-term and long-term exponents. This is in constrast to previous results for blood pressure time series, where α 1 was much larger than α 2, and therefore questions a very close relation between pulse transit times and blood pressure values. Nevertheless, very similar sleep-stage dependent differences are observed for the long-term fluctuation exponent α 2 in all considered signals including EEG alpha-band power. In conclusion, we found that the observed fluctuation exponents are very robust and hardly modified by body mass index, alcohol consumption, smoking, or sleep disorders. The long-term fluctuations of all observed systems seem to be modulated by patterns following sleep stages generated in the brain and thus regulated in a similar manner, while short-term regulations differ between the organ systems. Deviations from the reported dependence in any of the signals should be indicative of problems in the function of the particular organ system or its control mechanisms.

Density fluctuations of two-dimensional active-passive mixtures

Dimension-dependent giant density fluctuations are a typical feature of active matter systems. In this work, we study the density fluctuation in two-dimensional mixtures of active and passive particles by Brownian dynamics simulations. The boundary of motility-induced phase separation is determined by the transition from unimodal to bimodal density distribution. A rapid increase of the fluctuation exponent near the boundary of phase separation in the plane of density and Péclet number was observed. When phase separation occurs, the fluctuation exponent is an approximate constant of

Effect of compensation chamber structure on the hydraulic performance of pressure compensating drip emitters

Pressure compensating (PC) emitters are installed at the end of the drip irrigation system to compensate for pressure loss and deliver water evenly to each crop. The PC emitter with excellent hydraulic performance can improve the irrigation uniformity of drip irrigation system. In this paper, the effects of main structural parameters of compensation chamber on the hydraulic performance of PC emitters were investigated by response surface methodology, and multivariate regression models were built to obtain the best hydraulic performance. The results showed that the minimum compensating pressure of PC emitters increased with the increase of compensation chamber height, while it decreased with the increase of compensation chamber diameter. The flow fluctuation exponent of PC emitters was affected by all the four structural parameters, which were outflow groove depth, outflow groove width, compensation chamber height and compensation chamber diameter. The outflow groove depth had the greatest influence on flow fluctuation exponent, and with the increase of outflow groove depth, the flow fluctuation exponent first decreased and then increased. Moreover, the value of the outflow groove depth influenced the effect of the other three parameters on flow fluctuation exponent. The optimal combination of parameters was: outflow groove depth of 0.14 mm, outflow groove width of 0.37 mm, compensation chamber height of 0.83 mm and compensation chamber diameter of 6.91 mm. The corresponding optimal results were: minimum compensating pressure of 38.73 kPa and flow fluctuation exponent of 0.024. The research provides a reference for the design and optimisation of PC emitters. • Interactions between parameters studied by response surface methodology. • Optimization goal was to reduce minimum compensating pressure and flow fluctuation of PC emitters. • Compensation chamber height and diameter had the opposite effect on performance index. • Overflow groove depth was the most important parameter affecting the flow fluctuation of PC emitters.

Fluctuation lower bounds in planar random growth models

Nous montrons des bornes inferieures de $\sqrt{\log n}$ pour l’ordre des fluctuations de trois modeles planaires de croissance (percolation de premier passage, percolation de dernier passage et polymeres diriges) sans autre hypothese sur la loi des poids des sommets ou des aretes que les conditions minimales permettant d’eviter les cas pathologiques. De telles bornes etaient connues auparavant seulement pour certaines classes restreintes de lois. De surcroit, nous montrons que l’exposant des fluctuations autour de la forme limite pour la percolation de premier passage est au moins $1/8$, ce qui etend des resultats precedents a des lois plus generales.

