The attractive nonlinear delta-function potential

We solve the continuous one-dimensional Schr\"{o}dinger equation for the case of an inverted {\em nonlinear} delta-function potential located at the origin, obtaining the bound state in closed form as a function of the nonlinear exponent. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with the nonlinear exponent, becoming an almost completely extended state when this approaches two. At an exponent value of two, the bound state suffers a discontinuous change to a delta-like profile. Further increase of the exponent increases again the width of the probability profile, although the bound state is proven to be stable only for exponents below two. The transmission of plane waves across the nonlinear delta potential increases monotonically with the nonlinearity exponent and is insensitive to the sign of its opacity.Comment: submitted to Am. J. of Phys., sixteen pages, three figure

A two-state kinetic model for the unfolding of single molecules by mechanical force

We investigate the work dissipated during the irreversible unfolding of single molecules by mechanical force, using the simplest model necessary to represent experimental data. The model consists of two levels (folded and unfolded states) separated by an intermediate barrier. We compute the probability distribution for the dissipated work and give analytical expressions for the average and variance of the distribution. To first order, the amount of dissipated work is directly proportional to the rate of application of force (the loading rate), and to the relaxation time of the molecule. The model yields estimates for parameters that characterize the unfolding kinetics under force in agreement with those obtained in recent experimental results (Liphardt, J., et al. (2002) {\em Science}, {\bf 296} 1832-1835). We obtain a general equation for the minimum number of repeated experiments needed to obtain an equilibrium free energy, to within $k_BT$, from non-equilibrium experiments using the Jarzynski formula. The number of irreversible experiments grows exponentially with the ratio of the average dissipated work, \bar{\Wdis}, to $k_BT$.}Comment: PDF file, 5 page

Tests hidrodinàmics amb el codi multidimensional FLASH

El FLASH és un codi estable i flexible, desenvolupat a la Universitat de Chicago, que permet estudiar diferents fenòmens hidrodinàmics. L'objectiu futur és desenvolupar models multidimensionals centrats en el camp de les noves per a caracteritzar algunes fases complexes del fenomen, però degut a la complexitat del codi (més de 500000 línies de programació en C++ i Fortran90), la realització d'una simulació estel.lar necessita tenir un coneixement bàsic del FLASH. Per tant, l'objectiu d'aquest treball és la realització d'alguns tests numèrics per aprendre'n el funcionament. Però a la vegada, l'objectiu és doble, ja que per utilitzar una eina de treball com el FLASH, necessitem testejar el propi codi i comprovar que és capaç de resoldre correctament diferents problemes hidrodinàmics que ja s'han simulat prèviament. Així, s'han treballat els diferents tests i hem observat que el codi FLASH és eficaç per simular-los correctament i, per tant, podem esperar que serà un bon codi per a les futures simulacions estel.lars. D'altra banda, aquest primer contacte amb el codi ens ha permès familiaritzar-nos amb l'estructura, adquirir destresa en el seu ús i començar a entendre'l, tot deixant per a més endavant l'estudi detallat dels diferents models numèrics implementats en el codi i eines auxiliars de representació gràfica

Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem

Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to $4096^3$. The results are analyzed in terms of the classical analyticity strip method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of the time-decay of the width of the analyticity strip $\delta(t)$ is observed at the highest resolution for $3.7<t<3.85$ while preliminary 3D visualizations show the collision of vortex sheets. The BKM criterium on the power-law growth of supremum of the vorticity, applied on the same time-interval, is not inconsistent with the occurrence of a singularity around $t \simeq 4$. These new findings lead us to investigate how fast the analyticity strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A new simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if $\delta(t)$ vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for $\delta(t)$ then its exponent must be greater than some critical value, thus providing a new test that is applied to our $4096^3$ Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time-interval on which a well-resolved power-law behavior of $\delta(t)$ takes place, and check whether the new regime is genuine and not simply a crossover to a faster exponential decay

Identification of new transitional disk candidates in Lupus with Herschel

New data from the Herschel Space Observatory are broadening our understanding of the physics and evolution of the outer regions of protoplanetary disks in star forming regions. In particular they prove to be useful to identify transitional disk candidates. The goals of this work are to complement the detections of disks and the identification of transitional disk candidates in the Lupus clouds with data from the Herschel Gould Belt Survey. We extracted photometry at 70, 100, 160, 250, 350 and 500 $\mu$m of all spectroscopically confirmed Class II members previously identified in the Lupus regions and analyzed their updated spectral energy distributions. We have detected 34 young disks in Lupus in at least one Herschel band, from an initial sample of 123 known members in the observed fields. Using the criteria defined in Ribas et al. (2013) we have identified five transitional disk candidates in the region. Three of them are new to the literature. Their PACS-70 $\mu$m fluxes are systematically higher than those of normal T Tauri stars in the same associations, as already found in T Cha and in the transitional disks in the Chamaeleon molecular cloud. Herschel efficiently complements mid-infrared surveys for identifying transitional disk candidates and confirms that these objects seem to have substantially different outer disks than the T Tauri stars in the same molecular clouds.Comment: Accepted for publication in A&A. 16 pages, 9 figures, 7 table

Two-phase stretching of molecular chains

While stretching of most polymer chains leads to rather featureless force-extension diagrams, some, notably DNA, exhibit non-trivial behavior with a distinct plateau region. Here we propose a unified theory that connects force-extension characteristics of the polymer chain with the convexity properties of the extension energy profile of its individual monomer subunits. Namely, if the effective monomer deformation energy as a function of its extension has a non-convex (concave up) region, the stretched polymer chain separates into two phases: the weakly and strongly stretched monomers. Simplified planar and 3D polymer models are used to illustrate the basic principles of the proposed model. Specifically, we show rigorously that when the secondary structure of a polymer is mostly due to weak non-covalent interactions, the stretching is two-phase, and the force-stretching diagram has the characteristic plateau. We then use realistic coarse-grained models to confirm the main findings and make direct connection to the microscopic structure of the monomers. We demostrate in detail how the two-phase scenario is realized in the \alpha-helix, and in DNA double helix. The predicted plateau parameters are consistent with single molecules experiments. Detailed analysis of DNA stretching demonstrates that breaking of Watson-Crick bonds is not necessary for the existence of the plateau, although some of the bonds do break as the double-helix extends at room temperature. The main strengths of the proposed theory are its generality and direct microscopic connection.Comment: 16 pges, 22 figure