Damping effects autocorrelation

Due to the finite propagation time T of the wavepackets, the Fourier transformation causes artifacts known as the Gibbs phenomenon [122]. In order to reduce this effect, the autocorrelation function is first multiplied by a damping function cos jtt/IT) [81,123]. Furthermore, to simulate the experimental line broadening, the autocorrelation functions will be damped by an additional multiplication with a Gaussian function exp — t/xd)% where zj is the damping parameter. This multiplication is equivalent to a convolution of the spectrum with a Gaussian with a full width at half maximum (FWHM) of /xd- The convolution thus simulates... 

This surprising result prompted Mazenko, Ramaswamy and Toner to examine the anharmonic fluctuation effects in the hydrodynamics of smectics. We have already shown that the undulation modes are purely dissipative with a relaxation rate given by (5.3.39). To calculate the effect of these slow, thermally excited modes on the viscosities, we recall that a distortion u results in a force normal to the layers given by (5.3.32). This is the divergence of a stress, which, from (5.3.53), contains the non-linear term 0,(Vj uf. Thus, there is a non-linear contribution (Vj uf to the stress. Now the viscosity at frequency co is the Fourier transform of a stress autocorrelation function, so that At (co), the contribution of the undulations to the viscosity, can be evaluated. It was shown by Mazenko et that Atj(co) 1 /co. In other words, the damping of first and second sounds in smectics, which should go as >/(oo)oo , will now vary linearly as co at low frequencies. 

The spectra from the 4-mode model coupled to 0, 5, 10, and 20 bath modes are plotted in Figs. 9(a)-(d). The addition of the bath clearly results in the structure of the spectrum being washed out. The experimental spectrum is, however, not obtained. The effect of the bath modes is made clear in Figs. 10(a)-(d), in which the absolute values of the autocorrelation function for the 4-mode system with 0, 5, 10, and 20 bath modes is plotted. Just the strongest 5 bath modes lead to a significant damping of the oscillations in the function. This is simply due to the extra volume of phase-space available... 

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