Identification of short-term and long-term time scales in stock markets and effect of structural break

The paper presents the comparative study of the nature of stock markets in short-term and long-term time scales ( τ) with and without structural break in the stock data. Structural break point has been identified by applying Zivot and Andrews structural trend break model to break the original time series (TSO) into two time series: time series before structural break (TSB) and time series after structural break (TSA). In order to identify the τ of short-term and long-term market, the Hurst exponent (H) technique has been applied on the intrinsic mode functions (IMF) obtained from the TSO, TSB and TSA by using empirical mode decomposition method. H≈0.5 for all the IMFs of TSO, TSB and TSA having τ in the range of few days (D) to 3 months (M), and H≥0.75 for all the IMFs of TSO, TSB and TSA having τ≥5M. Based on the value of H, the market has been divided into two time horizons: short-term market having 3D≥τ≥3M and H≈0.5, and long-term market having τ≥5M and H≥0.75. As H≈0.5 in short-term and H≥0.75 in long-term, the market is random in short-term and has long-range correlation in long-term. Robustness of the results has also been verified by using detrended fluctuation exponent (ν) analysis and normalised variance (NV) techniques. We obtained ν≈0.5 for reconstructed short-term time series and ν≈1.68 for long-term reconstructed time series. Separation of short-term and long-term market are also identified using NV technique. The time scales for short-term and long-term markets are independent of structural break happened due to extreme event. The τ obtained using the proposed method for short-term and long-term market may be useful for investors to identify the investment time horizon, and hence to design the investment and trading strategies.

Transversal fluctuations for a first passage percolation model

We introduce a new first passage percolation model in a Poissonian environment on $\mathbb{R}^{2}$. In this model, the action of a path depends on the geometry of the path and the travel time. We prove that the transversal fluctuation exponent for point-to-line action minimizers is at least $3/5$.

The Competition of Roughness and Curvature in Area-Constrained Polymer Models

The competition between local Brownian roughness and global parabolic curvature experienced in many random interface models reflects an important aspect of the KPZ universality class. It may be summarised by an exponent triple $(1/2,1/3,2/3)$ representing local interface fluctuation, local roughness (or inward deviation) and convex hull facet length. The three effects arise, for example, in droplets in planar Ising models (Alexander, '01, Hammond, '11,'12). In this article, we offer a new perspective on this phenomenon. We consider directed last passage percolation model in the plane, a paradigmatic example in the KPZ universality class, and constrain the maximizing path under the additional requirement of enclosing an atypically large area. The interface suffers a constraint of parabolic curvature as before, but now its local structure is the KPZ fixed point polymer's rather than Brownian. The local interface fluctuation exponent is thus two-thirds rather than one-half. We prove that the facet lengths of the constrained path's convex hull are governed by an exponent of $3/4$, and inward deviation by an exponent of $1/2$. That is, the exponent triple is now $(2/3,1/2,3/4)$ in place of $(1/2,1/3,2/3)$. This phenomenon appears to be shared among various isoperimetrically extremal circuits in local randomness. Indeed, we formulate a conjecture to this effect concerning such circuits in supercritical percolation, whose Wulff-like first-order behaviour was recently established (Biskup, Louidor, Procaccia and Rosenthal, '12).

Applications of Variability Analysis Techniques for Continuous Glucose Monitoring Derived Time Series in Diabetic Patients.

