External Potential Research Articles

Density functional theory from spherically symmetric densities: Ground and excited states of Coulomb systems.

Recently, Theophilou [J. Chem. Phys. 149, 074104 (2018)] proposed a peculiar version of the density functional theory by showing that the set of spherical averages of the density around the nuclei determines uniquely the external potential in atoms, molecules, and solids. Here, this novel theory is extended to individual excited states. The generalization is based on the method developed in the series of papers by Ayers, Levy, and Nagy [Phys. Rev. A 85, 042518 (2012)]. Generalized Hohenberg-Kohn theorems are proved to the set of spherically symmetric densities using constrained search. A universal variational functional for the sum of the kinetic and electron-electron repulsion energies is constructed. The functional is appropriate for the ground state and all bound excited states. Euler equations and Kohn-Sham equations for the set are derived. The Euler equations can be rewritten as Schrödinger-like equations for the square root of the radial densities, and the effective potentials in them can be expressed in terms of wave function expectation values. The Hartree plus exchange-correlation potentials can be given by the difference of the interacting and the non-interacting effective potentials.

Generality and Exception in Islamic Legal Theory: Intent, Language, and the Jurist’s Role

Abstract Whenever discussions arise about the generality or specificity of law and legal provisions, they occur on at least two different levels. On the external level, there is an expectation that law should be general insofar as similar cases and factual situations are concerned, and specific with regards to different and exceptional cases. This is often invoked as a requirement for what constitutes good law or proper governance. On the internal level, generality refers to the expectation that law must offer coherent rules, principles, or standards of action in some form. General and specific language (al-ʿām wal-khāṣṣ) in Islamic legal theory (uṣūl al-fiqh) enjoyed a noteworthy degree of attention in modern scholarship. This attention largely centers on those concepts’ external potential for driving reform in specific areas of law. This Article, by contrast, focuses on the generality–specificity binary as a tool for ensuring the law’s internal cohesiveness, including its claims to legitimacy and its enforceability. To this end, the Article offers an overview of the examination of general and specific expressions in the thought of two influential twelfth-century C.E. (sixth-centuryA.H.) jurist-theologians of different juristic and theological affiliation: the Shāfiʿī-Ashʿarī Abū Ḥāmid al-Ghazālī (d. 1111 C.E./505 A.H.) and the Central Asian Ḥanafī-Māturīdī ʿAlāʾ al-Dīn al-Samarqandī (d. 1145 C.E./539 A.H.). The Article then goes on to argue that ensuring a coherent relation between intent, language, and norms was a central guiding purpose of these studies. We can see this with particular clarity in the fact that, despite reaching comparable practical positions concerning how norms ought to follow from general language, each jurist attained his conclusion through a different path, reflecting different understandings of the relation of norms to divine intent.

Efficient photoelectrochemical H2 generation via electrochemically modified TiO2/electrochemically reduced graphene oxide photoelectrode with Ti3+ defects and Mg(OH)2 nanoplates

This study investigates the electrochemical synthesis and electrochemical modification of a TiO2/Electrochemically reduced graphene oxide (ERGO) photoelectrode to improve its photoelectrochemical efficiency for hydrogen production. TiO2/ERGO nanocomposites were co-electrodeposited on fluorine-doped tin oxide (FTO) substrate. After electrochemical modification of TiO2/ERGO in MgCl2 solution, structural and morphological characterizations revealed that Mg(OH)2 nanoplates formed on the surface, and Mg2+ was doped into the TiO2/ERGO nanocomposite, leading to a formation of Ti3+ defects. Optical characterization revealed a redshift in band gap energy and enhanced absorbance due to Ti3+ defects and Mg(OH)2 nanoplates. The Mg(OH)2/TiO2/ERGO electrode exhibited a tenfold increase in photocurrent densities, reaching 1.20 mA cm−2. The hydrogen production rate reached 24.16 µmol cm−2h−1 without applying external potential and 26.02 µmol cm−2h−1 at an applied potential of 1.0 V. Stability tests showed promising long-term performance, making Mg(OH)2/TiO2/ERGO a potential candidate for photoelectrochemical applications, particularly in H2 generation.

