Markov assumption

Furthermore, the Markov assumption implies that the Wt satisfy dW(n, 11 m, 0)... 

Planck equation. In this way the actual equations of motion need be solved only during At, which can be done by some perturbation theory. The Fokker-Planck equation then serves to find the long-time behavior. This separation between short-time behavior and long-time behavior is made possible by the Markov assumption. 

The knowledge of the two-minima energy surface is sufficient theoretically to determine the microscopic and static rate of reaction of a charge transfer in relation to a geometric variation of the molecule. In practice, the experimental study of the charge-transfer reactions in solution leads to a macroscopic reaction rate that characterizes the dynamics of the intramolecular motion of the solute molecule within the environment of the solvent molecules. Stochastic chemical reaction models restricted to the one-dimensional case are commonly used to establish the dynamical description. Therefore, it is of importance to recall (1) the fundamental properties of the stochastic processes under the Markov assumption that found the analysis of the unimolecular reaction dynamics and the Langevin-Fokker-Planck method, (2) the conditions of validity of the well-known Kramers results and their extension to the non-Markovian effects, and (3) the situation of a reaction in the absence of a potential barrier. 

The Markov assumption asserts that the past and the future are statistically independent when the present is known this means that... 

Under the Markov assumption, the hierarchy of the joint probability densities [Eqs. (4.9)] describing the evolution of the system takes the following form ... 

The largest value for the PFD was obtained when using Markov analysis. We notice that for this method, we are not able to distinguish between independent and dependent failures. This follows as a natural consequence of the Markov assumption which states The probability of the system being in state i + 1 is only dependent ofstate i and independent ofstate i—, see Ross (99). 

The Markov assumption applies in the sense that the state of the system at a given time to + t, is prescribed entirely by its state at time to. Accordingly, the probability P(H , t + t) of state at time To + t is written in terms of the time-delayed conditional probability CfO, To -I- tlQp, To) of state at time To + T given state Cp at time To... 

Obviously this initial distribution is concentrated at the point n = no. The Markov assumption is now made that the conditional probability does not depend on the motion before time t which led to the initial configuration no. The expression (3.9) is zero by definition if no + or no contain negative numbers n ai + kai or n ah these are invalid configurations. 

This can now be reduced with the help of the Markov assumption. 

Markov Assumption The Maikov assumption, sometimes referred to as static world assumption, specifies that if one knows the robot s location, future measurements are independent of past ones (and vice versa). 

This is the simplest of the models where violation of the Flory principle is permitted. The assumption behind this model stipulates that the reactivity of a polymer radical is predetermined by the type of bothjts ultimate and penultimate units [23]. Here, the pairs of terminal units MaM act, along with monomers M, as kinetically independent elements, so that there are m3 constants of the rate of elementary reactions of chain propagation ka ]r The stochastic process of conventional movement along macromolecules formed at fixed x will be Markovian, provided that monomeric units are differentiated by the type of preceding unit. In this case the number of transient states Sa of the extended Markov chain is m2 in accordance with the number of pairs of monomeric units. No special problems presents writing down the elements of the matrix of the transitions Q of such a chain [ 1,10,34,39] and deriving by means of the mathematical apparatus of the Markov chains the expressions for the instantaneous statistical characteristics of copolymers. By way of illustration this matrix will be presented for the case of binary copolymerization ... 

If we will consider arbitrary random process, then for this process the conditional probability density W xn,tn x, t, ... x i,f i) depends on x1 X2,..., x . This leads to definite temporal connexity of the process, to existence of strong aftereffect, and, finally, to more precise reflection of peculiarities of real smooth processes. However, mathematical analysis of such processes becomes significantly sophisticated, up to complete impossibility of their deep and detailed analysis. Because of this reason, some tradeoff models of random processes are of interest, which are simple in analysis and at the same time correctly and satisfactory describe real processes. Such processes, having wide dissemination and recognition, are Markov processes. Markov process is a mathematical idealization. It utilizes the assumption that noise affecting the system is white (i.e., has constant spectrum for all frequencies). Real processes may be substituted by a Markov process when the spectrum of real noise is much wider than all characteristic frequencies of the system. 

The derivation of the terminal (or hrst-order Markov) copolymer composition equation (Eq. 6-12 or 6-15) rests on two important assumptions—one of a kinetic nature and the other of a thermodynamic nature. The Erst is that the reactivity of the propagating species is independent of the identity of the monomer unit, which precedes the terminal unit. The second is the irreversibility of the various propagation reactions. Deviations from the quantitative behavior predicted by the copolymer composition equation under certain reaction conditions have been ascribed to the failure of one or the other of these two assumptions or the presence of a comonomer complex which undergoes propagation. 

