Double Potential Research Articles

Separation property and asymptotic behavior for a transmission problem of the bulk-surface coupled Cahn–Hilliard system with singular potentials and its Robin approximation

We consider a class of bulk-surface coupled Cahn–Hilliard systems in a smooth, bounded domain [Formula: see text] [Formula: see text], where the bulk phase variable is connected to the surface phase variable via a Dirichlet boundary condition or its Robin approximation. For a general class of singular potentials (including the physically relevant logarithmic potential), we establish the regularity propagation of global weak solutions to the initial boundary value problem. In particular, when the spatial dimension is two, we prove the instantaneous strict separation property, which ensures that every global weak solution remains uniformly away from the pure states [Formula: see text] after any given positive time. In the three-dimensional case, we obtain the eventual strict separation property that holds for sufficiently large time. This strict separation property allows us to prove that every global weak solution converges to a single equilibrium as time goes to infinity using the Łojasiewicz–Simon approach. Finally, we study the double obstacle limit for the problem with logarithmic potentials in the bulk and on the boundary, showing that as the absolute temperature [Formula: see text] tends to zero, the corresponding weak solutions converge (for a suitable subsequence) to a weak solution of the problem with a double obstacle potential.

CONVERGENCE OF THE METHOD OF PIECEWISE LINEAR APPROXIMATIONS AND COLLOCATIONS FOR A TWO-DIMENSIONAL HYPERSINGULAR INTEGRAL EQUATION ON A SET WITH BOUNDARY

A hypersingular integral equation on a convex bounded set on the plane with an integral understood in the sense of a finite part in the sense of Hadamard is considered. Equations of this type, in particular, arise when solving the Neumann boundary value problem for the Lapalse and Helmholtz equations on a flat screen in the case where the solution is sought in the form of a double layer potential. To numerically solve the equation, a numerical scheme is used based on piecewise linear approximation of the unknown function on a triangular conformal mesh and the collocation method. The uniform convergence of numerical solutions to an exact solution on a grid when the maximum cell diameter tends to zero has been proven.

On the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions and non-smooth potentials: Well-posedness and asymptotic limits

We analyze a class of Cahn–Hilliard equations with kinetic rate dependent dynamic boundary conditions that describe possible short-range interactions between the binary mixture and the solid boundary. In the presence of surface diffusion on the boundary, the initial boundary value problem can be viewed as a transmission problem consisting of Cahn–Hilliard-type equations both in the bulk and on the boundary. We first establish the existence, uniqueness, and continuous dependence of global weak solutions. In the construction of weak solutions, an explicit convergence rate in terms of the parameter for the Yosida approximation is obtained. Under some additional assumptions, we further prove the existence and uniqueness of global strong solutions. Next, we study the asymptotic limit as the coefficient of the boundary diffusion goes to zero and show that the limit problem with a forward-backward dynamic boundary condition is well posed in a suitable weak formulation. At last, we investigate the asymptotic limits as the kinetic rate tends to zero and infinity, respectively. Our results are valid for a general class of bulk and boundary potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double obstacle potential.

Disentangling heterogeneous thermocatalytic formic acid dehydrogenation from an electrochemical perspective

Heterogeneous thermocatalysis of formic acid dehydrogenation by metals in solution is of great importance for chemical storage and production of hydrogen. Insightful understanding of the complicated formic acid dehydrogenation kinetics at the metal-solution interface is challenging and yet essential for the design of efficient heterogeneous formic acid dehydrogenation systems. In this work, formic acid dehydrogenation kinetics is initially studied from a perspective of electrochemistry by decoupling this reaction on Pd catalyst into two short-circuit half reactions, formic acid oxidation reaction and hydrogen evolution reaction and manipulating the electrical double layer impact from the solution side. The pH-dependences of formic acid dehydrogenation kinetics and the associated cation effect are attributed to the induced change of electric double layer structure and potential by means of electrochemical measurements involving kinetic isotope effect, in situ infrared spectroscopy as well as grand canonical quantum mechanics calculations. This work showcases how kinetic puzzles on some important heterogeneous catalytic reactions can be tackled by electrochemical theories and methodologies.

