Bose-Einstein condensation at finite temperatures: Mean field laws with periodic microstructure

At finite temperatures below the phase transition point, the Bose-Einstein condensation, the macroscopic occupation of a single quantum state by particles of integer spin, is not complete. In the language of superfluid helium, this means that the superfluid coexists with the normal fluid. Our goal is to describe this coexistence in trapped, dilute atomic gases with repulsive interactions via mean field laws that account for a {\em spatially varying} particle interaction strength. By starting with the $N$-body Hamiltonian, $N\gg 1$, we formally derive a system of coupled, nonlinear evolution equations in $3+1$ dimensions for the following quantities: (i) the wave function of the macroscopically occupied state; and (ii) the single-particle wave functions of thermally excited states. For stationary (bound) states and a scattering length with {\em periodic microstructure} of subscale $\epsilon$, we heuristically extract effective equations of motion via periodic homogenization up to second order in $\epsilon$.Comment: 28 page

On solutions of Maxwell's equations with dipole sources over a thin conducting film

We derive and interpret solutions of time-harmonic Maxwell's equations with a vertical and a horizontal electric dipole near a planar, thin conducting film, e.g. graphene sheet, lying between two unbounded isotropic and non-magnetic media. Exact expressions for all field components are extracted in terms of rapidly convergent series of known transcendental functions when the ambient media have equal permittivities and both the dipole and observation point lie on the plane of the film. These solutions are simplified for all distances from the source when the film surface resistivity is large in magnitude compared to the intrinsic impedance of the ambient space. The formulas reveal the analytical structure of two types of waves that can possibly be excited by the dipoles and propagate on the film. One of these waves is intimately related to the surface plasmon-polariton of transverse-magnetic (TM) polarization of plane waves.Comment: 48 pages, 4 figure

Unification of step bunching phenomena on vicinal surfaces

We unify step bunching (SB) instabilities occurring under various conditions on crystal surfaces below roughening. We show that when attachment-detachment of atoms at step edges is the rate-limiting process, the SB of interacting, concentric circular steps is equivalent to the commonly observed SB of interacting straight steps under deposition, desorption, or drift. We derive a continuum Lagrangian partial differential equation, which is used to study the onset of instabilities for circular steps. These findings place on a common ground SB instabilities from numerical simulations for circular steps and experimental observations of straight steps