Hamiltonian Equations Research Articles

Modelling of Cut-off Frequency of Armchair Graphene Nanoribbon Tunnel Field Effect Transistor Using Transfer Matrix Method

Graphene is considered as a promising two-dimensional material for electronic device applications due to its high charge carrier mobility and it is extremely thin in size. As a material, graphene has great potential for the development of high-speed nano electric devices. Graphene can be used as a channel material in Tunnel Field Effect Transistor (TFET). In this study, the cut-off frequency of the armchair graphene nanoribbon tunnel field effect transistor (AGNR-TFET) device was modelled quantum mechanically. The solution of the Dirac-like Hamiltonian and Poisson equations self-consistently is used to determine the potential profile of device and the transmittance is calculated using the transfer matrix method (TMM). The Landauer formulation with the help of Gauss Legendre Quadrature Method (GLQM) is used to calculate the tunnelling current and the cut-off frequency of AGNR-TFET. The calculation results showed that the cut-off frequency increased as the drain voltage increased, while the increase in channel length and the N-index graphene made the cut-off frequency decreased. Moreover, the thicker the oxide layer could increase the cut-off frequency and the higher the temperature on the device would reduce the cut-off frequency.

Stability Analysis and Optimal Control for Spreading Typhoid Fever Model with Direct and Indirect Transmissions

This article discusses a model of spreading typhoid fever with direct and indirect transmissions. There are five compartments included in the model, namely susceptible individuals, infected individuals, chronic carrier individuals, recovered individuals, and salmonella typhi bacteria in the environment. As an effort to control and minimize the numbers of infected individuals and chronic carrier individuals, we introduce simultaneously campaign for the susceptible individuals and treatment for the infected individuals and for chronic carrier individuals. The resulting model is a system of non-linear differential equations and quite challenging to analysis globally. Existence of both disease-free equilibrium point and endemic equilibrium point are analysed analytically. Stability of the equilibrium point is analysed locally by determining the eigenvalues of the associated Jacobian matrix, Routh-Hurwitz stability criteria, and basic reproduction number () via next generation matrix. The minimum Pontryagin principle and Hamiltonian equation are referred to minimize the numbers of infected and chronic carrier individuals. Simulation is carried out using suitable values of parameters. We found that the eigenvalues for endemic equilibrium point are all negative real numbers and . Optimal paths for state and constate variables are plotted using fourth order forward-backward Runge-Kutta method. In case there are no campaign and treatments, we found and endemic occurs. When campaign and treatments are included together in the model, we found optimal paths for all compartments that minimize the number of infected and chronic carrier individuals. Giving campaign and treatments increase significantly the susceptible and recovered individuals. At the same time, it reduces significantly the infected, chronic carrier individuals, and salmonella typhi bacteria in the environment. Involving campaign and treatment in the model of spreading typhoid fever can be considered as an effective strategy to minimize the numbers of infected and chronic carrier individuals.

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Classical energy-momentum methods study the existence and stability properties of solutions of t-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Ad}^*$$\\end{document}-equivariance of momentum maps. Eigenfunctions of t-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Reducibility of Linear Quasi-periodic Hamiltonian Derivative Wave Equations and Half-Wave Equations Under the Brjuno Conditions

Reducibility of Linear Quasi-periodic Hamiltonian Derivative Wave Equations and Half-Wave Equations Under the Brjuno Conditions

Thermomechanical buckling and vibration of composite sandwich doubly curved shells with carbon nanotube‐reinforced face layers

AbstractIn this paper, the thermomechanical buckling and vibration analysis are investigated for carbon nanotube‐reinforced composite sandwich doubly curved shells (CNCSDS) under the action of both thermal loads and in‐plane compressive loads. Based on the von‐Karman non‐linear kinematic relations, First Order Shear Theory, and Hamilton's principle, the vibration and stability equations for CNCSDS under thermomechanical loads are deduced. For obtaining the critical thermomechanical buckling load and vibration frequency, an exact method is adopted. Subsequently, the results are validated by comparing with the finding in published literature and simulation results. At last, the influence of system parameters on critical thermomechanical buckling load and vibration frequency is studied and displayed graphically. Meanwhile, some new results about thermomechanical buckling and vibration analysis of CNCSDS are proposed for the first time.Highlights Thermomechanical buckling and vibration of carbon nanotube‐reinforced composite sandwich doubly curved shells was studied. Vibration and stability equations under thermomechanical loads was deduced. The change rules of thermomechanical buckling load and frequency were obtained.