Large Energy Bubble Solutions for Supercritical Fractional Schrödinger Equation with Double Potentials

Large Energy Bubble Solutions for Supercritical Fractional Schrödinger Equation with Double Potentials

Activation patterns and electrophysiologic characteristics of Marshall bundle–related left atrial tachycardias after atrial fibrillation ablation

BackgroundEpicardial Marshall bundles (MBs) are frequently used in left atrial tachycardias (LATs) after atrial fibrillation (AF) ablation with pulmonary vein isolation and substrate modification. ObjectiveThis study sought to classify different activation patterns of MB-mediated LATs and the corresponding electrophysiologic characteristics. MethodsFrom 2019 to 2021, 28 cases of atrial tachycardias after AF ablation were diagnosed as MB-mediated LATs by ultrahigh-density mapping and entrainment. Cannulation and mapping in the vein of Marshall (VOM) and epicardial mapping in the MB region were also performed in selected cases to further prove the mechanism. ResultsThree activation patterns were identified with a critical isthmus through the MB: perimitral macroreentry (perimitral LAT; n = 20 [71.4%]); left atrial appendage–related reentry (n = 5 [17.9%]); and left pulmonary vein–related reentry (n = 3 [10.7%]). In 18 patients, a characteristic triple potential observed along the previously endocardial left atrial ridge block line was composed of near-field double potentials and far-field MB potential. These findings were further delineated in 24 patients with either cannulation in the VOM (19 patients) or epicardial mapping (5 patients). Ethanol infusion of the VOM resulted in atrial tachycardia termination in 20 of 28 patients. ConclusionDifferent types of MB-mediated LATs after AF ablation could be identified by ultrahigh-density mapping. Ethanol infusion within the VOM was effective in eliminating these tachycardias.

A note on vibrational perturbation theory for tunneling splitting in a symmetric double well potential.

The modified version of second and fourth order vibrational perturbation theory, whereby the Euclidean action for tunneling is computed on the inverted potential at a shifted energy that is ℏ2 dependent, is applied to a symmetric double well quartic potential. The mean energies of the doublets in each well are also computed using vibrational perturbation theory. Results show that the modified vibrational perturbation theory significantly improves the estimates of tunneling splitting energies both for the ground state and for excited state doublets.

Geometric topological effect on non-relativistic solution under pseudoharmonic and extended double ring-shaped potentials

Geometric topological effect on non-relativistic solution under pseudoharmonic and extended double ring-shaped potentials

Topological defects on solutions of the non-relativistic equation for extended double ring-shaped potential

In this study, we focus into the non-relativistic wave equation described by the Schrödinger equation, specifically considering angular-dependent potentials within the context of a topological defect background generated by a cosmic string. Our primary goal is to explore quasi-exactly solvable problems by introducing an extended ring-shaped potential. We utilize the Bethe ansatz method to determine the angular solutions, while the radial solutions are obtained using special functions. Our findings demonstrate that the eigenvalue solutions of quantum particles are intricately influenced by the presence of the topological defect of the cosmic string, resulting in significant modifications compared to those in a flat space background. The existence of the topological defect induces alterations in the energy spectra, disrupting degeneracy. Afterwards, we extend our analysis to study the same problem in the presence of a ring-shaped potential against the background of another topological defect geometry known as a point-like global monopole. Following a similar procedure, we obtain the eigenvalue solutions and analyze the results. Remarkably, we observe that the presence of a global monopole leads to a decrease in the energy levels compared to the flat space results. In both cases, we conduct a thorough numerical analysis to validate our findings.

Simulation of Gaussian wave packets used to illustrate elementary quantum mechanics scenarios

In this paper we numerically solve the time dependent Schrödinger equation for scenarios using wave packets. These examples include the free wave packet, which we use to show the difference between group and phase velocities, the packet in a harmonic oscillator potential with non-trivial initial conditions in one and two dimensions, which is compared with their classical analogs to show how Ehrenfest theorem holds. We also include simulations of the diffraction through the single and double slit potentials, the refraction with a step potential and the dispersion by a central potential. The aim of this paper is to illustrate with simulations, nowadays easy to implement, scenarios that can help explaining the basics of the wave-particle duality.

In Ukrainian

Brownian motors belong to the class of nanoscale devices that use the thermal noise of the environment as one of the necessary components in the mechanism of their operation. Today, there are a lot of practical implementations of such nanomachines, both inorganic, fairly simple mechanisms produced artificially, and more complex ones created from separate biological components available at the cellular level. One of the options for implementing the mechanism of straightening the chaotic thermal noise of the environment into unidirectional motion is the presence of a motor particle in the field of action of an asymmetric periodic stationary potential, which undergoes certain small disturbances (fluctuations) periodically over time. To describe such asymmetric one-dimensional structures (for example, dipole chains or fibers of the cytoskeleton) in the theory of Brownian motors, two model potentials are most often used: piecewise linear sawtooth and double sinusoidal. In this work, within the framework of the approximation of small fluctuations, a model of a pulsating Brownian motor with a stationary double sinusoidal potential and a disturbing small harmonic signal is considered. A new method of parametrization of such a problem is proposed, which allows to separate the contributions from various factors affecting the operation of the ratchet, and the numerical procedure for calculating the average speed of the directional movement of nanoparticles for the selected type of model potentials is specified. A number of numerical dependences of the average speed on the main parameters of the system were obtained. Peculiarities of the behavior of the motor as dependent on the parameter responsible for asymmetry and the number of potential wells on the spatial period of the stationary potential have been investigated. It is shown that the direction of the generated flux of nanoparticles depends not only on the phase shift between the stationary and fluctuating components of the potential, but also on the temperature of the system and the frequency of fluctuations, i.e., a possibility of temperature-frequency control of the direction of movement in the considered model has been found. Diagrams have been constructed that allow you to choose the ratio between the parameters of the nanomotor to create a flux of particles in the desired direction.