Anaerobic Process Intensification via Electro-methanogenesis − Advancing Biogas Upgradation through CO2 Utilization

This study investigates the enhancement of anaerobic processes through electro-methanogenesis (EM), focusing on biogas upgrading and in situ CO2 utilization. The effects of applied potential (AP), closed circuit (CC), open circuit (OC), and control (C; without electrode) conditions were evaluated in electro-methanogenic bioreactors operated at 30 °C and 48 °C. Using stainless-steel electrodes, eight reactor configurations were compared. Results showed that mesophilic conditions (48 °C) yielded higher biogas and methane production efficiencies than ambient conditions (30 °C). Among the electrode configurations, the AP condition achieved the highest biogas production, followed by CC, OC, and control. The integration of electrochemical conditions with elevated temperatures significantly improved the calorific value of the biogas, achieving a 1.6–2.08-fold increase. The R8/EMAP/48 condition demonstrated the highest methane equivalent utilization, with CH4 as the primary electron sink. In situ and external potentials enhanced bacterial metabolic activity and facilitated mediated interspecies electron transfer through soluble redox-active molecules such as quinones and NADH. This approach effectively overcomes the thermodynamic limitations of conventional anaerobic digestion, providing a method for increasing methane production by enhancing in situ electron flux while also supporting targeted decarbonization by upgrading the biogas energy value.

The invariant measure of a walking droplet in hydrodynamic pilot–wave theory

We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential.

The cubic-quintic nonlinear Schrödinger equation with inverse-square potential

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of [11] in the setting of the standard cubic-quintic NLS.

The Simplified Approach to the Bose Gas Without Translation Invariance

The simplified approach to the Bose gas was introduced by Lieb in 1963 to study the ground state of systems of interacting Bosons. In a series of recent papers, it has been shown that the simplified approach exceeds earlier expectations, and gives asymptotically accurate predictions at both low and high density. In the intermediate density regime, the qualitative predictions of the simplified approach have also been found to agree very well with quantum Monte Carlo computations. Until now, the simplified approach had only been formulated for translation invariant systems, thus excluding external potentials, and non-periodic boundary conditions. In this paper, we extend the formulation of the simplified approach to a wide class of systems without translation invariance. This also allows us to study observables in translation invariant systems whose computation requires the symmetry to be broken. Such an observable is the momentum distribution, which counts the number of particles in excited states of the Laplacian. In this paper, we show how to compute the momentum distribution in the simplified approach, and show that, for the simple equation, our prediction matches up with Bogolyubov’s prediction at low densities, for momenta extending up to the inverse healing length.

Enabling Ab Initio Molecular Dynamics under Bias: The CP2K+SMEAGOL Interface for Integrating Density Functional Theory and Non-Equilibrium Green Functions.

Density functional theory (DFT) combined with non-equilibrium Green's functions (NEGF) is a powerful approach to model quantum transport under external bias potentials at reasonable computational cost. In this work, we present a new interface between the popular mixed Gaussian/plane waves electronic structure package, CP2K, and the NEGF, code SMEAGOL, the most feature-rich implementation of DFT-NEGF available for CP2K to date. The CP2K+SMEAGOL interface includes the implementation of current induced forces. We verify this implementation for a variety of systems: an infinite 1D Au wire, a parallel-plate capacitor, and a Au-H2-Au junction. We find good agreement with SMEAGOL calculations performed with SIESTA for the same systems and with the example of a solvated Au wire demonstrating for the first time that DFT-NEGF can be used to perform molecular dynamics simulations under bias of large-scale condensed phase systems under realistic operating conditions.

Photoelectrocatalytic Synthesis of Urea from Carbon Dioxide and Nitrate over a Cu2O Photocathode.

Transformation of carbon dioxide and nitrate ions into urea offers an attractive route for both nitrogen fertilizer production and environmental remediation. However, achieving this transformation under mild conditions remains challenging. Herein, we report an efficient photoelectrochemical method for urea synthesis by co-reduction of carbon dioxide and nitrate ion over a Cu2O photocathode, delivering urea formation rate of 29.71±2.20 μmol g-1 h-1 and Faradaic efficiency (FE) of 12.90±1.15 % at low external potential (-0.017 V vs. reversible hydrogen electrode). Experimental data combined with theoretical calculations suggest that the adsorbed CO* and NO2* species are the key intermediates, and associated C-N coupling is the rate-determining step. This work demonstrates that Cu2O is an efficient catalyst to drive co-reduction of CO2 and NO3 - to urea under light irradiation with low external potential, showing great opportunity of photoelectrocatalysis as a sustainable tool for value-added chemical synthesis.

