Generalized Kramers equation

Extensive studies have been made of dilute, center-labeled chains in multiple solvents at a range of T and P. Experiment is consistent with a mean relaxation time that follows a generalized Kramers equation... 

Substantial studies have compared x of dilute chains with solvent viscosity, q being manipulated by changing the chemical identity of the solvent, the temperature, or the pressure. Experiment supports a generalized Kramers equation... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

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) ... 

I. Generalized Langevin equation. Zwanzig s Hamiltonian, n. Evaluation of quantum rates for multi-dimensional systems, ni. Beyond the Langevin equation/quantum Kramers paradigm ... 

Now we present the standard derivation of the Fokker-Planck equation for polymers in solution. (Terminology can often be confusing in the present instance, the equation of interest is also called the Smoluchowski equation, and may be regarded as a limiting case of a more general Fokker-Planck equation, or a Kramers equation.)... 

In the usual derivations of the Klein-Kramers equation, the moments of the velocity increments, Eq. (68), are taken as expansion coefficients in the Chapman-Kolmogorov equation [9]. Generalizations of this procedure start off with the assumption of a memory integral in the Langevin equation to finally produce a Fokker-Planck equation with time-dependent coefficients [67]. We are now going to describe an alternative approach based on the Langevin equation (67) which leads to a fractional IGein-Kramers equation— that is, a temporally nonlocal behavior. 

In the Kramers approach the friction models collisions between the particle and the surrounding medium, and it is assumed that the collisions occur instantaneously. There is a time-scale separation between the reactive mode and its thermal bath. The dynamics are described by the Langevin equation (4.141). The situation where the collisions do not occur instantaneously but take place on a time scale characterizing the interactions between the particle and its surrounding can be described by a generalized Langevin equation (GLE),158,187... 

The theory of Brownian motion is a particular example of an application of the general theory of random or stochastic processes [2]. Since Kramers approach is based on a more general stochastic equation than the Langevin equation, we have reviewed some of the fundamental ideas and methods of the theory of stochastic processes in Appendix H. 

The idea in Kramers theory is to describe the motion in the reaction coordinate as that of a one-dimensional Brownian particle and in that way include the effects of the solvent on the rate constants. Above we have seen how the probability density for the velocity of a Brownian particle satisfies the Fokker-Planck equation that must be solved. Before we do that, it will be useful to generalize the equation slightly to include two variables explicitly, namely both the coordinate r and the velocity v, since both are needed in order to determine the rate constant in transition-state theory. 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

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... 

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) ... 

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