And Langevin model

Three-Body Ion-Ion Recombination Thomson s Theory and Langevin Model... 

Comparison of the Thomson and Langevin models shows that recombination rate coefficients grow with pressme at moderate pressures and begin to decrease as... 

Thomson and Langevin Models of Three-Body Ion-Ion Recombination. Comparing Thomson s and Langevin s models for the three-body ion-ion recombination, (2-93) and (2-97), find a typical value of pressure when the recombination rate coefficient is the highest. Specifically consider oxygen plasma at room gas temperature. 

An alternative framework to Newtonian dynamics, namely Langevin dynamics, can be used to mask mild instabilities of certain long-timestep approaches. The Langevin model is phenomenological [21] — adding friction and random... 

The Langevin model has been employed extensively in the literature for various numerical and physical reasons. For example, the Langevin framework has been used to eliminate explicit representation of water molecules [22], treat droplet surface effects [23, 24], represent hydration shell models in large systems [25, 26, 27], or enhance sampling [28, 29, 30]. See Pastor s comprehensive review [22]. 

There are three issues of concern regarding the application of LI to biomolecular dynamics (1) the governing Langevin model, (2) the implications of numerical damping, and (3) the CPU performance, given that nonlinear minimization is required at each, albeit longer, timestep. 

From a mathematical perspective either of the two cases (correlated or non-correlated) considerably simplifies the situation [26]. Thus, it is not surprising that all non-adiabatic theories of rotational and orientational relaxation in gases are subdivided into two classes according to the type of collisions. Sack s model A [26], referred to as Langevin model in subsequent papers, falls into the first class (correlated or weak collisions process) [29, 30, 12]. The second class includes Gordon s extended diffusion model [8], [22] and Sack s model B [26], later considered as a non-correlated or strong collision process [29, 31, 32],... 

Fig. 6.8. The dependence of rj2 on x) by the Ivanov model (I) and friction model (F) in comparison with predictions of the extended. /-diffusion (ED) and Langevin (L) models for linear molecules. The line (H) corresponds to the Hubbard inverse proportionality between xgj and xj at very high densities. Experimental data from [81] are in rectangles around line G with the length of their vertical and horizontal sides being equal, correspondingly, to the experimental errors in x el and rj measurements. Experimental data from [270] (J) are shown both in original position and shifted down by a factor of four (broken line).

Laguerre polynomials 3-D. /-space 264 eigenfunctions 119, 262-3 Raman and CARS spectra 120 Landau-Teller formula 159 Langevin model equation 32... 

Haworth, D. C. and S. B. Pope (1986). A generalized Langevin model for turbulent flows. 

Wouters, H. A., T. W. J. Peters, and D. Roekaerts (1996). On the existence of a generalized Langevin model representation for second-moment closures. Physics of Fluids 8, 1702-1704. 

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]... 

We consider the so-called Langevin model for ion-molecule reactions. An ion and a neutral non-polar molecule interact—at large distances—through an ion-induced-dipole potential... 

Phase space theory considers the long-range polarisation between the associating ion and neutral and this interaction has generally been described on the basis of the Langevin model [808]. As has been realised for some time [482, 486, 547, 614], the phase space theory is equivalent to transition state theory in the extreme case of a loose transition state. The term loose as used here means that the incipient fragments can each freely rotate [287, 333] within the confines of conservation of angular momentum. 

Behavior of the Fluctuation-Dissipation Ratio and of the Effective Temperature in the Langevin Model... 

Since X(x,tw) does not depend on T, it can still be viewed as rescaling the temperature. The temperature rescaling factor now depends on both times x and tw. In the limit of large x and fw(yx 1, jtw 1), it tends toward the constant value 1/2 which corresponds to the viscous limit of the Langevin model [Eq. (71)]. As for the associated inverse effective temperature, since one has... 

In fact, the validity of Eqs. (90) and (91) is not restricted to the simple (i.e., nonretarded) Langevin model as defined by Eq. (73). These formulas can be applied in other classical descriptions of Brownian motion in which a time-dependent diffusion coefficient can be defined. This is for instance, the case in the presence of non-Ohmic dissipation, in which case the motion of the Brownian particle is described by a retarded Langevin equation (see Section V). 

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