Slow variable regime

In liquid phase chemical kinetics, the reactant potential of mean force W S x) does not play the same universal role that the reactant potential energy function U x) plays in gas phase kinetics. Rather the form of the driving potential of a liquid phase reaction depends on its timescale regime, with W(S x) driving only slow variable regime reactions. 

As a result, the traditional picture of liquid phase reaction dynamics —solute motion in the potential of mean force damped by friction— holds only for slow variable regime reactions. For fast variable regime reactions the nonclassical picture of Eq. (3.36) and Figure 3.5 applies. 

While it is not easy to fully solve this problem, it is relatively simple to solve it in two limiting timescale regimes. These are the traditional slow variable regime, and the opposite and nontraditional fast variable regime noted in Section I. 

We will proceed as follows. For the slow variable regime we will evaluate F[5 x(t)] as a corrected form of the force for the idealized infinity slow process of Case 1 after Eq. (3.14). Correspondingly, for the fast variable regime we will evaluate F[S T(t)] as a corrected form of the force for the idealized infinity fast process. 

Thus, recalling the argument of Section II.A.l, in the slow variable regime the potential lT(S x) that determines the particle s statics also drives its dynamics. 

We will first consider the slow variable regime. 

The slow variable regime applies to processes Xf S) —> xf=o S) throughout which the particle s velocity x t) is much less than the mean thermal velocity of a fluid molecule. 

We next point out that the near equilibrium physics of the slow variable regime gives rise to a very simple picture of particle motion and underlies the traditional concept of frictional damping. 

Thus, in the slow variable regime the following picture of particle motion applies ... 

We have also found that very different pictures of the motion and very different concepts of dissipation emerge in the two limiting regimes. These are summarized in Eqs. (3.29) and (3.30) and Eqs. (3.36) and (3.37). In particular, the potential energy function that drives the process xf S) Xf=o S) is different in the two regimes namely, it is the potential of mean force W S x) in the slow variable regime, and the instantaneous potential V S x) in the short time regime. 

These comments may be rephrased as follows. Just as Mori s Eq. (A.54) is formally exact, Eq. (3.43) is nearly formally exact close to the barrier top. However, as noted in Appendix Section F.15, formal exactness does not equate to physical content. Thus, just as Eq. (A.54) can be merely a nugatory definition of the memory kernel 3 (t) rather than a real solution of the problem of finding an equation of thermodynamic relaxation Eq. (3.43) is not necessarily a satisfactory solution of the central problem of determining the physical nature of the forces acting on the solute reaction coordinate. Rather far from the slow variable regime, Eq. (3.43) can be merely a formal definition of S t) and f S t) as those quantities that convert the slow variable force -dW[S x t)]/dx t) into the unrelated actual forces acting on the reaction coordinate. 

Thus for the systems in Tables III and IV, the agreement of kgh(T ) and Kst T) with simulations alone is consistent with three possibilities (1) the reaction occurs in the short time regime where V S x) drives the reaction and thus kst(T) provides a useful parameterization of k(T ) (2) the reaction occurs in the slow variable regime where W S x) drives the reaction and thus Kgh(T ) provides a useful parameterization and (3) the reaction occurs in an intermediate regime and neither expression has much physical content. 

---