The Burgers equation is the simplest nonlinear generalization of the diffusion equation. We present a detailed dynamical renormalization-group analysis of this equation subject to random noise. The noise itself can be the product of another stochastic process and is hence allowed to have correlations in space and/or time. In dimensions higher than a critical $d_c$ weak and strong noise lead to different scaling exponents, while for d<$d_c$ any amount of noise is relevant resulting in strong-coupling behavior. In the absence of temporal correlations we find two regimes for d<$d_c$: either the hydrodynamic behavior is determined by white noise and correlations are unimportant, or correlations dominate and the resulting scaling exponents can be obtained exactly. With temporal correlations present, the hydrodynamic behavior is much more complex, as renormalization predicts a complicated dependence of the effective noise spectrum on frequency in certain regimes. The relevance of these results to two interesting problems is discussed. One is the anomalous transverse fluctuations of a directed polymer in a random medium, and the other is a description of a growing interface. Various recent numerical simulations are reviewed in the light of these results. For example, we show that an exponent identity observed in all simulations so far follows simply from the Galilean invariance of the equation in the absence of temporal correlations.

We present a model for the transmission of spin-polarized electrons through oriented chiral molecules, where the chiral structure is represented by a helix. The scattering potential contains a confining term and a spin-orbit contribution that is responsible for the spin-dependent scattering of electrons by the molecular target. The differential scattering cross section is calculated for right- and left-handed helices and for arbitrary electron spin polarizations. We apply our model to explain chiral effects in the intensity of photoemitted polarized electrons transmitted through thin organic layers. These are molecular interfaces that exhibit spin-selective scattering with surprisingly large asymmetry factors as well as a number of remarkable magnetic properties. In our model, differences in intensity are generated by the preferential transmission of electron beams whose polarization is oriented in the same direction as the sense of advance of the helix. This model can be easily extended to the Landauer regime of conductance where conductance is due to elastic scattering, so that we can consider the conductance of chiral molecular junctions.

A minimal model is exactly solved for electron spin transport on a helix. Electron transport is assumed to be supported by well oriented p_z type orbitals on base molecules forming a staircase of definite chirality. In a tight binding interpretation, the spin-orbit coupling (SOC) opens up an effective π_z − π_z coupling via interbase p_x,y − p_z hopping, introducing spin coupled transport. The resulting continuum model spectrum shows two Kramers doublet transport channels with a gap proportional to the SOC. Each doubly degenerate channel satisfies time reversal symmetry; nevertheless, a bias chooses a transport direction and thus selects for spin orientation. The model predicts (i) which spin orientation is selected depending on chirality and bias, (ii) changes in spin preference as a function of input Fermi level and (iii) back-scattering suppression protected by the SO gap. We compute the spin current with a definite helicity and find it to be proportional to the torsion of the chiral structure and the non-adiabatic Aharonov-Anandan phase. To describe room temperature transport, we assume that the total transmission is the result of a product of coherent steps.