Non-Markovian effects

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 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. 

Anomalous Negative Magnetoresistance Caused by Non-Markovian Effects... 

Other reactions have been studied that appear to also require consideration of non-Markovian effects. For example, in a recent study of the photoisomerization of tra/u-stilbene and trdeviations from the Kramers rate in the case of /ranj-stilbene. These discrepancies were tentatively related to the larger flexibility of this molecule but appeared to be well simulated by the non-Markovian theory of Grote and Hynes. ... 

Figure 12 shows that this simple expression agrees fairly well with both the theory of Carmeli and Nitzan and the result of their purely numerical calculations. The plots in Fig. 12 show how well the non-Markovian effects on the rate may be simulated by a simple multiplicative factor (1 -I- UqT ). For the sake of comparison, we fitted an expression with this factor to Carmeli and Nitzan s results so as to include their accurate Markovian rate. 

The authors of Chapter IX use the theoretical methods developed in this book to illustrate the state of the art in the field of chemical reaction processes in the liquid state. The well-known Kramers theory can be properly generalized so as to deal successfully with non-Markovian effects of the liquid state. From a theoretical point of view the nonlinear interaction between reactive and nonreactive modes is still an open problem that touches on the subject of internal multiplicative fluctuations. 

The Need for Generalization of the Kramers Theory The Generalized Kramers Model Non-Markovian Effects in the One-Dimensional Case The Escape Rate of a Non-Markov Multidimensional Process... 

In this limit, however, non-Markovian effects are no longer important because co(E - Eg) 0 so that co(E) so the evolution becomes Markovian near the barrier. Eqs. (5.49) and (5.50) then imply that Eq. (5.54) is identical to Eq. (2.41) in this limit. 

For M = 1 this yields t = (/c R ie ) S which is exactly the low-friction generalized Kramers result. (The friction y is replaced here by the more general fcvR. which incorporates non-Markovian effects if present. In this low-friction limit i = Eg.) For intermediate large values of n Eq. (6.72) may be approximated by vibrational relaxation rate kyg(E) = dE/dt,... 

The theory outlined in Section II was based on the assumption that vibrational relaxation occurs on a time scale slow compared with the bath (translation and rotation) degrees of freedom. In this case, a Markov approximation (separation of time scales) can be made and the relaxation can be described through rate equations the resulting population decay is given by an exponential or a sum of exponentials. This time scale separation assumption is certainly valid for the small molecules that require a nanosecond or longer to relax, but in the picosecond or subpicosecond domain which applies to larger molecules non-Markovian effects may be present. In this section we outline the results of some theoretical studies of non-Markovian (nonexponential) relaxation. 

Abbott and Oxtoby have carried out exact quantum mechanical simulations of a two-level system strongly coupled to a heat bath in order to investigate non-Markovian effects. The hamiltonian has the form... 

To date, no experiments have provided unambiguous evidence for non-Markovian effects on vibrational relaxation in liquids. However, it seems likely that as time resolution and sensitivity are improved such effects will be observed. 

Up to the early 1970s a kinetic approach to the time-dependent properties of fluids was synonymous with a framework based on the Boltzmann equation and its extension by Enskog, in which a central role is played by those dynamical events referred to as uncorrelated binary collisions [29]. Because of this feature the Boltzmann equation is in general not applicable to dense fluids, where the collisions are so frequent that they are likely to interfere with each other. The uncorrelation ansatz is clearly equivalent to a loss of memory, or to a Markov approximation. As a result, for dense fluids the traditional kinetic approach should be critically revised to allow for the presence of non-Markovian effects. 

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. 

What is intended in the present contribution is to review a recently developed theory of the dynamics of molecular liquids which overcomes all these unwelcome features of currently available theories a self-contained theory that requires only the knowledge of parameters for potential functions and molecular geometry (such as bond lengths) as in simulation studies a theory whose memory function incorporates the non-Markovian effects a theory which accounts for all the characteristic features of collective excitations in molecular liquids. This will be accomplished by generalizing successful frameworks described so far in this section to molecular liquids based on the interaction-site representation. 

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