Function reaction probability density

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

Prompted by these considerations, Gillespie [388] introduced the reaction probability density function p (x, l), which is a joint probability distribution on the space of the continuous variable x (0 < x < oc) and the discrete variable l (1 = 1,..., to0). This function is used as p (x, l) Ax to define the probability that given the state n(t) at time t, the next event will occur in the infinitesimal time interval (t + x,t + x + Ax), AND will be an Ri event. Our first step toward finding a legitimate method for assigning numerical values to x and l is to derive, from the elementary conditional probability hi At, an analytical expression for p (x, l). To this end, we now calculate the probability p (x, l) Ax as the product po (x), the probability at time t that no event will occur in the time interval (t, t + x) TIMES a/ Ax, the subsequent probability that an R.i... 

Inserting the previous expression in (9.40), we arrive at the following exact expression for the reaction probability density function ... 

The quantum flux operator F measures the probability current density. The latter satisfies the continuity equation resulting from the invariance of the norm of the wave packet in the coordinate basis. For a stationary wave function, the probability density is independent of time and the flux is constant across any fixed hypersurface. In reaction dynamics the flux operator is most generally defined in terms of a dividing surface 0 which... 

A model must be introduced to simulate fast chemical reactions, for example, flamelet, or turbulent mixer model (TMM), presumed mapping. Rodney Eox describes many proposed models in his book [23]. Many of these use a probability density function to describe the concentration variations. One model that gives reasonably good results for a wide range of non-premixed reactions is the TMM model by Baldyga and Bourne [24]. In this model, the variance of the concentration fluctuations is separated into three scales corresponding to large, intermediate, and small turbulent eddies. 

The y-axis is the free energy. In general the free energy is related to the probability density function P( ) of the reaction coordinate through... 

The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase flow the reader is lead towards complex chemistry and two-phase flow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration fields. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-... 

The figure shows U >. S L in this region and Da is predominantly small. At the highest Reynolds numbers the region is entered only for very intense turbulence, U > SL. The region has been considered a distributed reaction zone in which reactants and products are somewhat uniformly dispersed throughout the flame front. Reactions are still fast everywhere, so that unbumed mixture near the burned gas side of the flame is completely burned before it leaves what would be considered the flame front. An instantaneous temperature measurement in this flame would yield a normal probability density function—more importantly, one that is not bimodal. 

The simplest physical picture for the tunneling of a hydrogen nucleus during a hydrogen-transfer reaction takes note of the nuclear probability-density function for... 

Then, the exponent is given by the number of degrees of freedom, s, minus 1 for the breaking bond. For a strict treatment of fragmenting ions by QET one would need to know the activation energies of all reactions accessible and the probability functions describing the density of energy levels. 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with... 

A chemical reaction, for instance, an intramolecular reaction R P, is viewed as a one-dimensional motion in the phase space of a particle in a double-well potential and undergoing Markovian random forces. The dynamics of the particle are described in terms of the Kramers equation (4.160), and the rate of reaction can, in principle, be calculated from the knowledge of the probability density function p(x0 t, x).16,147... 

Equation (10.40) is sometimes found in a simpler form at the cost of hiding the complexity of the terms involved. This form is based on the introduction of a probability density function for the reaction coordinate and the associated potential of mean force, in contrast to previously, where we considered the probability density of a particular arrangement of n atoms. Let I l(Q )d,Q be the probability of finding the reaction coordinate in the range Q, Q + dQ. Then, from equilibrium statistical mechanics (see Appendix A.2), the probability density function I l(Q ) is given by... 

Given these fundamental probability density functions, the following algorithm can be used to carry out the reaction set simulation ... 

TST, and/or MD simulations (the choice depends mainly on whether the process is activated or not). The creation of a database, a lookup table, or a map of transition probabilities for use in KMC simulation emerges as a powerful modeling approach in computational materials science and reaction arenas (Maroudas, 2001 Raimondeau et al., 2001). This idea parallels tabulation efforts in computationally intensive chemical kinetics simulations (Pope, 1997). In turn, the KMC technique computes system averages, which are usually of interest, as well as the probability density function (pdf) or higher moments, and spatiotemporal information in a spatially distributed simulation. 

For initially nonpremixed reactants, two limiting cases may be visualized, namely, the limit in which the chemistry is rapid compared with the fluid mechanics and the limit in which it is slow. In the slow-chemistry limit, extensive turbulent mixing may occur prior to chemical reaction, and situations approaching those in well-stirred reactors (see Section 4.1) may develop. There are particular slow-chemistry problems for which the previously identified moment methods and age methods are well suited. These methods are not appropriate for fast-chemistry problems. The primary combustion reactions in ordinary turbulent diffusion flames encountered in the laboratory and in industry appear to lie closer to the fast-chemistry limit. Methods for analyzing turbulent diffusion flames with fast chemistry have been developed recently [15], [20], [27]. These methods, which involve approximations of probability-density functions using moments, will be discussed in this section. 

Equation (42) cannot be used if NO concentrations approach their equilibrium values, since the net production rate then depends on the concentration of NO, thereby bringing bivariate probability-density functions into equation (40). Also, if reactions involving nitrogen in fuel molecules are important, then much more involved considerations of chemical kinetics are needed. Processes of soot production similarly introduce complicated chemical kinetics. However, it may be possible to characterize these complex processes in terms of a small number of rate processes, with rates dependent on concentrations of major species and temperature, in such a way that a function w (Z) can be identified for soot production. Rates of soot-particle production in turbulent diffusion flames would then readily be calculable, but in regions where soot-particle growth or burnup is important as well, it would appear that at least a bivariate probability-density function should be considered in attempting to calculate the net rate of change of soot concentration. 

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