On the Capability of the Generalized Gamma Function to Represent Spray Drop‐Size Distribution

The three‐parameter, Generalized Gamma function solution of a recent MEF formulation used to derive liquid spray drop‐size distribution, is applied to sprays resulting from three different atomization processes. The objectives of these applications are to determine the sign of the parameters for which this function reports a more reliable fit and to further understand the parameter stability problem reported elsewhere. It is found that the lack of stability of the parameters is related to a characteristic feature of the mathematical function and appears for a series of spray drop‐size distributions with constant shape. For each situation analyzed in the present study, the Generalized Gamma function provides a very good fit with parameters that are either constant or correlated to the working conditions. As far as the sign of the parameters is concerned, the results show that the best formulation is a function of the spray and that it is impossible to know, a priori, which parameter sign will report the best fit. Finally, for one situation, it is found that the Generalized Gamma function allows extrapolation of drop sizes outside the measured values. All of the results converge to conclude that the three‐parameter Generalized Gamma function, which is identical to the well‐known Nukiyama‐Tanasawa distribution, accumulates valuable attributes to represent liquid spray drop‐size distributions.     

Application of the Maximum Entropy Formalism on Sprays Produced by Ultrasonic Atomizers

A recent application of the Maximum Entropy Formalism on liquid atomization problems led to the development of a mathematical volume‐based drop‐size distribution. This function, which depends on three parameters, is a reduction of the four‐parameter generalized Gamma function. The aim of the present work is to investigate the relevance of the three parameters in the characterization of liquid atomization processes. To achieve this, a variety of experimental drop‐size distributions of ultrasonic sprays were analyzed with the mathematical function. Firstly, it is found that the mathematical drop‐size distribution is very suitable to represent the volume‐based drop‐size distribution of ultrasonic sprays. Furthermore, it is seen that when considering the three parameters introduced by the function, one of them is constant for all the situations investigated, and the other two are linked to a non‐dimensional group that includes the main parameters controlling the drop production. These results are very important, since they suggest a possible development of physical models of primary atomization based on the M.E.F., which would allow for the prediction of the spray drop‐size distribution. Thusfar, such a model does not exist.     

Use of the Maximum Entropy Formalism to Determine Drop Size Distribution Characteristics

The work reported in this paper is an extension of a previous study on the prediction of volume-based drop size distribution from the application of the maximum entropy formalism (M.E.F.). The procedure developed in that study led to the derivation of a two-parameter distribution. The present work investigated the problem of the determination of the two parameters (q and D_q0), i.e. the determination of the information required in the procedure. It was found that the relative span factor Δ_v and the mean drop diameter D₄₃ of the measured distribution constitute reliable information from which q and D_q0 can be calculated. It is shown that the distributions calculated in this way generally show a better fit than distributions obtained from different information or from an extended set of constraints. This work therefore illustrates the importance of the nature of the information to be introduced in the application of the M.E.F. Furthermore, it shows how the M.E.F. can be used to determine relevant spray drop size characteristics and the extent to which this formalism could provide relevant information to study and characterize atomization processes.     

Development of a Three-parameter Volume-based Spray Drop Size Distribution through the Application of the Maximum Entropy Formalism

This work is an extension of a previous investigation on the determination of mathematical volume-spray drop size distributions by the application of the maximum entropy formalism. A two-parameter drop size distribution was derived and was found to give reasonable fits with experimental distributions obtained under different experimental conditions. However, as it is discussed, this two-parameter distribution shows critical limitations and cannot be applied in any situations of interest as far as drop size distributions in liquid sprays are concerned. To overcome this problem, a third parameter, equivalent to a drop diameter, is introduced into the procedure. This correction leads to a three-parameter drop size distribution with independent mean, width and symmetry. This function is a generalized gamma distribution and it can cover more practical situations than the previous two-parameter distribution. Furthermore, it is found that, contrary to the two-parameter distribution, the new volume-based drop size distribution shows a corresponding number-based drop size distribution with a physical behavior as the drop diameter decreases. This last result shows the importance of using three parameters to describe spray drop size distributions and that one of these parameters must represent the population of small drops.     

