Path integrals and Liapunov functionals.
Liapunov functionals for time delay systems by path integrals in state space, using convolution equations involving distributions with compact support
Gruber, M.↗
SEARCH · Engineering Papers
Results for “PATH INTEGRALS”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
A path-integral approach to the problem of time
Quantum transition amplitudes are formulated for model systems with local internal time, using intuition from path integrals. The amplitudes are shown to be more regular near a turning point of internal time than could be expected based on existing canonical treatments. In particular, a successful transition through a turning point is provided in the model systems, together with a new definition of such a transition in general terms. Some of the results rely on a fruitful relation between the problem of time and general Gribov problems.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Path integrals and Lyapunov functionals
Liapunov functionals for time delay systems generated by path integrals in state space in terms of convolution equations
Gruber, M.↗
Squeezed states and path integrals
The continuous-time regularization scheme for defining phase-space path integrals is briefly reviewed as a method to define a quantization procedure that is completely covariant under all smooth canonical coordinate transformations. As an illustration of this method, a limited set of transformations is discussed that have an image in the set of the usual squeezed states. It is noteworthy that even this limited set of transformations offers new possibilities for stationary phase approximations to quantum mechanical propagators.
Daubechies, Ingrid↗
A path integral molecular dynamics study of the hyperfine coupling constants of the muoniated and hydrogenated acetone radicals
The on-the-fly ab initio density functional path integral molecular dynamics (PIMD) simulations, which can account for both the nuclear quantum effect and thermal effect, were carried out to evaluate the structures and “reduced” isotropic hyperfine coupling constants (HFCCs) for muoniated and hydrogenated acetone radicals (2-muoxy-2-propyl and 2-hydoxy-2-propyl) in vacuo. The reduced HFCC value from a simple geometry optimization calculation without both the nuclear quantum effect and thermal effect is −8.18 MHz, and that by standard ab initio molecular dynamics simulation with only the thermal effect and without the nuclear quantum effect is 0.33 MHz at 300 K, where these two methods cannot distinguish the difference between muoniated and hydrogenated acetone radicals. In contrast, the reduced HFCC value of the muoniated acetone radical by our PIMD simulation is 32.1 MHz, which is about 8 times larger than that for the hydrogenated radical of 3.97 MHz with the same level of calculation. We have found that the HFCC values are highly correlated with the local molecular structures; especially, the Mu—O bond length in the muoniated acetone radical is elongated due to the large nuclear quantum effect of the muon, which makes the expectation value of the HFCC larger. Although our PIMD result calculated in vacuo is about 4 times larger than the measured experimental value in aqueous solvent, the ratio of these HFCC values between muoniated and hydrogenated acetone radicals in vacuo is in reasonable agreement with the ratio of the experimental values in aqueous solvent (8.56 MHz and 0.9 MHz); the explicit presence of solvent molecules has a major effect on decreasing the reduced muon HFCC of in vacuo calculations for the quantitative reproduction.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Galerkin formulation of path integrals in lattice field theory
We present a mathematical framework for Galerkin formulations of path integrals in lattice field theory. The framework is based on using the degrees of freedom (DOFs) associated to a Galerkin discretization as the fundamental lattice variables. We formulate standard concepts in lattice field theory, such as the partition function and correlation functions, in terms of the DOFs. For example, using continuous finite element spaces, we show that the two-point spatial correlation function can be defined between any two points on the domain (as opposed to at just lattice sites) and furthermore, this two-point function satisfies a weak propagator (or Green’s function) identity, in analogy to the continuum case, as well as a convergence estimate obtained from the standard finite element techniques. Furthermore, this framework leads naturally to higher-order formulations of lattice field theories by considering higher-order finite element spaces for the Galerkin discretization. We consider analytical and numerical examples of scalar field theory to investigate how increasing the order of piecewise polynomial finite element spaces affect the approximation of lattice observables. Finally, we sketch an outline of this Galerkin framework in the context of gauge field theories.
97 MATHEMATICS AND COMPUTING↗
Path integral Monte Carlo simulations of warm dense aluminum
Here, we perform first-principles path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) calculations to explore warm dense matter states of aluminum. Our equation of state (EOS) simulations cover a wide density-temperature range of $0.1-32.4$~g$\,$cm$^{-3}$ and $10^4-10^8$~K. Since PIMC and DFT-MD accurately treat effects of the atomic shell structure, we find two compression maxima along the principal Hugoniot curve attributed to K-shell and L-shell ionization. The results provide a benchmark for widely used EOS tables, such as SESAME, QEOS, and models based on Thomas-Fermi and average-atom techniques. A subsequent multi-shock analysis provides a quantitative assessment for how much heating occurs relative to an isentrope in multi-shock experiments. Finally, we compute heat capacity, pair-correlation functions, the electronic density of states, and Z to reveal the evolution of the plasma structure and ionization behavior.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
A 2-Micron Pulsed Integrated Path Differential Absorption Lidar Development For Atmospheric CO2 Concentration Measurements
A 2-micron pulsed, Integrated Path Differential Absorption (IPDA) lidar instrument for ground and airborne atmospheric CO2 concentration measurements via direct detection method is being developed at NASA Langley Research Center. This instrument will provide an alternate approach to measure atmospheric CO2 concentrations with significant advantages. A high energy pulsed approach provides high-precision measurement capability by having high signal-to-noise level and unambiguously eliminates the contamination from aerosols and clouds that can bias the IPDA measurement.
