Instanton action

The dimensionless upside-down barrier frequency equals = 2(1 — and the transverse frequency Qf = Q. The instanton action at = oo in the one-dimensional potential (4.41) equals [cf. eq. (3.68)]... 

The situation changes when moving on to low temperature. Friction affects not only the prefactor but also the instanton action itself, and the rate constant depends strongly on rj. In what follows we restrict ourselves to the action alone, and for the calculation of the prefactor we refer the reader to the original papers cited. For the cusp-shaped harmonic potential... 

The instanton action behaves in accord with the scaling predictions and is independent of coq. Loosely speaking, the frequency ojq is replaced by the friction coefficient rj. Grabert et al. [1984b] have studied the energy loss A tunneling process and found that is... 

The symmetric coupling case has been examined by using Sethna s approximations for the kernel by Benderskii et al. [1990, 1991a]. For low-frequency bath oscillators the promoting effect appears in the second order of the expansion of the kernel in coj r, and for a single bath oscillator in the model Hamiltonian (4.40) the instanton action has been found to be... 

Figure 5.1. Dimensionless instanton action Sin

To obtain an expression for the multidimensional instanton action, we generalize [29] the analytical solutions obtained [21, 31] for two- and three-dimensional Hamiltonians in the form... 

Figure 29.5 Instanton actions at 7 = 0 along the one-dimensional instanton (—( 1 ) and the MBP (S2o) (solid lines, in units C), and their relation (dashed line) as function of the correlation parameter C. The vertical line divides the regions ofweak (C < 1 /2) and intermediate (1/2 < C < 1) coupling corresponding to surfaces (a) and (b) in Fig. 29.3, respectively.

To interpret pH jjj physical terms, we note that, for a barrier formed by the crossing of two equivalent harmonic potentials, the instanton action can be approximated by [21]... 

Fig. 32. Dimensionless instanton action S, ,

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