Calibration Invariance of the MaxEnt Distribution in the Maximum Entropy Principle

The maximum entropy principle consists of two steps: The first step is to find the distribution which maximizes entropy under given constraints. The second step is to calculate the corresponding thermodynamic quantities. The second part is determined by Lagrange multipliers’ relation to the measurable physical quantities as temperature or Helmholtz free energy/free entropy. We show that for a given MaxEnt distribution, the whole class of entropies and constraints leads to the same distribution but generally different thermodynamics. Two simple classes of transformations that preserve the MaxEnt distributions are studied: The first case is a transform of the entropy to an arbitrary increasing function of that entropy. The second case is the transform of the energetic constraint to a combination of the normalization and energetic constraints. We derive group transformations of the Lagrange multipliers corresponding to these transformations and determine their connections to thermodynamic quantities. For each case, we provide a simple example of this transformation.     

Drop‐coating deposition Raman spectroscopy of liposomes

Drop‐coating deposition Raman (DCDR) spectroscopy was tested as a potential technique for studying liposomes at very low sample concentrations. We used model liposomes prepared either from 1,2‐distearoyl‐sn‐glycero‐3‐phospocholine or from soybean asolectin, which is composed of various lipids and thus represents a good model of natural membranes. In both cases, deposited samples formed a dried drop with a circular shape with a ring of concentrated liposomes at the edge. Spectral mapping showed that maximum Raman intensity originated from the inner part of the edge ring, while Raman signal gradually decreased in both radial directions. The Raman spectra exhibited excellent reproducibility of spectral characteristics at different locations in the drop, indicating similar conformation and ordering of hydrocarbon lipid chains in the sample. Our results suggest that DCDR spectroscopy can be used for studying lipids in situ, and sensitivity of this technique is at least two orders of magnitude higher than that of conventional Raman microscopy. Copyright © 2011 John Wiley & Sons, Ltd.     

Retrieval of Optical Constants from Magnetoreflectance by Maximum Entropy Model

Phase retrieval, and thereby retrieval of optical constants, from magnetoreflectance data is performed using the maximum entropy model for spectra analysis. The applicability of the analysis is tested by using the Lorentz model for the permittivity of a nonmagnetic insulator in a case of Faraday configuration.     

Investigation on the Drop Size Distribution of SpraysProduced by a High‐Pressure Swirl Injector. Measurementsand Application of the Maximum Entropy Formalism

This paper reports an investigation on the volume‐based drop size distribution of sprays produced by swirl atomizers dedicated to direct‐injection spark‐ignited engines. Because of the use of high injection pressures to reduce the atomization time, the spatial density of the spray is high and prevents from classical measurements of spray drop size distribution. This problem is overcome by combining an experimental approach to the application of the maximum entropy formalism (M.E.F.). Based on the determination of correction factor series to correct the measurements from multiple light scattering, the experimental procedure allows obtaining some distribution characteristic features. According to a previous study, two of these characteristics are used as information in a M.E.F. procedure to derive the spray volume‐based drop size distribution. This characteristic is of paramount importance for evaluating the large drop population with accuracy. The overall procedure is presented in detail and discussed. It was applied to a series of four swirl atomizers in order to study the influence of the nozzle geometry and of the injection pressure on the injector performances. Conducted under both stationary and transient working conditions, this study allows a more precise understanding of the performances of GDI injectors as far as the spray drop size distributions are concerned.     

Deconvolution with Maximum Entropy Solution to Determine Local Extinction Coefficient and Local Volume Concentration Values from Laser Diffraction Data

The traditional use of the laser diffraction technique provides line‐of‐sight liquid spray drop‐size distribution. However, deconvolution of the measurements can be performed for axisymmetric spray in order to determine local spray characteristics. In a previous publication, a new deconvolution technique making use of the maximum entropy principle was established and applied to determine the local drop‐size distributions. The entire approach was experimentally validated. In this work, the technique is employed to determine local extinction coefficient values. As in the previous investigation, the measurement procedure consists of scanning a laser beam through the spray cross‐section from the center to the edge of the spray. By use of the transmittance theory, the local extinction coefficients allow the local volume concentrations to be calculated. This theory introduces the mean scattering coefficient. The results show that this coefficient must be determined as a function of the Sauter mean diameter in order to avoid overestimation of the volume concentration. Although no proper validation is presented, the coherence of the overall approach is discussed in detail and solutions for improving the spatial resolution are presented. Finally, the local volume concentrations are combined with the local drop‐size distribution to provide local volume‐weighted, drop‐size distributions. These distributions provide information on the localization of the drops according to their diameter as well as on the spatial liquid distribution. This work illustrates applications and performances of laser diffraction technique that are rarely used.     

