High friction limit

The specific form of the short-time transition probability depends on the type of dynamics one uses to describe the time evolution of the system. For instance, consider a single, one-dimensional particle with mass m evolving in an external potential energy V(q) according to a Langevin equation in the high-friction limit... 

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. 

In the high-friction limit, the Smoluchowski expression (4.152) can be used to determine the time evolution of the particle and can be written as... 

Numerical results have been given,173 174 based on the original model of Kramers, which reproduce the high-friction limit accurately. Recently, the Kramers method has been reformulated135,137,175 in order to emphasize the underlying assumptions, namely, the quasi-stationary (long-time) behavior of the system and the concept of a two-state system... 

In Kramers theory that is based on the Langevin equation with a constant time-independent friction constant, it is found that the rate constant may be written as a product of the result from conventional transition-state theory and a transmission factor. This factor depends on the ratio of the solvent friction (proportional to the solvent viscosity) and the curvature of the potential surface at the transition state. In the high friction limit the transmission factor goes toward zero, and in the low friction limit the transmission factor goes toward one. 

Note that D equals Dy in the high-friction limit. 

The high-friction limit A/cuy 1 is interesting because of the nonlinear density dependence of the rate. In this limit the general expression of the rate constant is... 

As an example for a stochastic process consider a system evolving according to the Langevin equation in the high friction limit where inertial effects can be neglected and momenta are not required for the description of the system ... 

As explained in Sect. 6.5, this equation arises from (21) in the high friction limit as 7 1. The equilibrium probability density associated with (112) is... 

The physical manifestation of friction is the relaxation of velocity. In the high friction limit velocity relaxes on a timescale much faster than any relevant observation time, and can therefore be removed from the dynamical equation, leading to a solvable equation in the position variable only, as discussed in Section 8.4.4. The Fokker-Planck or Kramers equation (14.41) then takes its simpler, Smoluchowski form, Eq. (8.132)... 

We do not, however, need to solve Eq. (14.45) to reach the most important conclusion about this high friction limit. The structure of Eq. (14.45) implies at the outset that the time can be scaled by the diffusion coefficient, so any calculated rate should be proportional to D, and by (14.46) inversely proportional to the friction y. The rate is predicted to vanish like as y oo. Recalling that the TST rate does not depend on y it is of interest to ask how the transition between these different modes of behavior takes place. We address this issue next. 

In the intermediate to high friction limit, the steady-state Fokker-Planck equation can be solved with the potential in (3.19) to obtain the following more general result ... 

In summary then, we expect the usual sink Smoluchowski description to be valid for a slow (on the order of the diffusion rate) reaction. The usual description also entails neglect of the operator character of D, (high friction limit) and assumes that velocity correlations relax rapidly so that the 2 = 0 limit of D, can be taken. The coupling between the center of mass and relative motion is also neglected in the usual formulations. These latter conditions reduce (9.38) to (now in /-space)... 

Shepherd, T.D., Hernandez, R. Ghemical reaction dynamics with stochastic potentials beyond the high-friction limit, J. Ghem. Phys. 2001,115,2430. 

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