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.↗
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Results for “Path integrals”
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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↗
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↗
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↗
Path integrals, Liapunov functions, and quadratic minimization.
Nonlinear DC circuits analyzed by digital computer for application to path integrals and stability problems
SYSTEM STABILITY↗
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↗
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↗
Airborne Demonstration of Atmospheric Oxygen Optical Depth Measurements with an Integrated Path Differential Absorption Lidar
We report on an airborne demonstration of atmospheric oxygen (O2) optical depth measurements with an Integrated Path Differential Absorption (IPDA) lidar using a fiber-based laser transmitter and photon counting detectors. Accurate atmospheric temperature and pressure measurements are needed for NASA's Active Sensing of CO2 Emissions over Nights, Days and Seasons (ASCENDS) mission. Since O2 is uniformly mixed in the atmosphere, its spectrum can be used to estimate the dry mixing ratio of CO2. In its airborne configuration, the IPDA lidar uses an Erbium Doped Fiber amplifier, a frequency doubler and single photon counting detectors to measure O2 absorption at multiple wavelengths near 765 nm. This instrument was deployed in 2013 and 2014 aboard NASA's DC-8 airborne laboratory as part of two campaigns to measure CO2 mixing ratios over a wide range of topography and weather conditions from altitudes ranging between 3 km and 13 km. In this paper we will review a summary of the results from our flights, discuss the errors that limit the precision and accuracy of the measurement and identify possible areas of improvement.
Methane↗
Path-Integrated Attenuation from Airborne X-Band Radar and Passive Radiometer Measurements: Implication for Rainfall Measurements
This study compares path-integrated attenuation (PIA), in precipitation over the ocean, derived from a single-frequency X-band radar, using the surface reference technique (SRT), with that deduced from a radiometer also operating at X band. The data were collected during TRMM field campaigns. The PIA derived from radar using the SRT does not involve any assumptions regarding the precipitation but it assumes that the cross-section of the surface is stable, that is, it is not significantly altered by factors such as surface roughness. The PIA deduced from the radiometer, however, involves assumptions regarding the temperature and emissivity of the surface and absorption and scattering by the intervening precipitation, which in turn depend upon the size, concentration and composition of the precipitation particles. The comparison of the PIA from the two instruments serves not only as a check between the radar and the radiometer but also may yield insights into the structure of the intervening precipitation. Such study can provide valuable information for TRMM in which both radar and radiometers are used for rain measurements. The radiometer PIA was first deduced from the brightness temperature using a simple one-layer radiative transfer model assuming no scattering, an isothermal atmosphere. The initial results show a general agreement between the PIAs deduced from the two instruments. Largo disagreement was found at high values of PIAs that may have been caused saturation of the X-band brightness temperature or by uncertainties in wind roughening of the sea surface that affects the SRT. Further results including the effects of scattering and a non-isothermal atmosphere will be shown at the conference.
Tian, Lin↗
Machine learning-accelerated path integral molecular dynamics simulations of reactive organic electrolytes
Hydrogen bonded electrolytes that exhibit accelerated proton transport via sequential reactive hops have drawn interest for their promise in clean energy applications. Molecular dynamics simulations of these electrolytes offer the opportunity to uncover microscopic mechanistic details that could be used to design and tune the properties of candidate electrolyte technologies. However, accurately modeling the proton transfer reactions and transport properties that give rise to high charge conductivites in these electrolytes proves computationally challenging because of the need to perform lengthy condensed phase simulations, treating both the electronic and nuclear degrees of freedom quantum mechanically. In this paper, we demonstrate that such a modeling task can be efficiently achieved with the use of density functional theory (DFT)-trained machine learning potentials (MLP) to accelerate path integral molecular dynamics (PIMD) simulations. We highlight the practical utility of this approach by using it to benchmark how closely PIMD simulations employing different DFT exchange–correlation functionals reproduce the composition-dependent densities, diffusion coefficients, and electrical conductivities of mixtures consisting of imidazole and levulinic acid. Even with the speedup afforded by our MLPs, PIMD simulations remain quite expensive. Furthermore, in order to render PIMD more computationally tractable, we introduce and benchmark the accuracy of a ring polymer contraction approach that leverages a computationally efficient short-range MLP to accelerate our PIMD simulations by an additional factor of four.
Chemical bonding↗
Error Reduction Methods for Integrated-path Differential-absorption Lidar Measurements
We report new modeling and error reduction methods for differential-absorption optical-depth (DAOD) measurements of atmospheric constituents using direct-detection integrated-path differential-absorption lidars. Errors from laser frequency noise are quantified in terms of the line center fluctuation and spectral line shape of the laser pulses, revealing relationships verified experimentally. A significant DAOD bias is removed by introducing a correction factor. Errors from surface height and reflectance variations can be reduced to tolerable levels by incorporating altimetry knowledge and "log after averaging", or by pointing the laser and receiver to a fixed surface spot during each wavelength cycle to shorten the time of "averaging before log".
averging before log↗
Development of an Airborne Triple-Pulse 2-Micron Integrated Path Differential Absorption Lidar (IPDA) for Simultaneous Airborne Column Measurements of Carbon Dioxide and Water Vapor in the Atmosphere
This presentation will provide status and details of an airborne 2-micron triple-pulse integrated path differential absorption (IPDA) lidar being developed at NASA Langley Research Center with support from NASA ESTO Instrument Incubator Program. The development of this active optical remote sensing IPDA instrument is targeted for measuring both atmospheric carbon dioxide and water vapor in the atmosphere from an airborne platform. This presentation will focus on the advancement of the 2-micron triple-pulse IPDA lidar development. Updates on the state-of-the-art triple-pulse laser transmitter will be presented including the status of seed laser locking, wavelength control, receiver and detector upgrades, laser packaging and lidar integration. Future plan for IPDA lidar system for ground integration, testing and flight validation will also be presented.