Yu, Jirong↗
Persistent Homology Metrics Reveal Quantum Fluctuations and Reactive Atoms in Path Integral Dynamics
Nuclear quantum effects (NQEs) are known to impact a number of features associated with chemical reactivity and physicochemical properties, particularly for light atoms and at low temperatures. In the imaginary time path integral formalism, each atom is mapped onto a “ring polymer” whose spread is related to the quantum mechanical uncertainty in the particle’s position, i.e., its thermal wavelength. A number of metrics have previously been used to investigate and characterize this spread and explain effects arising from quantum delocalization, zero-point energy, and tunneling. Many of these shape metrics consider just the instantaneous structure of the ring polymers. However, given the significant interest in methods such as centroid molecular dynamics and ring polymer molecular dynamics that link the molecular dynamics of these ring polymers to real time properties, there exists significant opportunity to exploit metrics that also allow for the study of the fluctuations of the atom delocalization in time. Here we consider the ring polymer delocalization from the perspective of computational topology, specifically persistent homology, which describes the 3-dimensional arrangement of point cloud data, (i.e. atomic positions). We employ the Betti sequence probability distribution to define the ensemble of shapes adopted by the ring polymer. The Wasserstein distances of Betti sequences adjacent in time are used to characterize fluctuations in shape, where the Fourier transform and associated principal components provides added information differentiating atoms with different NQEs based on their dynamic properties. We demonstrate this methodology on two representative systems, a glassy system consisting of two atom types with dramatically different de Broglie thermal wavelengths, and ab initio molecular dynamics simulation of an aqueous 4 M HCl solution where the H-atoms are differentiated based on their participation in proton transfer reactions.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Effect of the shape on the spin state and exchange in quantum dots. Feynman path integral analysis
The ab initio computer simulation of the mixed quantum states of 1–5-nm model ellipsoid quantum dots with “soft” walls containing two and three quantum-indistinguishable nonrelativistic electrons has been performed by the path integral method. The calculation has been carried out beyond the single-electron and mean-field approximations with the fundamentally exact inclusion of Coulomb and exchange correlations of all orders and the spin variable. Distributions over the eigenfunctions of the spin-squared operator, as well as the equilibrium spin numbers, have been obtained depending on the shape of a quantum dot and the temperature. The complete set of basis functions symmetrized in permutations according to the spin of the system has been obtained by application of the Young symmetry operators. The dependence of the energy on the shape of the quantum dot corresponds to the negative sign of the surface tension at its boundary. The calculation indicates that the spin magnetic susceptibility in the system of two electrons decreases strongly for spherical quantum dots (“pairing” of spins) and the temperature dependences have a pronounced maximum whose position depends on the shape of the quantum dot. For three electrons in an oblate quantum dot, the inversion of the energy levels of spin states is observed and affects the spin magnetic susceptibility. The results indicate a strong dependence of the energy of collective spin states of electrons on the detailed inclusion of exchange and Coulomb spatial correlations.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Path integral Monte Carlo simulations of H{sub 2} adsorbed to lithium-doped benzene: A model for hydrogen storage materials
Finite temperature quantum and anharmonic effects are studied in H{sub 2}–Li{sup +}-benzene, a model hydrogen storage material, using path integral Monte Carlo (PIMC) simulations on an interpolated potential energy surface refined over the eight intermolecular degrees of freedom based upon M05-2X/6-311+G(2df,p) density functional theory calculations. Rigid-body PIMC simulations are performed at temperatures ranging from 77 K to 150 K, producing both quantum and classical probability density histograms describing the adsorbed H{sub 2}. Quantum effects broaden the histograms with respect to their classical analogues and increase the expectation values of the radial and angular polar coordinates describing the location of the center-of-mass of the H{sub 2} molecule. The rigid-body PIMC simulations also provide estimates of the change in internal energy, ΔU{sub ads}, and enthalpy, ΔH{sub ads}, for H{sub 2} adsorption onto Li{sup +}-benzene, as a function of temperature. These estimates indicate that quantum effects are important even at room temperature and classical results should be interpreted with caution. Our results also show that anharmonicity is more important in the calculation of U and H than coupling—coupling between the intermolecular degrees of freedom becomes less important as temperature increases whereas anharmonicity becomes more important. The most anharmonic motions in H{sub 2}–Li{sup +}-benzene are the “helicopter” and “ferris wheel” H{sub 2} rotations. Treating these motions as one-dimensional free and hindered rotors, respectively, provides simple corrections to standard harmonic oscillator, rigid rotor thermochemical expressions for internal energy and enthalpy that encapsulate the majority of the anharmonicity. At 150 K, our best rigid-body PIMC estimates for ΔU{sub ads} and ΔH{sub ads} are −13.3 ± 0.1 and −14.5 ± 0.1 kJ mol{sup −1}, respectively.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Path integrals, Liapunov functions, and quadratic minimization.
