Correlation function displacement response

Because there is no general microscopic theory of liquids, the analysis of inelastic neutron scattering experiments must proceed on the basis of model calculations. Recently1 we have derived a simple interpolation model for single particle motions in simple liquids. This derivation, which was based on the correlation function formalism, depends on dispersion relation and sum rule arguments and the assumption of simple exponential decay for the damping function. According to the model, the linear response in the displacement, yft), satisfies the equation... 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

Equation (524) is an exact result for the displacement SA(r, t)) fromequilibriumarising from the external force F (f, t). As evident from Eq. (524), the response of the system to the external force F (r, t) is nonlocal in space and time. Moreover, the response is nonlinear with the nonlinear dependence parried by the nonequilibrimn time correlation function r t ) through pf). 

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