Some new aspects of dendrimer applications

Dendrimers are characterized by special features that make them promising candidates for many applications. Here we focus on two such applications: dendrimers as light harvesting antennae, and dendrimers as molecular amplifiers, which may serve as novel platforms for drug delivery. Both applications stem from the unique structure of dendrimers. We present a theoretical framework based on the master equation within which we describe these applications. The quantities of interest are the first passage time (FPT), probability density function (PDF) and its moments. We examine how the FPT PDF and its characteristics depend on the geometric and energetic structures of the dendrimeric system. In particular, we investigate the dependence of the FPT properties on the number of generations (dendrimer size) and the system bias. We present analytical expressions for the FPT PDF for very efficient dendrimeric antennae and for dendrimeric amplifiers. For these cases the mean FPT scales linearly with the system length, and fluctuations around the mean FPT are negligible for large systems. Relationships of the FPT to light harvesting process for other types of system-bias are discussed.     

Evolution of the Probability Measure for the Majda Model: New Invariant Measures and Breathing PDFs

In 1993, Majda proposed a simple, random shear model from which scalar intermittency was rigorously predicted for the invariant probability measure of passive tracers. In this work, we present an integral formulation for the tracer measure, which leads to a new, comprehensive study on its temporal evolution based on Monte Carlo simulation and direct numerical integration. An interesting, non-monotonic “breathing” phenomenon is discovered from these results and carefully defined, with a solid example for special initial data to predict such phenomenon. The signature of this phenomenon may persist at long time, characterized by the approach of the PDF core to its infinite time, invariant value. We find that this approach may be strongly dependent on the non-dimensional Péclet number, of which the invariant measure itself is independent. Further, the “breathing” PDF is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit. Rigorous asymptotic analysis is also performed to identify the Gaussian core of the invariant measures, and the critical rate at which the heavy, stretched exponential regime propagates towards the tail as a function of time is calculated.     

Turbulent dynamics of polymer solutions

We study properties of dilute polymer solutions. The probability density function (PDF) of polymer end-to-end extensions R in turbulent flows is examined. We show that if the value of the Lyapunov exponent lambda is smaller than the inverse molecular relaxation time 1/tau then the PDF has a strong peak at the equilibrium size R0 and a power tail at R>R0. This confirms and extends the results of J. L. Lumley [Symp. Math. 9, 315 (1972)]. There is no essential influence of polymers on the flow in this regime. At lambdatau>1 the majority of molecules is stretched to the linear size R(op)>R0, which can be much smaller than the maximal length of the molecules due to their back reaction.     

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 Péclet 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 Péclet), to two algebraic singularities at both unit and zero scalar values (large Péclet).     

Statistics of tumbling of a single polymer molecule in shear flow

We present experimental results on statistics of polymer orientation angles relative to the shear plane and tumbling times in shear flow with thermal noise. The strong deviation of the probability distribution functions (PDFs) of the orientation angles from Gaussian PDFs was observed in good accord with theory. A universal exponential PDF tail for the tumbling times and its predicted scaling with Wi (that is, the dimensionless shear rate normalized by the polymer relaxation time) are also tested experimentally against numerics. The scaling relations of PDF widths for both angles as a function of Wi are verified and compared with numerics.     

Intermittency in drift-wave turbulence: structure of the momentum flux probability distribution function

We analytically compute the probability distribution function (PDF) of the local Reynolds stress ( R) for forced Hasegawa-Mima turbulence. With the assumption that the PDF tail is due to an instanton with the spatial form given by the modon solution, the tail of the PDF of R is found to be a stretched, non-Gaussian exponential, with the specific form exp[-cR(3/2)] ( c is a constant). We relate the temporal localization of the instanton to the degree of "burstiness" of the momentum transport event.     

Photon time-interval statistics applied to the analysis of laser heterodyne signal with photon counter

In this paper, we report a mathematical derivation of probability density function (PDF) of time-interval between two successive photoelectrons of the laser heterodyne signal, and give a confirmation of the theoretical result by both numerical simulation and an experiment. The PDF curve of the beat signal displays a series of fluctuations, the period and amplitude of which are respectively determined by the beat frequency and the mixing efficiency. The beat frequency is derived from the frequency of fluctuations accordingly when the PDF curve is measured. This frequency measurement method still works while the traditional Fast Fourier Transform (FFT) algorithm hardly derives the correct peak value of the beat frequency in the condition that we detect 80 MHz beat signal with 8 Mcps (counts per-second) photons count rate, and this indicates an advantage of the PDF method.     

