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NASA NTRS1985

A review of path-independent integrals in elastic-plastic fracture mechanics, task 4

The path independent (P-I) integrals in elastic plastic fracture mechanics which have been proposed in recent years to overcome the limitations imposed on the J integral are reviewed. The P-I integrals considered herein are the J integral by Rice, the thermoelastic P-I integrals by Wilson and Yu and by Gurtin, the J* integral by Blackburn, the J sub theta integral by Ainsworth et al., the J integral by Kishimoto et al., and the delta T sub p and delta T* sub p integrals by Atluri et al. The theoretical foundation of these P-I integrals is examined with emphasis on whether or not path independence is maintained in the presence of nonproportional loading and unloading in the plastic regime, thermal gradients, and material inhomogeneities. The similarities, differences, salient features, and limitations of these P-I integrals are discussed. Comments are also made with regard to the physical meaning, the possibility of experimental measurement, and computational aspects.

Kim, K. S.↗

DOE OSTI2016

Stationary bubbles and their tunneling channels toward trivial geometry

In the path integral approach, one has to sum over all histories that start from the same initial condition in order to obtain the final condition as a superposition of histories. Applying this into black hole dynamics, we consider stable and unstable stationary bubbles as a reasonable and regular initial condition. We find examples where the bubble can either form a black hole or tunnel toward a trivial geometry, i.e., with no singularity nor event horizon. We investigate the dynamics and tunneling channels of true vacuum bubbles for various tensions. In particular, in line with the idea of superposition of geometries, we build a classically stable stationary thin-shell solution in a Minkowski background where its fate is probabilistically given by non-perturbative effects. Since there exists a tunneling channel toward a trivial geometry in the entire path integral, the entire information is encoded in the wave function. This demonstrates that the unitarity is preserved and there is no loss of information when viewed from the entire wave function of the universe, whereas a semi-classical observer, who can see only a definitive geometry, would find an effective loss of information. Ultimately, this may provide a resolution to the information loss dilemma.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1982

Creep crack-growth: A new path-independent integral (T sub c), and computational studies

The development of valid creep fracture criteria is considered. Two path-independent integral parameters which show some degree of promise are the C* and (Delta T)sub c integrals. The mathematical aspects of these parameters are reviewed by deriving generalized vector forms of the parameters using conservation laws which are valid for arbitrary, three dimensional, cracked bodies with crack surface tractions (or applied displacements), body forces, inertial effects, and large deformations. Two principal conclusions are that (Delta T)sub c has an energy rate interpretation whereas C* does not. The development and application of fracture criteria often involves the solution of boundary/initial value problems associated with deformation and stresses. The finite element method is used for this purpose. An efficient, small displacement, infinitesimal strain, displacement based finite element model is specialized to two dimensional plane stress and plane strain and to power law creep constitutive relations. A mesh shifting/remeshing procedure is used for simulating crack growth. The model is implemented with the quartz-point node technique and also with specially developed, conforming, crack-tip singularity elements which provide for the r to the n-(1+n) power strain singularity associated with the HRR crack-tip field. Comparisons are made with a variety of analytical solutions and alternate numerical solutions for a number of problems.

Stonesifer, R. B.↗

DOE OSTI2019

Applications of the worldline Monte Carlo formalism in quantum mechanics

Highlights: Universal approach to the numerical computation of the Euclidean path integral. Inspired by recent work in relativistic quantum field theory. Here adapted to non-relativistic quantum mechanics. Worked out for the computation of propagators and ground-state energies. Special smoothing procedure for singular potentials. -- Abstract: In recent years efficient algorithms have been developed for the numerical computation of relativistic single-particle path integrals in quantum field theory. Here, we adapt this “worldline Monte Carlo” approach to the standard problem of the numerical approximation of the non-relativistic path integral, resulting in a formalism whose characteristic feature is the fast, non-recursive generation of an ensemble of trajectories that is independent of the potential, and is thus universally applicable. The numerical implementation discretises the trajectories with respect to their time parametrisation but maintains a continuous spatial domain. In the case of singular potentials, the discretised action gets adapted to the singularity through a “smoothing” procedure. We show for a variety of examples (the harmonic oscillator in various dimensions, the modified Pöschl–Teller potential, delta-function potentials, the Coulomb and Yukawa potentials) that the method allows one to obtain fast and reliable estimates for the Euclidean propagator and use them in a certain time window suitable for extracting the ground state energy. Usually Monte Carlo calculations in quantum mechanics are performed using algorithms that involve a more specific adaptation to the potential, and we briefly compare our method with the most widely used such algorithms, which are of Metropolis type. As an aside, we apply it for studying the classical limit where nearly classical trajectories are expected to dominate in the path integral. We expect the advances made here to be useful also in the relativistic case.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Computing real time correlation functions on a hybrid classical/quantum computer

