Capture models Langevin model

The fact that the O2+ measurements exceed the Langevin-Giomousis-Stevens (LGS) capture cross section raise a suspicion that these GIB cross sections may be too high. On the other hand, the simple LGS capture model may not be applicable in a case where rapidly dissociating states are produced, as is the case in the 02" " + Na system. Inspection of the O2/O2+ vibrational coordinates on a charge transfer energy scale... 

The two models, the Langevin and the polarizable PB equation with a < 0, are designed to represent the same phenomena, the lowering of a dielectric constant as the hydration stractures form around dissolved ions. The results, however, are not precisely comparable. Decrement of a dielectric constant near a wall are captured by both models, but density profiles are not comparable even quahtatively. Counterion profiles of the Langevin model are more concentrated, while those of the polarizable model are more dilute. Without exact simulation results, it is hard to know which model is accurate, hi recent work by Ma et al. [39] the Langevin PB equation with correlations has been solved and it yields a non-monotonic density profile with a valley at a wall followed by a peak further away from a wall. The depletion is, therefore, captured but immediately at a wall and not for the entire profile. A possible weak point of the negative polarizability model is the linearity assumption, -e —c. For homogenous solutions linearity breaks down for... 

Over time, increasingly sophisticated modifications to the Langevin capture cross-section model have... 

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. 

The long-time balance between recombination and drift of carriers as expressed by the y/n ratio has been analyzed using a Monte Carlo simulation technique and shown to be independent of disorder [40]. Consequently, the Langevin formalism would be expected to obey recombination in disordered molecular systems as well. However, the time evolution of y is of crucial importance if the ultimate recombination event proceeds on the time scale comparable with that of carrier pair dissociation (Tc/Td l). The recombination rate constant becomes then capture—rather than diffusion-con-trolled, so that Thomson-like model would be more adequate than Langevin-type formalism for the description of the recombination process (cf. Sec. 5.4). 

The LB method and its improved versions are widely used for the effident treatment of polymer solution dynamics. In the application to polymer solution dynamics, the polymer itself is still treated on a partide-based CG level using, for example, a bead-spring model, while the solvent is treated on the level of a discretized Boltzmann equation. The two parts are coupled by a simple dissipative point-partide force, and the system is driven by Langevin stochastic forces added to both the fluid and the polymers. In this approach, the hydrodynamics of the low-molecular-weight solvent is correctly captured, and the HI between polymer segments, which is mediated by the hydrodynamic flow generated within the solvent through the motion of the polymer, is present in the simulation without explidt... 

Similar to the hard-sphere collision model, the capture colhsion cross section or the Langevin cross section is defined as ... 

The ion induced-dipole potential and the corresponding Langevin rate expression have proven to be of great value in the interpretation of numerous studies of ion-molecule reaction kinetics (see chapters 2 and 3 by Canosa et al, and by Gerlich). For neutral reactions, a f/i potential, which can be taken to represent dispersion and/or dipole induced-dipole interactions, has provided a similarly important reference potential. The Gorin model, which is based on PST like assumptions for this potential, provides a simple expression for the capture rate ... 

The signihcance of Langevin s and Einstein s contributions lies with the calculation of the displacement of Brownian particles. Whereas scientists before them were after the particle velocity of agitation, Einstein developed a sound theory for a quantity that could be measured with high precision, the particle diffusion coefficient. Langevin captured this in a generalized model and Perrin unambiguously validated... 

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