Velocity relaxation

If the timestep is short relative to the velocity relaxation time, the solvent does not play much part in the motion. Indeed, if y = 0, there are no solvent effects at all. A simple algorithm for advancing the position vector r and velocity v has been given by van Gunsteren (van Gunsteren, Berendsen and Rullmaim, 1981) ... 

If the velocity relaxation time is short relative to the integration timestep, the following result is obtained ... 

Kushick J., Berne B. J. Methods for experimentally determining the angular velocity relaxation in liquids, J. Chem. Phys. 59, 4486-90 (1973). 

In the stochastic approach, the Markovian random process is usually used for the description of the solvent, and it is assumed that the velocity relaxation is much faster than the coordinate relaxation.74 Such a description is applicable at long time intervals which considerably exceed the characteristic times of the electron... 

Equation (10.6) for the mobility in the two-state model implicitly assumes that the electron lifetime in the quasi-free state is much greater than the velocity relaxation (or autocorrelation) time, so that a stationary drift velocity can occur in the quasi-free state in the presence of an external field. This point was first raised by Schmidt (1977), but no modification of the two-state model was proposed until recently. Mozumder (1993) introduced the quasi-ballistic model to correct for the competition between trapping and velocity randomization in the quasi-free state. 

When the solute molecules are significantly larger than the solvent molecules, the detailed motion of individual solvent molecules around a solute molecule matters less the solvent may be better approximated as a hydrodynamic continuum. Both the velocity relaxation time Tiei = mj Girrja and the velocity autocorrelation time rc are larger for larger solute... 

In Sect. 2.1, the timescale over which the diffusion equation is not strictly valid was discussed. When using the molecular pair analysis with an expression for h(f) derived from a diffusion equation analysis or random walk approach, the same reservations must be borne in mind. These difficulties with the diffusion equation have been commented upon by Naqvi et al. [38], though their comments are largely within the framework of a random walk analysis and tend to miss the importance the solvent cage and velocity relaxation effects. 

In this chapter, the motion of solute and solvent molecules is considered in rather more detail. Previously, it has been emphasised that this motion approximates to diffusion only over times which are long compared with the velocity relaxation time (see Chap. 8, Sect. 2.1). At times comparable with or a little longer than the velocity relaxation time, the diffusion equation does not provide a satisfactory description of molecular motion. An alternative approach must be sought. This introduces considerable complications to a theoretical analysis of very fast reactions in solution. To develop an understanding of chemical reactions occurring over very short time intervals, several points need to be discussed. Which reactions might be of interest and over what time scale What is known of the molecular motion of solute and solvent molecules Why does the Markovian (hydrodynamic) continuum analysis fail and what needs to be done to develop a better theory These points will be considered in further detail in this chapter. 

The velocity relaxation time is m/ (and this is mD/kB T 0.01— 0.1 ps at room temperature for small molecules). It is much longer than the time scale of fluctuations of the force f(t). The time average of f(f ) in the integral above is zero. Taking the average over the ensemble gives... 

This shows how the average velocity relaxes from its initial velocity u0 and also how an external force develops a drift velocity in the direction of that force. In both terms, the velocity relaxes with a times scale m/f-... 

The velocity relaxation time is again f/rn and the mean square velocity (up = k T/m. Schell et al. [272] have used the Langevin equation to model recombination of reactants in solutions. Finally, from the properties of the fluctuating force (see above)... 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

The Rayleigh particle is the same particle as the Brownian particle, but studied on a finer time scale. Time differences At are regarded that are small compared to the time in which the velocity relaxes, but, of course, still large compared to the duration of single collisions with the gas molecules. Thus the stochastic function to be considered is the velocity rather than the position. It is sufficient to confine the treatment to one dimension this is sometimes emphasized by the name Rayleigh piston . 0... 

The quantity y 1 is sometimes called the velocity relaxation time it can be considered to be the time taken for the particle to forget its initial velocity. The Eangevin equation of motion for a particle is therefore... 

Therefore, the motion of the particle is determined by a one-variable stochastic differential equation. The physical contents of (m/C)dV(t)l dr— 0 can be related to the fact that the velocity relaxes on a time scale much shorter than the time scale characterizing variations in position. 

In this section we treat the problem of evaluating an orientational correlational function without the inertial approximation (which assumes the molecular velocity relaxed to thermal equilibrium) and determining the spectroscopic effects of molecular inertia on a spin system S = 2 whose Hamiltonian is described by an axially anisotropic Zeeman interaction. 

The friction coefficient y defines the timescale, y of thermal relaxation in the system described by (8.13). A simpler stochastic description can be obtained for a system in which this time is shorter than any other characteristic timescale of our system. This high friction situation is often referred to as the overdamped limit. In this limit of large y, the velocity relaxation is fast and it may be assumed to quickly reaches a steady state for any value of the applied force, that is, v = x = 0. This statement is not obvious, and a supporting (though not rigorous) argument is provided below. If true then Eqs (8.13) and (8.20) yield... 

The physical manifestation of friction is the relaxation of velocity. In the high friction limit velocity relaxes on a timescale much faster than any relevant observation time, and can therefore be removed from the dynamical equation, leading to a solvable equation in the position variable only, as discussed in Section 8.4.4. The Fokker-Planck or Kramers equation (14.41) then takes its simpler, Smoluchowski form, Eq. (8.132)... 

Variation principle 18, 154, 222 VB (valence bond) model 94 Vector 4 Vector docking 57 Vector potential 294 Vector space 220 Vector, cross product 6 Vector, dot product 5 Vectors, orthogonal 6 Velocity dipole operator 193 Velocity relaxation 253... 

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