Stochastic theory averaging

A stochastic theory provides a simple model for chromatography.11 The term stochastic implies the presence of a random variable. The model supposes that, as a molecule travels through a column, it spends an average time Tm in the mobile phase between adsorption events. The time between desorption and the next adsorption is random, but the average time is Tm. The average time spent adsorbed to the stationary phase between one adsorption and one desorption is rs. While the molecule is adsorbed on the stationary phase, it does not move. When the molecule is in the mobile phase, it moves with the speed ux of the mobile phase. The probability that an adsorption or desorption occurs in a given time follows the Poisson distribution, which was described briefly in Problem 19-21. 

Some progress has been made in the direction of applying the thermodynamic and stochastic theory of rate processes presented here to disordered systems. In some cases [35] it is possible to construct a stochastic potential with the properties the same as that for ordered systems discussed in Chaps. 2-11. A general set of fluctuation-dissipation relations has been derived that establishes a connection between the expression of the average kinetic curve,... 

The stochastic problem is to describe properly the time evolution of the Heisenberg operator d(t) averaged over all the realizations of collisional process in the interval (0,t). The averaging, performed in the impact theory, results in the phenomenological kinetic equation [170, 158]... 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

A process that appears to be random and is not explainable by mechanistic theory. A stochastic process typically refers to variations that can be characterized as a frequency distribution based on observable data but for which there is no useful or practical means to further explain or make predictions based upon the underlying but perhaps unknown cause of the variation. For example, air pollutant concentrations include a stochastic component because of turbulence in the atmosphere that cannot be predicted other than in the form of an average or in the form of a distribution. 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

Averaging over all different possible stochastic behaviors in SMS yields the master equation used in ensemble spectroscopy, but the averaged master equation does not determine the dynamics of the (pure) states of individual molecules. Certain attempts have been made to derive a proper theory of individual behavior of single quantum systems, but a rigorous interpretation is still lacking. 

The current burst model is potentially powerful in providing explanations for many mechanistic and morphological aspects involved in the formation of PS. However, as recognized by Foil et al. themselves, it would be extremely difficult for such a unified model to be expressed in mathematical form because it has to include all of the conditional parameters and account for all of the observed phenomena. Fundamentally, all electrochemical behavior is in nature the statistical averages of the numerous stochastic events at a microscopic scale and could in theory be described by the oscillation of the reactions on some microscopic reaction units which are temporally and spatially distributed. Ideally, a single surface atom would be the smallest dimension of such a unit and the integration of the contribution of all of the atoms in time and space would then determine a specific phenomenon. In reality, it is not possible because one does not know with any certainty the reactivity functions of each individual atoms. The difficulty for the current burst model would be the establishment of the reactivity functions of the individual reaction units. Also, some of the assumptions used in this model are questionable. For example, there is no physical and chemical foundation for the assumption that the oxide covering the reaction unit is... 

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