Nonlinear Probability Distribution Function Research Articles

One-point probability distribution function from spherical collapse: early dark energy versus ΛCDM

ABSTRACT We compute the one-point probability distribution function (PDF) of an initially Gaussian dark matter density field using spherical collapse (SC). We compare the results to other forms available in the literature and also compare the PDFs in the Λ-cold dark matter model with an early dark energy (EDE) model. We find that the skewed lognormal distribution provides the best fit to the non-linear PDF from SC for both cosmologies, from a = 0.1 to 1 and for scales characterized by the comoving width of the Gaussian: σG = 0.5, 1, and 2. To elucidate the effect of cosmology, we examine the linear and non-linear growth rates through test cases. For overdensities, when the two models have the same initial density contrast, the differences due to cosmology are amplified in the non-linear regime, whereas, if the two models have the same linear density contrast today, then the differences in cosmology are damped in the non-linear regime. This behaviour is in contrast with voids, where the non-linear growth becomes ‘self-regulatory’ and is less sensitive to cosmology and initial conditions. To compare the PDFs, we examine the difference of the PDFs and evolution of the width of the PDF. The trends with scale and redshift are as expected. A tertiary aim of this paper was to check if the fitting form for the non-linear density–velocity divergence relation, derived for constant equation of state (w) models by Nadkarni-Ghosh holds for the EDE model. We find that it does with an accuracy of 4 per cent, thus increasing its range of validity.

Semi-Analytic Probability Density Function for System Uncertainty

In general, the behavior of science and engineering is predicted based on nonlinear math models. Imprecise knowledge of the model parameters alters the system response from the assumed nominal model data. One proposes an algorithm for generating insights into the range of variability that can be expected due to model uncertainty. An automatic differentiation tool builds the exact partial derivative models required to develop a state transition tensor series (STTS)-based solution for nonlinearly mapping initial uncertainty models into instantaneous uncertainty models. The fully nonlinear statistical system properties are recovered via series approximations. The governing nonlinear probability distribution function is approximated by developing an inverse mapping algorithm for the forward series model. Numerical examples are presented, which demonstrate the effectiveness of the proposed methodology.

Statistical distribution of nonlinear random wave height in shallow water

Here we present a statistical model of random wave, using Stokes wave theory of water wave dynamics, as well as a new nonlinear probability distribution function of wave height in shallow water. It is more physically logical to use the wave steepness of shallow water and the factor of shallow water as the parameters in the wave height distribution. The results indicate that the two parameters not only could be parameters of the distribution function of wave height but also could reflect the degree of wave height distribution deviation from the Rayleigh distribution. The new wave height distribution overcomes the problem of Rayleigh distribution that the prediction of big wave is overestimated and the general wave is underestimated. The prediction of small probability wave height value of new distribution is also smaller than that of Rayleigh distribution. The effect of wave steepness in shallow water is similar to that in deep water; but the factor of shallow water lowers the wave height distribution of the general wave with the reduced factor of wave steepness. It also makes the wave height distribution of shallow water more centralized. The results indicate that the new distribution fits the in situ measurements much better than other distributions.

Distribution of the nonlinear random ocean wave period

Because of the intrinsic difficulty in determining distributions for wave periods, previous studies on wave period distribution models have not taken nonlinearity into account and have not performed well in terms of describing and statistically analyzing the probability density distribution of ocean waves. In this study, a statistical model of random waves is developed using Stokes wave theory of water wave dynamics. In addition, a new nonlinear probability distribution function for the wave period is presented with the parameters of spectral density width and nonlinear wave steepness, which is more reasonable as a physical mechanism. The magnitude of wave steepness determines the intensity of the nonlinear effect, while the spectral width only changes the energy distribution. The wave steepness is found to be an important parameter in terms of not only dynamics but also statistics. The value of wave steepness reflects the degree that the wave period distribution skews from the Cauchy distribution, and it also describes the variation in the distribution function, which resembles that of the wave surface elevation distribution and wave height distribution. We found that the distribution curves skew leftward and upward as the wave steepness increases. The wave period observations for the SZFII-1 buoy, made off the coast of Weihai (37°27.6′ N, 122°15.1′ E), China, are used to verify the new distribution. The coefficient of the correlation between the new distribution and the buoy data at different spectral widths (ν=0.3−0.5) is within the range of 0.968 6 to 0.991 7. In addition, the Longuet-Higgins (1975) and Sun (1988) distributions and the new distribution presented in this work are compared. The validations and comparisons indicate that the new nonlinear probability density distribution fits the buoy measurements better than the Longuet-Higgins and Sun distributions do. We believe that adoption of the new wave period distribution would improve traditional statistical wave theory.

The non-linear probability distribution function in models with local primordial non-Gaussianity

We use the spherical evolution approximation to investigate non-linear evolution from the non-Gaussian initial conditions characteristic of the local fnl model. We provide an analytic formula for the non-linearly evolved probability distribution function (PDF) of the dark matter which shows that the underdense tail of the non-linear PDF in the fnl model should differ significantly from that for Gaussian initial conditions. Measurements of the underdense tail in numerical simulations may be affected by discreteness effects, and we use a Poisson counting model to describe this effect. Once this has been accounted, our model is in good quantitative agreement with the simulations. In principle, our calculation is an important first step in programs which seek to reconstruct the shape of the initial PDF from observations of large-scale structures in the Lyα forest and the galaxy distribution at later times.

Statistical distribution of nonlinear random wave height

A statistical model of random wave is developed using Stokes wave theory of water wave dynamics. A new nonlinear probability distribution function of wave height is presented. The results indicate that wave steepness not only could be a parameter of the distribution function of wave height but also could reflect the degree of wave height distribution deviation from the Rayleigh distribution. The new wave height distribution overcomes the problem of Rayleigh distribution that the prediction of big wave is overestimated and the general wave is underestimated. The prediction of small probability wave height value of new distribution is also smaller than that of Rayleigh distribution. Wave height data taken from East China Normal University are used to verify the new distribution. The results indicate that the new distribution fits the measurements much better than the Rayleigh distribution.