Langevin generalized

Some features of late-stage interface dynamics are understood for model B and also for model A. We now proceed to discuss essential aspects of tiiis interface dynamics. Consider tlie Langevin equations without noise. Equation (A3.3.57) can be written in a more general fonn ... 

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) [ ]... 

Zwanzig R 1973 Nonlinear generalized langevin equations J. Stat. Phys. 9 215-20... 

Poliak E 1990 Variational transition state theory for activated rate processes J. Chem. Phys. 93 1116 Poliak E 1991 Variational transition state theory for reactions in condensed phases J. Phys. Chem. 95 533 Frishman A and Poliak E 1992 Canonical variational transition state theory for dissipative systems application to generalized Langevin equations J. Chem. Phys. 96 8877... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

In general, Langevin dynamics simulations run much the same as molecular dynamics simulations. There are differences due to the presence of additional forces. Most of the earlier discussions (see pages 69-90 and p. 310-327 of this manual) on simulation parameters and strategies for molecular dynamics also apply to Langevin dynamics exceptions and additional considerations are noted below. 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

Second, the classical dynamics of this model is governed by the generalized Langevin equation of motion in the adiabatic barrier [Zwanzig 1973 Hanggi et al. 1990 Schmid 1983],... 

This relation is as general as the Langevin equation itself, i.e., it holds for collisions of any strength. When deriving Eq. (1.23) from Eq. (1.26),... 

This conclusion can be confirmed by an alternative derivation of Eq. (1.71). According to Mori [52], Eq. (1.71) may be obtained from a generalized Langevin equation ... 

Relations (2.46) and (2.47) are equivalent formulations of the fact that, in a dense medium, increase in frequency of collisions retards molecular reorientation. As this fact was established by Hubbard within Langevin phenomenology [30] it is compatible with any sort of molecule-neighbourhood interaction (binary or collective) that results in diffusion of angular momentum. In the gas phase it is related to weak collisions only. On the other hand, the perturbation theory derivation of the Hubbard relation shows that it is valid for dense media but only for collisions of arbitrary strength. Hence the Hubbard relation has a more general and universal character than that originally accredited to it. 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Brownian motion theory may be generalized to treat systems with many interacting B particles. Such many-particle Langevin equations have been investigated at a molecular level by Deutch and Oppenheim [58], A simple system in which to study hydrodynamic interactions is two particles fixed in solution at a distance Rn- The Langevin equations for the momenta P, (i = 1,2)... 

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 ... 

Many solvents do not possess the simple structure that allows their effects to be modeled by the Langevin equation or generalized Langevin equation used earlier to calculate the TS trajectory [58, 111, 112]. Instead, they must be described in atomistic detail if their effects on the effective free energies (i.e., the time-independent properties) and the solvent response (i.e., the nonequilibrium or time-dependent properties) associated with the... 

C. C. Martens, Qualitative dynamics of generalized Langevin equations and the theory of chemical reaction rates, J. Chem. Phys. 116, 2516 (2002). 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

The probability of a complete Brownian path is then obtained as the product of such single-time-step transition probabilities. For other types of dynamics, such as Newtonian dynamics, Monte Carlo dynamics or general Langevin dynamics, other appropriate short-time-step transition probabilities need to be used [5, 8]. 

The effective frequencies that characterize solvent response can be characterized more quantitatively from several points of view, including generalized Langevin theory [367-372], Brownian oscillators [373, 374], and instantaneous normal modes [375],... 

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 treatment of the solute-solvent system with the classical Generalized Langevin equation formalism [127], with especial attention to the present problem, has been examined by us [6] a wealth of information can be found in references [128-131],... 

In order to proceed now to a statistical mechanical description of the corresponding relaxation process, it is convenient to solve the equation of motion for the creation and destruction operators and cast them in a form ressembling a Generalized Langevin equation. We will only sketch the procedure. 

The set of equations (50) can be formally considered as generalized Langevin equations if the operator Fj(t) can be interpreted as a stochastic quantity in the statistical mechanical sense. If the memory function does not correlate different solute modes, namely, if Kjj =Sjj Kj, then a Langevin-type equation follows for each mode ... 

Cortes, E., West, B. J. and Lindenberg, K. On the generalized Langevin equation classical and quantum mechanical, J.Chem.Phys., 82 (1985), 2708-2717... 

Adelman, S. A. Generalized Langevin equations and many-body problems in chemical physics,... 

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