The high friction limit

The friction coefficient y defines the timescale, of thermal relaxation in the system described by (8.13). A simpler stochastic description can be obtained for a system in which this time is shorter than any other characteristic timescale of our system. This high friction situation is often referred to as the overdamped limit. In this limit of large y, the velocity relaxation is fast and it may be assumed to quickly reaches a steady state for any value of the applied force, that is, v = x = 0. This statement is not obvious, and a supporting (though not rigorous) argument is provided below. If true then Eqs (8.13) and (8.20) yield 

as discussed in Section 8-2.1, not relative to the environmental relaxation time. 

This is a Langevin type equation that describes strong coupling between the system and its environment. Obviously, the limit y 0 of deterministic motion cannot be identified here. 

Why can we, in this limit, neglect the acceleration term in (8.13) Consider a particular realization of the random force in this equation and denote —dV/dx + R t) = F t). Consider then Eq. (8.13) in the form 

If F is constant then after some transient period (short for large y) the solution of (8.22) reaches the constant velocity state 

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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... 

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)... 

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

It was shown that Kramers theory, in the high-friction limit resulting in Eq. la with a= 1, cannot accurately describe the segmental dynamics of a synthetic polymer in dilute solutions. 

Kramers theoretically investigated the passage of a particle over an energetic barrier in the presence of fi iction caused by colhsions with the other molecules. He obtained an Arrhenius-type expression for the rate constant of this process, i.e., an exponential dependence on the height of the barrier a- The pre-exponential term contains the mass of the particle, the friction coefficient, and parameters describing the shape of the barrier. At the high friction limit, the equation simplifies and acquires the form... 

SCEATS - In your work you have considered the Kramers model for 1-dimensional motion in the high and low friction limits and the Smoluchowski model for 3-dimensional motion. In my paper at this conference, I will show that the 3-d motion can be reduced to a one-dimensional problem by use of the one dimensional potential V(R)-2kTlnR, and that application of Kramers theory to the dynamics on this potential yields the Smoluchowski-Debye result in the high friction limit. Hence unimolecular and bimolecular reactions can be compared using the same model. 

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