A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory.

Hamiltonian formulation of linear non-Hermitian systems

In the case of a linear non-Hermitian system, I prove that it's possible to construct a Hamiltonian in such a way that the equations governing the non-Hermitian system can be exactly expressed using Hamilton's canonical equations. Initially, I demonstrate this within the discrete representation framework and subsequently extend it to continuous representation. Through this formulation employing the Hamiltonian, I can pinpoint a conserved charge using Noether's theorem and identify adiabatic invariants. When this approach is applied to Hermitian systems, all the obtained results converge to the well-known outcomes associated with the Schrödinger equation.

Entropic uncertainty relations and quantum coherence in the two-dimensional XXZ spin model with Dzyaloshinskii–Moriya interaction

Quantum renormalization group (QRG) is a tractable method for studying the criticalities of one-dimensional (1D) and two-dimensional (2D) many-body systems. By employing the QRG method, we first derive the effective Hamiltonian and QRG equations of a 2D XXZ model with Dzyaloshinskii–Moriya (DM) interaction analytically. The linear-entropy-based uncertainty, the quantum discord (QD), and the multipartite quantum coherence based on the square root of the quantum Jensen–Shannon divergence of the 2D XXZ model are then studied as the indicators of quantum phase transitions (QPTs). The nonanalytic and scaling behaviors of the uncertainty, QD and quantum coherence are also analyzed through numerical calculations. Moreover, we investigate the effect of the easy-axis anisotropy parameter and DM interaction on the QPT. We find that the uncertainty, QD, and quantum coherence can all be utilized to detect QPTs. Our findings could shed new light on the observable of the QPT of the many-body system with the uncertainty and quantum coherence, and enrich the application of QRG method to Heisenberg spin models.

Nonexistence of integrable nonlinear magnetic fields with invariants quadratic in momenta

Nonlinear, completely integrable Hamiltonian systems that serve as blueprints for novel particle accelerators at the intensity frontier are promising avenues for research, as Fermilab’s Integrable Optics Test Accelerator (IOTA) example clearly illustrates. Here, we show that only very limited generalizations are possible when no approximations in the underlying Hamiltonian or Maxwell equations are allowed, as was the case for IOTA. Specifically, no such systems exist with invariants quadratic in the momenta, precluding straightforward generalization of the Courant–Snyder theory of linear integrable systems in beam physics. We also conjecture that no such systems exist with invariants of higher degree in the momenta. This leaves solenoidal magnetic fields, including their nonlinear fringe fields, as the only completely integrable static magnetic fields, albeit with invariants that are linear in the momenta. The difficulties come from enforcing Maxwell equations; without constraints, we show that there are many solutions. In particular, we discover a previously unknown large family of integrable Hamiltonians.

Dynamics of graphene as an open quantum system

We examine the imbalance thermodynamics of uncontaminated decoherence in graphene. In a uncontaminated decoherence process, no exchange of energy takes place directly between graphene and the surroundings. It is seen that due to decoherence heat dissolution takes place due to which energy of the environment cannot remain conserved. In this paper, we find out the energy eigenvalues and wave function in the graphene from the perturbed Hamiltonian to find out the density matrix and entropy of the system and their unique relation between the density matrix, momentum of the graphene and entropy where the graphene is taken as an open system. In graphene, the behavior of electrons is controlled by the massless Hamiltonian and ([Formula: see text]) Dirac equation.

