Probability density function reaction rate calculation

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 (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. 

One of the most important and elusive quantities in molecular djmamics calculations of reacting system is the intermolecular energy transfer probability density function P(E ,E) which is used in master equation calculations of reaction rate coefficients [1]. 

As mentioned in section 16.3.1, rate constant calculations often rest on the determination of the activation free-energy which is usually deduced from the free-energy profile of the system as a function of the reaction coordinate. The reaction coordinate x is a function xif) of the coordinates of the atoms of the system chosen so as to follow the reaction advancement. In an ensemble of protein + solvent systems in equilibrium at temperature T, the x value distribution is characterized by the probability density p(x) given by ... 

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

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