An analytical solution to the nonlinear evolutionary equations for nucleation and growth of particles

ABSTRACT

In this paper, we consider a mathematical model for the growing crystals in supersaturated or supercooled systems. An integro-differential equation of the Fokker-Planck type is analytically solved using the saddle point method for the Laplace integral. The solution describes an intermediate stage of the phase transition process with allowance for the power law of the growth rate of spherical particles and the nucleation mechanisms known as the Meirs and Weber-Volmer-Frenkel-Zeldovich kinetics. A dynamical dependencies for the supersaturation/supercooling and the distribution function of crystals by their size are obtained. The novelty of the theory is the use of a power law for the growth rate of crystals, which leads to new analytical solutions of the problem.     

Mathematical modelling of nucleation and growth of crystals with buoyancy effects

A complete analytical solution of the integro-differential model describing the nucleation of crystals and their subsequent growth in a binary system with allowance for buoyancy forces is constructed. An exact analytical solution of the Fokker-Planck-type equation for the three-parameter density distribution function is found for arbitrary nucleation kinetics. Two important cases of the Weber–Volmer–Frenkel–Zel’dovich and Meirs kinetics are considered in some detail. It is shown that the solute concentration decreases and the distribution function increases with increasing the melt supercooling (with increasing the depth of a metastable system). It is demonstrated that the distribution function attains its minimum at a certain size of crystals owing to buoyancy forces.     

Estimation of the Nonlinear Random Coefficient Model when Some Random Effects Are Separable

A method is presented for marginal maximum likelihood estimation of the nonlinear random coefficient model when the response function has some linear parameters. This is done by writing the marginal distribution of the repeated measures as a conditional distribution of the response given the nonlinear random effects. The resulting distribution then requires an integral equation that is of dimension equal to the number of nonlinear terms. For nonlinear functions that have linear coefficients, the improvement in computational speed and accuracy using the new algorithm can be dramatic. An illustration of the method with repeated measures data from a learning experiment is presented.     

Prediction in future samples studied in terms of the gain from selection

The gain from selection (GS) is defined as the standardized average performance of a group of subjects selected in a future sample using a regression equation derived on an earlier sample. Expressions for the expected value, density, and distribution function (DF) of GS are derived and studied in terms of sample size, number of predictors, and the prior distribution assigned to the population multiple correlation. The DF of GS is further used to determine how large sample sizes must be so that with probability .90 (.95), the expected GS will be within 90 percent of its maximum possible value. An approximately unbiased estimator of the expected GS is also derived.     

On the likelihood ratio test in structural equation modeling when parameters are subject to boundary constraints

The authors show how the use of inequality constraints on parameters in structural equation models may affect the distribution of the likelihood ratio test. Inequality constraints are implicitly used in the testing of commonly applied structural equation models, such as the common factor model, the autoregressive model, and the latent growth curve model, although this is not commonly acknowledged. Such constraints are the result of the null hypothesis in which the parameter value or values are placed on the boundary of the parameter space. For instance, this occurs in testing whether the variance of a growth parameter is significantly different from 0. It is shown that in these cases, the asymptotic distribution of the chi-square difference cannot be treated as that of a central chi-square-distributed random variable with degrees of freedom equal to the number of constraints. The correct distribution for testing 1 or a few parameters at a time is inferred for the 3 structural equation models mentioned above. Subsequently, the authors describe and illustrate the steps that one should take to obtain this distribution. An important message is that using the correct distribution may lead to appreciably greater statistical power.     

Structural Equation Modeling of Multivariate Time Series

The covariance structure of a vector autoregressive process with moving average residuals (VARMA) is derived. It differs from other available expressions for the covariance function of a stationary VARMA process and is compatible with current structural equation methodology. Structural equation modeling programs, such as LISREL, may therefore be employed to fit the model.

Particular attention is given to assumptions concerning the process before the first observation. An application to a repeated time series is used to demonstrate the effect of these assumptions on the structure of the reproduced covariance matrix.     

