Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions

We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the semi-analytical method, relative to the number of eigenvalues, is investigated and the effect of varying the interface conditions on the solution behaviour is explored. Numerical experiments demonstrate that solutions can be computed using the new semi-analytical method that are more accurate and more efficient than the unified transform method of Sheils [Appl. Math. Model., 46:450-464, 2017]. Furthermore, unlike classical analytical solutions and the unified transform method, only the new semi-analytical method is able to correctly treat problems with both time-varying external boundary conditions and a large number of layers. The paper is concluded by replicating solutions to several important industrial, environmental and biological applications previously reported in the literature, demonstrating the wide applicability of the work.Comment: 24 pages, 8 figures, accepted version of paper published in Applied Mathematics and Computatio

Integral equation for inhomogeneous condensed bosons generalizing the Gross-Pitaevskii differential equation

We give here the derivation of a Gross-Pitaevskii--type equation for inhomogeneous condensed bosons. Instead of the original Gross-Pitaevskii differential equation, we obtain an integral equation that implies less restrictive assumptions than are made in the very recent study of Pieri and Strinati [Phys. Rev. Lett. 91 (2003) 030401]. In particular, the Thomas-Fermi approximation and the restriction to small spatial variations of the order parameter invoked in their study are avoided.Comment: Phys. Rev. A (accepted

Fast computation of effective diffusivities using a semi-analytical solution of the homogenization boundary value problem for block locally-isotropic heterogeneous media

Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution method to a standard finite volume method and show that equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new solution method can be applied to complex heterogeneous geometries represented by a grid of blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up simulations of diffusion in heterogeneous media.Comment: 29 pages, 4 figures, 5 table

Sharp crossover from composite fermionization to phase separation in mesoscopic mixtures of ultracold bosons

We show that a two-component mixture of a few repulsively interacting ultracold atoms in a one-dimensional trap possesses very different quantum regimes and that the crossover between them can be induced by tuning the interactions in one of the species. In the composite fermionization regime, where the interactions between both components are large, none of the species show large occupation of any natural orbital. Our results show that by increasing the interaction in one of the species, one can reach the phase-separated regime. In this regime, the weakly interacting component stays at the center of the trap and becomes almost fully phase coherent, while the strongly interacting component is displaced to the edges of the trap. The crossover is sharp, as observed in the in the energy and the in the largest occupation of a natural orbital of the weakly interacting species. Such a transition is a purely mesoscopic effect which disappears for large atom numbers.Comment: 5 pages, 3 figure

Quantum correlations and spatial localization in one-dimensional ultracold bosonic mixtures

We present the complete phase diagram for one-dimensional binary mixtures of bosonic ultracold atomic gases in a harmonic trap. We obtain exact results with direct numerical diagonalization for small number of atoms, which permits us to quantify quantum many-body correlations. The quantum Monte Carlo method is used to calculate energies and density profiles for larger system sizes. We study the system properties for a wide range of interaction parameters. For the extreme values of these parameters, different correlation limits can be identified, where the correlations are either weak or strong. We investigate in detail how the correlation evolve between the limits. For balanced mixtures in the number of atoms in each species, the transition between the different limits involves sophisticated changes in the one- and two-body correlations. Particularly, we quantify the entanglement between the two components by means of the von Neumann entropy. We show that the limits equally exist when the number of atoms is increased, for balanced mixtures. Also, the changes in the correlations along the transitions among these limits are qualitatively similar. We also show that, for imbalanced mixtures, the same limits with similar transitions exist. Finally, for strongly imbalanced systems, only two limits survive, i.e., a miscible limit and a phase-separated one, resembling those expected with a mean-field approach.Comment: 18 pages, 8 figure

Power, norms and institutional change in the European Union: the protection of the free movement of goods

How do institutions of the European Union change? Using an institutionalist approach, this article highlights the interplay between power, cognitive limits, and the normative order that underpins institutional settings and assesses their impact upon the process of institutional change. Empirical evidence from recent attempts to reinforce the protection of the free movement of goods in the EU suggests that, under conditions of uncertainty, actors with ambiguous preferences assess attempts at institutional change on the basis of the historically defined normative order which holds a given institutional structure together. Hence, path dependent and incremental change occurs even when more ambitious and functionally superior proposals are on offer

Substructure Boosts to Dark Matter Annihilation from Sommerfeld Enhancement

The recently introduced Sommerfeld enhancement of the dark matter annihilation cross section has important implications for the detection of dark matter annihilation in subhalos in the Galactic halo. In addition to the boost to the dark matter annihilation cross section from the high densities of these subhalos with respect to the main halo, an additional boost caused by the Sommerfeld enhancement results from the fact that they are kinematically colder than the Galactic halo. If we further believe the generic prediction of CDM that in each subhalo there is an abundance of substructure which is approximately self-similar to that of the Galactic halo, then I show that additional boosts coming from the density enhancements of these small substructures and their small velocity dispersions enhance the dark matter annihilation cross section even further. I find that very large boost factors ($10^5$ to $10^9$) are obtained in a large class of models. The implications of these boost factors for the detection of dark matter annihilation from dwarf Spheroidal galaxies in the Galactic halo are such that, generically, they outshine the background gamma-ray flux and are detectable by the Fermi Gamma-ray Space Telescope.Comment: PRD in pres