An infinite double bubble theorem

We consider a nonlocal isoperimetric problem defined in the whole space \mathbb R^N , whose nonlocal part is given by a Riesz potential with exponent \alpha \in (0, N –1) . We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L^1 -norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer. Finally we deduce that for small masses the ball is also the unique global minimizer, and that for small exponents a in the nonlocal term the ball is the unique minimizer as long as the problem has a solution.

The free energy of a ternary system with a self-organization property includes an interface energy and a longer ranging, inhibitory interaction energy. In a planar domain, if the two energies are properly balanced and two of the three constituents make up an equal but small fraction, the free energy admits a local minimizer that is shaped like a perturbed double bubble. Most difficulties in the proof of this result are related to the triple junction phenomenon that the three constituents of the ternary system meet at a point. Two techniques are developed to deal with the triple junction. First, one defines restricted classes of perturbed double bubbles. Each perturbed double bubble in a restricted class is obtained from a standard double bubble by a special perturbation. The two triple junction points of the standard double bubble can only move along the line connecting them, in opposite directions, and by the same distance. The second technique is the use of the so called internal variables. These variables derive from the more geometric quantities that describe perturbed double bubbles in restricted classes. The advantage of the internal variables is that they are only subject to linear constraints, and perturbed double bubbles in a restricted class represented by internal variables are elements of a Hilbert space. A local minimizer of the free energy in each restricted class is found as a fixed point of a nonlinear equation by a contraction mapping argument. The second variation at the fixed point within its restricted class is positive. This perturbed double bubble satisfies three of the four equations for critical points of the free energy. The unsolved equation is the 120 degree angle condition at triple junction points. Then perform another minimization among the local minimizers from all restricted classes. A minimum of minimizers emerges and solves all the equations for critical points.

We study the effects of a finite cubic volume with twisted boundary conditions on pseudoscalar mesons. We apply Chiral Perturbation Theory in the p-regime and introduce the twist by means of a constant vector field. The corrections of masses, decay constants, pseudoscalar coupling constants and form factors are calculated at next-to-leading order. We detail the derivations and compare with results available in the literature. In some case there is disagreement due to a different treatment of new extra terms generated from the breaking of the cubic invariance. We advocate to treat such terms as renormalization terms of the twisting angles and reabsorb them in the on-shell conditions. We confirm that the corrections of masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. Furthermore, we show that the matrix elements of the scalar (resp. vector) form factor satisfies the Feynman–Hellman Theorem (resp. the Ward–Takahashi identity). To show the Ward–Takahashi identity we construct an effective field theory for charged pions which is invariant under electromagnetic gauge transformations and which reproduces the results obtained with Chiral Perturbation Theory at a vanishing momentum transfer. This generalizes considerations previously published for periodic boundary conditions to twisted boundary conditions. Another method to estimate the corrections in finite volume are asymptotic formulae. Asymptotic formulae were introduced by Luscher and relate the corrections of a given physical quantity to an integral of a specific amplitude, evaluated in infinite volume. Here, we revise the original derivation of Luscher and generalize it to finite volume with twisted boundary conditions. In some cases, the derivation involves complications due to extra terms generated from the breaking of the cubic invariance. We isolate such terms and treat them as renormalization terms just as done before. In that way, we derive asymptotic formulae for masses, decay constants, pseudoscalar coupling constants and scalar form factors. At the same time, we derive also asymptotic formulae for renormalization terms. We apply all these formulae in combination with Chiral Perturbation Theory and estimate the corrections beyond next-to-leading order. We show that asymptotic formulae for masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. A similar relation connects in an independent way asymptotic formulae for renormalization terms. We check these relations for charged pions through a direct calculation. To conclude, a numerical analysis quantifies the importance of finite volume corrections at next-to-leading order and beyond. We perform a generic Analysis and illustrate two possible applications to real simulations.

We present data for the axial coupling constant ${g}_{A}$ of the nucleon obtained in lattice QCD with two degenerate flavors of dynamical nonperturbatively improved Wilson quarks. The renormalization is also performed nonperturbatively. For the analysis we give a chiral extrapolation formula for ${g}_{A}$ based on the small scale expansion scheme of chiral effective field theory for two degenerate quark flavors. Applying this formalism in a finite volume, we derive a formula that allows us to extrapolate our data simultaneously to the infinite volume and to the chiral limit. Using the additional lattice data in finite volume, we are able to determine the axial coupling of the nucleon in the chiral limit without imposing the known value at the physical point.