Maximum likelihood estimation in multidimensional scaling

A variety of distributional assumptions for dissimilarity judgments are considered, with the lognormal distribution being favored for most situations. An implicit equation is discussed for the maximum likelihood estimation of the configuration with or without individual weighting of dimensions. A technique for solving this equation is described and a number of examples offered to indicate its performance in practice. The estimation of a power transformation of dissimilarity is also considered. A number of likelihood ratio hypothesis tests are discussed and a small Monte Carlo experiment described to illustrate the behavior of the test of dimensionality in small samples.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.     

Contributions to the mathematical theory of human relations viii: Size distribution of cities

An attempt is made to connect the distribution function of the sizes of the cities with the distribution functions of some other characteristics of the individuals in the society. Several theoretical possibilities are discussed and different relations are derived. A possible connection with some observed relations is discussed.     

Effects of qualitatively different reinforcers on the parameters of the response-strength equation.

This experiment examined the relationship between two qualitatively different reinforcers and the parameters of a quantitative model of reinforced responding, referred to as the response-strength equation or the Herrnstein equation. A group of rats was first food deprived and later water deprived. An 11.5% sucrose solution served as the reinforcer in the food-deprivation condition, and water was the reinforcer in the water-deprivation condition. Each experimental session consisted of a series of seven variable-interval schedules, providing reinforcement rates that varied between 20 and 1,200 reinforcers per hour. The response rates increased in a negatively accelerating function in a manner consistent with the response-strength equation. This equation has two fitted parameters, k and Re. According to one theory, the k parameter is a measure of motor performance, and Re is indicative of the relative reinforcement efficacy of the background uncontrollable sources of reinforcement in relation to the experimentally arranged reinforcer. In this study, k did not change as a result of the different reinforcers, but Re was significantly larger in the sucrose-reinforcement condition. These results are consistent with the interpretation that k and Re measure two independent and experimentally distinguishable parameters and provide further evidence that absolute response rate is a function of relative reinforcement rate, as implied by the derivation of the response-strength equation based on the matching law.     

The race model inequality: interpreting a geometric measure of the amount of violation

An inequality by J. O. Miller (1982) has become the standard tool to test the race model for redundant signals reaction times (RTs), as an alternative to a neural summation mechanism. It stipulates that the RT distribution function to redundant stimuli is never larger than the sum of the distribution functions for 2 single stimuli. When many different experimental conditions are to be compared, a numerical index of violation is very desirable. Widespread practice is to take a certain area with contours defined by the distribution functions for single and redundant stimuli. Here this area is shown to equal the difference between 2 mean RT values. This result provides an intuitive interpretation of the index and makes it amenable to simple statistical testing. An extension of this approach to 3 redundant signals is presented.     

Estimating averages from distributions of tone durations

We examined whether estimating average duration was influenced by the distribution peak location. We presented participants with samples of various tone durations and then presented comparison tone durations. Participants judged whether each comparison duration was longer than the average sample duration. Estimates of the averages were inferred from the psychophysical functions. The durations were sampled from three distributions: one positively skewed, one symmetric, and one negatively skewed. In Experiment 1, every participant was presented with every distribution. Estimates of the averages were unbiased for the symmetric distribution but were biased toward the long tail of each skewed distribution. This would occur if participants combined the sample to be judged with the previous, irrelevant samples, or with the comparison durations. In Experiment 2, each participant was presented with samples from only one of the distributions. Estimates of the averages were still biased toward the long tails of the skewed distributions. This would occur if participants combined the sample to be judged with the comparison durations, which were the same for the three distributions. In Experiment 3, each participant was presented with only one distribution, and each distribution was tested with its own comparison durations, selected as percentiles from the distribution. The estimates were accurate for the smallest population mean (positively skewed distribution) but underestimated the larger means. These results could be explained by subjective shortening of the durations in memory, with a simple equation from scalar timing theory. This equation correctly predicted two results: The estimated averages were a linear function of the stimulus means, and the variances were a linear function of the squared stimulus means. Neither prediction was dependent on the skewness of the stimulus durations.     

