Rate equation deterministic

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

Fig. 5. Steady-state solution of deterministic rate equations to which a stochastic term has been added. Low noise level (mean absolute magnitude of fluctuations). Note increase in noise level near lower marginal stability point.

We proceed with the consideration of a linear chain of coupled first- and second-order reactions (fig. 5.7). If species Xi is pulsed, then the relaxation of the various species is shown in fig. 5.8. There are interesting approximate relations for such systems among the amplitudes of changes of relative concentrations. Consider the variation of X2 upon a pulse of Xi administered to the system the deterministic rate equations are... 

The difficulties in this example arise from the self-inhibition of the enzyme catalysis by Xg. The rate coefficient kA first increases with increasing concentration of Xg and then decreases. The hypersurface formed by eliminating the time dependence from the set of deterministic rate equations, by dividing the equation for each but one of the species by the equation for that one species, is folded over due to the quadratic dependence of kA on Xg. In the simulation the concentration of Xg is varied randomly and the responses of the other species are calculated to give time series of 2,000 data points these series are the starting point for both the EMC and the CMC analysis. 

Simulations on the effect of step free energy on grain growth behaviour have also been made. Figure 15.11 shows the result of a Monte Carlo simulation made by Cho. For the simulation, Cho assumed that the grain network was a set of grains with a Gaussian size distribution (standard deviation of 0.1) located on vertices of a two-dimensional square lattice. Deterministic rate equations, Eq. (15.15) for v/> and Eq. (15.29) for v j, were... 

Although kinetics plays such an important role in catalsrsis, its theory has for a long time mainly been restricted to the use of macroscopic deterministic rate equations. These implicitly assume a random distribution of adsorbates on the catalyst s surface. Effects of lateral interactions, reactant segregation, site blocking, and defects have only been described ad hoc. With the advent of Dynamic Monte-Carlo simulations (DMC simulations), also called Kinetic Monte-Carlo simulations, it has become possible to follow reaction systems on an atomic scale, and thus to study these effects properly. 

Prom the master equation, we can derive the result that the average concentration, the average number of X in a volume V, obeys the deterministic rate equation in the limit of large numbers of molecules. 

We now consider reaction-diffusion systems with two intermediates and multiple stationary states, which may be nodes or foci. For a real eigenvalue that approach is monotonic for a complex eigenvalue with negative real part that approach is one of damped oscillations. In the absence of cross diffusion the deterministic rate equations in one dimension, 2, are... 

If the truncated form of eq. (14), in which all e-dependent terras are set equal to zero, predicted a finite negative value of a2 for all values of t, then the probability distribution would essentially be Gaussian, with a width given by a Additional terms in the expansion (12) would be superfluous, as they would give corrections to the moments of P vanishing as a power of e. The evolution of the system would thus be essentially deterministic, since the most probable value would remain uniquely defined and would evolve according to the deterministic rate equation. 

Since the system is closed, the total number N e N/ + Ng is constant, and it is easy to verify that in this case the deterministic rate equation for Ng reduces to the form of Eq. 4, where the constant N is absorbed into the pre-exponential factor Rq of the latter equation. 

Equation (41.11) represents the (deterministic) system equation which describes how the concentrations vary in time. In order to estimate the concentrations of the two compounds as a function of time during the reaction, the absorbance of the mixture is measured as a function of wavelength and time. Let us suppose that the pure spectra (absorptivities) of the compounds A and B are known and that at a time t the spectrometer is set at a wavelength giving the absorptivities h (0- The system and measurement equations can now be solved by the Kalman filter given in Table 41.10. By way of illustration we work out a simplified example of a reaction with a true reaction rate constant equal to A , = 0.1 min and an initial concentration a , (0) = 1. The concentrations are spectrophotometrically measured every 5 minutes and at the start of the reaction after 1 minute. Each time a new measurement is performed, the last estimate of the concentration A is updated. By substituting that concentration in the system equation xff) = JC (0)exp(-A i/) we obtain an update of the reaction rate k. With this new value the concentration of A is extrapolated to the point in time that a new measurement is made. The results for three cycles of the Kalman filter are given in Table 41.11 and in Fig. 41.7. The... 

