Slow variable regime physics

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. 

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. 

To start, we note that the short time Kst(T ) and Grote-Hynes kgh(T ) transmission coefficients are algebraically equivalent [23]. However, Kst(T) and Kgh(T ) are useful expressions in different physical regimes. Eqs. (3.50) and (3.51) for Kst(T) provide a useful parameterization of k(T ) only for reactions for which the rate constant k T) is determined by short time dynamics while Eqs. (3.46) and (3.47) provide a useful parameterization only for reactions for which k T) is determined by slow variable dynamics. Nearly equivalently, Eqs. (3.50) and (3.51) apply to sharp barrier reactions, where the sharp barrier limit is defined as comip oc while Eqs. (3.46) and (3.47) apply to flat barrier reactions, where the flat barrier limit is defined as (Ormf 0. (The sharp barrier limit is taken as comip oo, not as PMF oc as in Section III.B, isasmuch as sharp barrier reactions are short time, high-frequency processes for which oomip is the physical barrier frequency. The converse argument yields the flat barrier limit as copmf 0.)... 

The plan of Section IV is as follows In section IV.A, we qualitatively outline the general picture of reaction dynamics that emerges from fast variable physics. Next, in section IV.B, we examine liquid phase-activated barrier crossing in the short time regime of Section II.C. In Section IV.C we note that the fast variable/slow bath timescale separation also applies to liquid phase vibrational energy relaxation and then discuss that process from the fast variable standpoint. Finally, in Section IV.D, we discuss some related work of others. 

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