Markovian regime

For the interbipolycondensation the condition of quasiideality is the independence of the functional groups either in the intercomponent or in both comonomers. In the first case the sequence distribution in macromolecules will be described by the Bernoulli statistics [64] whereas, in the second case, the distribution will be characterized by a Markov chain. The latter result, as well as the parameters of the above mentioned chain, were firstly obtained within the framework of the simplified kinetic model [64] and later for its complete version [59]. If all three monomers involved in interbipolycondensation have dependent groups then, under a nonequilibrium regime, non-Markovian copolymers are known to form. 

As in the linear regime, consider the sequential transition X X2 —U X3. Again Markovian behavior is assumed and the second entropy is added separately for the two transitions. In view of the previous results, in the nonlinear regime the second entropy for this may be written... 

The bath response function is usually associated with a characteristic correlation or memory time, t, which separates the non-Markovian (t t ) and the Markovian (t t ) temporal regimes, for example, 0(t) a Within the bath memory time, the bath modes oscillate coherently and in unison, and maintain memory of their interaction with the system, whereas after the correlation time has passed, the modes lose their coherent oscillations and forget their prior interactions [94]. 

Equation (320) predicts the TST result for very weak friction (Ar to ) and predicts the Kramers result for low barrier frequency (i.e., (ob —> 0) so that (2r) can be replaced by (0) in Eq. (322). If die barrier frequency is large (ia>b > 1013 s 1) and the friction is not negligible ( (0)/fi — cob), then the situation is not so straightforward. In this regime, which often turns out to be the relevant one experimentally, the effective friction (2r) can be quite small even if the zero frequency (i.e., the macroscopic) friction (proportional to viscosity) is very large. The non-Markovian effects can play a very important role in this regime. 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

Interestingly enough, one sees differences between the various variants of Markovian and non-Markovian theories already in static linear absorption spectra. In the regime of second-order perturbation theory in the coupling to the electromagnetic field the linear absorption line-shape / (ui) can be calculated from the Fourier transform of the dipole-dipole correlation function as... 

The significance and extent of the saturation effect can be easily estimated in the stationary regime available for analytical investigation. The stationary solution of the Markovian equations (3.441) is trivial... 

In other words, if we are exploring the low-friction regime, the interplay of non-Markovian statistics and external field renders the system still more inertial, thereby widening the range of validity of the formula provided by Kramers for the low-friction regime provided that y be replaced by... 

The Markovian description, inherent in Eqs. (2.1) and (2.2), is unrealistic for most chemical situations as it assumes that the relaxation time of the medium surrounding the molecule is faster than all molecular time scales. This assumption obviously breaks down in the low-friction regime where the escape rate is dominated (or at least influenced) by the well dynamics (that is, intramolecular motion), since typical molecular frequencies are of the same order or larger than intermolecular (solvent) frequencies. Even for higher friction, where the escape is dominated by the dynamics in the barrier regime, the Markovian assumption can fail because the characteristic barrier time... 

Figure 1. Different regimes of the system-environment coupling. Here a represents the coupling strength between a quantum system and its environment and k is the environment relaxation rate. Note that, in the Markovian limit, the system decouples from the environment altogether for a/n — 0 unless a2/ remain finite.

Consider now the barrier crossing problem in the barrier controlled regime discussed in Section 14.4.3. The result, the rate expressions (14.73) and (14.74), as well as its non-Markovian generalization in which cor is replaced by k ofEq. (14.90), has the structure of a corrected TST rate. TST is exact, and the correction factor becomes 1, if all traj ectories that traverse the barrier top along the reaction coordinate (x ofEq. (14.39)) proceed to a well-defined product state without recrossing back. Crossing back is easily visualized as caused by collisions with solvent atoms, for example, by solvent-induced friction. 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

In the particular case of a stationary regime, the stochastic dependence among the random variables attached to the different carriers becomes Markovian and our formalism becomes much simpler, and we can express the transit time corresponding to a state u, 0)u, in terms of the lifetimes of the intermediates corresponding to the different pathways ... 

In many systems comprising a large number of particles, even though a detailed quantum treatment of all degrees of freedom is not necessary, there may exist subsets that have to be treated quantum mechanically under the influence of the rest of the system. If the typical timescales between system and bath dynamics are very different, Markovian models of quantum dissipation can successfully mimic the influence of the bath onto the system dynamics [2]. However, in the femtosecond regime studied with ultrashort laser pulses, the so-called Markov approximation is not generally valid [3]. Furthermore, very often the bath operators are assumed to be of a special form (harmonic, for instance) which are sometimes not realistic enough. 

Figure 9.20. Population of the reactant state as a function of time normalized with respect to its stationary value in the normal (AG = 0) and the inverted (AG = — 2 ,) regimes calculated on the basis of a stochastic Markovian model of reversible ET assisted by a fast vibrational mode with E = ) = 9k T, and KXi/k T= 10. In the normal regime, the reaction is always activated and well described by a single exponential. Dashed line corresponds to the long-time limit in the inverted regime, described by /c j. (Reproduced from [309] with permission. Copyright (2001) by the American Institute of Physics.)...

The SSSV theory is a generalization of the Smoluchowski-Vlasov theory of Calef and Wolynes [41] based on the interaction-site model, and its application has shown that the theory predicts some of the essential features of van Hove correlation functions of water [46]. However, the SSSV theory in its current form is valid only in the diffusion regime, and the non-Markovian effects in the memory function, which are important in the dense-liquid dynamics, cannot be properly taken into account. 

Note that the Markovian dissipative dynamical process is governed by a frequency -independent Il-dissipator in eqn (13.48) that also implies an 5-in-dependent /C-tensor here, while the Markovian kinetic rate process is governed by the constant rate matrix, Al(j) = iC(0). Equation (13.52) would indicate non-Markovian rates in general, even with Markovian dissipative dynamics. However, kinetic rates are physically concerned with post-coherence events, in which the coherence-to-coherence dynamics timescale, the magnitude of l ccl is short compared with the relevant of interest. Therefore, the kinetic rate matrix of eqn (13.52) in the kinetics regime is often of K s) K K 0) = - /Cpp -I- /Cpc cc cp, where /Cpp = 0 in the absence of level relaxations. 

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