Markov statistics configurational sequences

This software uses statistical probabilities to calculate ) configurational sequences of polymers from NMR data. ) First-order and Second-order Markov propagation models ) are developed from initial giiess values of the dyads ) which are fitted into the statistical equations. If the ) intensities of the triads are known, then the dyads are... 

MMA polymerization is one of the most studied systems and was thought to be explicable, within experimental error, in terms of Bemoullian statistics. Moad et al. have made precise measurements of the configurational sequence distribution for PMMA prepared from C-labeled monomer. It is clear that Bemoullian statistics do not provide a satisfactory description of the tactidty. This finding is supported by other work. First-order Markov statistics provide an adequate fit of the data. Possible explanations include (1) penpenulti-mate unit effects are important and/or (2) conformational equilibrium is slow (Section 3.04.3.1.1). At this stage, the experimental data do not allow these possibilities to be distinguished. 

While static Monte Carlo methods generate a sequence of statistically independent configurations, dynamic MC methods are always based on some stochastic Markov process, where subsequent configurations X of the system are generated from the previous configuration X —X —X" — > with some transition probability IF(X —> X ). Since to a large extent the choice of the basic move X —X is arbitrary, various methods differ in the choice of the basic unit of motion . Also, the choice of transition probability IF(X — > X ) is not unique the only requirement is that the principle... 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

The MC technique is a stochastic simulation method designed to generate a long sequence, or Markov chain of configurations that asymptotically sample the probability density of an equilibrium ensemble of statistical mechanics [105, 116]. For example, a MC simulation in the canonical (NVT) ensemble, carried out under the macroscopic constraints of a prescribed number of molecules N, total volume V and temperature T, samples configurations rp with probability proportional to, with, k being the Boltzmann constant and T the... 

To prove the existence of a Bernoullian sequence, only triad information is needed, but tetrad information is required to verify a first-order Markov sequence. More complicated stereochemical mechanisms are possible of course (such as reactions controlled by penultimate configurations), and these would have to be fitted by more complex statistical analyses requiring knowledge of more than two probabilities. However, most detailed analyses to date on the tactleities of polymers obtained in homogeneous free radical or ionic polymerization reactions have been found to conform to either Bernoullian or first-order Markovian statistics. 

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