Friction kernel

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

Because of the general difficulty encountered in generating reliable potentials energy surfaces and estimating reasonable friction kernels, it still remains an open question whether by analysis of experimental rate constants one can decide whether non-Markovian bath effects or other influences cause a particular solvent or pressure dependence of reaction rate coefficients in condensed phase. From that point of view, a purely... 

In this equation, m. is the effective mass of the reaction coordinate, q(t -1 q ) is the friction kernel calculated with the reaction coordinate clamped at the barrier top, and 5 F(t) is the fluctuating force from all other degrees of freedom with the reaction coordinate so configured. The friction kernel and force fluctuations are related by the fluctuation-dissipation relation... 

Friction interactions, multiparticle collision dynamics, single-particle friction and diffusion, 114—118 Friction kernel, transition state trajectory, colored noise, 209... 

These results suggest a computational strategy for the study of reactions in condensed phases. One starts from some realistic intermolecular potentials and performs a molecular-dynamics-Kramers-Grote-Hynes scheme that consists of the following steps.First, we fix the proton at the transition state and run a MD simulation. The friction kernel y(t) is calculated and along with Eqs. (7,8) enables the calculation of the Grote-Hynes rate. This scheme has also been used as a means of obtaining input for quantum calculations as well. ... 

A critical assumption in Eq. (2) is that the friction kernel y(t) is independent of the position s. However, it is now known from numerical simulations that for some reactions in solution this assumption is violated. 

We should point out that Eq. (42) indicates that the function G(s) can be obtained from the value of the friction kernel at t = 0. This is a consequence of the fact that the friction kernel is calculated in the clamping approximation. In any case, Eq. (42) allows for the calculation of G(s) without the numerical difficulties that plague the long-time tail of molecular dynamics simulations. 

Fix the proton at some position s and run a MD simulation. The friction kernel is calculated from the force-force correlation function. 

The friction kernel gs(t) is calculated and with the help of Eqs. (42)-(43) the coupling y(s) is obtained. Then one solves for the Hamiltonian Eq. (36) to obtain the effective Grote-Hynes rate. 

There are two primary associations that must be made between the polymer systems and the iGLE (i) the construction of the potential of mean force (PMF), and (ii) characterization of the nonstationary friction kernel by way of g(t). 

This phenomenological treatment, however, can be extended to include other quenching mechanisms. For example, living polymers are known to quench when the monomers are reacted to completion. In the context of the iGLE, a friction kernel that would simulate such a mechanism is the addition of the term. 

The iGLE also presents a novel approach for studying the reaction dynamics of polymers in which the chemistry is driven by a macroscopic force that is representative of the macroscopic polymerization process itself The model relies on a redefined potential of mean force depending on a coordinate R which corresponds locally to the reaction-path coordinate between an n-mer and an (n -t 1 )-mer for R = nl. The reaction is quenched not by a kinetic termination step, but through an (R(t))-dependent friction kernel which effects a turnover from energy-diffusion-limited to spatial-diffusion-Iimited dynamics. The iGLE model for polymerization has been shown to exhibit the anticipated qualitative dynamical behavior It is an activated process, it is autocatalytic, and it quenches... 

A major limitation of the dissipative mechanisms involving multiplicative noise —and by extension the iGLE and WiGLE models— is that they involve equilibrium changes only in the strength of the response with respect to the instantaneous friction kernel. They do not involve a change in the response time of the solvent at equilibrium limits. Presumably the response time also changes in some systems, and the inclusion of this variation is a necessary component of the minimal class of models for nonstationary stochastic dynamics. Plow this should be included, however, is an open problem which awaits an answer. 

The quantity 17(f) is the time-dependent friction kernel. It characterizes the dissipation effects of the solvent motion along the reaction coordinate. The dynamic solute-solvent interactions in the case of charge transfer are analogous to the transient solvation effects manifested in C(t) (see Section II). We assume that the underlying dynamics of the dielectric function for BA and other molecules are similar to the dynamics for the coumarins. Thus we quantify t](t) from the experimental C(t) values using the relationship discussed elsewhere [139], The solution to the GLE is in the form of p(z, t), the probability distribution function. 

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