Photoelectrocatalytic Synthesis of Urea from Carbon Dioxide and Nitrate over a Cu2O Photocathode

AbstractTransformation of carbon dioxide and nitrate ions into urea offers an attractive route for both nitrogen fertilizer production and environmental remediation. However, achieving this transformation under mild conditions remains challenging. Herein, we report an efficient photoelectrochemical method for urea synthesis by co‐reduction of carbon dioxide and nitrate ion over a Cu2O photocathode, delivering urea formation rate of 29.71±2.20 μmol g−1 h−1 and Faradaic efficiency (FE) of 12.90±1.15 % at low external potential (−0.017 V vs. reversible hydrogen electrode). Experimental data combined with theoretical calculations suggest that the adsorbed CO* and NO2* species are the key intermediates, and associated C−N coupling is the rate‐determining step. This work demonstrates that Cu2O is an efficient catalyst to drive co‐reduction of CO2 and NO3− to urea under light irradiation with low external potential, showing great opportunity of photoelectrocatalysis as a sustainable tool for value‐added chemical synthesis.

Symplectic Quantization III: Non-relativistic Limit

First of all we shortly illustrate how the symplectic quantization scheme (Gradenigo and Livi, Found Phys 51(3):66, 2021) can be applied to a relativistic field theory with self-interaction. Taking inspiration from the stochastic quantization method by Parisi and Wu, this procedure is based on considering explicitly the role of an intrinsic time variable, associated with quantum fluctuations. The major part of this paper is devoted to showing how the symplectic quantization scheme can be extended to the non-relativistic limit for a Schrödinger-like field. Then we also discuss how one can obtain from this non-relativistic theory a linear Schrödinger equation for the single-particle wavefunction. This further passage is based on a suitable coarse-graining procedure, when self-interaction terms can be neglected, with respect to interactions with any external field. In the Appendix we complete our survey on symplectic quantization by discussing how this scheme applies to a non-relativistic particle under the action of a generic external potential.

A crossed module representation of a 2-group constructed from the 3-loop group Ω3G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega ^3G$$\\end{document}

The quantization of chiral fermions on a 3-manifold in an external gauge potential is known to lead to an abelian extension of the gauge group. In this article, we concentrate on the case of Ω3G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega ^3 G$$\\end{document} of based smooth maps on a 3-sphere taking values in a compact Lie group G. There is a crossed module constructed from an abelian extension Ω3G^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{\\Omega ^3 G}$$\\end{document} of this group and a group of automorphisms acting on it as explained in a recent article by Mickelsson and Niemimäki. We shall construct a representation of this crossed module in terms of a representation of Ω3G^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{\\Omega ^3 G}$$\\end{document} on a space of functions of gauge potentials with values in a fermionic Fock space and a representation of the automorphism group of Ω3G^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{\\Omega ^3 G}$$\\end{document} as outer automorphisms of the canonical anticommutation relations algebra in the Fock space.

Numerical Analysis of Stainless Steel of Crevice Corrosion Initiation Behavior with Varying External Potentials and Initial Crevice Gaps

The potential, current density, pH, and crevice profile distribution within the crevice structure of stainless steel in seawater were numerically analyzed. The crevice corrosion phenomena were mathematically modeled with the initial boundary value problem of diffusion and static electrical field, respectively. This initial boundary value problem was discretized using a finite difference method. The predicts of the crevice corrosion behaviors, set for various external potentials and initial crevice gaps, were classified into three categories: the pitting type, the active type, and no corrosion. The pitting type crevice corrosion was predicted to occur in conditions where the external potentials were noble and the initial crevice gaps were narrow. The active type crevice corrosion was predicted to occur in conditions where the external potentials were less noble and the initial crevice gaps were narrow. The range of external potential conditions where crevice corrosion was predicted not to occur increased in conditions with wide initial crevice gaps. Crevice corrosion was predicted to occur at all external potentials in conditions where the initial crevice gaps were 2 μm or less.