This research aims to provide a model to investigate the impact of some parameters such as impeller speed, temperature, and solid concentration on mass transfer coefficient and the dissolution rate of urea fertilizer in the water. To study the effect of solid concentration two models are presented for finite and infinite volume fluids using mass balance. Then the urea-water mass transfer coefficient was calculated at various impeller speeds and temperatures by measuring the time to complete dissolution. To investigate the effect of impeller speed and turbulency on the mass transfer coefficient, the impeller speed and Reynolds number were set in a range of 10-50 [rpm] and 300-3000, respectively. The Schmidt number also was used to study the effect of temperature on mass transfer coefficient in the range of 5-25[°C]. The results show that in both finite and infinite fluid volumes, at a constant impeller speed with decreasing Schmidt number, and at a constant temperature with increasing Reynolds number, the mass transfer coefficient, and mass transfer rate increase. Furthermore, four models are presented for mass transfer coefficient in finite and infinite volume, that show the mass transfer coefficient and release rate in finite volume were lower than that of infinite volume at a constant impeller speed and temperature.

The interaction of the pseudoscalar meson and the baryon octet with strangeness S=−2 and isospin I=1/2 is investigated by solving the Bethe–Salpeter equation in the infinite and finite volume respectively. It is found that there is a resonance state generated dynamically, which owns a mass of about 1550 MeV and a large decay width of 120–200 MeV. This resonance state couples strongly to the π Ξ channel. Therefore, it might not correspond to the Ξ(1620) particle announced by Belle collaboration. At the same time, this problem is studied in the finite volume, and an energy level at 1570 MeV is obtained, which is between the πΞ and K¯Λ thresholds and independent of the cubic box size.

Analytic methods from finite to infinite volumes

A theoretical study of electronic excited state transport among molecules randomly distributed in a finite volume is carried out. Two special cases of the general transport and trapping problem are treated. A truncated series expansion in powers in the chromophore density is used as an approximation for one component systems (i.e., donor–donor transport only). In two component systems of donors and traps, the Förster limit, in which transfer can occur only from donors to traps due to low donor concentrations, is solved exactly for a finite spherical volume. In both cases, the results presented demonstrate that time-dependent observables can be significantly altered in finite volume systems relative to infinite volume systems. These calculations have implications for the interpretation of experiments performed on real finite volume systems, e.g., energy transport among the chromophores of an isolated polymer chain.

In present work, we explore and experiment with an alternative approach to studying resonance properties in a finite volume. By analytic continuing the finite lattice size $L$ into a complex plane, the oscillating behavior of the finite volume Green's function is mapped onto an infinite volume Green's function that is corrected by exponentially decaying finite volume effect. The analytic continuation technique thus can be applied to study resonance properties directly in finite volume dynamical equations.

We have an $\sf{M}\times\sf{N}$ real-valued arbitrary matrix $A$ (e.g., a dictionary) with $\sf{M}<\sf{N}$ and data $d$ describing the sought-after object with the help of $A$. This work provides an in-depth analysis of the (local and global) minimizers of an objective function ${\mathcal{F}}_d$ combining a quadratic data-fidelity term and an $\ell_0$ penalty applied to each entry of the sought-after solution, weighted by a regularization parameter $\beta>0$. For several decades, this objective has attracted a ceaseless effort to conceive algorithms approaching a good minimizer. Our theoretical contributions, summarized below, shed new light on the existing algorithms and can help in the conception of innovative numerical schemes. Solving the normal equation associated with any $\sf{M}$-row submatrix of $A$ is equivalent to computing a local minimizer $\hat u$ of ${\mathcal{F}}_d$. (Local) minimizers $\hat u$ of ${\mathcal{F}}_d$ are strict if and only if the submatrix, composed of those columns of $A$ whose indices form the support of $\hat u$, has full column rank. An outcome is that strict local minimizers of ${\mathcal{F}}_d$ are easily computed without knowing the value of $\beta$. Each strict local minimizer is linear in data. It is proved that ${\mathcal{F}}_d$ has global minimizers and that they are always strict. They are studied in more detail under the (standard) assumption that rank$(A)=\sf{M}<\sf{N}$. The global minimizers with $\sf{M}$-length support are seen to be impractical. Given $d$, critical values $\beta_{\sf{K}}$ for any ${\sf{K}}\leqslant\sf{M}-1$ are exhibited such that if $\beta>\beta_{\sf{K}}$, all global minimizers of ${\mathcal{F}}_d$ are ${\sf{K}}$-sparse. An assumption on $A$ is adopted and proved to fail only on a closed negligible subset. Then for all data $d$ beyond a closed negligible subset, the objective ${\mathcal{F}}_d$ for $\beta>\beta_{\sf{K}}$, ${\sf{K}}\leqslant\sf{M}-1$, has a unique global minimizer, and this minimizer is ${\sf{K}}$-sparse. Instructive small-size ($5\times 10$) numerical illustrations confirm the main theoretical results.