Dynamical Symmetry and B(E2) Transitions of (_42^94)Mo by Utilizing (IBM-1)

Background:In this study, we utilize the IBM-1 model to theoretically investigate the structural characteristics of , with a specific emphasis on a nucleus consisting of 5 bosons. Employing the IBM-1, an innovative interacting boson model, we analyze the nuclear structure. Materials and Methods: In our theoretical exploration, we delved into the nuclear architecture of Molybdenum. Employing the Interacting Boson Model (IBM-1), we identified the pertinent Hamiltonian to scrutinize the isotope in this research endeavor. Results: Our investigation delved into the symmetry limit of the proposed nucleus, unveiling its stable nature within the SU(5) dynamical symmetry domain. To optimize alignment with observed energy states in experiments, we meticulously determined the parameters in the IBM-1 Hamiltonian equation. Furthermore, our research extends to the computation of B(E2) transitions, quadrupole moments, and energy levels across all bands for the examined nucleus. Conclusions: Our investigation reveals the stability of the nucleus in the isotope, showcasing the presence of SU(5) dynamical symmetry. These findings offer fresh perspectives by suggesting additional electric quadrupole moments.

Nonmetric geometric flows and quasicrystalline topological phases for dark energy and dark matter in f(Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(Q)$$\\end{document} cosmology

We elaborate on nonmetric geometric flow theory and metric-affine gravity with applications in modern cosmology. Two main motivations for our research follow from the facts that (1) cosmological models for f(Q) modified gravity theories, MGTs, are efficient for describing recent observational data provided by the James Webb Space Telescope; and (2) the statistical thermodynamic properties of such nonmetric locally anisotropic cosmological models can be studied using generalizations of the concept of G. Perelman entropy. We derive nonmetric distorted R. Hamilton and Ricci soliton equations in such canonical nonholonomic variables when corresponding systems of nonlinear PDEs can be decoupled and integrated in general off-diagonal forms. This is possible if we develop and apply the anholonomic frame and connection deformation method involving corresponding types of generating functions and generating sources encoding nonmetric distortions. Using such generic off-diagonal solutions (when the coefficients of metrics and connections may depend generically on all spacetime coordinates), we model accelerating cosmological scenarios with quasi-periodic gravitational and (effective) matter fields; and study topological and nonlinear geometric properties of respective dark energy and dark matter, DE and DM, models. As explicit examples, we analyze some classes of nonlinear symmetries defining topological quasicrystal, QC, phases which can modified to generate other types of quasi-periodic and locally anisotropic structures. The conditions when such nonlinear systems possess a behaviour which is similar to that of the Lambda cold dark matter (Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM) scenario are stated. We conclude that nonmetric geometric and cosmological flows can be considered as an alternative to the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM concordance models and speculate on how such theories can be elaborated. This is a partner work with generalizations and applications of the results published by Bubuianu, Vacaru et all. EPJC 84 (2024) 211; 80 (2020) 639; 78 (2018) 393; 78 (2018) 969; and CQG 35 (2018) 245009.

Chaotic behavior, bifurcations, sensitivity analysis, and novel optical soliton solutions to the Hamiltonian amplitude equation in optical physics

This study, highlights the exact optical soliton solutions in the context of optical physics, centering on the intricate Hamiltonian amplitude equation with bifurcation and sensitivity analysis. This equation is pivotal in optics which underpins the understanding of optical manifestations, encompassing solitons, nonlinear consequences, and wave interactions. Applying an analytical expansion approach, we extract diverse optical solutions, having trigonometric, hyperbolic, and rational functions. Next, we utilize concepts from the principle of planar dynamical systems to investigate the bifurcation processes and chaotic behaviors present in this derived system. Additionally, we use the Runge–Kutta scheme to carry out a thorough sensitivity analysis of the dynamical system. It has been verified through this analytical process that small variations in beginning conditions have negligible effects on the stability of the solution using bifurcation analysis. Validation via Mathematica software ensures the accuracy of these findings. Furthermore, we employ dynamic visualizations, such as 2D, 3D, and contour plots, to illustrate various soliton patterns, including kink, multi-kink, single periodic, multi-periodic, singular, and semi-bell-shaped configurations. These visual representations provide a glimpse into the fascinating behavior of optical phenomena. The solutions obtained via this proposed method showcase its efficacy, dependability, and simplicity in comparison to various alternative approaches.

