Langevin relation

Setting W(t) = (2kBT )l/2W(t), the Langevin relation—describing the motion of the particle in the phase space—can be written as a two-variable stochastic differential equation system of the form of... 

Most of the exothermic ion—molecular reactions have no activation energy (Talrose, 1952). Quantum-mechanical repulsion between molecules, which provides the activation barrier even in the exothermic reactions of nentrals, can be suppressed by the charge-dipole attraction in the case of ion-molecular reactions. Thus, rate coefficients of the reactions are very high and often correspond to the Langevin relations (2 8)-(2-50), sometimes partially hmitedby quantum-mechanical factors (Su Bowers, 1975 Virin et al., 1978). The efiect obviously can be apphed to both positive and negative ions. 

An appropriate value of 7 for a system modeled by the simple Langevin equation can also be determined so as to reproduce observed experimental translation diffusion constants, Dt in the diffusive limit, Dt is related to y hy Dt = kgTmy. See [22, 36], for example. 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

This relation is as general as the Langevin equation itself, i.e., it holds for collisions of any strength. When deriving Eq. (1.23) from Eq. (1.26),... 

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. 

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 ... 

K. Lindenberg, K. E. Shuler, V. Seshadri, and B. J. West, Langevin equations with multiplicative noise theory and applications to physical processes, in Probabilistic Analysis and Related Topics, Vol. 3, A. T. Bharucha-Reid (ed.), Academic Press, San Diego, 1983, pp. 81-125. 

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

The Langevin dynamics method simulates the effect of individual solvent molecules through the noise W, which is assumed to be Gaussian. The friction coefficient r is related to the autocorrelation function of W through the fluctuation-dissipation theorem,... 

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... 

Third, Eq. (31) shows that A is nondistributive, and determines fluctuations. Since there is a flucmation, we can expect that the time evolution in Eq. (34) may be related to a stochastic process. Indeed, one can show that the time evolution (34) is identical to the time evolution generated by the set of Langevin equations for the stochastic operators aj(r), a (r) (see Ref. 14) ... 

In Table 1 (pp. 251-254), IM rate constants for reaction systems that have been measured at both atmospheric pressure and in the HP or LP range are listed. Also provided are the expected IM collision rate constants calculated from either Langevin or ADO theory. (Note that the rate constants of several IM reactions that have been studied at atmospheric pressure" are not included in Table I because these systems have not been studied in the LP or HP ranges.) In general, it is noted that pressure-related differences in these data sets are not usually large. Where significant differences are noted, the suspected causes have been previously discussed in Section IIB. These include the reactions of Hcj and Ne with NO , for which pressure-enhanced reaction rates have been attributed to the onset of a termolecular collision mechanism at atmospheric pressure and the reactions of Atj with NO and Cl with CHjBr , for which pressure-enhanced rate constants have been attributed to the approach of the high-pressure limit of kinetic behavior for these reaction systems. 

A completely unambiguous formulation of the Langevin equation may, however, be constructed by relating the random Langevin forces to random functions /m(f) of the type used in our discussion of the Stratonovich SDE, by taking... 

We now show that the same drift velocity is obtained from a Stratonovich interpretation of the Langevin equation with unprojected and projected random forces that have the same soft components but different hard components. We consider an unprojected random force ri (t) and a corresponding projected random force Tj (f) that are related by a generalization of Eq. (2.300), in which... 

A further development is possible by noting that the high frequency shear modulus Goo is related to the mean square particle displacement (m ) of caged fluid particles (monomers) that are transiently localized on time scales ranging between an average molecular collision time and the structural relaxation time r. Specifically, if the viscoelasticity of a supercooled liquid is approximated below Ti by a simple Maxwell model in conjunction with a Langevin model for Brownian motion, then (m ) is given by [188]... 

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) ... 

This effectively states that the probability of the final state (left-hand side) is equal to that of all the initial states transforming to the final state (with probability P). Chandrasekhar expanded out the infinitesimal velocity and time changes of these quantities as Taylor series and used the Langevin equation to relate 5u and 5f. He showed that if the probability of changing velocity and position is given by a Gaussian distribution, then the probability, W(u, r, t) that a Brownian particle has a velocity u at a position r and at time t is... 

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