Hamiltonian Formulation Research Articles

Signature of excitonic insulators in phosphorene nanoribbons.

Phosphorene is a recently developed two-dimensional (2D) material that has attracted tremendous attention because of its unique anisotropic optical properties and quasi-one-dimensional (1D) excitons. We use first-principles calculations combined with the maximally localized Wannier function tight binding Hamiltonian (MLWF-TB) and Bethe-Salpeter equation (BSE) formalism to investigate quasiparticle effects of 2D and quasi-1D blue and black phosphorene nanoribbons. Our electronic structure calculations shows that both blue and black monolayered phases are semiconductors. On the other hand black phosphorene zigzag nanoribbons are metallic. Similar behavior is found for very thin blue phosphorene zig-zag and armchair nanoribbon. As a general behavior, the exciton binding energy decreases as the ribbon width increases, which highlights the importance of quantum confinement effects. The solution of the BSE shows that the blue phosphorene monolayer has an exciton binding energy four times higher than that of the black phosphorene counterpart. Furthermore, both monolayers show a different linear optical response with respect to light polarization, as black phosphorene is highly anisotropic. We find a similar, but less pronounced, optical anisotropy for blue phosphorene monolayer, caused exclusively by the quasi-particle effects. Finally, we show that some of the investigated nanoribbons show a spin-triplet excitonic insulator behavior, thus revealing exciting features of these nanoribbons and therefore provides important advances in the understanding of quasi-one dimensional phosphorus-based materials.&#xD.

Asymptotic symmetries of SU(2) Yang-Mills-Higgs theory in Hamiltonian formulation

We investigate the asymptotic symmetry group of a SU(2)-Yang-Mills theory coupled to a Higgs field in the Hamiltonian formulation. This extends previous work on the asymptotic structure of pure electromagnetism by Henneaux and Troessaert, and on electromagnetism coupled to scalar fields and pure Yang-Mills fields by Tanzi and Giulini. We find that there are no obstructions to global electric and magnetic charges, though that is rather subtle in the magnetic case. Again it is the Hamiltionian implementation of boost symmetries that need a careful and technically subtle discussion of fall-off and parity conditions of all fields involved.

Stationary coupled KdV systems and their Stäckel representations

AbstractIn this article, we investigate stationary coupled Korteweg–de Vries (cKdV) systems and prove that every ‐field stationary cKdV system can be written, after a careful reparameterization of jet variables, as a classical separable Stäckel system in different ways. For each of these parameterizations, we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of Stäckel representations of the same stationary cKdV system, when considered in the phase space extended by Casimir variables, is connected by an appropriate finite‐dimensional Miura map, which leads to an ‐Hamiltonian formulation for the stationary cKdV system.

Passivity-based control of fluid flow networks with capacitance

Fluid flow networks (FFNs) widely exist in industrial energy systems tightly related with the industrial production and residents living, such as district heating networks (DHN), mine-ventilation systems and gas transportation systems. The FFN operation is to drive the flowrates and pressures to their desired values, and the flowrate-pressure coordinated control of FFN is crucial in providing safe, stable and efficient operation. Although fluid capacitance has the same importance as fluid resistance and fluid inductance in giving dynamical characteristics of FFN, fluid capacitance is still not fully considered in the flowrate and pressure control design of FFN. In this paper, a nonlinear dynamic model of general FFN with fluid resistance, inductance and capacitance is first given. By using this nonlinear model, it is shown that the general FFN dynamics is strictly passive, and the corresponding generalized Hamiltonian form (GHF) is also given. Based on the passivity of FFN, a decentralized adaptive flowrate-pressure coordinated control is newly proposed, taking a simple proportional-integral (PI) form while providing globally asymptotic regulation for network flowrates and pressures. This FFN control method is applied to design the flowrate-pressure controller of an auxiliary startup circuit (ASC) equipped to a modular high temperature gas-cooled reactor (mHTGR), and simulation results in the cases of heating-up and cooling-down show the feasibility of control design.