Methods from non-linear dynamics have enhanced understanding of functional dysregulation in various diseases but received less attention in diabetes. This retrospective cross-sectional study evaluates and compares relationships between indices of non-linear dynamics and traditional glycemic variability, and their potential application in diabetes control. Continuous glucose monitoring provided data for 177 subjects with type 1 (n = 22), type 2 diabetes (n = 143), and 12 non-diabetic subjects. Each time series comprised 576 glucose values. We calculated Poincaré plot measures (SD1, SD2), shape (SFE) and area of the fitting ellipse (AFE), multiscale entropy (MSE) index, and detrended fluctuation exponents (α1, α2). The glycemic variability metrics were the coefficient of variation (%CV) and standard deviation. Time of glucose readings in the target range (TIR) defined the quality of glycemic control. The Poincaré plot indices and α exponents were higher (p < 0.05) in type 1 than in the type 2 diabetes; SD1 (mmol/l): 1.64 ± 0.39 vs. 0.94 ± 0.35, SD2 (mmol/l): 4.06 ± 0.99 vs. 2.12 ± 1.04, AFE (mmol2/l2): 21.71 ± 9.82 vs. 7.25 ± 5.92, and α1: 1.94 ± 0.12 vs. 1.75 ± 0.12, α2: 1.38 ± 0.11 vs. 1.30 ± 0.15. The MSE index decreased consistently from the non-diabetic to the type 1 diabetic group (5.31 ± 1.10 vs. 3.29 ± 0.83, p < 0.001); higher indices correlated with lower %CV values (r = -0.313, p < 0.001). In a subgroup of type 1 diabetes patients, insulin pump therapy significantly decreased SD1 (-0.85 mmol/l), SD2 (-1.90 mmol/l), and AFE (-16.59 mmol2/l2), concomitantly with %CV (-15.60). The MSE index declined from 3.09 ± 0.94 to 1.93 ± 0.40 (p = 0.001), whereas the exponents α1 and α2 did not. On multivariate regression analyses, SD1, SD2, SFE, and AFE emerged as dominant predictors of TIR (β = -0.78, -1.00, -0.29, and -0.58) but %CV as a minor one, though α1 and MSE failed. In the regression models, including SFE, AFE, and α2 (β = -0.32), %CV was not a significant predictor. Poincaré plot descriptors provide additional information to conventional variability metrics and may complement assessment of glycemia, but complexity measures produce mixed results.

Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation

The first goal of this paper is to prove multiple asymptotic results for a time‐discrete and space‐continuous polymer model of a random walk in a random potential. These results include: existence of deterministic free energy density in the infinite‐volume limit for every fixed asymptotic slope, concentration inequalities for free energy implying a bound on its fluctuation exponent, and straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this program is almost sure existence and uniqueness of polymer measures on one‐sided infinite paths with given endpoint and slope, and interpretation of these infinite‐volume Gibbs measures as thermodynamic limits. Moreover, we prove that marginals of polymer measures with the same slope and different endpoints are asymptotic to each other.The second goal of the paper is to develop ergodic theory of the Burgers equation with positive viscosity and random kick forcing on the real line without any compactness assumptions. Namely, we prove a one force–one solution principle, using the infinite‐volume polymer measures to construct a family of stationary global solutions for this system, and proving that each of those solutions is a one‐point pullback attractor on the initial conditions with the same spatial average. This provides a natural extension of the same program realized for the inviscid Burgers equation with the help of action minimizers that can be viewed as zero temperature limits of polymer measures. © 2018 Wiley Periodicals, Inc.

Non-linear Dynamical Analysis of Intraspinal Pressure Signal Predicts Outcome After Spinal Cord Injury.

The injured spinal cord is a complex system influenced by many local and systemic factors that interact over many timescales. To help guide clinical management, we developed a technique that monitors intraspinal pressure from the injury site in patients with acute, severe traumatic spinal cord injuries. Here, we hypothesize that spinal cord injury alters the complex dynamics of the intraspinal pressure signal quantified by computing hourly the detrended fluctuation exponent alpha, multiscale entropy, and maximal Lyapunov exponent lambda. 49 patients with severe traumatic spinal cord injuries were monitored within 72 h of injury for 5 days on average to produce 5,941 h of intraspinal pressure data. We computed the spinal cord perfusion pressure as mean arterial pressure minus intraspinal pressure and the vascular pressure reactivity index as the running correlation coefficient between intraspinal pressure and arterial blood pressure. Mean patient follow-up was 17 months. We show that alpha values are greater than 0.5, which indicates that the intraspinal pressure signal is fractal. As alpha increases, intraspinal pressure decreases and spinal cord perfusion pressure increases with negative correlation between the vascular pressure reactivity index vs. alpha. Thus, secondary insults to the injured cord disrupt intraspinal pressure fractality. Our analysis shows that high intraspinal pressure, low spinal cord perfusion pressure, and impaired pressure reactivity strongly correlate with reduced multi-scale entropy, supporting the notion that secondary insults to the injured cord cause de-complexification of the intraspinal pressure signal, which may render the cord less adaptable to external changes. Healthy physiological systems are characterized by edge of chaos dynamics. We found negative correlations between the percentage of hours with edge of chaos dynamics (−0.01 ≤ lambda ≤ 0.01) vs. high intraspinal pressure and vs. low spinal cord perfusion pressure; these findings suggest that secondary insults render the intraspinal pressure more regular or chaotic. In a multivariate logistic regression model, better neurological status on admission, higher intraspinal pressure multi-scale entropy and more frequent edge of chaos intraspinal pressure dynamics predict long-term functional improvement. We conclude that spinal cord injury is associated with marked changes in non-linear intraspinal pressure metrics that carry prognostic information.

