Ohmic friction

The exact solution of the instanton equation in the large ohmic friction limit has been found by Larkin and Ovchinnikov [1984] for the cubic parabola (3.18). At T = 0... 

This semiclassical turnover theory differs significantly from the semiclassical turnover theory suggested by Mel nikov, who considered the motion along the system coordinate, and quantized the original bath modes and did not consider the bath of stable normal modes. In addition, Mel nikov considered only Ohmic friction. The turnover theory was tested by Topaler and Makri, who compared it to exact quantum mechanical computations for a double well potential. Remarkably, the results of the semiclassical turnover theory were in quantitative agreement with the quantum mechanical results. 

For ohmic friction 17(f) = r>5(f), A = [a>2 + (r /2)2]1/2 — ry/2, and (5.32) becomes the celebrated Kramers formula for classical escape out of a metastable well in the case of moderate and strong damping [Kramers, 1940]. In accord with the predictions of multidimensional theory, the crossover temperature should be... 

For 8=1, the noise spectral density is a constant (white noise), at least in the angular frequency range co oo, the Langevin force F(t) is delta-correlated, and the Langevin equation is nonretarded. The white noise case corresponds to Ohmic friction. The cases 0 < 8 < 1 and 8 > 1 are known respectively as the sub-Ohmic and super-Ohmic models. Here we will assume that 0 < 8 < 2, for reasons to be developed below [28,49-51]. 

Results presented here will be derived from the Hamiltonian representation. Although almost all of them may be derived using other methods, I find that the Hamiltonian approach is the simplest in the sense that memory friction is as easy to handle as ohmic friction. The central building block for the parabolic barrier case is the normal mode transformation of the Hamiltonian, discussed in detail in Sec. Ill. A. In Sec. III.B the normal mode transformation is used to construct normal mode free-energy surfaces. 

The Langevin equation, in which the time-dependent friction is ohmic, plays a special role in the theory of activated rate processes. Kramers originally formulated the problem in terms of ohmic friction. In most applications in chemistry the friction will not be ohmic however, the ohmic case is the simplest to analyze. Apart from the historic importance, the analytical simplicity helps in understanding and analyzing more difficult cases of space- and time-dependent friction. A summary of important results for the parabolic barrier and ohmic friction is presented in Sec. III.C. 

As mentioned in the Introduction, ohmic friction is (Dirac) 8 correlated ... 

Note that in contrast to the ohmic friction, the normal mode friction function has memory. The corresponding spectral density of normal modes /(X) (cf. Eq. (49)) decays as X4, in a sense it is much better behaved than the ohmic spectral density J(o>), which increases without bound with u>. Finally, the collective bath mode frequency, fl (cf. Eq. (59)),... 

It is easy to show that for ohmic friction this reduces to Kramers function. Note though that we have immediately derived the generalization of Kramers function which is valid for arbitrary memory friction. By demonstrating that Kramers function is the projection onto the physical phase space of Pechukas characteristic function, we have actually shown that Kramers function may be interpreted as the probability for reaction at the given system phase space point. This interpretation was subsequently further developed by Kohen and Tannor (70). 

Kramers (11) correction to the one-dimensional TST rate came from consideration of the properties of the Fokker-Planck equation in the vicinity of the barrier in the presence of ohmic friction. As noted in the previous section, if one considers only the parabolic barrier limit, the Fokker-Planck equation may be solved analytically. Grote and Hynes (23) and Hanggi and Mojtabai (65) generalized Kramers result to include the case of memory friction and the GLE. A different approach (31) would be to consider the Hamiltonian equivalent, Equation (29), for space-independent coupling (g(q) = q in Eq. (29)) in the parabolic barrier limit. 

To make contact with previous approaches to the multidimensional turnover problem it is useful to consider the special case of ohmic friction for both modes such that... 

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