Depopulation factor

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. [Pg.135]

The TST rate I/psT has aheady beea dehned above (Ecj. the ICranierS Grote Hynes spatial dififiision faetor is defined in Eqs. 7 and 18. The depopulation factor Y is found to be ... [Pg.19]

When the energy loss is small in comparison to ksT the depopulation factor reduces to y pA and one recovers Kramers estimate for the rate in the energy diffusion limit. When the energy loss is large compared to kBT the depopulation factor approaches unity exponentially fast, y Eq. 28 gives... [Pg.19]

In many cases, when the damping is weak there is hardly any difference between the unstable mode and the system coordinate, while in the moderate damping limit, the depopulation factor rapidly approaches imity. Therefore, if the memory time in the friction is not too long, one can replace the more complicated (but more accurate) PGH perturbation theory, with a simpler theory in which the small parameter is taken to be for each of the bath modes. In such a theory, the average energy loss has the much simpler form ... [Pg.20]

The expressions for the depopulation factor as given in Eqs. 29 and 30 for the single and double well potential cases respectively, remain unchanged. This version of the turnover theory for space and time dependent friction has been tested successfully against numerical simulation data, in Refs. 68,137. [Pg.20]

The quantum depopulation factor also differs from the classical and takes the form ... [Pg.22]

Here T1U is the so called one-dimensional TST estimate for the rate and is mainly determined by the one-dimensional potential of mean force w(q). The depopulation factor Y becomes much smaller than unity in the underdamped limit and is important when the rate is limited by the energy diffusion process. In the spatial-diffusion-limited regime, the depopulation factor Y is unity but the spatial diffusion factor becomes much smaller than unity. The major theme of this review is theoretical methods for estimating the depopulation and spatial diffusion factors. [Pg.620]

The theory for the depopulation factor Y for the GLE and the STGLE is discussed in Sec. VI. Multidimensional generalizations for the depopulation and the spatial diffusion factors are presented in Sec. VII. Extension of the theory to include motion on periodic potentials and surface diffusion is given in Sec. VIII. We end with a discussion of future directions and topics which remain unsolved at this point. [Pg.621]

Note that when the damping constant y - 0 also u, - 0, as may be seen from inspection of Eqs. (43) and (53). It is important to realize that in the presence of memory friction, there exist limits such that u, 0 but = to (12). Claims to the contrary not withstanding (87), using u, as the perturbation parameter leads therefore to a more general theory for the depopulation factor than any theory based on the weak damping limit which is defined by a small damping constant. [Pg.647]

Given the energy loss, the depopulation factor is found by solution of the integral equation, Eq. (142). A clear description of the necessary algebra may be found in the appendix of Ref. 67 here we just write the final result ... [Pg.650]

The result given in Eq. (155) is correct for a single-well potential. For a doublewell potential in which the energy loss in each of the two wells is Au, A, one must revise the integral equation to take into consideration the flux returning from each one of the wells. As shown by Melnikov (85,86), this then leads to the deceptively simple result for the depopulation factor ... [Pg.650]

In the Kramers turnover theory (12,67), the expression for the rate, valid for all values of the damping is still given as a product of three factors as in Eq. (6). As discussed in the previous section, the depopulation factor is determined uniquely by the reduced energy loss parameter 8 = (3A (12,67), the explicit dependence is given in Eq. [Pg.652]

In the underdamped limit, for which the energy loss 8 1 the depopulation factor Y = 8 and the expression for the rate... [Pg.656]

In the one-dimensional case, the depopulation factor in the underdamped limit is linearly proportional to while in the two-degrees-of-freedom case studied here, it is quadratic. The rate increases much faster with damping in the multidimensional case. The reason is that both modes are activated by their respective baths. Since they are strongly coupled, the spectator mode z will supply the energy it picks up from its bath to the reaction coordinate thus enhancing the possibility of activation. The added enhancement of the rate in the underdamped limit is critically dependent on the strong coupling between the modes. [Pg.656]

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