We perform equilibrium molecular dynamics simulations for nanoscale fluids confined between two parallel walls and investigate how the autocorrelation function of force acting on one wall is related to the slip length. We demonstrate that for atomically smooth surfaces, the autocorrelation function is accurately described by linearized fluctuating hydrodynamics (LFH). Excellent agreement between the simulation and the LFH solution is found over a wide range of scales, specifically, from the time scale of fluid relaxation even to that of molecular motion. Fitting the simulation data yields a reasonable estimation of the slip length. Thus, LFH provides a starting point for examining the relationship between the slip length and the fluctuations of the force acting on the wall.

We provide general derivations of the partial slip boundary condition from microscopic dynamics and linearized fluctuating hydrodynamics. The derivations are based on the assumption of separation of scales between microscopic behavior, such as collision of particles, and macroscopic behavior, such as relaxation of fluid to global equilibrium. The derivations lead to several statistical mechanical expressions of the slip length, which are classified into two types. The expression in the first type is given as a local transport coefficient, which is related to the linear response theory that describes the relaxation process of the fluid. The second type is related to the linear response theory that describes the non-equilibrium steady state and the slip length is given as combination of global transport coefficients, which are dependent on macroscopic lengths such as a system size. Our derivations clarify that the separation of scales must be seriously considered in order to distinguish the expressions belonging to two types. Based on these linear response theories, we organize the relationship among the statistical mechanical expressions of the slip length suggested in previous studies.

We study boundary conditions applied to the macroscopic dynamics of Newtonian liquids from the view of microscopic particle systems. We assume the existence of microscopic boundary conditions that are uniquely determined from a microscopic description of the fluid and the wall. By using molecular dynamical simulations, we examine a possible form of the microscopic boundary conditions. In the macroscopic limit, we may introduce a scaled velocity field by ignoring the higher order terms in the velocity field that is calculated from the microscopic boundary condition and standard fluid mechanics. We define macroscopic boundary conditions as the boundary conditions that are imposed on the scaled velocity field. The macroscopic boundary conditions contain a few phenomenological parameters for an amount of slip, which are related to a functional form of the given microscopic boundary condition. By considering two macroscopic limits of the non-equilibrium steady state, we propose two different frameworks for determining macroscopic boundary conditions.

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