Molecular Dynamics Study on Wave Equation of Liquid

Abstract

The equation of wave propagating in fluid is described as a differential equation of the velocity or small element density from the Navier-Stokes (NS) equation. On the other hand, the molecular dynamics (MD) equation expresses the motion of a particle constituting the fluid and shows that the particle is always in motion regardless of the existence of the wave. In this study, we discuss in what way meanings of the velocity and density in the wave equation can be adopted in the MD system. What are the differentials with respect to time and space in the NS equation for the system of MD particles? Ordinarily, the physical quantities in the NS equation are obtained as the ensemble and time averages over the MD system. We investigate the number of particles and duration of time that are sufficient for the averages and simultaneously confirm whether the averaged values satisfy the differential equation. The two-dimensional MD method is used for the qualitative understanding. The fluid is assumed to consist of particles connected by the Lennard-Jones potential. The satisfaction of the differential equation by the MD averaged values is shown by the propagation velocity in the wave equation. The propagation velocity can be also obtained in another way i.e. by theµ observation∂ of the wave fronts motion of velocity, density or total energy wave in the fluid. The propagation velocity that has resulted from the wave equation is strongly affected by the ensemble and time averages. On the other hand, when it is obtained from the wave front, it is independent of the ensemble and time averages. We can have the former propagation velocity close to the latter one when al ong time average is used for al arge ensemble and as hort time average for as mall ensemble, i.e. the product of both averages can be considered changed at same rate as if by a scaling coefficient.