Friction, limiting

The exact solution of the instanton equation in the large ohmic friction limit has been found by Larkin and Ovchinnikov [1984] for the cubic parabola (3.18). At T = 0... 

The specific form of the short-time transition probability depends on the type of dynamics one uses to describe the time evolution of the system. For instance, consider a single, one-dimensional particle with mass m evolving in an external potential energy V(q) according to a Langevin equation in the high-friction limit... 

We want to analyze here the effect of these long-range Coulomb forces in the large friction limit (396) we shall thus consider the Brownian-dynamic approximation, which, as we shall see presently, gives exactly the same result as the classical semi-phenomenological theory developed in Section V-A. 

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. 

In the high-friction limit, the Smoluchowski expression (4.152) can be used to determine the time evolution of the particle and can be written as... 

Numerical results have been given,173 174 based on the original model of Kramers, which reproduce the high-friction limit accurately. Recently, the Kramers method has been reformulated135,137,175 in order to emphasize the underlying assumptions, namely, the quasi-stationary (long-time) behavior of the system and the concept of a two-state system... 

In Kramers theory that is based on the Langevin equation with a constant time-independent friction constant, it is found that the rate constant may be written as a product of the result from conventional transition-state theory and a transmission factor. This factor depends on the ratio of the solvent friction (proportional to the solvent viscosity) and the curvature of the potential surface at the transition state. In the high friction limit the transmission factor goes toward zero, and in the low friction limit the transmission factor goes toward one. 

Instead of the quasi-stationary state assumption of Kramers, he assumed only that the density of particles in the vicinity of the top of the barrier was essentially constant. Visscher included in the Foldcer-Planck equation a source term to accoimt for the injection of particles so as to compensate those escaping and evaluated the rate constant in the extreme low-friction limit. Blomberg considered a symmetric, piecewise parabolic bistable potratial and obtained a partial solution of the Fokker-Hanck equation in terms of tabulated functions by requiring this piecewise analytical solution to be continuous, the rate constant is obtained. The result differs from that of Kramers only when the potential has a sharp, nonharmonic barrier. 

Very recently, Lavenda devised an interesting method of solution of the Kramers problem in the extreme low-friction limit. He was able to show that it could be reduced to a formal Schrddinger equation for the radial part of the hydrogen atom and thus be solved exactly. One particular form of the long-time behavior of the rigorous rate equation coincides with that obtained by Kramers with the quasi-stationary hypothesis and may thus clarify the implications of this hypothesis. The method of Lavenda is reminiscent of that used by van Kampen but applied to a Smoluchowski equation for the diffusion of the energy. 

The results illustrated on the right-hand side of Fig. 4 show that in this region the increase of k is much more sensitive to the increases in than it is in the high-friction region, thereby corroborating our statements about the role of inertia. This trend is especially emphasized in the limit y 0 and is better seen in Fig. 5. As remarked above, the reaction rate stays finite in this zero-friction limit, counter to Kramers prediction. 

In Section IV A the model is applied to diatomic dissociation on a Morse potential in the low-friction limit. This models diatomic dissociation in a low-density gas in which the collisional excitation is impulsive, being modeled by a zero-frequency friction, that is, the duration of the collision is assumed to be small relative to all other time scales. The objective of the section is to test whether the reduction to a one-dimensional effective potential P (l ) leads to an accurate formula for the dissociation rate. This is done by comparison... 

Note that D equals Dy in the high-friction limit. 

In this section the question of the accuracy of using the effective potential of Eq. (2.26) in dissociation dynamics of a diatomic is studied in the low-friction limit, which corresponds to dissociation in a low-density bath gas. Comparison with experiment will be considered later in this section, but this comparison is difficult because of complexities in real systems, such as adiabatic effects arising from the friction kernel and multiple electronic surfaces. In this... 

The dissociation rate in the low-friction limit is determined by the rate of accummulation of energy to the dissociation limit and is proportional to the friction i bb// bb- H is convenient to write in the form... 

The validity of the conclusions was checked by evaluation of the dissociation rate from the A state in the gas phase in which the low-friction limit of Eq. (4.3) applies. The agreement with exjjeriment was extremely good, indicating that the use of a constant friction is reasonable for these shallow potentials. Vibrational relaxation of the 12 molecule in the X state will be much slower... 

In the case of polar molecules an analytical expression for Rj cannot be found. As shown in Fig. 21, the transition state in is well defined for all values of the dipole-locking parameter C = Ho/2ctkT). The transition state rate of the stochastic theory is plotted against C in Fig. 22, where a comparison is made with other theories. The important point is that the result of this theory is identical to that of CVTST and, of free-energy VTST. When corrected for access to bound states, the result is close to that of i-VTST. This shows that use of the free energy surface for systems with internal degrees of freedom does not introduce significant errors in the evaluation of the low-friction limit. 

The high-friction limit A/cuy 1 is interesting because of the nonlinear density dependence of the rate. In this limit the general expression of the rate constant is... 

The bimolecular reaction rate for particles constrained on a planar surface has been studied using continuum diffusion theory " and lattice models. In this section it will be shown how two features which are not taken account of in those studies are incorporated in the encounter theory of this chapter. These are the influence of the potential K(R) and the inclusion of the dependence on mean free path. In most instances it is expected that surface corrugation and strong coupling of the reactants to the surface will give the diffusive limit for the steady-state rate. Nevertheless, as stressed above, the initial rate is the kinetic theory, or low-friction limit, and transient exp)eriments may probe this rate. It is noted that an adaptation of low-density gas-phase chemical kinetic theory for reactions on surfaces has been made. The theory of this section shows how this rate is related to the rate of diffusion theory. 

Escape in the Presence of External Periodic Force The Low-Friction Limit... 

The product J Eg)coQ is equal to Eg for a harmonic oscillator potential truncated at = Eg, and to 2Eg for a Morse potential with dissociation energy equal to Eg. Equation (2.41) is the low-friction limit result of Kramers. There are other methods to derive the results obtained in the previous section. One is to look for the eigenvalue with smallest positive real part of the ojjerator L defined so that dP/dt = — LP is the relevant Fokker-Planck or Smoluchowski equation. Under the usual condition of time scale separation this smallest real part is the escapie rate for a single well potential. Another way uses the concept of mean passage time. For the one-dimensional Fokker-Planck equation of the form... 

The subscript E on the right-hand side denotes fixed E (undamped trajectory). Computing x(t)x(0) without damping is consistent with the low-friction limit where damping is assumed to be small on the time scale associated with Z t) [see Eq. (5.29)]. Comparing the two results for (dE/dt)j.=o using Eq. (5.49) we get the result Eq. (5.57). Equation (5.57) provides a convenient numerical way to compute e(E) all one needs is to run a trajectory over the undisturbed molecular motion at the given E for a time of several t. ... 

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