Evolution Of Probability Density Function Research Articles

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Peclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Peclet), to two algebraic singularities at both unit and zero scalar values (large Peclet).

Universal properties of two-port scattering, impedance, and admittance matrices of wave-chaotic systems

Statistical fluctuations in the eigenvalues of the scattering, impedance, and admittance matrices of two-port wave-chaotic systems are studied experimentally using a chaotic microwave cavity. These fluctuations are universal in that their properties are dependent only upon the degree of loss in the cavity. We remove the direct processes introduced by the nonideally coupled driving ports through a matrix normalization process that involves the radiation-impedance matrix of the two driving ports. We find good agreement between the experimentally obtained marginal probability density functions (PDFs) of the eigenvalues of the normalized impedance, admittance, and scattering matrix and those from random matrix theory (RMT). We also experimentally study the evolution of the joint PDF of the eigenphases of the normalized scattering matrix as a function of loss. Experimental agreement with the theory by Brouwer and Beenakker for the joint PDF of the magnitude of the eigenvalues of the normalized scattering matrix is also shown.

Evolving densities in continuous strategy games through particle simulations

Many cases of strategic interaction between agents involve a continuous set of choices. It is natural to model these problems as continuous space games. Consequently, the population of agents playing the game will be represented with a density function defined over the continuous set of strategy choices. Simulating evolutionary dynamics on continuous strategy spaces is a challenging problem. The classic approach of discretizing the strategy space is ineffective for multidimensional strategy spaces. We present a principled approach to simulation of adaptive dynamics in continuous space games using sequential Monte Carlo methods. Sequential Monte Carlo methods use a set of weighted random samples, also named particles to represent density functions over multidimensional spaces. Sequential Monte Carlo methods provide computationally efficient ways of computing the evolution of probability density functions. We employ resampling and smoothing steps to prevent particle degeneration problem associated with particle estimates. The resulting algorithm can be interpreted as an agent based simulation with elements of natural selection, regression to mean and mutation. We illustrate the performance of the proposed simulation technique using two examples: continuous version of the repeated prisoner dilemma game and evolution of bidding functions in first-price closed-bid auctions.

Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.

Stochastic Models for Prodrug Targeting. 1. Diffusion of the Efflux Drug

In modeling prodrug targeting using the stochastic approach, we first modeled diffusion of the efflux drug. Drug efflux is one of the major reasons for the failure of prodrug strategy: the active agent is pumped out the membrane ("efflux"), causing an insufficient amount to be delivered to the targeted sites and thus diminishing the efficacy of chemotherapy. Because the biological body is a nonlinear nonequilibrium complex system, the molecular transport taking place in vivo often showed stochasticity. The model described here for diffusion of the efflux drug is basically a diffusion process with reflecting/absorbing boundary conditions, divided into two distinctive regions with one allowing the particles to jump to the origin as a result of efflux pumping. We study discrete time birth-death Markov chain and compute the time-dependent spatial probability density function (PDF) of particles. The results showed that the jumping probability, although small, has a significant impact on the evolution of PDF of the efflux drug. The implications of this model were discussed.

USING PATH INTEGRAL SHORT TIME PROPAGATORS FOR NUMERICAL ANALYSIS OF STOCHASTIC HYBRID SYSTEMS

Algorithms to approximate the evolution of probability density functions for stochastic hybrid systems rely on the knowledge of appropriate short time propagators. It is shown that a path integral propagator known for continuous stochastic systems can be adapted to the hybrid case. With this propagator, the HybPathTree algorithm performs well concerning precision and computational effort, e.g. in reachability analysis.

Modeling of Turbulent Diffusion Combustion Using a Combined Probability Density Function/Moment Method

A combined probability density function (PDF) /moment method has been developed to improve the accuracy of the Monte-Carlo method which is employed to simulate the evolution of PDFs in the PDF approach. The accuracy of Monte-Carlo method depends strongly on the number of Monte-Carlo particles and declines with the decrease in the PDF transport probability. The present method solves simultaneously a modeled PDF transport equation and moment equations of mean and variance. Both the mean and variance values obtained from the moment equations are just used as the reference values to match to the corresponding values of PDFs. In the frame of the conserved scalar concept, the usefulness of the present method is evaluated in the configuration of a turbulent jet diffusion flame, with a comparison with other methods; the PDF method and the flamelet model method. The results have showed good agreement between the present PDF method and the flamelet method, highlighting the improvement from the PDF method. Especially, the agreement in terms of PDF shape has verified that the beta function approximation in the flamelet model has high accuracy.

