The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy

This is the second in a series of papers in which we derive a $\Gamma$-expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the $\Gamma$-limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order $\Gamma$-limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of $\Gamma$-equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers

The Γ\mathbf \Gamma-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density

This is the first in a series of two papers in which we derive a $\Gamma$-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In this paper we show that an appropriate setting for $\Gamma$-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional $\Gamma$-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain a characterization of the limit charge density for the energy minimizers in the diffuse interface model

Using Technology to Support At-Risk Students' Learning

A new report finds that technology - when implemented properly -can produce significant gains in student achievement and boost engagement, particularly among students most at risk

Space-Time Symmetries: P and CP Violation

We begin with a few remarks on an explicit construction of a Bargmann-Wightman-Wigner-type quantum field theory [Phys. Lett. B {\bf 316}, 102 (1993)] in which bosons and associated antibosons have opposite relative intrinsic parities. We then construct $(1,0)\oplus(0,1)$ Majorana ($CP$ self conjugate) and Majorana-like ($C\Gamma^5$ self conjugate, $\Gamma^5=$ chirality operator) fields. We point out that this new structure in the space time symmetries may be relevant to $P$ and $CP$ violation.Comment: Talk presened by D. V. Ahluwalia at the III International Wigner Symposium, Christ Church, Oxford, September 1993

Selective excitation of homogeneous spectral lines

It is possible, for homogeneously broadened lines, to excite selectively the response signals, which are orders of magnitude narrower than the original lines. The new type of echo, which allows detecting such signals, and the formalism, useful for understanding the phenomenon, as well as the experimental examples from NMR spectroscopy are presented.Comment: 19 pages, 8 figure

Parallax measurements of cool brown dwarfs

Accurate parallax measurements allow us to determine physical properties of brown dwarfs, and help us to constrain evolutionary and atmospheric models, break the age-mass degeneracy and reveal unresolved binaries. We measured absolute trigonometric parallaxes and proper motions of 6 cool brown dwarfs using background galaxies to establish an absolute reference frame. We derive the absolute J-mag. The six T brown dwarfs in our sample have spectral types between T2.5 and T7.5 and magnitudes in J between 13.9 and 18.0, with photometric distances below 25 pc. The observations were taken in the J-band with the Omega-2000 camera on the 3.5 m telescope at Calar Alto, during a time period of 27 months, between March 2011 and June 2013. The number of epochs varied between 11 and 12 depending on the object. The reduction of the astrometric measurements was carried out with respect to the field stars. The relative parallax and proper motions were transformed into absolute measurements using the background galaxies in our fields. We obtained absolute parallaxes for our six brown dwarfs with a precision between 3 and 6 mas. We compared our results in a color-magnitude diagram with other brown dwarfs with determined parallax and with the BT-Settl 2012 atmospheric models. For four of the six targets we found a good agreement in luminosity with objects of similar spectral types. We obtained an improved accuracy in the parallaxes and proper motions in comparison to previous works. The object 2MASS J11061197+2754225 is more than 1 mag overluminous in all bands pointing to binarity or high order multiplicity.Comment: 9 pages, 15 figures. Astronomy & Astrophysics, 201