Langevin formalism

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 Berendsen [82] and Gauss [83] thermostats are also among other methods used. The Berendsen thermostat [82] was developed starting from the Langevin formalism by eliminating the random forces and replacing the friction term with one that depends on the ratio of the desired temperature to current kinetic temperature of the system. The resulting equation of motion takes the same form as the Nose-Hoover equation with... 

Such a distribution is never found in an experimental situation and, even if generated initially, would never be maintained. The rate constant k(vj) may, however, be applied in other situations under the very restrictive condition that it is independent off v) in that case, it is generally applicable. Such a procedure can only be justified by the experimental demonstration of this independence arguments based on the Langevin formalism constitute no justification, merely an unsubstantiated assumption. 

If the Langevin formalism is valid, the point to be stressed is that it will apply irrespective of the collision mechanism. Indeed, it has been judged to be valid at the lowest energies for both reactions such as Ar" (D2,D)ArD, which proceeds by a direct mechanism, and CD4, (C )4,CD3)CD5, which proceeds by complex mechanism in this energy range. ... 

The above discussion merely emphasizes that if the Langevin formalism may be used, it may only be used at thermal energies. Further, for many polyatomic nonpolar neutrals, the polarization potential is inexact for the treatment of collision processes at thermal energy, since the ion-quadrupole potential there plays a nonnegligible role for neutrals possessing a quadrupole moment. 

When the solvent is treated as a continuous dielectric background that interacts stochastically with the mobile ions, the ionic trajectories can be modeled with the Langevin formalism. " In particular, the strict or full Lange-vin equation can be used, which assumes Markovian random forces and neglects correlations (both spatially and temporally) of the ionic motion ... 

The simplest Langevin formalism is the single-variable expression this now becomes... 

Now we proceed to examine a consequence of the hig-fre-quency motion upon the overall dynamics. We shall begin with the remark that functions like C (t) play the part of a generalized friction coefficient in a Langevin formalism of rotational diffusion, and therefore the spectrum measured in the FIR is nothing but the spectral density of this rotational friction coefficient. 

For a comprehensive account of Langevin formalism we recommend one of the recent textbooks (21). In the rotational dif-... 

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