Examining mixed unknown distributions (mud)

A function, written in R, for testing whether the distribution of responses in one condition can be considered a combination of the distributions from two other conditions is described. The important aspect of this function is that it does not make any assumptions about the shape of the distributions. It is based o nthe Kolmogorov-Smirnov D statistic. The function also allows the user to test more specific and, hence, more statistically powerful hypotheses. One hypothesis, that the mixture does not capture the middle third of the distribution, is included as a built-in option, and code is provided so that other alternatives can easily be run. A power analysis reveals that the function is most likely to detect a difference between the combined conditions' distribution and the other distribution when the center of the other distribution is near the midpoint of the two original distributions. Critical p values are estimated for each set of distributions, using bootstrap methods. An example from human memory research, exploring the blending hypothesis of the misinformation effect, is used for illustrative purposes.     

Initial sintering of cylindrical particles of different sizes by surface diffusion

Sintering of two cylindrical particles of different sizes by surface diffusion at the early stage is theoretically analysed. Formulae for the neck growth rate are derived using a variational principle and the continuity equation of vacancies. The size difference between two particles is found to influence the growth rate of the neck only when the difference is large.     

Maximum Likelihood Analysis of Nonlinear Structural Equation Models With Dichotomous Variables

In this article, a maximum likelihood approach is developed to analyze structural equation models with dichotomous variables that are common in behavioral, psychological and social research. To assess nonlinear causal effects among the latent variables, the structural equation in the model is defined by a nonlinear function. The basic idea of the development is to augment the observed dichotomous data with the hypothetical missing data that involve the latent underlying continuous measurements and the latent variables in the model. An EM algorithm is implemented. The conditional expectation in the E-step is approximated via observations simulated from the appropriate conditional distributions by a Metropolis-Hastings algorithm within the Gibbs sampler, whilst the M-step is completed by conditional maximization. Convergence is monitored by bridge sampling. Standard errors are also obtained. Results from a simulation study and a real example are presented to illustrate the methodology.     

Does the ADF fit function decrease when the kurtosis increases?

In this study we demonstrate how the asymptotically distribution‐free (ADF) fit function is affected by (excessive) kurtosis in the observed data. More specifically, we address how different levels of univariate kurtosis affect fit values (and therefore fit indices) for misspecified factor models. By using numerical calculation, we show (for 13 factor models) that the probability limit F₀ of F for the ADF fit function decreases considerably as the kurtosis increases. We also give a formal proof that the value of F₀ decreases monotonically with the kurtosis for a whole class of structural equation models.     

Variation of the standard error of measurement

As usually interpreted, the standard error of measurement is assumed to be constant throughout the test-score range. In this investigation the standard error of measurement was assumed to be not higher than a second-degree function of the test score. By conceiving a test score to be made up of the scores on two parallel tests, an equation was derived for predicting the standard error of measurement from the test score. In the derivation the corresponding first four moments of the score distributions for the parallel tests were assumed to be identical, and certain errors of estimate involved in predicting the second test score from the first were assumed to be uncorrelated with powers of the score on the first test. An empirical verification was carried out, using nine synthetic tests and a 1000-case sample, and showed good agreement between predicted and observed results. The findings indicated that the standard error of measurement was constant only for a symmetrical, mesokurtic distribution of scores.This study was carried out while the author was a National Research Council Predoctoral Fellow in psychology at Princeton University.The author wishes to express his appreciation for the guidance given by his thesis adviser, Professor Harold Gulliksen. He wishes also to acknowledge his gratitude to the Educational Testing Service for extensive assistance in the empirical phase of the study, and to Dr. Ledyard Tucker for suggesting efficient methods of handling special computational problems.     