The presence of long-range interactions violates a condition necessary to relate the energy of two particles in a finite volume to their S-matrix elements in the manner of Luscher. While in infinite volume, QED contributions to low-energy charged particle scattering must be resummed to all orders in perturbation theory (the Coulomb ladder diagrams), in a finite volume the momentum operator is gapped, allowing for a perturbative treatment. The leading QED corrections to the two-particle finite-volume energy quantization condition below the inelastic threshold, as well as approximate formulas for energy eigenvalues, are obtained. In particular, we focus on two spinless hadrons in the A1+ irreducible representation of the cubic group, and truncate the strong interactions to the s-wave. These results are necessary for the analysis of Lattice QCD+QED calculations of charged-hadron interactions, and can be straightforwardly generalized to other representations of the cubic group, to hadrons with spin, and to include higher partial waves.

Lattice simulations of light nuclei necessarily take place in finite volumes, thus affecting their infrared properties. These effects can be addressed in a model-independent manner using Effective Field Theories. We study the model case of three identical bosons (mass m) with resonant two-body interactions in a cubic box with periodic boundary conditions, which can also be generalized to the three-nucleon system in a straightforward manner. Our results allow for the removal of finite volume effects from lattice results as well as the determination of infinite volume scattering parameters from the volume dependence of the spectrum. We study the volume dependence of several states below the break-up threshold, spanning one order of magnitude in the binding energy in the infinite volume, for box side lengths L between the two-body scattering length a and L = 0.25a. For example, a state with a three-body energy of -3/(ma^2) in the infinite volume has been shifted to -10/(ma^2) at L = a. Special emphasis is put on the consequences of the breakdown of spherical symmetry and several ways to perturbatively treat the ensuing partial wave admixtures. We find their contributions to be on the sub-percent level compared to the strong volume dependence of the S-wave component. For shallow bound states, we find a transition to boson-diboson scattering behavior when decreasing the size of the finite volume.

Using effective-range expansion for the two-body amplitudes may generate spurious sub-threshold poles outside of the convergence range of the expansion. In the infinite volume, the emergence of such poles leads to the inconsistencies in the three-body equations, e.g., to the breakdown of unitarity. We investigate the effect of the spurious poles on the three-body quantization condition in a finite volume and show that it leads to a peculiar dependence of the energy levels on the box size L. Furthermore, within a simple model, it is demonstrated that the procedure for the removal of these poles, which was recently proposed in ref. [1] in the infinite volume, can be adapted to the finite-volume calculations. The structure of the exact energy levels is reproduced with an accuracy that systematically improves order by order in the EFT expansion.

We develop a theoretical framework to quantify the structure of unstable hadron resonances. With the help of the corresponding system in a finite volume, we define the compositeness of resonance states which can be interpreted as a probability. This framework is used to study the structure of the scalar mesons f_0(980) and a_0(980). In both mesons, the Kbar K component dominates about a half of the wave function. The method is also applied to the Lambda(1405) resonance. We argue that a single energy level in finite volume represents the two eigenstates in infinite volume. The KbarK N component of Lambda(1405), including contributions from both eigenstates, is found to be 58%, and the rest is composed of the pi Sigma and other channels.

A ternary inhibitory system is a three component system characterized by two properties: growth and inhibition. A deviation from homogeneity has a strong positive feedback on its further increase. In the meantime a longer ranging confinement mechanism prevents unlimited spreading. Together they lead to a locally self-enhancing and self-organizing process. The model considered here is a planar nonlocal geometric problem derived from the triblock copolymer theory. An assembly of perturbed double bubbles is mathematically constructed as a stable stationary point of the free energy functional. Triple junction, a phenomenon in which the three components meet at a single point, is a key issue addressed in the construction. Coarsening, an undesirable scenario of excessive growth, is prevented by a lower bound on the long range interaction term in the free energy. The proof involves several ideas: perturbation of double bubbles in a restricted class; use of internal variables to remove nonlinear constraints, local minimization in a restricted class formulated as a nonlinear problem on a Hilbert space; and reduction to finite dimensional minimization. This existence theorem predicts a new morphological phase of a double bubble assembly.