This thesis is concerned with the use of the continuum Navier-Stokes (NS) equation’s eigenfunctions and eigenvalues for a two-dimensional channel to delineate the thermal fluctuations and their consequences, i.e., the fluctuation-dissipation theorem, that usually belong to the realm of kinetic theory and molecular dynamics, built upon the mathematics of discrete molecules. Furthermore, since the eigenfunctions of the NS equation are inherently descriptive of collective fluid motions over extended spatial distances, it is shown that such nonlocal correlation in the velocity field can be manifest in thermal fluctuations within a mesoscopic channel, with periodically modulated boundary conditions. We confirmed such nonlocal correlations in thermal fluctuations by using molecular dynamics, the first ever to be observed and in sharp contrast to the usual expectations.
Thermal fluctuation is a fundamental equilibrium phenomenon which depicts the random deviations of an observable from its average. In a fluid comprising a large number of interactive molecules, temporal and spatial correlations are usually governed by the relevant scales of molecular collisions, i.e., the Brownian motion of individual molecules. Complementary to the molecular point of view, hydrodynamic modes (HMs) are the eigenfunctions of the continuum NS equation under the appropriate boundary conditions. In contrast to the motions of individual molecules, a HM represents a collective motion of the fluid. From a mathematical point of view, the thermal fluctuations in a fluid can also be regarded as comprising a multitude of hydrodynamic modes (HMs) with random phases, each one having one degree of freedom. In this research work, we raise a new perspective of representing the thermal fluctuations in a fluid by using HMs. In this thesis we first show that the solution of the 2D channel HMs, under the Navier slip boundary condition, can be obtained analytically, and that the thermal fluctuations can be mathematically decomposed into HMs. A new expression of the fluctuation-dissipation theorem is obtained in terms of the eigenvalues of the HMs. However, since the HMs represent collective fluid motions, it is an intriguing question whether such non-local correlations in the velocity field may appear in thermal fluctuations under specified conditions. If so, that would introduce new elements into the statistical ensemble averaging of physical parameters. We show that by periodically modulating the slip length in the Navier boundary condition along the walls of a 2D mesoscopic channel, the nonlocal correlation of phase-locked HMs has indeed become partially detectable in molecular dynamics, in agreement with the prediction of the continuum perspective with the HMs.
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