Schauder Estimates up to the boundary on H-type groups: an approach via the double layer potential

Schauder Estimates up to the boundary on H-type groups: an approach via the double layer potential

Effect of atrial extrasystoles in electrophysiological properties in heart failure patients

Abstract Background Patients with heart failure (HF) most likely have an extensive arrhythmogenic substrate which increases vulnerability to atrial fibrillation (AF). Conduction disorders underlying this increased AF vulnerability -hidden during sinus rhythm- may be unmasked by spontaneous atrial extrasystole (AES). Objective The aim of this study is to investigate the impact of AES on potential arrhythmogenic electrophysiological properties in patients with HF and AES. Methods Intraoperative epicardial mapping of the right and left atrium (RA, LA), Bachmann’s bundle (BB) and pulmonary vein area (PVA) was performed in 132 patients (10 with HF and 122 without HF). Differences in unipolar potential morphologies and conduction disorders during SR and spontaneous AES were determined. Results A median of 1 (1–3) AES per patient were included. In patients with HF, AES were associated with lower median potential voltages (1.94mV versus 4.02mV, P<0.001), more low voltage areas (10.98% versus 4.97%, P=0.004) and a higher number of short double potentials (28.5% versus 13.0% of short double potentials, P<0.001) compared to patients without HF. Comparing differences (AES - SR) in parameters between patients with and without HF, there was a larger increase in long double potentials (+5.32% versus +0.31%, P<0.001), fractionated potential (+4.18% versus +0.49%, P=0.001), conduction block (+3.39% versus +0.63%, P=0.002) and continuous conduction delay and block (+6.72% versus +1.74%, P=0.001) in HF patients. Conclusions In HF patients, AES cause lower unipolar potentials voltages and more conduction disorders than in patients without HF. Differences in arrhythmogenic properties between sinus rhythm and atrial extrasystoles are more pronounced in HF patients. These results suggest that HF patients have hidden arrhythmogenic substrates that can be uncovered by AES and may explain the increased AF vulnerability.

Construction of infinitely many solutions for fractional Schrödinger equation with double potentials

Construction of infinitely many solutions for fractional Schrödinger equation with double potentials

Transmission in graphene through a double laser barrier

We study the tunneling behavior of Dirac fermions in graphene subjected to a double barrier potential profile created by spatially overlapping laser fields. By modulating the graphene sheet with an oscillating structure formed from two laser barriers, we aim to understand how the transmission of Dirac fermions is influenced by such a light-induced electric potential landscape. Using the Floquet method, we determine the eigenspinors of the five regions defined by the barriers applied to the graphene sheet. Applying the continuity of the eigenspinors at barrier edges and using the transfer matrix method, we establish the transmission coefficients. These allow us to show that oscillating laser fields generate multiple transmission modes, including zero-photon transmission aligned with the central band ε and photon-assisted transmission at sidebands ε + l ϖ, with l = 0, ± 1, ⋯ and frequency ϖ. For numerical purposes, our attention is specifically directed towards transmissions related to zero-photon processes (l = 0), along with processes involving photon emission (l = 1) and absorption (l = − 1). We find that transmission occurs only when the incident energy is above the threshold energy ε > k y + 2ϖ, with transverse wave vector k y . We find that the variation in distance d 1 separating two barriers of widths d 2 − d 1 suppresses one transmission mode. Additionally, we show that an increase in laser intensity modifies transmission sharpness and amplitude.

