Langevin interaction

This charge-induced dipole force is the most fundamental interaction between a charged particle and a neutral species with no permanent dipole moment of its own. It is known as the Langevin interaction. It is a l/r4-dependent force through which neutral and charged species can interact. There is, however, another way in which an interaction of this kind can occur. In the example just given, the polarisation of the neutral species can only be induced, because Hj has no permanent dipole moment. More complex molecules can, however, possess permanent dipole moments, which leads to a different situation. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The molecular dynamics method is useful for calculating the time-dependent properties of an isolated molecule. However, more often, one is interested in the properties of a molecule that is interacting with other molecules. With HyperChem, you can add solvent molecules to the simulation explicitly, but the addition of many solvent molecules will make the simulation much slower. A faster solution is to simulate the motion of the molecule of interest using Langevin dynamics. 

The classical motion of a particle interacting with its environment can be phenomenologically described by the Langevin equation... 

Approximating the intermoleculai interactions to only include two-body effects, e.g. electrostatic forces are only calculated between pairs of fixed atomic chai ges in force field techniques. Or the discrete interactions between molecules may be treated only in an average fashion, by using Langevin dynamics instead of molecular dynamics. 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

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. 

Intramolecular Isotope Effects. The data in Figure 2 clearly illustrate the failure of the experimental results in following the predicted velocity dependence of the Langevin cross-section. The remark has been frequently made that in the reactions of complex ions with molecules, hydrocarbon systems etc., experimental cross-sections correlate better with an E l than E 112 dependence on reactant ion kinetic energy (14, 24). This energy dependence of reaction presents a fundamental problem with respect to the nature of the ion-molecule interaction potential. So far no theory has been proposed which quantitatively predicts the E l dependence, and under these circumstances interpreting the experiment in these terms is questionable. 

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)... 

These results show that hydrodynamic interactions and the spatial dependence of the friction tensor can be investigated in regimes where continuum descriptions are questionable. One of the main advantages of MPC dynamics studies of hydrodynamic interactions is that the spatial dependence of the friction tensor need not be specified a priori as in Langevin dynamics. Instead, these interactions automatically enter the dynamics from the mesoscopic particle-based description of the bath molecules. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

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... 

Relaxation processes are probably the most important of the interactions between electric fields and matter. Debye [6] extended the Langevin theory of dipole orientation in a constant field to the case of a varying field. He showed that the Boltzmann factor of the Langevin theory becomes a time-dependent weighting factor. When a steady electric field is applied to a dielectric the distortion polarization, PDisior, will be established very quickly - we can say instantaneously compared with time intervals of interest. But the remaining dipolar part of the polarization (orientation polarization, Porient) takes time to reach its equilibrium value. When the polarization becomes complex, the permittivity must also become complex, as shown by Eq. (5) ... 

The role of instabilities involving confined impurity atoms has been investigated by Mtiser using a model in which two one-dimensional (1-D) or 2-D surfaces were separated by a very low concentration of confined atoms and slid past one another.25 The motion of the confined atoms was simulated with Langevin dynamics where the interactions between these atoms were neglected and the atom-wall interactions were described by... 

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