Markovian approximation

In Markovian approximation (zj =0) this quantity approaches the famous Debye plateau shown in Fig. 2.3 whereas non-Markovian absorption coefficient (2.56) tends to 0 when ft) — 0 as it is in reality. This is an advantage of the Rocard formula that eliminates the discrepancy between theory and experiment by taking into account inertial effects. As is seen from Eq. (2.56) and the Hubbard relation (2.28)... 

Fig. 4.5. The broadening of the P-R doublet (Atc = n/2, V2f = n/8) in the integral non-Markovian theory (solid line) and in the Markovian approximation (dotted line).

Previously, stochastic Schrodinger equations for a quantum Brownian motion have been derived only for the particle component through approximated equations, such as the master equation obtained by the Markovian approximation [18]. In contrast, our stochastic Schrodinger equation is exact. Moreover, our stochastic equation includes both the particle and the field components, so it does not rely on integrating out the field bath modes. 

G. R. Fleming Using a Markovian approximation for the bath is always likely to be a very poor approximation for electronic dephasing. For vibrational dynamics, while the issue needs careful examination,... 

Here an important definition for the rate constant of free carrier production, k, is given. The latter differs from the ionization rate constant by a multiplier equal to the charge separation quantum yield tpm, obtained in the Markovian approximation. This difference indicates that the number of photogenerated ions that avoid geminate recombination and become free is less than their total amount, cpm < 1. 

Let us note that in these dimensionless variables c plays the role of barrier height, while y = A /Xj is the effective friction constant. This can be shown by following the heuristic argument of Section I of Chapter II. Let us assume that Ai relaxes so fast that A, Eq. (56c), is approximately zero the system of Eqs. (3) will be recovered provided y = Ai/X. Moreover, in Section V.C of Chapter II it is shown that the AEP corrections to the trivial Markovian approximation of Eq. (58), n =1, are perturbation terms in the parameter... 

It is easy to determine Q from the scattering matrix S and of its derivative dS/dE. Simple expressions in closed form can be obtained when D does not depend on the energy (Markovian approximation). This situation is usually encountered far from the energy thresholds. In this case, the choice D = 1 leads to Go(z) = —Itt 1 and from Eqs. (65) and (67) one gets the exact formula... 

A = and M.22 z) is the matrix element (2 M(z) 2). All the other matrix elements ot the memory operator are zero [63]. The appearance ot the tactor i corresponds to the Hermitian definition ot the Liouvillian which is not universal. Equation (98) is an exact z-dependent expression. This z-dependence can be eliminated by approximating M22(z) by a positive constant r (Markovian approximation). This approximation is justified since the microscopic correlation time is much shorter than the regression time ot the fluctuation. Then Eq. (98) transtorms into the z-independent effective Liouvillian... 

Perhaps, it was Hynes who initiated two of the most popular so far semi classical non-Markovian approximations [84]. The first approximation was inspired by the success of the [1,0]-Pade approximant, which turns out to be exact in the Markovian limit. This approximation is sometimes referred to as the substitution approximation, because effectively one substitutes non-Markovian two-point distribution function (9.46)-(9.47) into the Markovian expressions (9.50)-(9.51) for the rate kernel. The substitution approximation was shown to work rather well for the case of biexponential relaxation with similar decay times [102]. However, as Bicout and Szabo [142] recently demonstrated, it considerably overestimates the reaction rate when the two relaxation timescales become largely different (see Fig. 9.14). They also showed that for a non-Markovian process with a multiexponential correlation function, which can be mapped onto a multidimensional Markovian process [301], the substitution approximation is equivalent to the well-known Wilemski-Fixman closure approximation [302-304]. A more serious problem arises when we try to deal with the... 

In a first-order Markovian approximation, there are two irdeper nt conditional xobabUities PA/A(fh4 probability of finding A to the r t of A) and PA/s(the probability of fintUng A to the ri t of B), and all other dharacteridics of the dudn structure are expressed in terms of them ... 

The soluticm of the system (37) with initial conditions Pa/a = Pa/b 1 at t = 0 g ves a complete description of the chain structure in a first-order Markovian approximation. 

In a second-order Markovian approximation there are four independent conditional probabilities Pa/aa. Pa/ba. Pa/ab.Pa/bb a third-order, there are ei t, and so on. 

