Langevin equation relaxation times

Since l is proportional to and q is proportional to 1/L, i is proportional to. Substitution of Eq. (67) into Eq.(62) gives the Langevin equation for the Rouse modes of the chain within the approximations of preaveraging for hydrodynamic interactions and mode-mode decoupling for intersegment potential interactions. Equation (62) yields the following results for relaxation times and various dynamical correlation functions. 

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)... 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

In the adiabatic regime, the solvent relaxation time rc reaction coordinate. This limit corresponds to (t) = 5(t), so the power spectrum (Eq. (11.87)) is equal to , that is, to white noise . The GLE is reduced to the simple Langevin equation with a time-local friction force — x. Xr is found from Eq. (11.85) ... 

Equations [13], [14], and [15] involve the assumption that the time scale of the process is large compared to the relaxation time t of the velocity distribution of particles, hence that this distribution reaches equilibrium rapidly In each of the points of the system. A measure of this relaxation time is the reciprocal of the friction coefficient obtained from the Langevin equation for the Brownian motion of a free particle (t = M/Ctoi/ ), where Mis the mass of the particle. If this condition is not satisfied, the Fokker-Planck equation (8) should be the starting point of the analysis. 

The Langevin Equation of Motion and the Spectrum of the Relaxation Times... 

In order to understand how the algorithm actually works and to construct an explicit expression for the error it is not convenient to work with the metadynamics equations (12) in their full generality. Instead, we notice that the finite temperature dynamics of the collective variables satisfies, under rather general conditions, a stochastic differential equation [54,55]. Furthermore, in real systems the quantitative behavior of metadynamics is perfectly reproduced by the Langevin equation in its strong friction limit [56]. This is due to the fact that all the relaxation times are usually much smaller than the typical diffusion time in the CV space. Hence, we model the CVs evolution with a Langevin t3rpe dynamics ... 

Here the ratio Xri/xb represents the coupling between the magnetic and mechanical motions arising from the nonseparable namre of the Langevin equations, Lqs. (121) and (122). Thus the correction to the solid-state result imposed by the fluid is once again of the order 10 Hence we may conclude, despite the iionseparability of the equations of motion, that the Neel relaxation time of the ferrofluid particle should still be accurately represented in the IHD and VLD limits by the solid-state relaxation time formulae, Eqs. (87) and (90). Furthermore, Eq. (122) should be closely approximated by the solid-state relaxation equation... 

In the Markovian Kramers model discussed in Section 14.4, the friction coefficient y describes the coupling of the reaction coordinate to the thermal environment. In the low friction (underdamped) limit it is equal to the thermal relaxation rate in the reactant well, which is equivalent in the present case to the solvation well of the initial charge distribution. More generally, this rate should depend also on the frequency >s of this well. The theory of solvation dynamics, Chapter 15, does not use a Langevin equation such as (14.39) as a starting point, however it stiU yields an equivalent relaxation rate, the inverse solvation time (tl) , which is used in the present discussion. 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

No calculations yet have tackled this problem in all its complexity. Typically, it is assumed that the photodissociation produces some initial distribution of pairs, and then the subsequent time evolution of the unreacted pair probability is calculated. Even this more modest program has been carried out only at the diffusion and Langevin equation levels. We briefly comment on these results, since they indicate the magnitude of the solvent and velocity relaxation effects. 

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