A CMOS-compatible oscillation-based VO2 Ising machine solver

Phase-encoded oscillating neural networks offer compelling advantages over metal-oxide-semiconductor-based technology for tackling complex optimization problems, with promising potential for ultralow power consumption and exceptionally rapid computational performance. In this work, we investigate the ability of these networks to solve optimization problems belonging to the nondeterministic polynomial time complexity class using nanoscale vanadium-dioxide-based oscillators integrated onto a Silicon platform. Specifically, we demonstrate how the dynamic behavior of coupled vanadium dioxide devices can effectively solve combinatorial optimization problems, including Graph Coloring, Max-cut, and Max-3SAT problems. The electrical mappings of these problems are derived from the equivalent Ising Hamiltonian formulation to design circuits with up to nine crossbar vanadium dioxide oscillators. Using sub-harmonic injection locking techniques, we binarize the solution space provided by the oscillators and demonstrate that graphs with high connection density (η > 0.4) converge more easily towards the optimal solution due to the small spectral radius of the problem’s equivalent adjacency matrix. Our findings indicate that these systems achieve stability within 25 oscillation cycles and exhibit power efficiency and potential for scaling that surpasses available commercial options and other technologies under study. These results pave the way for accelerated parallel computing enabled by large-scale networks of interconnected oscillators.

A canonical Hamiltonian formulation of the Navier–Stokes problem

This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ( $u_{i}$ , $p$ ) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$ . Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$ . If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

Linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries equations

In this paper, second-order and fourth-order linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries (KdV) equations are proposed. First, the KdV equation is semi-discretized into an oscillatory Hamiltonian system of Ordinary Differential Equations (ODEs) by semi-discretizations of the skew adjoint operator and the Hamiltonian in the Hamiltonian form of the KdV equation. Then the resulting semi-discrete system is integrated by exponential average vector field integrator. The order of convergence and the conservation properties are established for the introduced full-discrete procedures. It shows that the proposed methods are energy-preserving, mass-preserving and qualitatively preserve the dispersive relation well. Numerical experiments are carried out to validate the efficiency and the accuracy of the proposed schemes.

Noninvertible symmetries and anomalies from gauging 1-form electric centers

We devise a general method for obtaining 0-form noninvertible discrete chiral symmetries in 4-dimensional SU(N)/ℤp and SU(N) × U(1)/ℤp gauge theories with matter in arbitrary representations, where ℤp is a subgroup of the electric 1-form center symmetry. Our approach involves placing the theory on a three-torus and utilizing the Hamiltonian formalism to construct noninvertible operators by introducing twists compatible with the gauging of ℤp. These theories exhibit electric 1-form and magnetic 1-form global symmetries, and their generators play a crucial role in constructing the corresponding Hilbert space. The noninvertible operators are demonstrated to project onto specific Hilbert space sectors characterized by particular magnetic fluxes. Furthermore, when subjected to twists by the electric 1-form global symmetry, these surviving sectors reveal an anomaly between the noninvertible and the 1-form symmetries. We argue that an anomaly implies that certain sectors, characterized by the eigenvalues of the electric symmetry generators, exhibit multi-fold degeneracies. When we couple these theories to axions, infrared axionic noninvertible operators inherit the ultraviolet structure of the theory, including the projective nature of the operators and their anomalies. We discuss various examples of vector and chiral gauge theories that showcase the versatility of our approach.

IR absorption band of weak H-Bonded systems: Showcasing the signature of the quantum indirect damping within the framework of a new approach using non-Hermitean Hamiltonian formalism

Herein, we investigate the effects of the irreversible action of the medium on the theoretical elucidation of the IR spectrum of weakly H-bonded systems, a prototypical model which provides a rigorous treatment of the relaxation mechanisms impacts on the IR spectral density. The attention will be focused particularly on the effect of indirect damping on the IRυS(X−H⃗) spectrum beyond the harmonic and adiabatic approximations. The ultimate objective of the present investigation is the treatment of the action of the surrounding of the intermonomer modes of H-bonded systems, which must induce a broadening of the Dirac delta peaks, the nature of which, as shown by Maréchal and Witkowski theory, is a Franck–Condon progression. The quantum treatment of the IR absorption band reveals that quantum relaxation of the intermonomer mode of H-bonded complexes could be successfully approached by a non-Hermitian Hamiltonian formalism. Motivated by development of a second method that will be able to validate the first approach, a computationally efficient algorithm was proposed for elucidating the quantum indirect relaxation using Hermitean Hamiltonians. The real eigenvalues, corresponding to different energies of the system are considered to be complex by adjunction of the imaginary parts, which reflects the action of the indirect irreversible action of the medium. These two crude approaches may pave the way for the incorporation of the mechanism of indirect relaxation in more physical and complex situations dealing, particularly, with tunnelling effects in strong H-bonded species, Fermi resonances, and Davydov coupling for cyclic H-bonds dimers beyond the harmonic and adiabatic assumptions.