Singh, Upendra N.↗
Measurement of Rainfall Path Integrated Attenuation at Nadir: A Comparison of Radar and Radiometer Methods at 13.8GHz
An approach for reducing the ambiguity in the retrieved rainfall profile from spaceborne rain radar is to use the path integrated attenuation as a constraint.
radar surface reference technique microwave radiom↗
Imaginary time path integral Monte Carlo route to rate coefficients for nonadiabatic barrier crossing
Nonadiabatic transitions are central to many areas of chemical and condensed matter physics, ranging from biological electron transfer to the optical properties of one-dimensional conductors. Here, a path integral Monte Carlo method is used to simulate such transitions, based on the observation that nonadiabatic rate coefficients are often dominated by saddle point trajectories that correspond to an imaginary time. Simple analytic theories can be used to continue these imaginary time correlation functions to determine rate coefficients. The advantages and drawbacks of this approach are discussed.
Wolynes, Peter G.↗
Integrated Path Differential Absorption Lidar Optimizations Based on Pre-Analyzed Atmospheric Data for ASCENDS Mission Applications
In this paper a modeling method based on data reductions is investigated which includes pre analyzed MERRA atmospheric fields for quantitative estimates of uncertainties introduced in the integrated path differential absorption methods for the sensing of various molecules including CO2. This approach represents the extension of our existing lidar modeling framework previously developed and allows effective on- and offline wavelength optimizations and weighting function analysis to minimize the interference effects such as those due to temperature sensitivity and water vapor absorption. The new simulation methodology is different from the previous implementation in that it allows analysis of atmospheric effects over annual spans and the entire Earth coverage which was achieved due to the data reduction methods employed. The effectiveness of the proposed simulation approach is demonstrated with application to the mixing ratio retrievals for the future ASCENDS mission. Independent analysis of multiple accuracy limiting factors including the temperature, water vapor interferences, and selected system parameters is further used to identify favorable spectral regions as well as wavelength combinations facilitating the reduction in total errors in the retrieved XCO2 values.
Pliutau, Denis↗
2-Micron Triple-Pulse Integrated Path Differential Absorption Lidar Development for Simultaneous Airborne Column Measurements of Carbon Dioxide and Water Vapor in the Atmosphere
For more than 15 years, NASA Langley Research Center (LaRC) has contributed in developing several 2-micron carbon dioxide active remote sensors using the DIAL technique. Currently, an airborne 2-micron triple-pulse integrated path differential absorption (IPDA) lidar is under development at NASA LaRC. This paper focuses on the advancement of the 2-micron triple-pulse IPDA lidar development. Updates on the state-of-the-art triple-pulse laser transmitter will be presented including the status of wavelength control, packaging and lidar integration. In addition, receiver development updates will also be presented, including telescope integration, detection systems and data acquisition electronics. Future plan for IPDA lidar system for ground integration, testing and flight validation will be presented.
Singh, Upendra N.↗
Development of a Pulsed 2-Micron Integrated Path Differential Absorption Lidar for CO2 Measurement
Atmospheric carbon dioxide (CO2) is an important greenhouse gas that significantly contributes to the carbon cycle and global radiation budget on Earth. Active remote sensing of CO2 is important to address several limitations that contend with passive sensors. A 2-micron double-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 active remote sensing instrument will provide an alternate approach of measuring atmospheric CO2 concentrations with significant advantages. A high energy pulsed approach provides high-precision measurement capability by having high signal-to-noise ratio level and unambiguously eliminates the contamination from aerosols and clouds that can bias the IPDA measurement. Commercial, on the shelf, components are implemented for the detection system. Instrument integration will be presented in this paper as well as a background for CO2 measurement at NASA Langley research Center
Singh, Upendra N.↗
The gravitational path integral from an observer’s point of view
One of the fundamental problems in quantum gravity is to describe the experience of a gravitating observer in generic spacetimes. In this paper, we develop a framework for describing non-perturbative physics relative to an observer using the gravitational path integral. We apply our proposal to an observer that lives in a closed universe and one that falls behind a black hole horizon. We find that the Hilbert space that describes the experience of the observer is much larger than the Hilbert space in the absence of an observer. In the case of closed universes, the Hilbert space is not one-dimensional, as calculations in the absence of the observer suggest. Rather, its dimension scales exponentially with ${G}_{N}^{-1}$. Similarly, from an observer’s perspective, the dimension of the Hilbert space in a two-sided black hole is increased. We compute various observables probing the experience of a gravitating observer in this Hilbert space. We find that an observer experiences non-trivial physics in the closed universe in contrast to what it would see in a one-dimensional Hilbert space. In the two-sided black hole setting, our proposal implies that non-perturbative corrections to effective field theory for an infalling observer are suppressed until times exponential in the black hole entropy, resolving a recently-raised puzzle in black hole physics. While the framework that we develop is exemplified in the toy-model of JT gravity, most of our analysis can be extended to higher dimensions and, in particular, to generic spacetimes not admitting a conventional holographic description, such as cosmological universes or black hole interiors.
2D gravity↗