Double-well potential, fluctuating

Another approach to solvent fluctuation control of reactions in solution based on the Kramer model (Kramer, 1940 Sumi, 1999 and references therein). According to this model a transition over a double-well potential W(q) occurs as a result of zigzag diffusion. An important parameter of the theory is the relaxation time of the average motion of the medium... 

The second term is responsible for the fluctuations of and the symbol R denotes the set of variables necessary to assign to the variable S, the proper intermittent properties. This is a crucial assumption. The model might rest, for instance, on a double-well potential, within which the variable E, moves, virtually attaining only the values corresponding to the bottoms of the two wells. The crucial issue is to make the distributions of time of sojourn at the bottom of these two wells distinctly non-Poisson and renewal at the same time. Here we limit ourselves to assuming that the theoretical waiting time distribution i(t) has the form of Eq. (92) and that /exp(f) is related to it via Eq. (73). In the specific case that we are here describing, a convenient form for the projection operator P is... 

Although the equilibrium properties of this equation are well known/ their time behavior for e = 0 is still the object of controversy. When the additive stochastic force is present (e 0) and double-well potential, Eq. (1.1) describes the process of escape from a well of a fluctuating poten-... 

IV. Fluctuating Double-Well Potential A Comparison of Theory and Experiment.457... 

The first theory giving the tj -induced decrease of the rate constant is the Kramers theory presented as early as in 1940. He explicitly treated dynamical processes of fluctuations in the reactant state, not assuming a priori the themud equilibrium distribution therein. His reaction scheme can be understood in Fig. 1 which shows, along a reaction coordinate X, a double-well potential VTW composed of a reactant and a product well with a transition-state barrier between them. Reaction takes place as a result of diffusive Brownian motions of reactants surmounting... 

Figure 17. Two-dimensional double-well potential for reaction in the Sumi-Marcus model spanned by slow (diffusive) molecular-arrangement fluctuations in solvents on the abscissa for the coordinate X and fast (ballistic) intrasolute vibrational fluctuations on the ordinate. Also shown is a reactive trajectory surmounting the transition-state barrier on the line C with an A -dependent rate constant...

Mobile protons could be transferred in two types of motions (1) Protons could rattle back and forth between symmetric minima of the effective substrate potential energy. The minima are located at hydrogen bond distance from either of the two neighboring SGs. The double well potential, experienced by the intermittent proton, depends on the equilibrium separation of SGs and on their fluctuations. Similar to what happens in the formation of a Zundel ion in water, the double well potential may transform into a single well potential upon close approach of neighboring SGs. Spontaneous symmetry-breaking, associated with these proton motions, leads to the... 

Figure 1 A model double-well potential. The stable state is denoted by A and the metastable one is denoted by B. The gray shaded regions are those where the system fluctuates because of its thermal energy. The region in between is very unlikely to be explored, therefore making the transition from A to B less probable to occur.

Fig. 2. The fluctuating difference between the proton potential at the product side relative to that at the reactant side (the difference between the two wells in a double-well proton potential). Whenever this difference is close to zero, tunneling conditions are favourable.

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