Kramers rate

Christov, S. G. Two types of Kramers rate equations for reactions in condensed media, IntJ.Quantum Chem., 52(1994), 1219-1228... 

Figure 6. Potential well in the Kramers rate model. Initially the particle is assumed to be caught in the potential hole of depth AV = V(xraax) — V( min)- The x-axis corresponds to a reaction coordinate.

According to Kramers treatment, the proportionality of the Kramers rate to tj in the low viscosity limit turns over to the inverse proportionality in the high viscosity. The interpolating behavior for arbitrary rj was studied by Mel nikov and Meshkov [104]. 

The fraction rj/rja= d is thus the rescaling of the classical Kramers rate according to the parameters classifying the multiple trapping system with broadly distributed waiting times. Similarly, in the underdamped case, one finds the fractional Kramers rate... 

Other reactions have been studied that appear to also require consideration of non-Markovian effects. For example, in a recent study of the photoisomerization of tra/u-stilbene and trdeviations from the Kramers rate in the case of /ranj-stilbene. These discrepancies were tentatively related to the larger flexibility of this molecule but appeared to be well simulated by the non-Markovian theory of Grote and Hynes. ... 

Finally, we would like to mention the work of Fauve and Heslot, who studied the contemporary action of a periodic and stochastic force on a bistable electric syston. They found that applied force and noise exhibit resonant behavior. Resonance takes place when the external force time period is equal to the Kramers rate of esoq)e. ... 

The transition rates from the left to the right well and 7 from the right to the left well, which are both equal for the symmetric doublewell potential, are Kramers rates for the excitation over a potential barrier due to white noise. They can be calculated as [7, 8]... 

Fig. 2.8. Mean frequency v (top) and number of locked cycles / flock (bottom) of the dichotomically periodically driven bistable system eq. (2.6) with a = 6 = 1 (x symbols) for different values of the driving amplitude A and a driving frequency v = 0.001. The solid lines show the dependence according to eqs. (2.40) and (2.41). Kramers rates ro and n following eq. (2.7) has been inserted. The + symbols present results from simulations of the discrete model. [13]...

The theory proposed by Halperin and Alexander (H-A theory) [60] is based on the structural scaling description of polymeric micelles outlined in Sect. 2.1.2. Using a combination of scaling theory and Kramers rate theory for diffusirai in an external potential [61], the expulsion rate for both crew-cut and star-like spherical micelles was derived. Moreover, Halperin and Alexander discussed different scenarios of chain exchange between micelles. 

For one-dimensional stationary flow in a potential well, Kramers rate theory gives the following expression for the outgoing flux, / ... 

The elements of the transition rate matrix A in the DRIS model are estimated from the multidimensional energy surface associated with the interdependent rotation of neighboring bonds u g Kramers rate theory, as described above. Accordingly, the probability of occurrence of a given isomeric state for a bond depends on the state of its first neighbors aloi the chain. Likewise, in the kinetic Ising model the transition rate Wi(Oi) of the i-th spin is assumed to be coupled to the state of its first iKighbors by the relation... 

If the friction goes to zero, all particles reaching the reactant surface with sufficient velocity - calculated from the equilibrium distribution in the well - make it to the top and over, and the Kramers rate reduces to the TST rate. In the case of high friction, we expand the solution in a series in to obtain... 

---