Decomposing thermal fluctuations with hydrodynamic modes

We have obtained analytically the complete set of hydrodynamic modes (HMs) for a two-dimensional (2D) fluid confined within a channel with the Navier slip boundary condition at the hydrodynamic boundary. The HMs are orthogonal to each other and hence each represents an independent degree of freedom. We show that the HMs can be used to recursively generate a time series of random thermal fluctuations of displacement velocity, with identical statistical distributions as those obtained from MD simulations. By projecting the HMs onto molecular dynamics (MD) configurations and evaluating the resulting decay time from the autocorrelation function, we obtain from MD the eigenvalues of the HMs. Multiplying two different HMs and integrating as a function of z from center of the channel towards the fluid-solid interface, the position of the hydrodynamic boundary (HDB) is unambiguously identified as the point at which the integral vanishes. Invariably the HDB is located inside the fluid domain and not on the liquid-solid interface. With the knowledge of the HDB position, the value of the slip length can be obtained directly from HM's dispersion relation. We show that in terms of the complete set of HMs, the fluctuation-dissipation theorem may be expressed in a simple expression involving the average of the inverse of the eigenvalues. Besides offering an alternative perspective on thermal fluctuations and hydrodynamic boundary, the present work opens the possibility of using modulated boundary conditions to manipulate thermal fluctuations in mesoscopic channels, which can lead to interesting statistical mechanical consequences.