Fluctuation exponents for stationary exactly solvable lattice polymer models via a Mellin transform framework

We develop a Mellin transform framework which allows us to simultaneously analyze the four known exactly solvable 1+1 dimensional lattice polymer models: the log-gamma, strict-weak, beta, and inverse-beta models. Using this framework we prove the conjectured fluctuation exponents of the free energy and the polymer path for the stationary point-to-point versions of these four models. The fluctuation exponent for the polymer path was previously unproved for the strict-weak, beta, and inverse-beta models.

Thoron Mitigation System based on charcoal bed for applications in thorium fuel cycle facilities (part 1): Development of theoretical models for design considerations

Regulating the environmental discharge of 220Rn (historically known as thoron) and its decay products from thorium processing facilities is important for protection of environment and general public living in the vicinities. Activated charcoal provides an effective solution to this problem because of its high adsorption capacity to gaseous element like radon. In order to design and develop a charcoal based Thoron Mitigation System, a mathematical model has been developed in the present work for studying the 220Rn transport and adsorption in a flow through charcoal bed and estimating the 220Rn mitigation factor (MF) as a function of system and operating parameters. The model accounts for inter- and intra-grain diffusion, advection, radioactive decay and adsorption processes. Also, the effects of large void fluctuation and wall channeling on the mitigation factor have been included through a statistical model. Closed form solution has been provided for the MF in terms of adsorption coefficient, system dimensions, grain size, flow rate and void fluctuation exponent. It is shown that the delay effects due to intra grain diffusion plays a significant role thereby rendering external equilibrium assumptions unsuitable. Also, the application of the statistical model clearly demonstrates the transition from the exponential MF to a power-law form and shows how the occurrence of channels with low probability can lower mitigation factor by several orders of magnitude. As a part of aiding design, the model is further extended to optimise the bed dimensions in respect of pressure drop and MF. The application of the results for the design and development of a practically useful charcoal bed is discussed.

Macroweather precipitation variability up to global and centennial scales

AbstractWe investigate precipitation variability in the “macroweather” regime—the intermediate regime between the familiar weather and climate regimes—which is associated to time scales from about 10 days to 30–100 years. Macroweather precipitation is characterized by negative fluctuation exponents. This implies—contrary to the weather regime—that fluctuations tend to cancel each other out, they diminish with time scale, this is important for seasonal, annual, and decadal forecasts. Aiming at a wide‐scale range space‐time statistical description of macroweather precipitation, we study the scaling of three centennial, global‐scale precipitation products (one gauge based, one reanalysis based, and one satellite based) and systematically compare them over wide ranges of time and space scales. Although these products have very similar temporal statistics, at 5° resolution, they only agree with each other after being averaged over scales of several years, at scales larger than 2–3 decades, they disagree again. In space, there is less agreement on the statistics but—since the data have low resolutions (mostly 5° × 5°)—the disagreement is only over a small overall range of scales: the monthly data agree fairly well at scales 20°–30° and larger. Moreover, we quantify the outer scale limit of the temporal scaling (20–40 years, depending on the product, on the spatial scale, pixel, or global). Overall, results show that precipitation can be modeled with space‐time scaling processes. The improved understanding of the space‐time macroweather precipitation variability and the limitations of precipitation products provided by this work opens new perspectives to the stochastic modeling and forecasting of macroweather precipitation as well as separating natural and anthropogenic precipitation.

