Langevin equation potential

Huller and Baetz [1988] have undertaken a numerical study of the role played by shaking vibrations. The vibration was supposed to change the phase of the rotational potential V (p — a(t)). The phase a(t) was a stochastic classical variable subject to the Langevin equation... 

The specific form of the short-time transition probability depends on the type of dynamics one uses to describe the time evolution of the system. For instance, consider a single, one-dimensional particle with mass m evolving in an external potential energy V(q) according to a Langevin equation in the high-friction limit... 

Since l is proportional to and q is proportional to 1/L, i is proportional to. Substitution of Eq. (67) into Eq.(62) gives the Langevin equation for the Rouse modes of the chain within the approximations of preaveraging for hydrodynamic interactions and mode-mode decoupling for intersegment potential interactions. Equation (62) yields the following results for relaxation times and various dynamical correlation functions. 

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) ... 

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

A slightly different technique was used by Allen [536]. A large number of solvent molecules were allowed to move each according to a Langevin equation in a force field of all the other molecules (which interact by Leonard-Jones potentials). In this system, there are two reactants, AB and C, and the reaction is of the type... 

Hynes et al. [298] and later Schell et al. [272] have developed a numerical simulation method for the recombination of iodine atoms in solution. The motions of iodine atoms was governed by a Langevin equation, though spatially dependent friction coefficients could be introduced to increase solvent structure. The force acting on iodine atoms was obtained from the mutual potential energy of interaction, represented by a Morse potential and the solvent static potential of mean force. The solvent and iodine atoms were regarded as hard spheres. The probability of reaction was calculated by following many trajectories until reaction had occurred or was most improbable. The importance of the potential of... 

Remark. Normally the name quantum Langevin equation is used for equations that are the direct analog of the classical Langevin equation e.g., in the case of a one-dimensional particle of unit mass in a potential V,... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

In Kramers theory that is based on the Langevin equation with a constant time-independent friction constant, it is found that the rate constant may be written as a product of the result from conventional transition-state theory and a transmission factor. This factor depends on the ratio of the solvent friction (proportional to the solvent viscosity) and the curvature of the potential surface at the transition state. In the high friction limit the transmission factor goes toward zero, and in the low friction limit the transmission factor goes toward one. 

As before, (Av) and ((Av)2) may be determined from the Langevin equation. It differs slightly from the Langevin equation in Eq. (11.5), since the motion takes place in an external potential U(r), as sketched in Fig. 11.0.2. This is the potential from the gas phase modified by the interactions from the solvent molecules. 

On the other hand, if the position r(n) of the particle is smaller than the decay length of the interaction potential, the external foroe experienced by the particle during the time interval t < t < cannot be assumed constant because the interaction potential is a strong function of position. In order to discretize the generalized Langevin equation, the time interval t < t < t +i is further subdivided into K subintervals such that the... 

---