Laplace transform friction

The normal mode transformation implies that q = uP + Ej Ujoyj and that p = u0oq + 2j UojXj. One can show,50,88 that the matrix element uoo may be expressed in terms of the Laplace transform of the time dependent friction and the barrier frequency A ... 

We see that a calculation of Ar involves a Laplace transform of the time-dependent friction kernel. This may typically be determined in a molecular dynamics (MD) simulation, where the autocorrelation function of the random force (R(O)R(t)) may be determined, which then allows us to determine (f) using the fluctuation-dissipation theorem in Eq. (11.58). Note that Eq. (11.85) is an implicit equation for Ar that in general must be solved by iteration. In the absence of friction we see from Eq. (11.85)... 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

Equations (25) and (26) are Grote-Hynes key results. They show that k is determined by Xr and that, in its turn, this reactive frequency Ar is determined both by the barrier frequency Wb and by the Laplace transform frequency component of the friction (see eq.(27)). 

Finally, the Laplace transform of the time-dependent friction is given in discretized form as... 

The parabolic barrier plays a special role in rate theory. The GLE (with space-independent friction) may be solved analytically using Laplace transforms. The two-dimensional Fok-ker-Planck equation derived from the Langevin equation may be solved analytically, as was done by Kramers in his famous paper of 1940. In this section we present some of the analytic results for the parabolic barrier dynamics. These results are important from both a conceptual and a practical point of view. Later we shall see how one returns to the parabolic barrier case as a source of comprehension, approximation, etc. 

The Laplace transform of the friction function K(t) is known in the continuum limit (cf. Eq. (48)) so that one may now directly relate the spectral density of the normal modes /(A) to the spectral density 7(w). One finds... 

The second equality on the right-hand side follows directly from Eq. (46) for e = 0. This allows the collective bath mode frequency to be expressed in terms of the Laplace transform of the time-dependent friction (cf. Eqs. (39) and (43)), so it is well defined in the continuum limit. 

The results for the transformation coefficients, Eqs. (196) and (197), enable us to express the various parameters in terms of the barrier frequency, the Laplace transforms of the two time-dependent friction kernels, and the coefficients a() and b(). The explicit relations are... 

In Eq. [7], the frequency-dependent friction is the Laplace transform of the time-dependent friction The presence of the Laplace transform means that the time-dependence of the friction must be known in order to determine the Laplace transform. This friction can be readily determined from molecular dynamics simulations in the approximation where the motion along the reaction coordinate is fixed at x = 0. (A discussion of some subtle, but important, aspects of this approximation is given by Carter et al. ) In that case, the random force R(t) can be calculated from equilibrium dynamics in the presence of this one constraint. From R(t), the time-dependent friction (t) can be calculated and the implicit Eq. [7] solved. The result gives the Grote—Hynes value of the transmission coefficient for that system. 

Here, denotes the Laplace transform of the memory friction. The above equation ean be easily transformed into... 

The temprature dependence of the friction can be obtained by reverse Fourier transform of Eq. (134), which after the Laplace transform gives... 

Fig. 7 Decay of translational (U) and rotational (12) velocity correlations of a suspended sphere. The time-dependent velocities of the sphere are shown as solid symbols the relaxation of the corresponding velocity autocorrelation functions are shown as open symbols (with statistical error bars). A sufficiently large fluid volume was used so that the periodic boundary conditions had no effect on the numerical results for times up to r = 1,000 in lattice units (h = b = 1). The solid lines are theoretical results, obtained by an inverse Laplace transform of the frequency-dependent friction coefficients [175] of a sphere of appropriate size (a = 2.6) and mass (pj/p = 12) the kinematic viscosity of the pure fluid = 1/6...

---