The non-equivalence of the statistical and kinetic methods Is given by the fact that the statistical generation Is always a Markovian process yielding a Markovian distribution, e.g. In case of a blfunc-tlonal monomer the most probable or pseudo-most probable distributions. The kinetic generation Is described by deterministic differential equations. Although the Individual addition steps can be Markovian, the resulting distribution can be non-Markovian. An Initiated step polyaddltlon can be taken as an example the distribution Is determined by the memory characterized by the relative rate of the Initiation step ( ). ... 

If the solution of a deterministic reaction rate equation differs from the first moment corresponding to the solution of the master equation, it can generally be considered as a differently conditioned average of the same random variable.144... 

For this simple two-state transition, the traditional deterministic chemical kinetics (see Chapter 3) is based on rate equations for the concentration of A ... 

It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... 

The deterministic population balance equations governing the description of mass transfer with reaction in liquid-liquid dispersions present a framework for analysis. However, signiflcant difficulties exist in obtaining solutions for realistic problems. No analytical solutions are available for even the simplest cases of interest. Extension of the solution to multiple reactants for uniform drops is possible using a method of moments but the solution is limited to rate equations which are polynomials (E3). Solutions to the population balance equations for spatially nonhomogeneous dispersions were only treated for nonreacting dispersions (P4), and only a simple case was solved for a spray column (B19). Treatment of unmixed feeds presents a problem. 

Recent measurements have shown that laser ablation (or negatively spoken damage) becomes more deterministic when femtosecond laser pulses are applied [22, 23, 40, 41]. This observation is due to the generation of conduction band (seed) electrons by means of multiphoton ionization (MPI). Based on this knowledge, a model for optical breakdown that takes into account avalanche ionization and MPI was developed [40], The temporal behavior of the free electron density in the conduction band n(t) can be described by a rate equation... 

We wish to introduce next a topic of increasing importance to chemical engineers, stochastic (random) simulation. In stochastic models we simulate quite directly the random nature of the molecules. We will see that the deterministic rate laws and material balances presented in the previous sections can be captured in the stochastic approach by allowing the numbers of molecules in the simulation to become large. From this viewpoint, deterministic and stochastic approaches are complementary. Deterministic models and solution methods are quite efficient when the numbers of molecules are large and the random behavior is not important. The numerical methods for solution of the nonlinear differential equations of the deterministic models are... 

To describe the dynamic system behavior, deterministic kinetic rate equations of the form... 

As was mentioned earlier, deterministic models of chemical reactions might be identified with eqn (1.3). However, not all kinds of systems of differential equations, not even all those with a polynomial right-hand side can be considered as reaction kinetics equations. Trivially, the term -kc2 t)c t) cannot occur in a rate equation referring to the velocity of Cj, since the quantity of a component cannot be reduced in a reaction in which the component in question does not take place. Putting it another way, the negative cross-effect is excluded. A necessary and sufficient condition is required to restrict eqn (1.3) to be able to be a kinetic equation. 

Oppenheim, L, Shuler, K. E. Weiss, G. M. (1969). Stochastic and deterministic formulation of chemical rate equations. J. Chem. Phys., 50, 460-6. 

In the first reaction glucose reacts with ATP to produce ADP and PEP the enzyme for this first step is hexokinase the notation is similar for the remaining reaction steps, with the enzymes as indicated. The rate of influx of glucose into the system is constant. Due to the feedback mechanisms in both the PFK and PK reactions chemical oscillations of some species may occur, see Fig. 16.4. These oscillations have been observed [1] and are also obtained from numerical solutions of the deterministic mass action rate equations of the model in Fig. 16.1 for given glucose inflow conditions, see Fig. 16.4. 

The deterministic kinetic equations for this system are given in [13], but we need not repoduce them here they are nonlinear and for experimental values of the rate coefficients represent a damped oscillator. 

Stochastic simulation has also demonstrated that it is impossible to find a single rate equation of the deterministic kind the rate equation appears to be dependent on the method of variation of the number of particles inside the... 

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