Analytical Expressions for Formal Kinetics

2012 ◽  
Vol 715-716 ◽  
pp. 971-976 ◽  
Author(s):  
Paulo Rangel Rios ◽  
Weslley L.S. Assis ◽  
Tatiana C. Salazar ◽  
Elena Villa
Keyword(s):  
Cellular Automata ◽  
Microstructural Evolution ◽  
Analytical Solutions ◽  
Stochastic Geometry ◽  
Poisson Point Process ◽  
Point Processes ◽  
Formal Kinetics ◽  
Bulk Nucleation ◽  
Inhomogeneous Poisson Point Process ◽  
Analytical Expressions

In recent papers Rios and Villa resorted to developments in stochastic geometry to revisit theclassical KJMA theory and generalize it for situations in which nuclei were located in space accordingto both homogeneous and inhomogeneous Poisson point processes as well as according to Materncluster process and surface and bulk nucleation in small specimens. Rigorous mathematical methodswere employed to ensure the reliability of the new expressions. These results are briefly described.Analytical expression for inhomogeneous Poisson point process nucleation gives very good agreementwith Cellular Automata simulations. Cellular Automata simulations complement the analyticalsolutions by showing the corresponding microstructural evolution. These new results considerablyexpand the range of situations for which analytical solutions are available.

scholarly journals ON MODELLING RECRYSTALLIZATION PROCESSES WITH RANDOM GROWTH VELOCITIES OF THE GRAINS IN MATERIALS SCIENCE

2012 ◽  
Vol 31 (3) ◽  
pp. 149 ◽  
Author(s):  
Elena Villa ◽  
Paulo R. Rios
Keyword(s):  
Stochastic Geometry ◽  
Point Processes ◽  
Moving Boundary ◽  
Materials Science ◽  
Measure Theory ◽  
Practical Interest ◽  
Geometric Measure Theory ◽  
Nucleation And Growth ◽  
Analytical Expressions

Heterogeneous transformations (or reactions)  may be defined as those transformations in which there is a sharp moving boundary between the transformed and untransformed region. In Materials Science such transformations are normally called nucleation and growth transformations, whereas birth-and-growth processes is the preferred denomination in Mathematics. Recently, the present authors in a series of papers have derived new analytical expressions for nucleation and growth transformations with the help of stochastic geometry methods. Those papers focused mainly on the role of nuclei location in space, described by point processes, on transformation kinetics.  In this work we focus on the effect that a random velocity of the moving boundaries of the grains has in the overall kinetics. One example of a practical situation in which such a model may be useful is that of recrystallization.  Juul Jensen and Godiksen reviewed recent 3-d experimental results  on recrystallization kinetics and concluded that there is compelling evidence  that  every  grain has its own distinct growth rate. Motivated by this practical application we present here new general kinetics expressions for various situations of practical interest, in which a random distribution of growth velocities is assumed. In order to do this, we  make use of tools from Stochastic Geometry and Geometric Measure Theory. Previously known results follow here as particular cases. Although the motivation for this paper was recrystallization the expressions derived here may be applied to nucleation and growth reactions in general.

scholarly journals Inhomogeneous Poisson point process nucleation: comparison of analytical solution with cellular automata simulation

2009 ◽  
Vol 12 (2) ◽  
pp. 219-224 ◽  
Author(s):  
Paulo Rangel Rios ◽  
Douglas Jardim ◽  
Weslley Luiz da Silva Assis ◽  
Tatiana Caneda Salazar ◽  
Elena Villa
Keyword(s):  
Cellular Automata ◽  
Analytical Solution ◽  
Point Process ◽  
Poisson Point Process ◽  
Inhomogeneous Poisson Point Process

scholarly journals A numerical method for computing interval distributions for an inhomogeneous Poisson point process modified by random dead times

2021 ◽  
Vol 115 (2) ◽  
pp. 177-190
Author(s):  
Adam J. Peterson
Keyword(s):  
Point Process ◽  
Time Distribution ◽  
Poisson Point Process ◽  
Point Processes ◽  
Dead Time ◽  
Closed Form Expression ◽  
Arbitrary Distribution ◽  
Observation Window ◽  
The Dead ◽  
Inhomogeneous Poisson Point Process

AbstractThe inhomogeneous Poisson point process is a common model for time series of discrete, stochastic events. When an event from a point process is detected, it may trigger a random dead time in the detector, during which subsequent events will fail to be detected. It can be difficult or impossible to obtain a closed-form expression for the distribution of intervals between detections, even when the rate function (often referred to as the intensity function) and the dead-time distribution are given. Here, a method is presented to numerically compute the interval distribution expected for any arbitrary inhomogeneous Poisson point process modified by dead times drawn from any arbitrary distribution. In neuroscience, such a point process is used to model trains of neuronal spikes triggered by the detection of excitatory events while the neuron is not refractory. The assumptions of the method are that the process is observed over a finite observation window and that the detector is not in a dead state at the start of the observation window. Simulations are used to verify the method for several example point processes. The method should be useful for modeling and understanding the relationships between the rate functions and interval distributions of the event and detection processes, and how these relationships depend on the dead-time distribution.

