Laplace transform time-dependent 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)... 

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

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. 

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...

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