We discuss an application of extended hydrodynamics to a model of water, in a range of wave numbers k, where the effect of single-molecule modes must be taken into account together with the collective phenomena underlying sound propagation and dispersion. The calculation of the density-density, energy-density, energy-energy, and longitudinal and transverse current correlation functions from a molecular dynamics simulation of the transferable intermolecular potential with four points (TIP4P) model of water, allows us to obtain the k dependence of the generalized hydrodynamic coefficients. In particular, we have found that the ratio of generalized heat capacities γ(k)=cp(k)/cv(k)1 up to k1 -1 and that the correlation between temperature and density fluctuations is negligible at all times, while there is an important frequency dependence of the transport coefficients. This leads to a remarkable simplification of the expression of the Laplace transform of the correlation functions, although models for the transport coefficients are still necessary at the present state of the theory. The frequency dependence of the transport coefficients is necessary to describe correctly the behavior of the density-density and temperature-temperature autocorrelation functions (ACF's). A model for the frequency dependence of the generalized viscosity φ(k,z) and thermal diffusivity DT(k,z) is proposed here. In addition to the correct short-time behavior of the correlation functions of the memory kernel, this model is able to account satisfactorily for the effects of the acoustic mode and the single-molecule modes, in particular, that related to the oscillation in the nearest neighbor cage (45 THz). A simple polynomial extrapolation to k=0 of the parameters of the model gives values consistent with the large sound dispersion observed in water. In the supercooled region, the shape of the predicted dispersion curve shows that there are two k ranges, 0.01-0.03 and 0.2-0.5 -1, which account for most of the dispersion. When the temperature increases the contribution to the lower k range is less apparent and shifted to higher k, but the behavior of the 0.2-0.5 -1 range does not change. The model also predicts an acoustic mode frequency ωmax(k)/k, 2-3 times larger, and a bandwidth Δω1/2(k)/k2, almost an order of magnitude smaller than those in the hydrodynamic regime. Moreover, ωmax(k) and Δω1/2(k) are in quantitative agreement with the neutron scattering data at T=298 K. The location and height of the first step of the dispersion curve are related to the long-time tail of generalized viscosity, while its size is determined by the anomalous value of the second moment of the longitudinal current ω(k) as compared to that of the density-density ACF ω0(k). The analysis of the transverse current ACF with the same model and the value of the transport coefficients obtained confirm that the TIP4P model potential leads to a shear and bulk viscosity in satisfactory agreement with the experimental data at 298 K. In the supercooled region, conversely, the dynamics obtained with the TIP4P potential is 2-3 times faster than that of real water at the same temperature, as already noted for the self-diffusion coefficient and dielectric relaxation times. © 1995 The American Physical Society.
