| One of the research interests being addressed is to provide a new approach to quantum mechanical many-body systems, specifically concentrating on the fermion system. A fermion ground state energy functional is set up in terms of particle density; relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a, lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact stun rules. Prototypical application is made to several one-dimensional spinless non-interacting models. The effectiveness of "atomic" constraints on model "molecules" is observed, as well as the structure of systems with only finitely many bound states. Extension to the more realistic systems---three-dimensional fermion systems with interactions---are also presented in the developing stage, to the system with spherical symmetry.; Another problem being studied is in the classical regime, a multi-body interaction mediated by turbulence. We looked into a mono-layer of spheres sedimenting in a convective fluid, in which the spheres behaves like a deformable patch collectively. In this table size experiment, we observe a cyclic occurrence of two dynamic states---packing and unpacking---from the coupling between the convective flow and the irregular transportation of the spheres due turbulence. There appears a fastest oscillation with a moderate choice of size of the spheres, Rayleigh number and the coverage ratio. We attempt to give both qualitative explanations to the cause of fastest oscillation and a self-excited bi-stable oscillator, which are derivable from first-principle fluid mechanics, is providing us with a phenomenological model to simplify the realistic system. With a proper implementation of the physical parameters, such as the size and temperature difference, this phenomena might be related to other industrial and natural process with the similar dimensionless Rayleigh number, Prandtl number and Reynolds number. |
