Langevin theory

This universality is peculiar for the high-temperature approximation, which is valid for //J < 1 only. For sufficiently high temperature the quantum theory confirms the classical Langevin theory result of J-diffusion, also giving xj = 2xE (see Chapter 1). This relation results from the assumed non-adiabaticity of collisions and small change of rotational energy in each of them ... 

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 effective frequencies that characterize solvent response can be characterized more quantitatively from several points of view, including generalized Langevin theory [367-372], Brownian oscillators [373, 374], and instantaneous normal modes [375],... 

A particular question of interest is whether the DNA torsional motions observed on the nanosecond time scale are overdamped, as predicted by simple Langevin theory, and as observed for Brownian motions on longer time scales, or instead are underdamped, so that damped oscillations appear in the observed correlation functions. A related question is whether the solvent water around the DNA exhibits a normal constant viscosity on the nanosecond time scale, or instead begins to exhibit viscoelastic behavior with a time-, or frequency-, dependent complex viscosity. In brief, are the predictions for... 

Additional evidence on electron-cloud radii is given by diamagnetic susceptibility and by refractive index. For the well-known Larmor-Langevin theory of diamagnetism (11—13) gives for the molecular diamagnetic susceptibility —Xm the formula... 

The fluctuations of isolated steps have been studied, both theoretically - using Langevin theory, Monte Carlo simulations of SOS models, as well as exact methods, and experimentally by scanning tunneling microscopy (caution is needed in the measurements to avoid artefacts of tip assisted motions of the steps O-... 

Now note that the right hand side of (85) is the same as the expression (81)(with t —> 2f), so that within Langevin theory, the equilibrium fluetuations G(2t) have the same time dependence as the non-equilibrium growth of fluetuations w (f). 

In terms of the energies defined in Fig. 5, the rate F appearing in the Langevin theory, is given by. 

LANGEVIN THEORY OF POLYMER DYNAMICS IN DILUTE SOLUTION ... 

Dynamics, Polymer, Langevin Theory of, in Dilute Solution (Zwanzig) 15 325... 

Langevin Theory of Polymer Dynamics in Dilute Solution (Zwanzig) Large Tunnelling Corrections in Chemical Reaction Rates (Johnston) Lattices, Linear, Reversible Kinetics on, with Neighbor Effects... 

In the Langevin theory the memory function is proportional to the time-correlation function for the random force. 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

Consider the case of a single collective solvent coordinate y. This coordinate is linearly coupled to the solute at the transition state by generalized Langevin theory [71-75], (It is not necessary to couple the solvent to the solute for the calculation of reactant properties because we retain the equilibrium-reactant approximation.) The form of the coupling is [60,61]... 

In conventional semiconductors, the situation differs in many ways In such materials, transport in extended states is the rule, and the mean free paths are much larger than atomic dimensions Langevin theory does not... 

The behavior of ferrofluid particles subject to a constant magnetizing field is adequately described by this Langevin theory of paramagnetism suitably modified to take account of a distribution of particle sizes and particle interactions. Thus ferrofluids have a magnetization curve which does not exhibit hysteresis. 

Zwanzig, R. Langevin theory of polymer dynamics in dilute solution. Adv. Chem. Phys. 15, 325-331 (1969). 

The term AWi (= Ajr ) is the first-order energy shift, and AW, (= Cjr ) is the second-order shift. Equation 16 has the form of (2), and in the region of validity of the rotational Stark effect the simple Langevin theory applies. However, the effective angular momentum is determined by the first two terms of (16), and the effective polarizability is determined by the last two terms. 

In the Langevin orbiting theory, only the ion-induced dipole interaction was considered as a long-range force operative between the ion—molecule pair. Thus the theory applies only to the reactions of ions with non-polar molecules. In fact, it has been pointed out that some ion—molecule reactions in which the neutral molecule has a permanent dipole have reaction cross-sections greater than those predicted by the Langevin theory [56—63]. Such ion—polar molecule reactions have also been treated classical mechanically by several authors [57, 58, 61, 64—68]. 

Brooks, C.L., Berkowitz, M., Adelman, S.A. Generalized Langevin theory for many-body problems in chemical-dynamics—Gas-surface collisions, vibrational-energy relaxation in solids, and recombination reactions in liquids. J. Chem. Phys. 1980, 73,4353-64. 

Fig. 13. Excitation function for the reaction CH4 (CH4,CH3)CH5 compared with the prediction of the Langevin theory. Horizontal bars indicate the FWHM energy dispersion of the reactant ion swarm resulting from the impulsive acceleration. In these experiments, a short path length (0.3 cm) was used, under which conditions the CHj product ions are detected efficiently. The Langevin model is not applicable above thermal energies (see Section 4.2.2d).

Fig. 22. Excitation function for the total charge-transfer cross section for the reactants Ar + CH4. Open circles refer to data obtained by the longitudinal tandem/pulsed ejection technique illustrated in Fig. 3. Solid circles refer to data obtained by the single-source impulse technique discussed in Section 3.4.4c. The relative excitation function of Koski is also shown and is normalized to Masson s absolute excitation function at 10 eV. Shown as a dashed line is the close-collision cross section predicted from the Langevin theory.

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