Structural, topological, vibrational, and electronic analysis, and ADMET study of methyl-2-(4-isobutylphenyl)propanoate

In this work, the methyl ibuprofen ester, named methyl-2-(4-isobutylphenyl)propanoate (C14H20O2, hereinafter MET-IBU).was synthesized and characterized by 1D- and 2D-NMR, ATR-FTIR, FT-Raman, and UV–Vis spectroscopy, while their structural, QTAIM topological and spectroscopic properties were investigated using DFT calculations. Through the analysis of the topological properties of the electron density computed at bonding critical points (BCP's) for MET-IBU molecule were determined the electron density, the Laplacian of electron density, the eigenvalues of Hessian matrix, the ellipticity of electron density, the potential energy density, the kinetic energy density in the Lagrangian and Hamiltonian forms, the total energy density, the Lagrangian density, and the electron delocalization index. Density Functional Theory (DFT) calculations were also used to determine the vibrational normal modes, the frontier molecular orbitals (FMOs), the dipole moment, and the molecular electrostatic potential (MEP) of MET-IBU. Theoretical UV absorption spectrum was also evaluated from time-dependent density functional theory (TD-DFT) providing the electronic transitions, which are expected for this compound. The molecular descriptors of the properties of absorption, distribution, metabolism and excretion, and toxicity (ADMET) were obtained. Empirical models for estimating ADME attributes showed that the MET-IBU has a distribution volume that allows permeability and activity in the Central Nervous System - CNS, with structural contributions from the side chain of isobutyl and methoxy that favor liver metabolism and excretion with a low toxic response. Furthermore, the MET-IBU does not have ionizable centers compared to its analog Ibuprofen constituting a more lipophilic chemical product.

Thermocapillary thin films: periodic steady states and film rupture

We study stationary, periodic solutions to the thermocapillary thin-film model 0,\\ x\\in \\mathbb{R},$?> ∂th+∂xh3∂x3h−g∂xh+Mh21+h2∂xh=0,t>0, x∈R, which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value M∗ , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

Thermodynamics-Informed Graph Neural Networks

In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting integration scheme. The first is achieved with Graph Neural Networks, which induces a non-Euclidean geometrical prior with permutation invariant node and edge update functions. The second bias is forced by learning the GENERIC structure of the problem, an extension of the Hamiltonian formalism, to model more general non-conservative dynamics. Several examples are provided in both Eulerian and Lagrangian description in the context of fluid and solid mechanics respectively, achieving relative mean errors of less than 3% in all the tested examples. Two ablation studies are provided based on recent works in both physics-informed and geometric deep learning.

Optical ladder operators in the Glauber-Fock oscillator array

In this study, we investigate the stationary states of the Glauber-Fock oscillator waveguide array. We begin by transforming the associated Hamiltonian into the form of a quantum harmonic oscillator Hamiltonian, allowing the implementation of a supersymmetric (SUSY) approach. By considering the simplest case for the intertwining operator, the optical ladder operators are straightforwardly constructed and shown to map eigensolutions into eigensolutions of the corresponding Hamiltonian operator, in pretty much the same manner as it is done for the quantum harmonic oscillator case. The ladder of the corresponding (eigen) supermodes is then easily established.

The Dirac equation across the horizons of the 5D Myers–Perry geometry: separation of variables, radial asymptotic behaviour and Hamiltonian formalism

We analytically extend the 5D Myers–Perry metric through the event and Cauchy horizons by defining Eddington–Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster–Röken and Stone’s formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green’s functions of the radial ODE.

On affine Toda field theories related to D r algebras and their real Hamiltonian forms

The paper deals with affine 2-dimensional Toda field theories related to simple Lie algebras of the classical series D r . We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of 𝔤. This is illustrated on the examples Dr+1(1)→Br(1) and D4(1)→G2(1) using external automorphisms of the corresponding extended Dynkin diagrams.

