The semiclassical path integral (SPI) method has been applied for studying spin relaxation in a narrow 2D strip with the Rashba spin-orbit interaction. Our numerical calculations show a good agreement with the experimental data, although some features of experimental results are not clear yet. We also calculated the relaxation of a uniform spin-density distribution in the ballistic regime of very narrow wires. With the decreasing wire width, the spin polarization exhibits a transition from the exponential decay to the oscillatory Bessel-like relaxation. The SPI method has been also employed to calculate the relaxation of the particularly long-lived helix mode. A good agreement has been found with calculations based on the diffusion theory.

Bacterial carpets consist of randomly anchored uni-polar-flagellated sodium-motive bacterial matrix are prepared by flow deposition. Collective flow dynamics across the bacterial carpets are probed with optical tweezers-microsphere assay. Around the center of a uniform bacterial cluster, collective forces that pull microsphere towards carpet surface are detected at a distance of 10 μm away from carpets. At sodium-motive driving over a critical value, the force magnitudes increase abruptly, suggesting a threshold-like transition of hydrodynamic synchronization across bacterial carpet. The abrupt force increase is explained in term of bifurcation to phase synchronization in a noisy non-linearly coupled rotor array mediated by hydrodynamic interactions.

We generalized the semiclassical path integral method originally used in the D'yakonov-Perel' mechanism to study the spin relaxation of the Elliott-Yafet mechanism in low-dimensional systems. In quantum wells, the spin properties calculated by this method confirmed the experimental results. In two-dimensional narrow wires, size and impurity effects on the Elliott-Yafet relaxation were predicted, including the wire-width-dependent relaxation time, the polarization evolution on the sample boundaries, and the relaxation behavior during the diffusive-ballistic transition. These properties were compared with those of the D'yakonov-Perel' relaxation calculated under similar conditions. For ballistic narrow wires, we derived an exact relation between the Elliott-Yafet relaxation time and the wire width, which confirmed the above simulations.

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