A 3D view on the local gravitational instability of cold gas discs in star-forming galaxies at 0 ≲ z ≲ 5

Local gravitational instability (LGI) is considered crucial for regulating star formation and gas turbulence in galaxy discs, especially at high redshift. Instability criteria usually assume infinitesimally thin discs or rely on approximations to include the stabilising effect of the gas disc thickness. We test a new 3D instability criterion for rotating gas discs that are vertically stratified in an external potential. This criterion reads Q3D < 1, where Q3D is the 3D analogue of the Toomre parameter Q. The advantage of Q3D is that it allows us to study LGI in and above the galaxy midplane in a rigorous and self-consistent way. We apply the criterion to a sample of 44 star-forming galaxies at 0 ≲ z ≲ 5 hosting rotating discs of cold gas. The sample is representative of galaxies on the main sequence at z ≈ 0 and includes massive star-forming and starburst galaxies at 1 ≲ z ≲ 5. For each galaxy, we first apply the Toomre criterion for infinitesimally thin discs, finding ten unstable systems. We then obtain maps of Q3D from a 3D model of the gas disc derived in the combined potential of dark matter, stars and the gas itself. According to the 3D criterion, two galaxies with Q < 1 show no evidence of instability and the unstable regions that are 20% smaller than those where Q < 1. No unstable disc is found at 0 ≲ z ≲ 1, while ≈60% of the systems at 2 ≲ z ≲ 5 are locally unstable. In these latter, a relatively small fraction of the total gas (≈30%) is potentially affected by the instability. Our results disfavour LGI as the main regulator of star formation and turbulence in moderately star-forming galaxies in the present-day Universe. LGI likely becomes important at high redshift, but the input by other mechanisms seems required in a significant portion of the disc. We also estimate the expected mass of clumps in the unstable regions, offering testable predictions for observations.

Particle and wave dynamics of nonlocal solitons in external potentials

We study nonlocal bright solitons subject to external spatially nonuniform potentials. If the potential is slowly varying on the soliton scale, we derive analytical soliton solutions behaving like Newtonian particles. If the potential has the form of an attractive delta-like point defect, we identify different dynamical regimes, defined by the relative strength of the nonlocality and the point defect. In these regimes, the soliton can be trapped at the defect's location, via a nonlinear resonance with a defect mode –which is found analytically– reflected by or transmitted through the defect, featuring a wave behavior. Our analytical predictions are corroborated by results of direct numerical simulations.

Thomas–Fermi theory of out-of-plane charge screening in graphene

This paper provides a variational treatment of the effect of external charges on the free charges in an infinite free-standing graphene sheet within the Thomas–Fermi theory. We establish existence, uniqueness and regularity of the energy minimizers corresponding to the free charge densities that screen the effect of an external electrostatic potential at the neutrality point. For the potential due to one or several off-layer point charges, we also prove positivity and a precise universal asymptotic decay rate for the screening charge density, as well as an exact charge cancellation by the graphene sheet. We also treat a simpler case of the non-zero background charge density and establish similar results in that case.

Autonomous dynamics of two-dimensional insulating domain with superclimbing edges

Superclimbing dynamics is the signature feature of transverse quantum fluids describing wide superfluid one-dimensional interfaces and/or edges with negligible Peierls barrier. Using Lagrangian formalism, we show how the essence of the superclimb phenomenon—dynamic conjugation of the fields of the superfluid phase and geometric shape—clearly manifests itself via characteristic modes of autonomous motion of the insulating domain (“droplet”) with superclimbing edges. In the translation invariant case and in the absence of supercurrent along the edge, the droplet demonstrates ballistic motion with the velocity-dependent shape and zero bulk currents. In an isotropic trapping potential, the droplet features a doubly degenerate sloshing mode. The period of the ground-state evolution of the superfluid phase (dictating the frequency of the AC Josephson effect) is sensitive to the geometry of the droplet. The supercurrent along the edge dramatically changes the droplet dynamics: The motion acquires features resembling that of a two-dimensional charged particle interacting with a perpendicular magnetic field. In a linear external potential (uniform force field), the state with a supercurrent demonstrates a spectacular gyroscopic effect—uniform motion in the perpendicular to the force direction. Published by the American Physical Society 2024

Manipulating circular Airy beam dynamics with quadratic phase modulation in fractional systems under some diffraction modulations and potentials