Jacobi group symmetry of Hamilton's mechanics

We show that diffeomorphisms of an extended phase space with time, energy, momentum and position degrees of freedom leaving invariant a symplectic 2-form and a degenerate orthogonal metric dt2, corresponding to the Newtonian time line element, locally satisfy Hamilton's equations up to the usual canonical transformation on the position-momentum subspace.

A class of germs arising from homogenization in traffic flow on junctions

We consider traffic flows described by conservation laws. We mainly study a 2:1 junction (with two incoming roads and one outgoing road). At the mesoscopic level, the priority law at the junction is given by traffic lights, which are periodic in time; the traffic can also be slowed down by periodic in time flux-limiters. Looking at long-time behavior and on large space scale, we intuitively expect an effective junction condition to emerge, thus deriving a macroscopic model from the mesoscopic one. At the limit of the rescaling, we show rigorous homogenization of the problem and identify the effective junction condition, which belongs to a general class of germs (in the terminology of [B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of [Formula: see text]-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal. 201 (2011) 27–86; U. S. Fjordholm, M. Musch and N. H. Risebro, Well-posedness and convergence of a finite volume method for conservation laws on networks, SIAM J. Numer. Anal. 60(2) (2022) 606–630; M. Musch, U. S. Fjordholm and N. H. Risebro, Well-posedness theory for nonlinear scalar conservation laws on networks, Netw. Heterog. Media 17 (2022) 101–128]). The proof of homogenization requires two steps: our first key result is the identification of this germ and of a characteristic subgerm which determines the whole germ. The second key result is the construction of a family of correctors whose values at infinity are related to each element of the characteristic subgerm. This construction is indeed explicit at the level of some mixed Hamilton–Jacobi equations for concave Hamiltonians (i.e. fluxes). The solutions are found in the spirit of representation formulas for optimal control problems.

Exact global control of small divisors in rational normal form *

Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schrödinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

Dynamical quantum phase transitions following a noisy quench

We study how time-dependent energy fluctuations impact the dynamical quantum phase transitions (DQPTs) following a noisy ramped quench of the transverse magnetic field in a quantum Ising chain. By numerically solving the stochastic Schrödinger equation of the mode-decoupled fermionic Hamiltonian of the problem, we identify two generic scenarios: Depending on the amplitude of the noise and the rate of the ramp, the expected periodic sequence of noiseless DQPTs may either be uniformly shifted in time or else replaced by a disarray of closely spaced DQPTs. Guided by an exact noise master equation, we trace the phenomenon to the interplay between noise-induced excitations which accumulate during the quench and the near-adiabatic dynamics of the massive modes of the system. Our analysis generalizes to any one-dimensional fermionic two-band model subject to a noisy quench. Published by the American Physical Society 2024

Propagation of dark solitons of DNLS equations along a large-scale background

We study dynamics of dark solitons in the theory of the derivative nonlinear Schrödinger equations by the method based on imposing the condition that this dynamics must be Hamiltonian. Combining this condition with Stokes’ remark that relationships for harmonic linear waves and small-amplitude soliton tails satisfy the same linearized equations, so the corresponding solutions can be converted one into the other by replacement of the packet’s wave number k by iκ, κ being the soliton’s inverse half-width, we find the Hamiltonian and the canonical momentum of the soliton’s motion. The Hamilton equations are reduced to the Newton equation whose solutions for some typical situations are compared with exact numerical solutions of the Kaup-Newell DNLS equation.

Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics.

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon-Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.