Scaling behavior of EEG amplitude and frequency time series across sleep stages

We study short-term and long-term persistence properties (related with auto-correlations) of amplitudes and frequencies of EEG oscillations in 176 healthy subjects and 40 patients during nocturnal sleep. The amplitudes show scaling from 2 to 500 seconds (depending on the considered band) with large fluctuation exponents during (nocturnal) wakefulness (0.73–0.83) and small ones during deep sleep (0.50–0.69). Light sleep is similar to deep sleep, while REM sleep (0.64–0.76) is closer to wakefulness except for the EEG γ band. Some of the frequency time series also show long-term scaling, depending on the selected bands and stages. Only minor deviations are seen for patients with depression, anxiety, or Parkinson's disease.

On the scaling of the solar incident flux

Abstract. It was recently found that spectral solar incident flux (SIF) as a function of the ultraviolet wavelengths exhibits 1/f-type power-law correlations. In this study, an attempt was made to explore the residues of the SIF with respect to the Planck law over a wider range of wavelengths, from 115.5 to 629.5 nm. Using spectral, Haar and Detrended Fluctuation analyses, we show that over the range from 10–20 nm to the maximum lag (≈ 500 nm), the SIF residues have a scaling regime with fluctuation exponent H &amp;amp;approx; 0.37 but with high intermittency (C1 &amp;amp;approx; 0.16, multifractal index α&amp;amp;approx; 1.7) and spectral exponent &amp;amp;approx; 1.46. Over the shorter wavelengths range we found on the contrary low intermittency (C1 &amp;amp;approx; 0) with spectral exponent &amp;amp;approx; 1 and H &amp;amp;approx; 0.

The joint space-time statistics of macroweather precipitation, space-time statistical factorization and macroweather models.

Over the range of time scales from about 10 days to 30-100 years, in addition to the familiar weather and climate regimes, there is an intermediate "macroweather" regime characterized by negative temporal fluctuation exponents: implying that fluctuations tend to cancel each other out so that averages tend to converge. We show theoretically and numerically that macroweather precipitation can be modeled by a stochastic weather-climate model (the Climate Extended Fractionally Integrated Flux, model, CEFIF) first proposed for macroweather temperatures and we show numerically that a four parameter space-time CEFIF model can approximately reproduce eight or so empirical space-time exponents. In spite of this success, CEFIF is theoretically and numerically difficult to manage. We therefore propose a simplified stochastic model in which the temporal behavior is modeled as a fractional Gaussian noise but the spatial behaviour as a multifractal (climate) cascade: a spatial extension of the recently introduced ScaLIng Macroweather Model, SLIMM. Both the CEFIF and this spatial SLIMM model have a property often implicitly assumed by climatologists that climate statistics can be "homogenized" by normalizing them with the standard deviation of the anomalies. Physically, it means that the spatial macroweather variability corresponds to different climate zones that multiplicatively modulate the local, temporal statistics. This simplified macroweather model provides a framework for macroweather forecasting that exploits the system's long range memory and spatial correlations; for it, the forecasting problem has been solved. We test this factorization property and the model with the help of three centennial, global scale precipitation products that we analyze jointly in space and in time.

The fluctuation function of the detrended fluctuation analysis — investigation on the AR(1) process

We derive an analytical expression for the fluctuation function of the first order autoregressive process AR(1) by means of the detrended fluctuation analysis (DFA). This process is short-range correlated and therefore the fluctuation exponent should be α = 1/2. However, the fluctuation function exhibits a crossover between a region with α > 1/2 and the expected 1/2. We calculate the crossover point and compare it with the characteristic correlation time of the process. We conclude that DFA is data consuming and requires one to two orders of magnitude more data than the estimation of the autocorrelation function.

Random geometry and the Kardar–Parisi–Zhang universality class

We consider a model of a quenched disordered geometry in which a random metric is defined on , which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius R scales as , with a fluctuation exponent , while the lateral spread of the minimizing geodesic between two points at a distance L grows as , with wandering exponent value . Results on related first-passage percolation problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar–Parisi–Zhang universality class of surface kinetic roughening, with ξ and χ relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy–Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW–GUE statistics with good accuracy in arrival times.