Based on a split-step Fourier algorithm, the transmission of circular Airy beams with quadratic phase modulation (QPM) is investigated in the fractional Schrödinger equation (FSE) under diffraction modulations (periodic modulation, linear modulation and power function modulation) and external potentials (parabolic potential and linear potential). The results show that QPM is able to change the focusing position and intensity, as well as the transmission trajectory of the beam. In a periodic modulation, the circular Airy beam (CAB) exhibits periodic variation characteristics, and the beam splitting is retarded under the action of the QPM. The self-focusing distance of the beam is significantly reduced, and its transmission trajectory and beam width are altered by the QPM under the linear modulation. The CAB progressively evolves into a non-diffraction beam under the power function modulation, and the QPM is able to reduce the light intensity and increase the beam width as the Lévy index decreases. In a parabolic potential, CABs display autofocusing and defocusing behavior, and the QPM affects the intensity distribution and optical width of the beam. The CAB is deflected and evolves periodically in a linear potential. The beam width increases and gradually stabilizes with the addition of the QPM. The propagation of CABs controlled with QPM in parabolic and linear potentials is also analyzed in the frequency domain. The results demonstrate that we can control the transmission of CABs in an FSE optical system by rationally setting parameters such as QPM, modulation coefficients, and external potentials.

Scalar field dark matter: impact of supernova-driven blowouts on the soliton structure of low-mass dark matter haloes

ABSTRACT We present the first study on the gravitational impact of supernova feedback in an isolated soliton and a spherically symmetric dwarf scalar field dark matter (SFDM) halo of virial mass $1\times 10^{10}\,\mathrm{M_\odot }$. We use a boson mass $m=10^{-22}\,\mathrm{eV\,c^{-2}}$ and a soliton core $r_\mathrm{ c} \approx 0.7$ kpc, comparable to typical half-light radii of Local Group dwarf galaxies. We simulate the rapid gas removal from the centre of the soliton by a concentric external time-dependent Hernquist potential. We explore two scenarios of feedback blowouts: (i) a massive single burst and (ii) multiple consecutive blowouts injecting the same total energy to the system, including various magnitudes for the blowouts in both scenarios. In all cases, we find one single blowout has a stronger effect on reducing the soliton central density. Feedback leads to central soliton densities that oscillate quasi-periodically for an isolated soliton and stochastically for an SFDM halo. The range in the density amplitude depends on the strength of the blowout; however, we observe typical variations of a factor of $\geqslant$2. One important consequence of the stochastic fluctuating densities is that, if we had no prior knowledge of the system evolution, we can only know the configuration profile at a specific time within some accuracy. By fitting soliton profiles at different times to our simulated structures, we found the (1$\sigma$) scatter of their time-dependent density profiles. For configurations within the 1$\sigma$ range, we find the inferred boson mass is typically less than 20 per cent different from the real value used in our simulations. Finally, we compare the observed dynamical masses of field dwarf galaxies in our Local Group with the implied range of viable solitons from our simulations and find good agreement.

Fabrication of CdSe/Si nanostructure for self-powered visible light photodetector

Self-powered visible light photodetector based nanostructured CdSe/Si geometry was fabricated using pulsed laser deposition (PLD). Herein, a detailed microstructural investigation was demonstrated in conjunction with the overall device photoresponsive performance. Particularly, an average particle diameter of 66.62 nm along optical band gap of 2.25 eV were attained. This was, subsequently, perceived in the fabricated device spectral response performance during which photocurrent (Iph) of 47 µA was attained under incident light of 575 nm and 6 mW/cm2, the nearest incident wavelength to the obtained optical band gap. At the reported incident light features, the proposed arrangement exhibited responsivity (Rλ) of 7.85 mA/W with light to dark current ratio (Iph/ID) of 998 %, respectively. Herein, the incident power variation suggested a linear increment (R2≈ 0.94) in both Iph and Iph/ID profiles, with values of 71 µA and 1501 %, respectively, at 575 nm and 26 mW/cm2. The fabricated device exhibited a pronounced time-resolved feature with rise/fall periods of 0.31 and 0.34 sec., while the power dependance based time-resolved behavior indicated a linear correlation between the stated factors with R2 value of 0.924, both of which were carried out at 0 bias voltage. The dual-singularity evidenced the self-powered feature of the proposed photodetector, where zero external potential is required.