Application of stochastic geometry to problems in plankton ecology

1992 ◽  
Vol 336 (1277) ◽  
pp. 225-237 ◽  
Keyword(s):  
Optical Sensors ◽  
Stochastic Geometry ◽  
Point Processes ◽  
Random Distribution ◽  
Interparticle Distance ◽  
Ecosystem Dynamics ◽  
Order Moment ◽  
Patch Structure ◽  
Covariance Functions ◽  
Plankton Ecology

The most fundamental linkages in ecosystem dynamics are trophodynamic. A trophodynamic theory requires a framework based upon inter-organism or interparticle distance, a metric important in its own right, and an essential component relating trophodynamics and the kinetic environment. It is typically assumed that interparticle distances are drawn from a random distribution, even though particles are known to be distributed in patches. Both random and patch-structure interparticle distance are analysed using the theory of stochastic geometry. Aspects of stochastic geometry - point processes and random closed sets (RCS) - useful for studying plankton ecology are presented. For point-process theory, the interparticle distances, random -distribution order statistics, transitions from random to patch structures, and second-order-moment functions are described. For RCS-theory, the volume fractions, contact distributions, and covariance functions are given. Applications of stochastic-geometry theory relate to nutrient flux among organisms, grazing, and coupling between turbulent flow and biological processes. The theory shows that particles are statistically closer than implied by the literature, substantially resolving the troublesome issues of autotroph-heterotroph nutrient exchange; that the microzone notion can be extended by RCS; that patch structure can substantially modify predator-prey encounter rates, even though the number of prey is fixed; and that interparticle distances and the RCS covariance function provide a fundamental coupling with physical processes. In addition to contributing to the understanding of plankton ecology, stochastic geometry is a potentially useful for improving the design of acoustic and optical sensors

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (03) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k } are the points of a renewal process and {Ak } are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

Analytical solutions for turbulent Boussinesq fountains in a linearly stratified environment

2011 ◽  
Vol 691 ◽  
pp. 487-497 ◽  
Author(s):  
Rabah Mehaddi ◽  
Olivier Vauquelin ◽  
Fabien Candelier
Keyword(s):  
Theoretical Model ◽  
Asymptotic Analysis ◽  
Flow Rate ◽  
Vertical Velocity ◽  
Analytical Solutions ◽  
Boussinesq Approximation ◽  
Singular Case ◽  
Maximal Height ◽  
Analytical Integration ◽  
Analytical Expressions

AbstractThis paper theoretically investigates the initial up-flow of a vertical turbulent fountain (round or plane) in a linearly stratified environment. Conservation equations (volume, momentum and buoyancy) are written under the Boussinesq approximation assuming an entrainment proportional to the vertical velocity of the fountain. Analytical integration leads to exact values of both density and flow rate at the maximal height reached by the fountain. This maximal height is expressed as a function of the release conditions and the stratification strength and plotted from a numerical integration in order to exhibit overall behaviour. Then, analytical expressions for the maximal height are derived from asymptotic analysis and compared to experimental correlations available for forced fountains. For weak fountains, these analytical expressions constitute a new theoretical model. Finally, modified expressions are also proposed in the singular case of an initially non-buoyant vertical release.

scholarly journals Simple and Accurate Analytical Solutions of the Electrostatically Actuated Curled Beam Problem

2014 ◽  
Author(s):  
Mohammad I. Younis
Keyword(s):  
Analytical Solutions ◽  
Galerkin Approximation ◽  
Single Mode ◽  
Beam Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Analytical Formulas ◽  
Euler Bernoulli Beam ◽  
Analytical Expressions ◽  
Multi Mode

We present analytical solutions of the electrostatically actuated initially deformed cantilever beam problem. We use a continuous Euler-Bernoulli beam model combined with a single-mode Galerkin approximation. We derive simple analytical expressions for two commonly observed deformed beams configurations: the curled and tilted configurations. The derived analytical formulas are validated by comparing their results to experimental data in the literature and numerical results of a multi-mode reduced order model. The derived expressions do not involve any complicated integrals or complex terms and can be conveniently used by designers for quick, yet accurate, estimations. The formulas are found to yield accurate results for most commonly encountered microbeams of initial tip deflections of few microns. For largely deformed beams, we found that these formulas yield less accurate results due to the limitations of the single-mode approximations they are based on. In such cases, multi-mode reduced order models need to be utilized.

Analytical solutions and expressions of the propagator for Bloch equations

2020 ◽  
Vol 34 (18) ◽  
pp. 2050158
Author(s):  
Heung-Ryoul Noh
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Secular Equation ◽  
Magnetization Vector ◽  
Bloch Equations ◽  
Optical Bloch Equations ◽  
Parameter Spaces ◽  
Pulse Spectroscopy ◽  
Analytical Expressions ◽  
Composite Laser

In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.

Extremes for shot noise processes with heavy tailed amplitudes

1997 ◽  
Vol 34 (3) ◽  
pp. 643-656 ◽  
Author(s):  
William P. McCormick
Keyword(s):  
Point Process ◽  
Shot Noise ◽  
Renewal Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extremal Index ◽  
Two Dimensional ◽  
Limiting Behavior ◽  
Local Maxima ◽  
Heavy Tailed

Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where {τ k} are the points of a renewal process and {Ak} are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.

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