Langevin equation generalized coordinates

Following the general procedure of geometric TST, we start by discussing the linearized dynamics in relative coordinates. If the definition (41) is substituted into the linearized Langevin equation (13), it yields an equation of motion for the relative coordinate ... 

In the GH theory, it is assumed that the reaction barrier is parabolic in the neighborhood of x and that the solute reactive coordinate satisfies a generalized Langevin equation (GLE),... 

The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

The first analysis of the constrained Langevin equation was given by Fixman [9], who worked primarily in generalized coordinates. In order to resolve a discrepancy between Fixman s results and those of Hinch [10], we now follow Fixman by considering a Langevin equation for the soft coordinates, ...,... 

An alternative view of the same physical process is to model the interaction of the reaction coordinate with the environment as a stochastic process through the generalized Langevin equation (GEE)... 

The starting point is the well-known generalized Langevin equation (GLE) as adopted for stochastic motions involving coupling to a solvent coordinate. We employ the notation of Hynes [63] which is consistent with the discussion of solvation in Section II. 

The rate theory of Grote and Hynes [149] included the non-Markovian (memory) effects by considering the following generalized Langevin equation (GLE) for the dynamics along the reaction coordinate ... 

In the absence of the external potential V, Eqs. (52) can be given a rigorous derivation from a microscopic Liouville equation (see Chapter I). We make the naive assumption that when an external potential driving the reaction coordinate is present, the two contributions (the deterministic motion resulting from the external potential and the fluctuation-dissipation process described by the standard generalized Langevin equation) can simply be added to each other. 

The current attempts at generalizing the Kramers theory of chemical reactions touch two major problems The fluctuations of the potential driving the reaction coordinate, including the fluctuations driven by external radiation fields, and the non-Markovian character of the relaxation process affecting the velocity variable associated to the reaction coordinate. When the second problem is dealt with within the context of the celebrated generalized Langevin equation... 

In the Markovian Kramers model discussed in Section 14.4, the friction coefficient y describes the coupling of the reaction coordinate to the thermal environment. In the low friction (underdamped) limit it is equal to the thermal relaxation rate in the reactant well, which is equivalent in the present case to the solvation well of the initial charge distribution. More generally, this rate should depend also on the frequency >s of this well. The theory of solvation dynamics, Chapter 15, does not use a Langevin equation such as (14.39) as a starting point, however it stiU yields an equivalent relaxation rate, the inverse solvation time (tl) , which is used in the present discussion. 

This model is based on the following generalized Langevin equation for the reaction coordinate... 

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