arXiv subjects

Teepanis Chachiyo

Publications and source records attributed to Teepanis Chachiyo.

5 recordsLinked to original sources

arXiv2026

Spectral taxonomy for quartic systems: fundamental clock, parity, and continuum

A symmetric quartic potential is a physics motif with incredibly expansive applications, ranging from broadband energy harvesters, quantum tunneling in molecules and the early universe, to torque-free spacecraft rotation. For nearly two centuries, its rich dynamics have been classified into regimes and expressed as disjointed time-domain solutions. Here we build a taxonomy for this broad class of motions and discover that their regimes exhibit a universal spectral structure: they share a fundamental clock, obey parity selection, and dissolve into the separatrix through a discrete-to-continuum transition. Applied to the famous Dzhanibekov effect where a rotating body (e.g., a spacecraft) periodically undergoes rapid 180-degree flips in its attitude, the taxonomy reveals its spectral anatomy. The three principal-axis rotations share a common clock while occupying distinct parity channels, with stable-axis branches exchanging DC bias across the separatrix. This converts the torque-free tumbling from a purely time-domain crisis into a frequency-domain design opportunity. By presenting the exact spectral solutions and their taxonomy, we offer a new frequency-aware framework by which physical systems can be characterized, designed, and controlled. We discuss a case study where the three spectral pillars: clock, parity, and continuum, survive the Wick rotation from real-time into imaginary-time kinematics. The persistent characteristics also invite the possibility that the universal spectral structure encompasses an entire class of major physics motifs -- a possible canonical behavior in conservative 1D dynamics.

physics.class-ph↗

arXiv2025

Simple and accurate complete elliptic integrals for the full range of modulus

The complete elliptic integral of the first and second kind, K(k) and E(k), appear in a multitude of physics and engineering applications. Because there is no known closed-form, the exact values have to be computed numerically. Here, approximations for the integrals are proposed based on their asymptotic behaviors. An inverse of K is also presented. As a result, the proposed K(k) and E(k) reproduce the exact analytical forms both in the zero and asymptotic limits, while in the mid-range of modulus maintain average error of 0.06% and 0.01% respectively. The key finding is the ability to compute the integrals with exceptional accuracy on both limits of elliptical conditions. An accuracy of 1 in 1,000 should be sufficient for practical or prototyping engineering and architecture designs. The simplicity should facilitate discussions of advanced physics topics in introductory physics classes, and enable broader collaborations among researchers from other fields of expertise. For example, the phase space of energy-conserving nonlinear pendulum using only elementary functions is discussed. The proposed inverse of K is shown to be Never Failing Newton Initialization and is an important step for the computation of the exact inverse. An algorithm based on Arithmetic-Geometric Mean for computing exact integrals and their derivatives are also presented, which should be useful in a platform that special functions are not accessible such as web-based and firmware developments. Comparisons with sharp bounds from the mathematical inequalities literature further highlight the competitiveness of the proposed approximations.

physics.gen-ph↗

arXiv2025

Universal spectral structure in pendulum-like systems

Pendulum-like dynamics is a universal motif across many areas of physics, underlying systems ranging from classical nonlinear oscillators to superconducting qubits and cold-atom tunneling platforms. Here we present an exact frequency-domain formulation of the pendulum equation that applies uniformly across oscillatory, separatrix, and rotational regimes. The resulting spectral representation reveals a previously hidden unification: all regimes share the same analytic spectral structure and characteristic frequency scale. We discover that all regimes arise from a single universal spectral kernel, with parity selection distinguishing the periodic motions and the separatrix representing their discrete-to-continuum limit. Regime changes thus correspond to symmetry-driven reorganizations in frequency space rather than changes in the underlying spectral structure, with the stopping trajectory representing the continuum limit reached without system-size scaling. The spectral structure can be derived via a spectral discretization approach starting from the separatrix solution, without relying on the classical Jacobi elliptic formulation. Beyond providing closed-form solutions, the framework reveals a transparent spectral structure underlying a broad class of classical and quantum pendulum-like systems.

physics.class-ph↗

arXiv2020

Simple and accurate exchange energy for density functional theory

A non-empirical exchange functional based on an interpolation between two limits of electron density: slowly varying limit and asymptotic limit, is proposed. In the slowly varying limit, we follow the study by Kleinman in 1984 which considered the response of a free-electron gas to an external periodic potential, but further assume that the perturbing potential also induces Bragg diffraction of the Fermi electrons. The interpolation function is motivated by the exact exchange functional of a hydrogen atom. Combined with our recently proposed correlation functional, tests on 56 small molecules show that, for the first-row molecules, the exchange-correlation combo predicts the total energies four times more accurate than presently available Quantum Monte Carlo results. For the second-row molecules, errors of the core electrons exchange energies can be corrected, leading to the most accurate molecular total energy predictions to date despite minimal computational efforts. The calculated bond energies, zero point energies, and dipole moments are also presented.

cond-mat.mtrl-sci↗

arXiv2019

Understanding electron correlation energy through density functional theory

A curious behavior of electron correlation energy is explored. Namely, the correlation energy is the energy that tends to drive the system toward that of the uniform electron gas. As such, the energy assumes its maximum value when a gradient of density is zero. As the gradient increases, the energy is diminished by a gradient suppressing factor, designed to attenuate the energy from its maximum value similar to the shape of a bell curve. Based on this behavior, we constructed a very simple mathematical formula that predicted the correlation energy of atoms and molecules. Combined with our proposed exchange energy functional, we calculated the correlation energies, the total energies, and the ionization energies of test atoms and molecules; and despite the unique simplicities, the functionals' accuracies are in the top tier performance, competitive to the B3LYP, BLYP, PBE, TPSS, and M11. Therefore, we propose that, as guided by the simplicities and supported by the accuracies, the correlation energy is the energy that locally tends to drive the system toward the uniform electron gas.

cond-mat.mtrl-sci↗