Once a connection is made it is discontinued at a random moment, rn = an. (This somewhat unrealistic assumption is needed for the Markov character.) The master equation is... 

Conclusion. In classical statistical mechanics the evolution of a many-body system is described as a stochastic process. It reduces to a Markov process if one assumes coarse-graining of the phase space (and the repeated randomness assumption). Quantum mechanics gives rise to an additional fine-graining. However, these grains are so much smaller that they do not affect the classical derivation of the stochastic behavior. These statements have not been proved mathematically, but it is better to say something that is true although not proved, than to prove something that is not true. 

More convenient and entirely sufficient for the present purpose is the calculation of ratio of rate constants. The calculation will be reviewed for a one-way first-order Markov process. A one-way mechanism is chosen because it is intuitively the most appropriate model for a free radical mechanism. Furthermore it has some experimental support. The assumption of a first-order Markov process does not rule out higher Markov processes. The differentiation between a first-order Markov process and higher order Markov processes is however possible experimentally in very rare cases because it involves the determination of tetrad, pentad, etc. fractions (11, 12, 13, 14). A Bemoullian process is ruled out by the analysis of the data of Table I. 

The Markov model is a special case of the semi-Markov model in which all the retention-time variables are exponentially distributed, Ai Exp(/v(), and Ki is the parameter of the exponential. In this case, the semi-Markov model parameters are k, = ha and ujtl = h,l j//q for j / i and i,j = 1,..., m. This results from the assumption of the Markov model given in (9.1), which implies that the conditional transfer probability from i to j in a time increment At is time-invariant, or in other words is independent of the age of the particle in the compartment. Particles with such a constant flow rate, or hazard rate, are said to lack memory of their past retention time in the compartment. 

Levy diffusion is a Markov process corresponding to the conditions established by the ordinary random walk approach with the random walker making jumps at regular time values. To explain why the GME, with the assumption of Eq. (112), yields Levy diffusion, we notice [50] that the waiting time distribution is converted into a transition probability n(x) through... 

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. 

When the assumptions (4.19) are valid the Markov chain describing a sequence distribution in macromolecules is found to be reversible. It follows from the chain definition and relationships that ... 

Our objective is to understand the statistical properties of the reactive trajectories in the ensemble (2). We will try to do so under minimum assumptions about the dynamics of x t), but we have to require the following from the start. First we require that the dynamics be Markov, i.e. given x t), its future x t ) . t > t and its past x t ) . t probability density m(x), i.e. given a suitable observable F x) and a generic trajectory x t) —00 [Pg.456]

The above description can be framed by a Markov process in the following way. A reptile was defined as system and the following states shown in the figure were selected, i.e. Si = reptile at position i, i = 1, 2,..., 11. On the basis of above states, the following matrix may be established. Some assumptions made were ... 

The models discrete in space and continuous in time as well as those continuous in space and time, led many times to non-linear differential equations for which an analytical solution is extremely difficult or impossible. In order to solve the equations, simplifications, e.g. linearization of expressions and assumptions must be carried out. However, if this is not sufficient, one must apply numerical solutions. This led the author to a major conclusion that there are many advantages of using Markov chains which are discrete in time and space. The major reason is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Additional reasons are detailed in Chapter 1. It will be shown later that this presentation coincides also with the fact that it yields the finite difference equations of the process under consideration on the basis of which the differential equations have been derived. 

The simplest model [73, p.l80] of a two impinging-stream reactor is shown schematically in Fig.4.S-l. On the LHS is demonstrated the actual configuration of the reactor and on the RHS the Markov-chain model. The latter employs the following considerations and assumptions ... 

Theoretical interpretation of the concentration dependence of equivalent conductivity for simple binary mixtures was first presented by Markov and Shumina (1956). It should be emphasized that this theory, even when considering simple structural aspects, represents rather a method of interpretation of the experimental data than a genuine picture of the structure of the melt. In molten salts generally only ions and not molecules are present, hence the conception of Markov and Shumina (1956) is to be considered also from this aspect. Their theory is based on the assumption that the electrical conductivity of a mixture of molten salts varies with temperature like pure components. In this respect, general character of the electrical conductivity dependence on composition, indicating the interaction of components in an ideal solution, could be expected. 

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