Spatial diffusion limit

Rayleigh quotient method has been used only in the spatial diffusion limited regime but not in the energy diffusion limited regime (see the next Seetion). 

The quantum thermodynamic factor S is the quantum correction to the Kramers-Grote-Hynes classical result in the spatial diffusion limited regime, derived by Wolynes " ... 

Eqs. 42 and 43 provide a uniform expression for the partial rates, the decay rate and the diffusion coefficient in terms of the energy loss 5, the quantum parameter a and the rate expression in the spatial diffusion limit. The mean squared traversal distance may be obtained directly from the ratio of the diffusion coefficient to the escape rate. 

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 iGLE also presents a novel approach for studying the reaction dynamics of polymers in which the chemistry is driven by a macroscopic force that is representative of the macroscopic polymerization process itself. The model relies on a redefined potential of mean force depending on a coordinate R which corresponds locally to the reaction-path coordinate between an n-mer and an (n + 1 )-mer for R = nl. The reaction is quenched not by a kinetic termination step, but through an (R(t))-dependent friction kernel which effects a turnover from energy-diffusion-limited to spatial-diffusion-limited dynamics. The iGLE model for polymerization has been shown to exhibit the anticipated qualitative dynamical behavior It is an activated process, it is autocatalytic, and it quenches... 

Kramers derived an expression for the rate in the underdamped and spatial diffusion limits. He did not derive a uniform expression for the rate valid for all values of the damping strength. This is the Kramers turnover problem, which was solved only in the late eighties by Poliak, Grabert, and Hanggi (12) and is known as PGH theory. 

Kramers model is simplistic. It is one-dimensional and assumes that the friction is Markovian—uncorrelated in time. A multidimensional generalization of Kramers problem in the spatial diffusion limit was proposed and solved by Langer (18). The multidimensional energy diffusion limit was solved by Matkowsky, Schuss, and coworkers (19,20). A multidimensional turnover theory has been recently formulated (21). 

Kramers approach to rate theory in the underdamped and spatial-diffusion-limited regimes spurred extensions which were applicable to the much more complex STGLE. Grote and Hynes (23) used a parabolic barrier approximation to derive the rate expression for the GLE in the spatial diffusion limit. Carmeli and Nitzan derived expressions for the rate of the GLE (24) and the STGLE (25) in the underdamped limit. The overdamped limit for the rate in the presence of delta correlated friction was solved using the mean first passage time expression (26,27). A turnover theory, valid for space- and time-dependent friction, has only been recently presented by Haynes, Voth, and Poliak... 

Here T1U is the so called one-dimensional TST estimate for the rate and is mainly determined by the one-dimensional potential of mean force w(q). The depopulation factor Y becomes much smaller than unity in the underdamped limit and is important when the rate is limited by the energy diffusion process. In the spatial-diffusion-limited regime, the depopulation factor Y is unity but the spatial diffusion factor becomes much smaller than unity. The major theme of this review is theoretical methods for estimating the depopulation and spatial diffusion factors. 

Truncating the potential of mean force as a parabolic barrier allows diagonalization of the Hamiltonian, as discussed in the previous section. Motion is separable along the generalized reaction coordinate p. TST will therefore be exact (in the parabolic barrier limit) if one chooses the dividing surface / = p — p. Inserting this choice into the TST expression for the rate (31) leads to the well-known KGH expression for the rate in the spatial diffusion limit ... 

The addition of the extra degree of freedom leads to a correction to the one-dimensional result in the form of a prefactor which is just the reduced barrier frequency at the saddle point. This result is very general. The KGH expression for the rate in the spatial diffusion limit (Eq. (78)) is just a special case in which the bath modes are harmonic and the coupling between the system and the bath is bilinear. However, Eq. (118) is much more general, in fact it is not yet completely defined since we have not yet shown how to determine the transformation coefficients a0, j = 1,. . ., N. 

The rate formula Eq. (141) is exact. Approximations enter because the nonequili-brium probability f(E) is not known exactly. Note though that in the equilibrium limit, replacing f E) by /eq( ) in Eq. (141) immediately leads to the KGH estimate for the rate. In the strong coupling limit, energy diffusion is fast and equilibrium is maintained throughout. In this limit, PGH theory reduces to the correct spatial diffusion limited expression. 

We will consider only one-dimensional surface diffusion. The potential w(q) is assumed to be periodic with a spacing a between wells. In the spatial diffusion limit, the particle might escape from a well, get trapped in an adjacent well, and after a long time escape with equal probability in either direction. In this case, the diffusion coefficient is proportional to the product of the spatial diffusion rate of escape from the well and the distance squared between the wells (116) ... 

The difference between the periodic potential and the single- or double-well potentials in the spatial diffusion limit, is that in the periodic potential the particle may escape from either side of the well. The spatial diffusion rate is therefore twice as large. 

Kramers final coup d etat in this work was a recasting of his rate expressions in terms of the then newly developed transition-state theory [8, 9], which has since become the most prominent rate theory in chemistry. In both limits Kramers was able to cast his result in terms of a multiplicative prefactor to the transition-state theory result. I note that the transition-state method to which Kramers compared takes only the solute degrees of freedom into consideration. Only some 40 years later was it recognized that multidimensional variational-transition-state theory [10], inclusive of all the solvent degrees of freedom, can reproduce the Kramers result in the high-viscosity, spatial-diffusion-limited regime [11-13]. 

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