Computationally-efficient and accurate particle PDF simulations of turbulent combustion using coupled pre-partitioned adaptive chemistry and tabulation

LES/PDF methods are known to provide accurate results for challenging turbulent combustion configurations with strong turbulence-chemistry interactions. These methods are generally applicable as they do not make any assumptions on the topology of the underlying flame structure. However, this added generality comes at an increased computational cost. To mitigate this added cost, the majority of the LES/PDF computations performed to date utilize reduced mechanisms. We recently presented a coupled pre-partitioned adaptive chemistry (PPAC) and tabulation (ISAT) methodology (Newale et al., Comb. Th. Mod., 2019), which retains the fidelity of the detailed mechanism, while keeping the computational cost affordable. This methodology was tested in a partially-stirred reactor configuration. In this work, we describe the developments required for a holistic integration of PPAC-ISAT with a LES/PDF framework. We examine the performance of this coupled methodology in two LES/PDF configurations of Sandia flame D. A smaller simulation domain is initially utilized to characterize the efficiency and accuracy of standalone PPAC and coupled PPAC-ISAT in detail. Then, the performance of PPAC-ISAT is examined in a full-scale LES/PDF simulation. We show that the coupled PPAC-ISAT LES/PDF captures the resolved mean and RMS profiles of temperature and major species mass fractions to within 2% and OH to within 5%, with a reduction in the average simulation wall clock time per time step of 39% over an ISAT implementation using the detailed mechanism.     

The probabilistic solution of stochastic oscillators with even nonlinearity under poisson excitation

The probabilistic solutions of nonlinear stochastic oscillators with even nonlinearity driven by Poisson white noise are investigated in this paper. The stationary probability density function (PDF) of the oscillator responses governed by the reduced Fokker-Planck-Kolmogorov equation is obtained with exponentialpolynomial closure (EPC) method. Different types of nonlinear oscillators are considered. Monte Carlo simulation is conducted to examine the effectiveness and accuracy of the EPC method in this case. It is found that the PDF solutions obtained with EPC agree well with those obtained with Monte Carlo simulation, especially in the tail regions of the PDFs of oscillator responses. Numerical analysis shows that the mean of displacement is nonzero and the PDF of displacement is nonsymmetric about its mean when there is even nonlinearity in displacement in the oscillator. Numerical analysis further shows that the mean of velocity always equals zero and the PDF of velocity is symmetrically distributed about its mean.     

水平管段塞流压力/压差波动特性分析

本文在对水平管段塞流压力波动特性进行理论分析的基础之上,对其压力/压差波动特性进行了试验研究。结果表明,折算液速对段塞频率的影响作用远大于折算气速的影响作用;分别增加管线中的气量、液量,或者气液量同时增加,均会造成管线运行过程中均值压力/压差和最大压力/压差的增加;压力信号的概率密度分布大部分呈双峰分布,但其中也存在单峰和多峰分布;压差信号的概率密度符合正态分布;压力信号的功率谱密度具有频率波动范围窄、幅值大的特点;与同工况压力信号的功率谱密度相比,压差信号的功率谱密度具有频率波动范围宽、幅值小的特点。     

Steady-state probability density function in wave turbulence under large volume limit

We investigate the possibility for two-mode probability density function(PDF) to have a non-zero flux steady state solution.We take the large volume limit so that the space of modes becomes continuous.It is shown that in this limit all the steady-state two-or higher-mode PDFs are the product of one-mode PDFs.The flux of this steady-state solution turns out to be zero for any finite mode PDF.     

Asymptotic solution of the Wang-Uhlenbeck recurrence time problem

A Langevin particle is initiated at the origin with positive velocity. Its trajectory is terminated when it returns to the origin. In 1945, Wang and Uhlenbeck posed the problem of finding the joint probability density function (PDF) of the recurrence time and velocity, naming it "the recurrence time problem". We show that the short-time asymptotics of the recurrence PDF is similar to that of the integrated Brownian motion, solved in 1963 by McKean. We recover the long-time t(-3/2) decay of the first arrival PDF of diffusion by solving asymptotically an appropriate variant of McKean's integral equation.     

Nonergodic Brownian Motion

The long-time behavior of a system is suggested to confirm nonergodicity of non-Markovian Brownian dynamics, namely, whether the stationary probability density function （PDF） of the system characterized mainly by low moments of variables depends on the initial preparation. Thus we classify nonergodic Brownian motion into two classes： the class-I is that the PDF of a force-free particle depends on the initial velocity and the equilibration can be recovered through a bounded potential; while the PDF in the class-H depends on the initial coordinate and the equilibration can not be approached by introducing any potential. We also compare our result with the conditions of three kinds for ergodicity.     

