Classical correlation functions. Redfield

One issue that arises when classical correlation functions are used is that they do not satisfy the detailed-balance relation Eq. (14) (because they are even with respect to i = 0) and hence cannot produce a Redfield tensor that lets the subsystem come to thermal equilibrium. Therefore, before being introduced into Eq. (18), the classical results must be modified to satisfy detailed balance. Unfortunately, there is no unique way to accomplish this, and a handful of different approaches are found in the literature. 

The strength of the bath coupling to each system variable is described by the coupling constants / and, because they enter at second order, the rate constant for the dissipation process arising from each term in Eq. (38) will be proportional to f I- The only important properties of the F t) are their autocorrelation and cross-correlation functions, (FJfi)F t)) and F (0)Fi,(t)), which enter the definition of the Redfield tensor in Eq. (18). These, like the classical correlation functions discussed earlier, do not satisfy the detailed-balance relation and must be corrected in the same way. It is convenient, but not necessary, that the variables be chosen to be independent, so that the cross-correlation functions vanish. 

This relation is, itself, sufficient to satisfy the detailed-balance-condition and define a Redfield tensor that allows the system to come to thermal equilibrium. It has thus been used directly by some as an ad hoc correction for classical or stochastic correlation functions [38]. 

Several remarks are in order here. First, the fact that the correlation functions need to be evaluated at the transition frequency reminds us of the earlier classical results for barrier transitions. A friction coefficient is also related to solvent fluctuations in fact, it can be written as a velocity-velocity correlation function, and as such the Redfleld theory is not much different from classical theories. This is to be expected because these can be derived from a very similar formalism based on the classical Liouville equation, albeit without the problems encountered here. In the Kramers case, we needed to evaluate the friction at zero frequency, which sometimes severely overestimates its role. It would be better to evaluate it at the barrier frequency. In the theory developed by Grote and Hynes, the friction needs to be evaluated not at the original barrier frequency but at the frequency the barrier transition actually takes place as Eq. (9.13) shows. The reason is simple in the latter case the back reaction of the system on the solvent is taken into account, something that was not allowed in the Redfield theory. 

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