On Covariant and Canonical Hamiltonian Formalisms for Gauge Theories

The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms, have received a lot of attention in the literature. However, a full understanding of the relation between them is not available, especially when the gauge theories are defined over regions with boundaries. Here, we consider this issue, by first making it precise what we mean by equivalence between the two formalisms. Then, we explore several first-order gauge theories and assess whether their corresponding descriptions satisfy the notion of equivalence. We shall show that, even when in several cases the two formalisms are indeed equivalent, there are counterexamples that signal that this is not always the case. Thus, non-equivalence is a generic feature of gauge field theories. These results call for a deeper understanding of the subject.

Duality in the problems of optimal control described by Darboux-type differential inclusions

AbstractThis paper is devoted to the optimization of the Mayer problem with hyperbolic differential inclusions of the Darboux type and duality. We use the discrete approximation method to get sufficient conditions of optimality for the convex problem given by Darboux differential inclusions and the polyhedral problem for a hyperbolic differential inclusion with state constraint. We formulate the adjoint inclusions in the Euler-Lagrange inclusion and Hamiltonian forms. Then, we construct the dual problem to optimal control problem given by Darboux differential inclusions with state constraint and prove so-called duality results. Moreover, we show that each pair of primal and dual problem solutions satisfy duality relations and that the optimal values in the primal convex and dual concave problems are equal. Finally, we establish the dual problem to the polyhedral Darboux problem and provide an example to demonstrate the main constructions of our approach.

On CCGG, the De Donder‐Weyl Hamiltonian formulation of canonical gauge gravity

AbstractThis short paper gives a brief overview of the manifestly covariant canonical gauge gravity (CCGG) that is rooted in the De Donder‐Weyl Hamiltonian formulation of relativistic field theories, and the proven methodology of the canonical transformation theory. That framework derives, from a few basic physical and mathematical assumptions, equations describing generic matter and gravity dynamics with the spin connection emerging as a Yang Mills‐type gauge field. While the interaction of any matter field with spacetime is fixed just by the transformation property of that field, a concrete gravity ansatz is introduced by the choice of the free (kinetic) gravity Hamiltonian. The key elements of this approach are discussed and its implications for particle dynamics and cosmology are presented. New insights: Anomalous Pauli coupling of spinors to curvature and torsion of spacetime, spacetime with (A)dS ground state, inertia, torsion and geometrical vacuum energy, Zero‐energy balance of the Universe leading to a vanishing cosmological constant and torsional dark energy.

Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group

A number of approaches to gravitation have much in common with the gauge theories of the standard model of particle physics. In this paper, we develop the Hamiltonian formulation of a class of gravitational theories that may be regarded as spontaneously-broken gauge theories of the complexified Lorentz group SO(1,3)C with the gravitational field described entirely by a gauge field valued in the Lie algebra of SO(1,3)C and a ‘Higgs field’ valued in the group’s fundamental representation. The theories have one free parameter β which appears in a similar role to the inverse of the Barbero–Immirzi parameter of Einstein–Cartan theory. However, contrary to that parameter, it is shown that the number of degrees of freedom (DOFs) crucially depends on the value of β. For non-zero values of β, it is shown the theories possesses three complex DOFs, and for the specific values β=±i an extension to general relativity is recovered in a symmetry-broken regime. For the value β = 0, the theory possesses no local DOFs. A non-zero value of β corresponds to the self-dual and anti-self-dual gauge fields appearing asymmetrically in the action, therefore in these models, the existence of gravitational DOFs is tied to chiral asymmetry in the gravitational sector.

Finite‐time robust simultaneous stabilization control for two nonlinear singular time‐delay systems

AbstractIn this paper, the finite‐time simultaneous stabilization problem of two nonlinear singular time‐delay Hamiltonian systems is investigated. Firstly, the two general nonlinear singular time‐delay Hamiltonian systems are transformed into their Hamiltonian differential‐algebraic forms by equivalent transformation, respectively, they are then converted to two strictly dissipative Hamiltonian forms via the state feedback method, and two strictly dissipative Hamiltonian systems are extended into a Hamiltonian one. Based on which, a suitable Hamiltonian function is constructed to analyze the finite‐time simultaneous stability and finite‐time robust simultaneous stabilization problem. Finally, an illustrative example is given to show the effectiveness of the results obtained. It should be pointed out that since there does not exist a finite‐time stability criterion on the nonlinear singular delay system, we adopt a new double compound asymptotic method.