Large eddy simulation of turbulent premixed combustion using tabulated detailed chemistry and presumed probability density function

A method of chemistry tabulation combined with presumed probability density function (PDF) is applied to simulate piloted premixed jet burner flames with high Karlovitz number using large eddy simulation. Thermo-chemistry states are tabulated by the combination of auto-ignition and extended auto-ignition model. To evaluate the predictive capability of the proposed tabulation method to represent the thermo-chemistry states under the condition of different fresh gases temperature, a-priori study is conducted by performing idealised transient one-dimensional premixed flame simulations. Presumed PDF is used to involve the interaction of turbulence and flame with beta PDF to model the reaction progress variable distribution. Two presumed PDF models, Dirichlet distribution and independent beta distribution, respectively, are applied for representing the interaction between two mixture fractions that are associated with three inlet streams. Comparisons of statistical results show that two presumed PDF models for the two mixture fractions are both capable of predicting temperature and major species profiles, however, they are shown to have a significant effect on the predictions for intermediate species. An analysis of the thermo-chemical state-space representation of the sub-grid scale (SGS) combustion model is performed by comparing correlations between the carbon monoxide mass fraction and temperature. The SGS combustion model based on the proposed chemistry tabulation can reasonably capture the peak value and change trend of intermediate species. Aspects regarding model extensions to adequately predict the peak location of intermediate species are discussed.     

Pair distribution function analysis:Fundamentals and application to battery materials

Battery materials are of vital importance in powering a elean and sustainable society.Improving their performance relies on a clear and fundamental understanding of their properties,in particular,structural properties.Pair distribution function(PDF) analysis,which takes into account both Bragg scattering and diffuse scattering,can probe structures of both crystalline and amorphous phases in battery materials.This review first introduces the principle of PDF,followed by its application in battery materials.It shows that PDF is an effective tool in studying a series of key scientific topics in battery materials.They range from local ordering,nano-phase quantification,anion redox reaction,to lithium storage mechanism,and so on.     

Gaussian models for the distribution of Brownian particles in tilted periodic potentials The limit of high barriers

We present two Gaussian approximations for the time-dependent probability density function (PDF) of an overdamped Brownian particle moving in a tilted periodic potential. We assume high potential barriers in comparison with the noise intensity. The accuracy of the proposed approximated expressions for the time-dependent PDF is checked with numerical simulations of the Langevin dynamics. We found a quite good agreement between theoretical and numerical results at all times.     

小长径比垂直管气液两相流动特性分析

实验观察了小长径比垂直上流管内流型及特点,并对管入口处的压力波动特性和系统的压差波动特性进行了试验研究.结果表明:小长径比(L/D)垂直管内流型表现为泡状流、塞状流、乳沫状流、环状流和液束环状流;分别增加管线中的气量、液量,或者同时增加气液流量,均会造成垂直管入口处压力波动的均值和最大压力的增加;压力信号的概率密度(PDF)大部分呈双峰分布,也存在单峰和多峰分布;差压信号的概率密度符合正态分布,其功率谱密度同压力信号相比具有频率波动范围宽、幅值小的特点.     

Curvature of lagrangian trajectories in turbulence

We report measurements of the curvature of Lagrangian trajectories in an intensely turbulent laboratory water flow measured with a high-speed particle-tracking system. The probability density function (PDF) of the instantaneous curvature is shown to have robust power-law tails. We propose a model for the instantaneous curvature PDF, assuming that the acceleration and velocity are uncorrelated Gaussian random variables, and show that our model reproduces the tails of our measured PDFs. We also predict the scaling of the most probable vorticity magnitude in turbulence, assuming Heisenberg-Yaglom scaling. Finally, we average the curvature along trajectories and show that, by removing the effects of large-scale flow reversals, the filtered curvature reveals the turbulent features.     

Entanglement of two-qubit photon beam by magnetic field

We show how the photon input parton distribution function (PDF) may be calculated with good accuracy and used in an extended DGLAP global parton analysis in which the photon is treated as an additional point-like parton. The uncertainty of the input photon PDF is relatively small, since the major part of the distribution (which is produced by the coherent emission of the photon from a proton that remains intact) is well known. We present the expected photon PDFs and compare the predictions with ZEUS data for isolated photon electroproduction at negative rapidities.