Deterministic Trajectory Models

If we express the density of the particle phase in terms of the product of the particle number density n and the mass of a particle m, i.e., appp = nm, and noting that rp = n(dm/dt), we have 

integrating the preceding equation over any cross section of a stream tube of the particle phase and using the Gauss theorem, we can obtain 

Fap is the drag force between the gas and the solids, which can be expressed as 

Note that /ep in Eq. (5.238) is replaced with /Ep for Eq. (5.240), where /Ep is the heat generated by thermal radiation per unit volume and Qap is the heat transferred through the interface between gas and particles. Thus, once the gas velocity field is solved, the particle velocity, particle trajectory, particle concentration, and particle temperature can all be obtained directly by integrating Eqs. (5.235), (5.237), (5.231), and (5.240), respectively. Since the equations for the gas phase are coupled with those for the solid phase, final solutions of the governing equations may have to be obtained through iterations between those for the gas and solid phases. 

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Two basic trajectory models, i.e., the deterministic trajectory model [Crowe etal., 1977] and the stochastic trajectory model [Crowe, 1991], are introduced in this section. The deterministic approach, which neglects the turbulent fluctuation of particles, specifically, the turbulent diffusion of the mass, momentum, and energy of particles, is considered the most... 

Molecular dynamics-deterministic molecular modeling tool that evaluates the forces on individnal atoms using an energy forcefield, then uses Newton s classical eqnation of motion to compute new atomic positions after a short time interval (on the order of a femtosecond) snccessive evaluation for a large number of time steps provides a time-dependent trajectory of all atomic motions. 

Particle trajectories can be calculated by utilizing the modern CFD (computational fluid dynamics) methods. In these calculations, the flow field is determined with numerical means, and particle motion is modeled by combining a deterministic component with a stochastic component caused by the air turbulence. This technique is probably an effective means for solving particle collection in complicated cleaning systems. Computers and computational techniques are being developed at a fast pace, and one can expect that practical computer programs for solving particle collection in electrostatic precipitators will become available in the future. 

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

Having specified the interactions (i.e., the model of the system), the actual simulation then constructs a sequence of states (or the system trajectory) in some statistical mechanical ensemble. Simulations can be stochastic (Monte Carlo (MC)) or deterministic (MD), or they can combine elements of both, such as force-biased MC, Brownian dynamics, or generalized Lan-gevin dynamics. It is usually assumed that the laws of classical mechanics (i.e., Newton s second law) may adequately describe the atoms and molecules in the physical system. 

Therefore, oscillations of K (t) result in the transition of the concentration motion from one stable trajectory into another, having also another oscillation period. That is, the concentration dynamics in the Lotka-Volterra model acts as a noise. Since along with the particular time dependence K — K(t) related to the standing wave regime, it depends also effectively on the current concentrations (which introduces the damping into the concentration motion), the concentration passages from one trajectory onto another have the deterministic character. It results in the limited amplitudes of concentration oscillations. The phase portrait demonstrates existence of the distinctive range of the allowed periods of the concentration oscillations. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

MD allows the study of the time evolution of an V-body system of interacting particles. The approach is based on a deterministic model of nature, and the behavior of a system can be computed if we know the initial conditions and the forces of interaction. For a detail description see Refs. [14,15]. One first constructs a model for the interaction of the particles in the system, then computes the trajectories of those particles and finally analyzes those trajectories to obtain observable quantities. A very simple method to implement, in principle, its foundations reside on a number of branches of physics classical nonlinear dynamics, statistical mechanics, sampling theory, conservation principles, and solid state physics. 

Often, the strict definition of ergodicity must be relaxed to accommodate the realistic modelling setting for example only some portion of the eno gy surface may be accessible by trajectories in the time-interval available, due to the cost of computing steps with a numerical method. For a more detailed discussion of these and other ergodicity issues in the context of deterministic molecular dynamics, see the articles of Tupper [379, 381]. 

FIG. 11 Spatially periodic suspensions of fractal aggregates. The aggregate in (a) contains 1024 cubic particles of size a- it was built with the hierarchical model with linear trajectories. The deterministic self-similar flake in (b) is at the third-generation stage with b = 5. 

This model with only three variables, whose only nonlinearities are xy and xz, exhibited dynamic behavior of unexpected complexity (Fig. 7.2). It was especially surprising that this deterministic model was able to generate chaotic oscillations. The corresponding limit set was called the Lorenz attractor and limit sets of similar type are called strange attractors. Trajectories within a strange attractor appear to hop around randomly but, in fact, are organized by a very complex type of stable order, which keeps the system within certain ranges. 

To implement the deterministic optimization approach, the sequential strategy proposed by Li et al. (1998) is used, where the whole algorithm is divided into one optimization layer with SQP as a standard NLP solver and one simulation layer, where all dependent variables are computed through an integration step. The model is a large scale DAE system which is discretized with collocation on finite elements. The whole batch time is discretized into 30 time intervals. The control variables are set as piecewise constants. The computed trajectories of the control variables for the optimal operation are illustrated in Fig. 1. 

The fiuctuational trajectory away from a stationary state to a given point in concentration space (X, Y) in general differs from the deterministic path from that point back to the stationary state for systems without detailed balance. We show this in some calculations for the Selkov model in (3.38) we take m = n = r = l,s = 3 other parameters are given in [1], p. 4555. Figure 3.1 gives some results of these calculations. 

Fig. 4.4. Plot in the concentration space of the variables x,y of the Selkov model (a) optimal fluctuational trajectory from a stable stationary state (x,y)st to a given point (x,y)p and (b) the deterministic return to the stationary state. Prom [3]...

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