In Fig. 1 the results of Markovian approximations are compared with the results of exact andytical calculation. One can see that the use of Markovian approximations can be rather efficient in the case of a retarding nei boring-groups effect and in the case of small accelerations. The most optimal is evidently a second-order Markovian approximation, as the accuracy of a first-order approximation is not sufficient, while the accuracy of a third-order approximation is the same as the second order, but twice as many equations are required. The Markovian approximatirms are not very useful for greater accelerations, as they permit only the calMation of the probabilities of very diort sequences with sufficient accuracy. 

S 100 broken curve, first-oiOer Markovian approximation solid curve, modiHed flist-ordei Markovian apMoximation points, Monte Carlo calculation ... 

Pa/a arid Pa/6> obtained from the solution of the system (37), allow the cdculation of the function and the disperdon of composition distribution in the first-order Markovian approximation "In Figs. 3,4 the resultsofatthcalculations are compared with the results of Monte Cado computa m. It can be seen that in this case (ko ki kj = 1 5 100), the deviation is very strong, indicating tiie inaccuracy of this approximation. 

Plat, Utmanovich and Noah proposed a modified form of a first-order Markovian approximation. In this approach, the noticm of a Gaussian distribution form is maintained, and its disperdon can be calculated according to Eq. (S). But the Markovian transitional probabilities, Pa/a and Pa/b, are proposed for csdculating by the following expressions (36) ... 

In other words, when a modified first-order Markovian approximation is formulated, one assumes that at the time under consideration, the d n is a first-order Markovian one, but dl its previous history is described by the accurate equations. 

For the concrete calculations it is easily observed that the self-consistency of the involved parameters requires extra-input trial information. This supplementary knowledge can be avoided if we consider an additional limit. It is introduced the so-called Markovian approximation that regards the Eq. (2.122) limit (see discussion of Section 2.5.5) hp 0. We should mention here that this limit corresponds with the ultra-short correlation of the involved electrons with the apphed external potential. This can be motivated by remembering the temporal nature of the quantum statistical quantity hficcAf, see for instance Eq. (3.36). This means that assuming initially (Ar = 0 = 0) the electronic system in the free... 

In order to obtain the Markovian approximations to G3 in (6.16) we remember that, apart from a normalization, Gj(RR 0 LsO) represents the number of chain configurations that pass through 00, R s, and RL. Hence the desired Markovian approximation to Gj is the number of chains running from 00 to R s times the number running from R s to RL or... 

The Edwards SCF therefore results from introducing the Markovian approximation to the lowest member of the hierarchy (6.16). This approximation can be introduced into higher members of the hierarchy also. If this is done in the equation for G in terms of G4, we obtain a result that is almost identical to the corrected Reiss result. Specifically, if we use Edwards formal approach (see next section for details) to find the best approximation to G3(RR 0 LsO) and therefore to P(R J[RT]) via (2.18)... 

Here, can be easily evaluated via Eq. (3.7), but with TZs there being replaced by TZu [cf. Eq. (B.12b)] that may be considered as the Markovian dissipation superoperator. This statement may be supported by the arguments that the Markovian approximation amounts to the following two conditions (i) The bath correlation time is short compared with the reduced system dynamics (ii) The correlated effects of driving and dissipation can be neglected so that the Green s function G(t r) in the memory kernel can be replaced by its field-free counterpart Gs t t) = In this case, Eq. (B.9) reduces to... 

AE = [ 4 H ) 4> H 4>) of the initial state must be much smaller than the half-width in energy W of the doorway state QQH correlation time U/W s much smaller than the fluctuation time h/AE (see Section IV D in Ref. [9]). Using (7), Eq. (50) can be written also as... 

Markovian approximation in a coarse-grained description of atomic systems. J. Chem. Phys., 125, 204101. 

In order to circumvent the difficulties in dealing with Eq. (548), investigators often introduce an ansatz for the memory functions, i.e., an analytic form is assumed for the memory functions on the basis of mathematical convenience and physical intuition. The simplest and most widely used memory function model is the Markovian approximation In essence, one neglects non-Markovian retardation by assiuning that the memory function Kjk t) can be written as... 

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