A Comparative Study of Various Size Distribution

One of the major product specifications of a crystalline material is the crystal size distribution (CSD). In order to monitor and control the CSD in an industrial crystallization process, on‐line sensors are required. Over the years, a number of techniques to measure the CSD have been established. In this paper, three instruments operated in an on‐line fashion and an off‐line method are compared. The instruments were the OPUS, a HELOS/VARIO (both manufactured by Sympatec) and a Malvern 2600c (manufactured by Malvern). They were implemented on an 1100‐l evaporative draft tube baffle (DTB) crystallizer producing ammonium sulfate crystals from aqueous solution. Samples from this crystallizer were also analyzed offline by wet sieving. The results show reasonably good agreement between the different on‐line techniques and the wet sieving technique concerning the shape of the distribution. However, there is a discrepancy regarding the absolute values, which can be explained by the fact that the techniques used are based on different measuring principles.     

A Maximum Entropy Model of Bounded Rational Decision-Making with Prior Beliefs and Market Feedback

Bounded rationality is an important consideration stemming from the fact that agents often have limits on their processing abilities, making the assumption of perfect rationality inapplicable to many real tasks. We propose an information-theoretic approach to the inference of agent decisions under Smithian competition. The model explicitly captures the boundedness of agents (limited in their information-processing capacity) as the cost of information acquisition for expanding their prior beliefs. The expansion is measured as the Kullblack–Leibler divergence between posterior decisions and prior beliefs. When information acquisition is free, the homo economicus agent is recovered, while in cases when information acquisition becomes costly, agents instead revert to their prior beliefs. The maximum entropy principle is used to infer least biased decisions based upon the notion of Smithian competition formalised within the Quantal Response Statistical Equilibrium framework. The incorporation of prior beliefs into such a framework allowed us to systematically explore the effects of prior beliefs on decision-making in the presence of market feedback, as well as importantly adding a temporal interpretation to the framework. We verified the proposed model using Australian housing market data, showing how the incorporation of prior knowledge alters the resulting agent decisions. Specifically, it allowed for the separation of past beliefs and utility maximisation behaviour of the agent as well as the analysis into the evolution of agent beliefs.     

Kaniadakis Entropy Leads to Particle–Hole Symmetric Distribution

We discuss generalized exponentials, whose inverse functions are at the core of generalized entropy formulas, with respect to particle–hole (KMS) symmetry. The latter is fundamental in field theory; so, possible statistical generalizations of the Boltzmann formula-based thermal field theory have to take this property into account. We demonstrate that Kaniadakis’ approach is KMS ready and discuss possible further generalizations.     

Exploring the Origin of Maximum Entropy States Relevant to Resonant Modes in Modern Chladni Plates

The resonant modes generated from the modern Chladni experiment are systematically confirmed to intimately correspond to the maximum entropy states obtained from the inhomogeneous Helmholtz equation for the square and equilateral triangle plates. To investigate the origin of maximum entropy states, the inhomogeneous Helmholtz equation is modified to consider the point interaction coming from the driving oscillator. The coupling strength associated with the point interaction is characterized by a dimensionless factor α. The δ potential of the point interaction is numerically modelled by a truncated basis with an upper index N. The asymptotic behavior for the upper index N is thoroughly explored to verify that the coupling strength of α = 1.0 can make the theoretical resonant modes agree excellently with the maximum entropy states as N→∞. It is further authenticated that nearly the same resonant modes can be obtained by using a larger coupling strength α when a smaller upper index N is exploited in the calculation.     

Modelling of Drop Size Distribution of Rain from Rain Rate and Attenuation Measurements at Millimeter and Optical Wavelengths

This paper describes a technique for modelling of rain drop size distributions at Calcutta in terms of negative exponential function, from the measurements of rain rate and attenuation over a dual wavelength LOS link at millimeter and optical frequencies. The DSD model obtained is then used to determine the attenuation at 94 GHz, for comparison with experimentally obtained attenuation at 94 GHz. This is also compared with the attenuation calculated by considering other experimentally obtained DSD models. The best fit negative exponential distribution function (modified M-P model) is presented along with some other experimentally obtained and reference models.     

