Kramers theory generalization

E. Poliak The RRKM theory and Kramers theory and its later generalizations by Grote, Hynes, and other are two sides of the same coin. In the spatial diffusion limit, one can show that Kramers s rate expression is identical in form to the RRKM expression, that is, a ratio of equilibrium unidirectional flux and density of reactants. The difficult problem in the application of RRKM theory to the stilbene molecule with a few attached benzenes is whether the equilibration of energy occurs fully on the time scale of the isomerization. One should also... 

The idea in Kramers theory is to describe the motion in the reaction coordinate as that of a one-dimensional Brownian particle and in that way include the effects of the solvent on the rate constants. Above we have seen how the probability density for the velocity of a Brownian particle satisfies the Fokker-Planck equation that must be solved. Before we do that, it will be useful to generalize the equation slightly to include two variables explicitly, namely both the coordinate r and the velocity v, since both are needed in order to determine the rate constant in transition-state theory. 

In this section we shall explore a different kind of generalization of the Kramers theory to take into account the problems resulting from the breakdown of the time-scale separation between the reactive mode and its thermal bath. This problem may also be found in the multimodal theories in Section III when the nonreactive modes are not much faster than the motion along the reaction coordinate. 

The current attempts at generalizing the Kramers theory of chemical reactions touch two major problems The fluctuations of the potential driving the reaction coordinate, including the fluctuations driven by external radiation fields, and the non-Markovian character of the relaxation process affecting the velocity variable associated to the reaction coordinate. When the second problem is dealt with within the context of the celebrated generalized Langevin equation... 

The region ranging from y = oo to y can also be explored using the RMT. In Section IV we showed that the basic ideas of the RMT supplemented by the generalization of the Kramers theory to the multidimensional case allows us to recover the simple expression first derived by Grote and Hynes. This quite interesting formula reads... 

The generalization of the Kramers theory involving the problem of multiplicative fluctuations is still an open field of investigation. A large part of the discrepancies between the AEP and the other approaches do certainly derive from the fact that this theory is applied to a set of differential equations, the formal expression of which seems to be not completely legitimate. For instance, a rigorous microscopic derivation certainly cannot result in formal expressions such as those of Eqs. (38) and (44). 

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. 

In the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

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... 

This chapter reviews the generalizations of the Kramers model that were develojjed during the past few years. The result of this effort, which we may call the generalized Kramers theory, provides a useful framework for the theoretical description of activated rate processes in general and of chemical reaction rates in condensed phases in particular. Some applications of this framework as well as its limitations are also discussed. In the last few years there has also been substantial progress in the study of the quantum mechanical Kramers model, which may prove useful for condensed phase tunneling reactions. This aspect of the problem is not covered by the present review. 

In the following we review a generalized Kramers theory which overcomes many of the problems mentioned. Some of the ingredients of this generalized theory were developed many years ago, while may others are results obtained in recent years by us and other workers. 

The Kramers theory and its extensions have found many applications since the original work by Kramers. Recent application of the non-Markovian theory in the low-friction limit to thermal desorption was described by Nitzan and Carmeli. Another novel application of the Markovian theory is to transition from a nonequilibrium state of a Josephson junction. In what follows we shall briefly review the recent application of the generalized Kramers theory to chemical rate processes. More detailed reviews of the exjjerimental and theoretical status of this field may be found in Hynes. ... 

The characteristic time scale for the motion of the particle in the parabolic top barrier is the inverse barrier frequency, the sharper is the barrier, the faster is the motion. Typically, atom transfer barrier are quite sharp therefore the key time scale is very short, and the short-time solvent response becomes relevant instead of the long-time overall response given by the ( used in Kramers theory (see eq.(20)). To account for this critical feature of reaction problems, Grote and Hynes (1980) introduce the generalized Langevin equation (GLE) ... 

The approach adopted by Hoogendam et al. [14] for R is based on the general Kramers theory for reaction rates, in the isothermal limit, i.e., considering polymer chains (with fixed charge density) with fully relaxed conformations. The starting equation reads ... 

Until the mid-eighties it was generally accepted that the STGLE was a reasonable representation of the dynamics which allowed for dynamically induced corrections to the rate predicted by the TST method. Kramers theory was considered to be complementary but essentially different from TST. It was well understood that the STGLE is a continuum limit of a Hamiltonian in which the solute interacts nonlinearly with a harmonic bath... 

Schmickler has treated the ion-transfer process within the framework of a lattice-gas model [66]. In this model the interface is treated as an extended region in which the two solvents mix. The reaction rate is generally obtained using the transition state theory in combination with Kramers theory depending on whether frictional forces are important. 

This result is the generalization to the multivariate problem of the Kramers theory as found by Langer, and it confirms the validity of the site function method. 

The first theoretical description of chemical reartions was the van t Hoff-Arrhenius law, which describes the exponential dependence of the reaction rate on temperature. Its prefactor was then related to the theory of Brownian motion, which is the starting point for the transition state theory and Kramers theory. Both theories give the lifetime of a bond with an energy barrier Ay (which in general is itself temperature dependent) in the absence of an external force as... 

Berezhkovskii A M and Zitserman V Yu 1992 Generalization of the Kramers-Langer theory decay of the metastable state in the case of strongly anisotropic friction J. Phys. A Math. Gen. 25 2077-92... 

Berezhkovskii A M, Poliak E and Zitserman V Y 1992 Activated rate processes generalization of the Kramers-Grote-Hynes and Langer theories J. Chem. Phys. 97 2422... 

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