Composite Markov process

Solving this equation means determining the probability to find at t > 0 the system in (i, r) when it was at t = 0 in (i0, r0). This problem decomposes into two successive steps first find how the molecule jumps among the levels regardless of r, and subsequently add on the behavior in r. This is the reason why we use the name composite Markov process for any random process obeying a master equation of the type (7.4). ... 

Exercise. The solution of the ordinary M-equation for non-composite Markov processes, such as (V.1.5), can also be written as a sum over realizations. The result is, in analogy with (7.10),... 

Exercise. Formulate the process described by the V and X of a Rayleigh particle as a composite Markov process in the sense of VII.7. 

The joint process is a composite Markov process as defined in VII.7. 

Remark. The essential feature of our composite process is that i is an independent process by itself, while the transition probabilities of r are governed by i. This situation can be formulated more generally. Take a Markov process Y(t), discrete or continuous, having an M-equation with kernel... 

Thus, copolymers of the same composition can have qualitatively different sequence distributions depending on the solvent in which the chemical transformation is performed. In a solvent selectively poor for modifying agent, hydrophobically-modified copolymers were found to have the sequence distribution with LRCs, whereas in a nonselective (good) solvent, the reaction always leads to the formation of random (Bernoullian) copolymers [52]. In the former case, the chemical microstructure cannot be described by any Markov process, contrary to the majority of conventional synthetic copolymers [ 10]. 

In order to obtain the expression for the components of the vector of instantaneous copolymer composition it is necessary, according to general algorithm, to firstly determine the stationary vector ji of the extended Markov chain with the matrix of transitions (13) which describes the stochastic process of conventional movement along macromolecules with labeled units and then to erase the labels. In this particular case such a procedure reduces to the summation ... 

Figure 9.13. Schematic representation of matrix polymerization of urea and formaldehyde in the presence of PAA (a) moderately concentrated solution of PAA and monomers (monomer molecules are not indicated), (b) 1st step of the process - gel formation (composite polycomplex + excess of PAA), (c) polycomplex PAA-PFU =1 1, (d) composite polycomplex + excess of PFU. Reprinted from I. M. Papisov, 0. E. Kuzovleva, S. V. Markov and A. A. Litmanovich, Eur. PoZy/n. J.,20,195(1984),...

It should be emphasized that for the Markovian copolymers, the knowledge of these structure parameters will suffice for finding the probabilities of any sequences LZ, i.e., for a comprehensive description of the structure of the chains of such copolymers at their given average composition. As for the CD of the Markovian copolymers, for any fraction of Z-mers it is described at Z 1 by the normal Gaussian distribution with covariance matrix, which is controlled along with Z only by the values of structure parameters (Lowry, 1970). The calculation of their dependence on time and on the kinetic parameters of a reaction system enables a complete statistical description of the chemical structure of a Markovian copolymer. It is obvious therewith to which extent a mathematical modeling of the processes of the synthesis of linear copolymers becomes simpler when the sequence of units in their macromolecules is known to obey Markov statistics. 

The active centers in this process are free radicals, whose reaction with double bonds of monomers leads to the growth of a polymer chain. In the framework of the ideal kinetic model, the reactivity of a macroradical is exclusively governed by the type of its terminal unit. According to this model, the sequence distribution in macromolecules formed at any moment is described by the Markov chain with elements controlled by the instantaneous composition of the monomer mixture in the reactor as... 

The Markovian character of the sequence distribution statistics in the macromolecules results [6, 94] from assumption about the steady-state of the radical concentrations, which usually holds with a high degree of accuracy in the copolymerization processes [6, 95], It is worth mentioning that along with such kinetic stationarity one should usually speak about the statistical stationarity. It means that when the number of the units in copolymer molecules exceeds 10-15, their composition practically becomes independent on degree of polymerization and is indistinguishable from the value predicted by the stationary Markov chain theory. This conclusion is supported by the theoretical [96,97,6] and experimental [98] evidence. 

For this simple case, the sdution of the kinetic equation also gives an evaluation of the composition hetetogeiwity of the reactimi products. Because of the stodiastic and independent nature of the siibstitutimi processes A B, the process can be regarded as a Markov chain of the zeroth order. From the general theory of regular Markov chains, it is known that the compositimi distribution is Bemoiillian in this case, with dispersion defined as... 

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