Bistable potential

For a one-dimensional bistable potential with the transition state positioned along the reaction coordinate x x = x, the TST rate for forward reaction is defined as... 

Another widely used approximate approach for obtaining transition rates is the method of eigenfunction analysis. As an example, let us consider the symmetric bistable potential, depicted in Fig. 2. 

For a small noise intensity, the double integral may be evaluated analytically and finally we get the following expression for the escape time (inverse of the eigenvalue yj) of the considered bistable potential ... 

The obtained escape time iy for the bistable potential is two times smaller than the Kramers time (3.10) Because we have considered transition over the barrier top x = 0, we have obtained only a half. 

Here we are interested in escape out of the domain L specified by a single cycle of the potential that is out of a domain of length n that is the domain of the well. Because the bistable potential of Eq. (5.42) has a maximum at x = n/2 and minima at x = 0, x = 7t, it will be convenient to take our domain as the interval —7t/2 < x < n/2. Thus we will impose absorbing boundaries at x = —n/2, x = n/2. Next we shall impose a second condition that all particles are initially located at the bottom of the potential well so that x0 = 0. The first boundary condition (absorbing barriers at —n/2, n/2) implies that only odd terms in p in the Fourier series will contribute to Y (x). While the second ensures that only the cosine terms in the series will contribute because there is a null set of initial values for the sine terms. Hence... 

For a symmetric bistable potential as in fig. 37 another treatment is possible, based on the eigenfunction expansion of (1.8) as given in V.7. The successive eigenfunctions [Pg.335]

Exercise. A typical bistable potential is ( Landau-Ginzburg potential )... 

Exercise. Formulate the escape problem for the bistable potential in fig. 37. Show that the result is again (6.13). )... 

In this case, for a constant intensity of the input signal, /jn(f) = Im = constant, Eq. (1) describes the Brownian motion of the phase [Pg.478]

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

The applicability of the reaction-rate formula in the intermediate-friction regime to the interpretation of conformational changes of bi-naphtyl in solution has been discussed.47 The intramolecular motion is governed by a bistable potential depending on a single variable, which is identified with the dihedral angle between the naphthalene moieties in binaphtyl. The influence of hydrodynamic interactions and coupling with the other coordinates of the molecules also are evoked. 

The eigenmode expansion was also used to determine the time-dependent solution of the Smoluchowski equations for diverse bistable potentials.185... 

In our model with the aid of parameter e we continuously pass from a zero bistable potential (magnetically isotropic particle) to a pair of symmetric wells of infinite depth (highly anisotropic particle). For the magnetic case, as for those of Refs. 21 and 22, a crucial circumstance enabling the harmonic suppression is that an antisymmetric contribution (bias) should be present in the potential. On the other hand, the presence of a symmetric contribution turns out to be an... 

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