Integration in the GHP Formalism IV: A New Lie Derivative Operator Leading to an Efficient Treatment of Killing Vectors

In order to achieve efficient calculations and easy interpretations of symmetries, a strategy for investigations in tetrad formalisms is outlined: work in an intrinsic tetrad using intrinsic coordinates. The key result is that a vector field is a Killing vector field if and only if there exists a tetrad which is Lie derived with respect to ; this result is translated into the GHP formalism using a new generalised Lie derivative operator with respect to a vector field . We identify a class of it intrinsic GHP tetrads, which belongs to the class of GHP tetrads which is generalised Lie derived by this new generalised Lie derivative operator in the presence of a Killing vector field . This new operator also has the important property that, with respect to an intrinsic GHP tetrad, it commutes with the usual GHP operators if and only if is a Killing vector field. Practically, this means, for any spacetime obtained by integration in the GHP formalism using an intrinsic GHP tetrad, that the Killing vector properties can be deduced from the tetrad or metric using the Lie-GHP commutator equations, without a detailed additional analysis. Killing vectors are found in this manner for a number of special spaces.     

Using the Semantic Information G Measure to Explain and Extend Rate-Distortion Functions and Maximum Entropy Distributions

In the rate-distortion function and the Maximum Entropy (ME) method, Minimum Mutual Information (MMI) distributions and ME distributions are expressed by Bayes-like formulas, including Negative Exponential Functions (NEFs) and partition functions. Why do these non-probability functions exist in Bayes-like formulas? On the other hand, the rate-distortion function has three disadvantages: (1) the distortion function is subjectively defined; (2) the definition of the distortion function between instances and labels is often difficult; (3) it cannot be used for data compression according to the labels’ semantic meanings. The author has proposed using the semantic information G measure with both statistical probability and logical probability before. We can now explain NEFs as truth functions, partition functions as logical probabilities, Bayes-like formulas as semantic Bayes’ formulas, MMI as Semantic Mutual Information (SMI), and ME as extreme ME minus SMI. In overcoming the above disadvantages, this paper sets up the relationship between truth functions and distortion functions, obtains truth functions from samples by machine learning, and constructs constraint conditions with truth functions to extend rate-distortion functions. Two examples are used to help readers understand the MMI iteration and to support the theoretical results. Using truth functions and the semantic information G measure, we can combine machine learning and data compression, including semantic compression. We need further studies to explore general data compression and recovery, according to the semantic meaning.     

Particle Size Distribution of Nanospheres by Monte Carlo Fitting of Small Angle X‐Ray Scattering Curves

A method for recovering the size distribution of spherical particles from small angle scattering data by using a Monte Carlo interference function fitting algorithm is presented. The method is based on the direct simulation of the small angle scattering data upon the assumption of non‐interacting hard sphere ensembles (“dilute” solution approximation). The algorithm for retrieving the particle size distribution does not require any additional parameters apart from the input of the scattering data. The fitting strategy necessarily implies positive particle size distributions, while preserving the advantage of the indirect transformation method for data desmearing. Furthermore, the present approach does not use any regularisation procedures of the best fit solution and favours smooth particle size distributions. The Monte Carlo procedure has been tested against several simulated cases with various types of mono‐ and bi‐modal size distributions and different noise levels. In the special case of non‐interacting spheres, the Monte Carlo fitting algorithm had the same retrieving ability as the well assessed indirect transformation, structure interference and maximum entropy methods. Finally, the algorithm was applied to retrieve the distribution of spherical nanopowders produced by gas‐to‐particle conversion both as free powder and as reinforcing second‐phase agent in polymer nanocomposites.     

气溶胶抛撒过程中首次破碎液滴的尺寸分布

对气溶胶爆炸抛撒过程中,首次破碎液滴的尺寸分布进行理论和数值研究.基于热力学第二定律的最大熵增理论,建立首次破碎过程满足的方程组约束条件,给出破碎后液滴尺寸分布的确定方法,并对Air-blast喷管实验和Samirant相关实验进行数值模拟预测,与实验数据符合较好.