Modeling return to isotropy using kinetic equations

Kinetic equations modeling the behavior of the velocity probability density function (PDF) in homogeneous anisotropic decaying turbulence are hypothesized and their implications for return-to-isotropy are investigated. Anisotropic turbulent decay is a parametrically simple but theoretically complex turbulent flow that is dominated by nonlinear interactions. The physical implications of the Bhatnagar–Gross–Krook model, a relaxation model, and the Fokker–Planck model for the “collision” term in the PDF evolution equation are analyzed in detail. Using fairly general assumptions about the physics, three different parameter-free return-to-isotropy models are proposed. These models are compared with experimental data, classical models, and analytical limits. The final model expression is particularly interesting, and can easily be implemented in classic Reynolds stress transport models.

Joint-scalar transported PDF modeling of soot formation and oxidation

The ability of the transported probability density function (PDF) approach to reproduce the evolution of mean, rms fluctuations, and conditional PDFs of soot is explored in the context of two turbulent ethylene diffusion flames at Reynolds numbers of 11,800 and 15,600. The chemical similarity between surface reactions and PAH formation is explored on the basis of a second ring PAH analogy, and soot oxidation is accounted for through reactions with O, OH, and O 2. The method of moments is used to account for coagulation and agglomeration in the coalescent and fractal aggregate limits. The soot model is coupled with a transported PDF approach closed at the joint-scalar level to directly account for interactions between turbulence, and the solid and gas phase chemistry. The latter is represented by a systematically reduced reaction mechanism for ethylene featuring 144 reactions, 15 solved and 14 steady-state species. Radiation from soot and gas phase species is accounted for through the RADCAL method and the inclusion of enthalpy into the joint-scalar PDF. Predicted temperature and soot statistics compare well with experimental data indicating the practical potential of the approach and the importance of turbulence-chemistry interactions in the context of soot formation and burnout.

Critical Scale Invariance in a Healthy Human Heart Rate

We demonstrate the robust scale-invariance in the probability density function (PDF) of detrended healthy human heart rate increments, which is preserved not only in a quiescent condition, but also in a dynamic state where the mean level of the heart rate is dramatically changing. This scale-independent and fractal structure is markedly different from the scale-dependent PDF evolution observed in a turbulentlike, cascade heart rate model. These results strongly support the view that a healthy human heart rate is controlled to converge continually to a critical state.

Supersensitivity due to uncertain boundary conditions

AbstractWe study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi‐polynomial‐chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar ‘stochastic supersensitivity’ for the mean location of the transition layer. Subsequently, we employ up to fourth‐order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a ‘truncated’ Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied. Copyright © 2004 John Wiley & Sons, Ltd.

PDF modelling of turbulent non-premixed combustion with detailed chemistry

Although the significant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions in turbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practical calculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consisting of 53 species and 325 elemental reactions) is firstly incorporated into the PDF calculation of a turbulent non-premixed jet flame (Sandia Flame D). The flow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale (MTS) k– ε turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled by the Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm for turbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms are integrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial profiles of mean and rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The flame structures are well represented by the present calculation, including intermediate species (e.g. CO and H 2) mass fractions, pollutant NO emission and local extinction.

A unified pdf-flamelet model for turbulent premixed combustion

The calculation of turbulent premixed flames in the framework of the Probability Density Function (PDF) approach remains quite a challenging modeling problem because of the strong coupling that exists between micromixing and reaction in premixed flamelets. In the flamelet regime of turbulent premixed combustion, instantaneous progress variable gradients are essentially fixed by these propagating flamelets. Consequently, the PDF evolution due to the micromixing term cannot be handled without taking into account the reactive contribution itself. In this study, a generalization of the work of Pope and Anand (1984; Flamelet and Distributed Combustion in Premixed Turbulent Flames, Proc. Combust. Inst. , vol. 20, pp. 403-410) is put forward to take into account the consequence of flamelet structure, but only when and where this structure is not perturbed by turbulence. In this manner, a criterion based on the ratio of the Kolmogorov length scale m K to laminar flame width i L allows turbulent premixed combustion to be described with a unique model from flamelet to distributed combustion regime. The new unified model is applied to the propagation of one-dimensional turbulent premixed flame in prescribed turbulent flow conditions and to the case of a high-velocity piloted Bunsen burner.