Nonlinear DC circuits analyzed by digital computer for application to path integrals and stability problems
SYSTEM STABILITY↗
Integrated path planning and control through proximal policy optimization for a marine current turbine
This paper presents an integrated path planning and tracking control framework for a marine current turbine (MCT), where the MCT is treated as an energy-harvesting autonomous underwater vehicle (AUV). Considering the ocean (space of action) is continuous, the proposed framework employs two modules to address path planning and path tracking enabled by the proximal policy optimization (PPO) algorithm, which is a policy gradient deep reinforcement learning (RL) method. Further, to enable fully autonomous operation in a stochastic oceanic environment, the proposed path planning seeks a primary objective of maximizing the harvested energy; then, the path tracking module is designed to minimize the tracking error and avoid collisions with static and dynamic obstacles. Using field-collected acoustic Doppler current profiler (ADCP) data, the performance of the proposed framework is evaluated. Comparative studies with baseline algorithms in three different scenarios of path planning, path tracking without an obstacle, and path tracking with collision avoidance verify the effectiveness of our proposed approach.
16 TIDAL AND WAVE POWER↗
Path-integral isomorphic Hamiltonian for including nuclear quantum effects in non-adiabatic dynamics
We describe a path-integral approach for including nuclear quantum effects in non-adiabatic chemical dynamics simulations. For a general physical system with multiple electronic energy levels, a corresponding isomorphic Hamiltonian is introduced such that Boltzmann sampling of the isomorphic Hamiltonian with classical nuclear degrees of freedom yields the exact quantum Boltzmann distribution for the original physical system. In the limit of a single electronic energy level, the isomorphic Hamiltonian reduces to the familiar cases of either ring polymer molecular dynamics (RPMD) or centroid molecular dynamics Hamiltonians, depending on the implementation. An advantage of the isomorphic Hamiltonian is that it can easily be combined with existing mixed quantum-classical dynamics methods, such as surface hopping or Ehrenfest dynamics, to enable the simulation of electronically non-adiabatic processes with nuclear quantum effects. As a result, we present numerical applications of the isomorphic Hamiltonian to model two- and three-level systems, with encouraging results that include improvement upon a previously reported combination of RPMD with surface hopping in the deep-tunneling regime.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Path integrals, differential renormalization-group, and stochastic systems near criticality
It is demonstrated, using the techniques of path integrals and renormalization-group, that nonlinear stochastic systems near criticality (including self-organized criticality) generally exhibit low-dimensional behavior. The symmetry which characterizes a particular criticality can be broken by the appearance of relevant scaling fields. A connection is made between the fractal dimensions of finite-dimensional chaotic systems and the anomalous dimensions in stochastic systems near criticality. The effect of additional random noise on stochastic systems is also delineated.
Chang, Tom↗
Nucleon clustering at kinetic freezeout of heavy-ion collisions via path-integral Monte Carlo
Clustering of the four-nucleon system at kinetic freezeout conditions is studied using path-integral Monte Carlo techniques. This method seeks to improve upon previous calculations which relied on approximate semiclassical methods or few-body quantum mechanics. Estimates are given for the decay probabilities of the 4N system into various light nuclei decay channels and the strength of spatial correlations is characterized. Additionally, a simple model is presented to describe the impact of this clustering on nucleon multiplicity distributions. Additionally, the effects of a possible modification of the inter-nucleon interaction due to the close critical line (and hypothetical QCD critical point) on the clustering are also studied.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Path Planning: Differential Dynamic Programming and Model Predictive Path Integral Control on VTOL Aircraft
This paper explores two optimal control approaches, widely used in robotics, to establish their viability as real-time trajectory planners for vehicle configurations envisioned for the emerging aviation sector of Urban Air Mobility (UAM). Differential Dynamic Programming (DDP) enables planning over highly nonlinear dynamics using second-order approximations along a nominal trajectory, and displays quadratic convergence to a local solution. Model Predictive Path Integral (MPPI) is a stochastic sampling-based algorithm that can optimize for general cost criteria, including potentially highly nonlinear formulations, and supports parallel computation through the use of modern GPU hardware. In this work, DDP and MPPI were implemented using model predictive control (MPC), and the results indicate they are able to successfully transition the aircraft over different flight envelopes and generate trajectories unique to UAM vehicles.
Differential Dynamic Programming↗
Path integral quantization corresponding to the deformed Heisenberg algebra
In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity show very much different result.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