The Discrete General Dynamic Equation

A spatially homogeneous aerosol of uniform chemical composition can be fully characterized by the number concentrations of particles of various sizes Nk t) as a function of time. These particle concentrations may undergo changes due to coagulation, condensation, evaporation, nucleation, emission of fresh particles, and removal. The dynamic equation governing Nk(t) for k 2 can be developed as follows. 

Consider first the contribution of coagulation to this concentration change. We showed in Section 13.3.2 that the production rate of k-mers due to coagulation ofy-mers with (,k — j)-mers is given by 

To calculate the rate of change of Nk(t) due to evaporation, let y k be the flux of monomers per unit area leaving a k-mer. Then, the overall rate of escape of monomers from a k-mer, yk, with surface area ak will be given by 

The rate of loss of k-mers from evaporation can then be written as 

The rate of formation of k-mers by evaporation of (k + l)-mers is then just 

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An aerosol distribution can be described by the number concentrations of particles of various sizes as a function of time. Let us define Nk(t) as the number concentration (cm-3) of particles containing k monomers, where a monomer can be considered as a single molecule of the species representing the particle. Physically, the discrete distribution is appealing since it is based on the fundamental nature of the particles. However, a particle of size 1 pm contains on the order of 1010 monomers, and description of the submicrometer aerosol distribution requires a vector (N2, N-j,..., N10io) containing 1010 numbers. This makes the use of the discrete distribution impractical for most atmospheric aerosol applications. We will use it in the subsequent sections for instructional purposes and as an intermediate step toward development of the continuous general dynamic equation. 

General Dynamic Equation for the Discrete Distribution Function 3Q7... 

GENERAL DYNAMIC EQUATION FOR THE DISCRETE DISTRIBUTION FUNCTION... 

The electronic states associated to the cyclic and oscillatory reactions, reactions with instabilities, etc. are obeying for the discrete state dynamics to the quantum general evolution equation (QME quantum master equation) (see Gray Scott, 1990 van Kampen, 1987 Gardiner, 1994 Risken, 1984 Haken, 1978, 1987, 1988) ... 

The general strategy of attack to the molecular dynamics outlined by this and earlier chapters appears especially promising to shed further light into this stimulating field of research In the special case of H-bonded liquids, a natural development of the ideias outlined here implies the replacement of the discrete variable t) with a continuous variable, which in turn involves the replacement of the master equation method with a suitable Fokker-PIanck equation. Moreover, this improvement of the theory is fundamental to exploring the short-time dynamics when the details of the correlation functions on the time scale of structure V of water must be accounted for. 

The prediction horizon is discretized in cycles, where a cycle is a switching time tshift multiplied by the total number of columns. Equation 9.1 constitutes a dynamic optimization problem with the transient behavior of the process as a constraint f describes the continuous dynamics of the columns based on the general rate model (GRM) as well as the discrete switching from period to period. To solve the PDE models of columns, a Galerkin method on finite elements is used for the liquid... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

To numerically solve equations of the above mathematical models, the general computational gas dynamics is adopted in the present work. The general differential equations (2.7) and (2.31) are then discretized by the control volume-based finite difference method, and the resulting set of algebraic equations is iteratively solved. The numerical solver for the general differential equations can be repeatedly appUed for each scale variable over a controlled volume mesh. This process must be conducted extremely carefully to avoid the influence of slight changes in the accuracy of discretization. 

The category of algebraic equation models is quite general and it encompasses many types of engineering models. For example, any discrete dynamic model described by a set of dijference equations falls in this category for parameter estimation purposes. These models could be either deterministic or stochastic in nature or even combinations of the two. Although on-line techniques are available for the estimation of parameters in sampled data systems, off-line techniques... 

To study the dynamic behavior of the BZ gels, we numerically integrate Eqs (8.1 -8.3) in two [1, 2] or three [3] dimensions using our recently developed gLSM. This method combines a finite-element approach for the spatial discretization of the elastodynamic equations and a finite-difference approximation for the reaction and diffusion terms. We used the gLSM approach to examine 2D confined films and 3D bulk samples here, we briefly discuss the more general 3D formulation [3]. 

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment. 

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