Markovian property

Exploiting the Markovian property of random process. Rib, it is possible to derive in a standard way the expression for an arbitrary chemical correlator. In particular, for the three-point correlator the expression... 

This is the memory-less exact time evolution equation. In the limit when the Markovian property is expected to hold, we should obtain by comparison with (64) the result... 

What is the significance of the Markovian property of a physical process Note that the Newton equations of motion as well as the time-dependent Schrodinger equation are Markovian in the sense that the future evolution of a system described by these equations is fully determined by the present ( initial ) state of the system. Non-Markovian dynamics results from reduction procedures used in order to focus on a relevant subsystem as discussed in Section 7.2, the same procedures that led us to consider stochastic time evolution. To see this consider a universe described by two variables, zi and z, which satisfy the Markovian equations of motion... 

Therefore the Markovian property (7.48) is satisfied provided that... 

If 7 (Z) satisfies the Markovian property (8.20), it follows from the Wiener-Khintchine theorem (7.76) that its spectral density is constant... 

The model of the reaction under discussion can be regarded as a Markovian process with locdly interacting components . The process is Markovian in the sense that the state of the chain at the time t + At depends stochastically on the state at the time t but not on the previous states. (This Markovian property in time should not be confused with the Markovian property in space, which is characteristic for the copolymerization proces). 

The Markovian property of vectorial processes can be lost by reducing the dimensionality of the process. One cannot expect that the knowledge of only a few components is again sufficient to predict the future probability (van Kampen, 1981, p. 80). 

Unperturbed Dimenaons.—The most fundamental model of the statistics of a polymer in solution is the linear Gaussian chain, which exhibits the Markovian property that the mean square end-to-end distance o is directly proportional to the number of chain segments i.e. ounperturbed dimensions , and b is the (Kuhn) statistical segment length. 

Exact solutions to the problem of determining Pf are not available for most cases of practical interest. One takes recourse to approximate analytical solutions or to Monte Carlo simulation-based methods. The analytical methods can be broadly classified into those based on level crossing statistics and those based oti Markovian property of system responses. The simulation-based methods typically employ suitable strategies to control the sampling variance. A discussion of these topics forms the subject of the following sections. 

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