DELAY‐AMOUNT TRADEOFFS IN CHOICES BY PIGEONS AND RATS: HYPERBOLIC VERSUS EXPONENTIAL DISCOUNTING

An adjusting‐delay procedure was used to study the choices of pigeons and rats when both delay and amount of reinforcement were varied. In different conditions, the choice alternatives included one versus two reinforcers, one versus three reinforcers, and three versus two reinforcers. The delay to one alternative (the standard alternative) was kept constant in a condition, and the delay to the other (the adjusting alternative) was increased or decreased many times a session so as to estimate an indifference point—a delay at which the two alternatives were chosen about equally often. Indifference functions were constructed by plotting the adjusting delay as a function of the standard delay for each pair of reinforcer amounts. The experiments were designed to test the prediction of a hyperbolic decay equation that the slopes of the indifference functions should increase as the ratio of the two reinforcer amounts increased. Consistent with the hyperbolic equation, the slopes of the indifference functions depended on the ratios of the two reinforcer amounts for both pigeons and rats. These results were not compatible with an exponential decay equation, which predicts slopes of 1 regardless of the reinforcer amounts. Combined with other data, these findings provide further evidence that delay discounting is well described by a hyperbolic equation for both species, but not by an exponential equation. Quantitative differences in the y‐intercepts of the indifference functions from the two species suggested that the rate at which reinforcer strength decreases with increasing delay may be four or five times slower for rats than for pigeons.     

Multiple bi-serial and multiple point bi-serial correlation

Normal equations, using data in various forms, are presented for securing the regression weights for prediction of a dichotomized criterion, and a simplified equation for the estimation of the multiple bi-serial or multiple point bi-serial, depending upon the proper assumption as to the nature of the distribution of the criterion, on the basis of these maximal weights is given also. The weights, unaffected by the assumption as to the nature of the criterion, are identical (or proportional) to those found by the discriminant function approach based upon analysis of variance. The author holds that the present multiple correlation approach is both easier and more informative than the discriminant function (analysis of variance) approach and suggests that the discriminant function be abandoned in favor of multiple bi-serial and/or multiple point bi-serial correlation and regression.     

On the consistency of intramodal intensity matching in olfaction

Magnitude matchings of odor intensity were obtained for three chemical compounds (hydrogen sulfide, pyridine, and dimethyl disulfide) in four experiments. An equal-sensation function in the form of a power function described the data well. This equation can be written neglecting the multiplicative constant, Ф_i = Ф_k ^b _ik, where i and k represent two different odorants. The exponent b_ik, where the first index stands for matching continuum and the second index for reference continuum, was fairly well predicted according to the formulae, (ie, 264, 2) where i, j, and k are three different odorants. The results validate the equal-sensation function and indicate implicitly that intensity matchings between different intramodal continua are approximately transitive and symmetric with respect to odor.     

Attention control and the antisaccade task: a response time distribution analysis

In three experiments response time (RT) differences between correct prosaccade and antisaccade trials were examined via distribution analyses by fitting an ex-Gaussian function to individual RT distributions. Experiment 1 demonstrated that antisaccades are slower than prosaccades and this difference is due to both a shift in the overall distribution as well as a lengthening of the tail of the distribution. Experiment 2 demonstrated that manipulating foreperiod duration led to changes in both accuracy and RT for antisaccades but not prosaccades. Furthermore, the change in RT for antisaccades resulted in a lengthening in the tail of the distribution but not a shift in the distribution. Finally, Experiment 3 demonstrated that with sufficient practice performance on antisaccades was equated with performance on prosaccades in terms of both accuracy and RT. An examination of the RT distributions suggested that practice led to parallel changes in both the mean of the distribution and the tail of the antisaccade distribution. These results are interpreted within a two-factor theory of attention control that suggests that performance on antisaccades is driven by both competition resolution and goal-maintenance abilities.