The Generalized Langevin Equation GLE

Polymers can be described adequately in terms of classical formalisms. The set of all position vectors and momentums of all segments in the system to be treated is denoted by y, that is a point in phase space. In principle any physical property of the system can be described by an appropriate function A y) of phase variables. Consider now two arbitrary physical quantities corresponding to the functions A y) and B y). The scalar product of these functions is given by 

One is now interested in the kinetics of a certain set of physical quantities Ai,A2,. .. A , the so-called quantities of interest. The set of all linear combinations of these quantities forms the linear subspace L [A, L. The projection operator with respect to L A, is defined by 

The Mori transformation can be used to formally obtain an exact system of equations for quantities of interest  

Equation 84 is often called the generalized Langevin equation (GLE) [109, 115]. 

Note that the time evolution of experimental observables zl (t) is governed by real dynamics which is determined according to Eq. 83 by the real propagator exp fit. The situation with the time evolution of the stochastic force Fn t) and the memory matrix Knk t) is much more complicated. According to Eq. 85, their evolution is governed by projected dynamics the propagator of which is given by exp iQLQt.  

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In Kramers classical one dimensional model, a particle (with mass m) is subjected to a potential force, a frictional force and a related random force. The classical equation of motion of the particle is the Generalized Langevin Equation (GLE) ... 

If Xe is somewhat larger, then there may arise an effective time scale Xr > Xe, with 5, < Xr sueh that the environment has some memory of the particle s previous history and therefore responds accordingly. This is the regime of the generalized Langevin equation (GLE) with colored friction. - In all these cases, the environment is sufficiently large that the particle is unable to affect the environment s equilibrium properties. Likewise, the environment is noninteracting with the rest of the universe such that its properties are independent of the absolute time. All of these systems, therefore, describe the dynamics of a stochastic particle in a stationary —albeit possibly colored— environment. 

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 (GLE)... 

The characteristic time scale for the motion of the particle in the parabolic top barrier is the inverse barrier frequency, the sharper is the barrier, the faster is the motion. Typically, atom transfer barrier are quite sharp therefore the key time scale is very short, and the short-time solvent response becomes relevant instead of the long-time overall response given by the ( used in Kramers theory (see eq.(20)). To account for this critical feature of reaction problems, Grote and Hynes (1980) introduce the generalized Langevin equation (GLE) ... 

In this section the generalized Langevin equation (GLE) for density correlation functions for molecular liquids is derived based on the memory-function formalism and on the interaction-site representation. In contrast to the monatomic liquid case, all functions appearing in the GLE for polyatomic fluids take matrix forms. Approximation schemes are developed for the memory kernel by extending the successful frameworks for simple liquids described in Sec. 5.1. 

Of course, the problem of determining the time dependence of B(, p ) remains. At present, the physically interesting variables can a best be approximately modelled, for dense phases. However, many of these models are (or can be) obtained by simplifyin a rigorous, general equation of motion known as the generalized Langevin equation (GLE). We now show how this expression can be derived from the Liouville equation. 

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