Study on Electrical Characteristics and Formation Mechanism of Micro-Ion Capacitor in Shale Pore and Fracture Structure

Due to the micro/nanoscale and intricate pore structure, poor connectivity, complex pathways, the presence of microfractures, and the coexistence of organic and inorganic pores, shale and other tight reservoirs exhibit increasingly complex conductive characteristics. This paper mainly studied the electrical properties of shale based on experimental test data, including scanning electron microscope (SEM) thin-section and microtomography CT (μ-CT) images, and analyzed the shale pore type, pore structure characteristics, and development of fracture. Then, the distribution of pyrite, content, and graphitization of organics and their influence on the electrical properties are discussed. Furthermore, the double electrical layer and zeta potential in the shale zone were discussed in depth. The results revealed that, within the content of pyrite, organics, and its graphitization, the vitrinite maturity are inversely proportional to shale resistivity. It was also found that in the presence of an external electromagnetic field, the fluid in shale pores is subjected to the combined strength of pore pressure and external field potential difference. Thus, its response equation should be an improved Navier-Stokes equation, which considers pore pressure, zeta potential, and Coulomb force. When shale is subjected to an external electromagnetic field, due to the complex pores structure and organic and inorganic minerals, it will represent more of a dielectric-like property than electricity. So, it will form special microscopic ionic capacitors, which are different from common plate capacitors. There are three special kinds of microscopic ionic capacitors, they are (I) the intergranular pore microscopic ionic capacitor model, (II) the particle with isolated pore microscopic ionic capacitor model, and (III) the pyrite or graphite or other organics microscopic ionic capacitor model. Finally, the characteristics of microscopic ion capacitors are summarized: irregular polar area and varying distance between poles, varying charges with time, and salinity of the formation water.

COLQ-Congenital myasthenic syndrome in an Iranian cohort: the clinical and genetics spectrum

BackgroundCongenital myasthenic syndrome (CMS) is a group of neuromuscular disorders caused by abnormal signal transmission at the motor endplate. Mutations in the collagen-like tail subunit gene (COLQ) of acetylcholinesterase are responsible for recessive forms of synaptic congenital myasthenic syndromes with end plate acetylcholinesterase deficiency. Clinical presentation includes ptosis, ophthalmoparesis, and progressive weakness with onset at birth or early infancy.MethodsWe followed 26 patients with COLQ-CMS over a mean period of 9 years (ranging from 3 to 213 months) and reported their clinical features, electrophysiologic findings, genetic characteristics, and therapeutic management.ResultsIn our population, the onset of symptoms ranged from birth to 15 years. Delayed developmental motor milestones were detected in 13 patients (\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim$$\\end{document} 52%), and the most common presenting signs were ptosis, ophthalmoparesis, and limb weakness. Sluggish pupils were seen in 8 (\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim$$\\end{document} 30%) patients. All patients who underwent electrophysiologic study showed a significant decremental response (> 10%) following low-frequency repetitive nerve stimulation. Moreover, double compound muscle action potential was evident in 18 patients (\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim$$\\end{document} 75%). We detected 14 variants (eight novel variants), including six missense, three frameshift, three nonsense, one synonymous and one copy number variation (CNV), in the COLQ gene. There was no benefit from esterase inhibitor treatment, while treatment with ephedrine and salbutamol was objectively efficient in all cases.ConclusionDespite the rarity of the disease, our findings provide valuable information for understanding the clinical and electrophysiological features as well as the genetic characterization and response to the treatment of COLQ-CMS.

The Lavrentiev phenomenon in calculus of variations with differential forms

In this article we study convex non-autonomous variational problems with differential forms and corresponding function spaces. We introduce a general framework for constructing counterexamples to the Lavrentiev gap, which we apply to several models, including the double phase, borderline case of double phase potential, and variable exponent. The results for the borderline case of double phase potential provide new insights even for the scalar case, i.e., variational problems with 0-forms.

CONTINUITY OF THE DOUBLE LAYER POTENTIAL OF A SECOND ORDER ELLIPTIC DIFFERENTIAL OPERATOR IN SCHAUDER SPACES ON THE BOUNDARY

AbstractWe prove the validity of a regularizing property on the boundary of the double layer potential associated with the fundamental solution of anonhomogeneoussecond order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M. Günter, S. Mikhlin, V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.V. Burchuladze, U. Heinemann and extends the work of A. Kirsch who has considered the case of the Helmholtz operator.

Solution to the schrödinger equation for bound states of polar molecules using shallow neural networks

The time independent Schrödinger equation can be solved analytically in very few cases. This is why scientists rely on approximate numerical methods. One that is yet to be a part of the everyday toolbox for physicists are neural networks. With it, it is possible to calculate eigenfunctions and the energy spectra of bound states. A double exponential central potential is used to describe the interaction between two electric dipoles that form through van der Waals interactions. These systems can be important when the electric dipoles are very large and differ from covalent, ionic, and other kinds of bonds, as is the case of the monomers M−C60, where M=Cs,Li or other alkali metals. Based in statistical physics, it has been shown that a dimer can be formed through a spontaneous and exothermic reaction. The ground state and the first two excited states of this system were calculated, also the solutions and their associated energies were plotted. The result for the ground state is compared to the shooting method and the direct variational method. The numerical value of the ground states obtained suggests that the dimers can be dissociated by electromagnetic waves with wavelengths in the far infrared regime, very close to microwaves.