Assumed PDF turbulence-chemistry closure with temperature-composition correlations

An assumed probability density function (PDF) approach that accounts for temperature and composition fluctuations, including effects of temperature-composition correlations, has been developed. Parametric studies were performed to determine the gross behavior of the assumed PDF and provide insight into the expected trends of the model for a given kinetic mechanism. The model was implemented into an existing Navier-Stokes solver and applied to a supersonic, co-axial H 2/air burner. Proper prediction of the turbulence effects on the chain initiating step of the kinetic process is a first-order concern in supersonic combustion applications because of small flow residence times. Temperature-composition correlations were extracted and compared with results obtained by direct integration of a PDF evolution equation. The results of this comparison showed that the model properly predicted both the magnitude and sign of the cross-correlation terms associated with the chain initiating step of the H 2/air kinetic mechanism. Correlations for all but one of the remaining kinetic steps qualitatively reproduced those obtained by direct integration.

Binomial mixing model for premixed reacting turbulent flows applied to autoignition

A turbulent mixing model based on a stochastic process using a binomially distributed random variable is presented and tested in a premixed decaying turbulent flow. Attention is focused on the joint probability density function (PDF) of reactive and inert scalar that are calculated by the PDF transport equation. The numerical work under study is applied to a field similar to autoignition in which the initial medium is composed of partially burnt gas distributed within the fresh mixture. Three dimensional direct numerical simulation has been performed in order to check the prediction capability of the model whose results showed an excellent agreement with the data from numerical experiments. The joint PDF distribution calculated by the binomial model demonstrates that binomial sampling process successfully describes the realistic features of the PDF evolution.

Analysis of a Nonlinear System Exhibiting Chaotic, Noisy Chaotic, and Random Behaviors

This study presents a stochastic approach for the analysis of nonchaotic, chaotic, random and nonchaotic, random and chaotic, and random dynamics of a nonlinear system. The analysis utilizes a Markov process approximation, direct numerical simulations, and a generalized stochastic Melnikov process. The Fokker-Planck equation along with a path integral solution procedure are developed and implemented to illustrate the evolution of probability density functions. Numerical integration is employed to simulate the noise effects on nonlinear responses. In regard to the presence of additive ideal white noise, the generalized stochastic Melnikov process is developed to identify the boundary for noisy chaos. A mathematical representation encompassing all possible dynamical responses is provided. Numerical results indicate that noisy chaos is a possible intermediate state between deterministic and random dynamics. A global picture of the system behavior is demonstrated via the transition of probability density function over its entire evolution. It is observed that the presence of external noise has significant effects over the transition between different response states and between co-existing attractors.

Assumed and evolution probability density functions in supersonic turbulent combustion calculations

The objective of this investigation is to compare the use of assumed probability density function (PDF) approaches for modeling supersonic turbulent reacting flowfields with the more elaborate approach where the PDF evolution equation is solved. Previous calculations using assumed PDFs have shown modest improvements in the prediction of mean flow variables when compared with experimental data. However, these PDFs were unable to predict the higher order correlations with any reasonable degree of accuracy. Solving the evolution equation for the PDF did show slight improvements in these correlations when compared with experiment, but at a substantial cost in computer storage and time. Both approaches yielded comparable mean flow quantities.

A closed probability density function evolution equation for stochastic diffusion

A closed probability density function (PDF) evolution equation is obtained for a homogeneous scalar field undergoing stochastic diffusion. The closure assumes that the scalar field is made up of an ensemble of one-dimensional periodic waves.

Probability density function approach for compressible turbulent reacting flows

The objective of the present work is to extend the probability density function (PDF) tubulence model to compressible reacting flows. The proability density function of the species mass fractions and enthalpy are obtained by solving a PDF evolution equation using a Monte Carlo scheme. The PDF solution procedure is coupled with a compression finite-volume flow solver which provides the velocity and pressure fields. A modeled PDF equation for compressible flows, capable of treating flows with shock waves and suitable to the present coupling scheme, is proposed and tested. Convergence of the combined finite-volume Monte Carlo solution procedure is discussed. Two super sonic diffusion flames are studied using the proposed PDF model and the results are compared with experimental data; marked improvements over solutions without PDF are observed.

Total linearization of probability density evolution equations

Total linearization of the Boltzmann and Vlasov equations is used to present a technique applicable to equations which have polynomial-type nonlinearities and describe the evolution of probability density functions or distributions. It is an extension of an earlier result, where a method yielding a linear equation with a nonlinear constraint was presented.