Evolution Of Probability Distribution Function Research Articles

On the statistical evolution of the large-scale structure and the effective dynamics

In this paper, we study the statistical evolution of the large-scale structure (LSS), focusing on the joint probability distribution function (PDF) of the coarse-grained cosmic field and its role in constructing effective dynamics.As the most comprehensive statistics, this PDF encodes all cosmological information of large-scale modes, therefore, could serve as the basis in the LSS modelling. Following the so-called PDF-based method from turbulence, we write down this PDF's evolution equation, which describes the probability conservation.We show that this conservation equation's characteristic curves follow the same PDF history and could be considered as an effective dynamics of the coarse-grained field.Unlike the EFT of LSS, which conceptually would work at both realization and statistics level, this effective dynamics is valid only statistically. However, this `statistical equivalence' also provides valuable insight into scale interactions at the statistical level. It also enables predicting a wide variety of statistics beyond the typical N-point polyspectra, including, e.g. topologies, density PDF and non-linear covariance matrices etc. Our formula expresses the small-scale effect as the ensemble average of their interactions conditional on the large-scale modes. This suggests an interesting way to measure effective terms directly from simulation. By applying the Gram-Charlier expansion, we demonstrate a different structure of these effective terms.This formalism is a natural framework for discussing the evolution of statistical properties of large-scale modes, and provides an alternative view for understanding the relationship between general effective dynamics and standard perturbation theory.

Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation.

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.

The effect of magnetic fields and ambipolar diffusion on the column density probability distribution function in molecular clouds

Simulations generally show that non-self-gravitating clouds have a lognormal column density ($\Sigma$) probability distribution function (PDF), while self-gravitating clouds with active star formation develop a distinct power-law tail at high column density. Although the growth of the power law can be attributed to gravitational contraction leading to the formation of condensed cores, it is often debated if an observed lognormal shape is a direct consequence of supersonic turbulence alone, or even if it is really observed in molecular clouds. In this paper we run three-dimensional magnetohydrodynamic simulations including ambipolar diffusion with different initial conditions to see the effect of strong magnetic fields and nonlinear initial velocity perturbations on the evolution of the column density PDFs. Our simulations show that column density PDFs of clouds with supercritical mass-to-flux ratio, with either linear perturbations or nonlinear turbulence, quickly develop a power-law tail such that $dN/d \log \Sigma \propto \Sigma^{-\alpha}$ with index $\alpha \simeq 2$. Interestingly, clouds with subcritical mass-to-flux ratio also proceed directly to a power-law PDF, but with a much steeper index $\alpha \simeq 4$. This is a result of gravitationally-driven ambipolar diffusion. However, for nonlinear perturbations with a turbulent spectrum ($v_{k}^{2} \propto k^{-4}$), the column density PDFs of subcritical clouds do retain a lognormal shape for a major part of the cloud evolution, and only develop a distinct power-law tail with index $\alpha \simeq 2$ at greater column density when supercritical pockets are formed.

Anomalous kinetics in diffusion limited reactions linked to non-Gaussian concentration probability distribution function

We investigate anomalous reaction kinetics related to segregation in the one-dimensional reaction-diffusion system A + B → C. It is well known that spatial fluctuations in the species concentrations cause a breakdown of the mean-field behavior at low concentration values. The scaling of the average concentration with time changes from the mean-field t(-1) to the anomalous t(-1/4) behavior. Using a stochastic modeling approach, the reaction-diffusion system can be fully characterized by the multi-point probability distribution function (PDF) of the species concentrations. Its evolution is governed by a Fokker-Planck equation with moving boundaries, which are determined by the positivity of the species concentrations. The concentration PDF is in general non-Gaussian. As long as the concentration fluctuations are small compared to the mean, the PDF can be approximated by a Gaussian distribution. This behavior breaks down in the fluctuation dominated regime, for which anomalous reaction kinetics are observed. We show that the transition from mean field to anomalous reaction kinetics is intimately linked to the evolution of the concentration PDF from a Gaussian to non-Gaussian shape. This establishes a direct relationship between anomalous reaction kinetics, incomplete mixing and the non-Gaussian nature of the concentration PDF.

A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [15,14], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker–Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker–Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker–Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [15] to further simplify the kinetic theory.

CMP pad wear and polish-rate decay modeled by asperity population balance with fluid effect

An important task in chemical–mechanical polishing (CMP) is to determine when the pad should be changed or reconditioned. A model which can predict the pad asperity probability distribution function (PDF) during polishing and conditioning is valuable for this purpose. Previous work has been done without incorporating fluid mechanics into the model in L.J. Borucki, J. Eng. Math. 43 (2002) 105–114, but that will overestimate the pad wear because the fluid reduces the load applied on the individual asperities. This work models the wear of pad asperities and polish-rate decay in CMP by coupling the population balance model with fluid mechanics. Modeling results with and without fluid effect are compared. Polish-rate model results are compared with experimental data in D. Stein et al., J. Electron. Mater. 25 (10) (1996) 1623–1627, and the results agree with experimental results for both cases by using different wear rate coefficients to fit experimental data. A lesser wear rate coefficient must be used to fit Stein’s data for the fluid case compared to the case without fluid. The wear rate of the pad is calculated from the rate of change of the pad–wafer separation distance during polishing because only asperities above this distance will be in contact and worn down and that portion will be piled up at the pad–wafer separation distance on the PDF curve of the pad asperities. The PDF evolution model results with fluid show much less pad wear compared to the case without fluid. Also if fluid is neglected, pad–wafer separation distance will continue to drop until it is zero, while for the fluid case, it will eventually reach a dynamic steady-state with no pad wear indicating that the load is entirely supported by the fluid. Accurate predicting the interface gap (pad wear) is critical for controlling both mechanical and chemical removal rate and within wafer non-uniformity (WIWNU). These results show that fluid is an important factor which needs to be included into the model in predicting the interface gap and surface profile evolution of the pad during polishing and conditioning.

Is Turbulent Mixing a Self-Convolution Process?

Experimental results for the evolution of the probability distribution function (PDF) of a scalar mixed by a turbulent flow in a channel are presented. The sequence of PDF from an initial skewed distribution to a sharp Gaussian is found to be nonuniversal. The route toward homogeneization depends on the ratio between the cross sections of the dye injector and the channel. In connection with this observation, advantages, shortcomings, and applicability of models for the PDF evolution based on a self-convolution mechanism are discussed.

Dynamical stochastic processes of returns in financial markets

We study the evolution of probability distribution functions of returns, from the tick data of the Korean treasury bond (KTB) futures and the S&P 500 stock index, which can be described by means of the Fokker–Planck equation. We show that the Fokker–Planck equation and the Langevin equation from the estimated Kramers–Moyal coefficients can be estimated directly from the empirical data. By analyzing the statistics of the returns, we present quantitatively the deterministic and random influences on financial time series for both markets, for which we can give a simple physical interpretation. We particularly focus on the diffusion coefficient, which may be important for the creation of a portfolio.

Dynamics of inviscid truncated model of two-dimensional turbulent shear flow

The method of classical statistical mechanics is applied to an inviscid truncated model system of two-dimensional turbulent shear flow. The idea of canonical equilibrium distribution is extended to treat a time-dependent Liouville equation governing the evolution of probability distribution function in the phase space. Results of numerical simulations supporting the theoretical conjectures based on the canonical distribution are shown.