Quantum devices may overcome limitations of classical computers in studies of nuclear structure functions and parton Wigner distributions of protons and nuclei. In this talk, we discuss a worldline approach to compute nuclear structure functions in the high energy Regge limit of QCD using a hybrid quantum computer, by expressing the fermion determinant in the QCD path integral as a quantum mechanical path integral over 0 + 1-dimensional fermionic and bosonic world-lines in background gauge fields. Our simplest example of computing the well-known dipole model result for the structure function F 2 in the high energy Regge limit is feasible with NISQ era technology using few qubits and shallow circuits. We note this example can be scaled up in complexity and extended in scope to compute structure functions, scattering amplitudes and other real-time correlation functions in QCD, relevant for example to describe non-equilibrium transport of quarks and gluons in a Quark-Gluon-Plasma.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Real-time spin systems from lattice field theory

We construct a lattice field theory method for computing the real-time dynamics of spin systems in a thermal bath. This is done by building on previous work of Takano with Schwinger-Keldysh and functional differentiation techniques. We derive a Schwinger-Keldysh path integral for generic spin Hamiltonians, then demonstrate the method on a simple system. Our path integral has a sign problem, which generally requires exponential run time in the system size, but requires only linear storage. The latter may place this method at an advantage over exact diagonalization, which is exponential in both. Our path integral is amenable to contour deformations, a technique for reducing sign problems.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

The wavefunction of a quantum S 1 × S 2 universe

We study quantum gravity corrections to the no-boundary wavefunction describing a universe with spatial topology S 1 × S 2 . It has been suggested that quantum effects become increasingly important when the size of the circle is large relative to the sphere. In this paper, we confirm this claim by an explicit four-dimensional one-loop calculation of the gravitational path integral preparing such a state. In the process, we clarify some aspects of the gravitational path integral on complex spacetimes. These quantum corrections play a crucial role in ensuring that the norm of the wavefunction is naturally expressed in terms of a path integral over S 2 × S 2 at the classical level. We extend some of the analysis to more general spatial topologies, as well as to the inclusion of fermions.

cosmological models↗

DOE OSTI2016

Worldline approach for numerical computation of electromagnetic Casimir energies: Scalar field coupled to magnetodielectric media

Here, we present a worldline method for the calculation of Casimir energies for scalar fields coupled to magnetodielectric media. The scalar model we consider may be applied in arbitrary geometries, and it corresponds exactly to one polarization of the electromagnetic field in planar layered media. Starting from the field theory for electromagnetism, we work with the two decoupled polarizations in planar media and develop worldline path integrals, which represent the two polarizations separately, for computing both Casimir and Casimir-Polder potentials. We then show analytically that the path integrals for the transverse-electric polarization coupled to a dielectric medium converge to the proper solutions in certain special cases, including the Casimir-Polder potential of an atom near a planar interface, and the Casimir energy due to two planar interfaces. We also evaluate the path integrals numerically via Monte Carlo path-averaging for these cases, studying the convergence and performance of the resulting computational techniques. Lastly, while these scalar methods are only exact in particular geometries, they may serve as an approximation for Casimir energies for the vector electromagnetic field in other geometries.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1988

A review of path-independent integrals in elastic-plastic fracture mechanics

The objective of this paper is to review the path-independent (P-I) integrals in elastic plastic fracture mechanics which have been proposed in recent years to overcome the limitations imposed on the J-integral. The P-I integrals considered are the J-integral by Rice (1968), the thermoelastic P-I integrals by Wilson and Yu (1979) and Gurtin (1979), the J-integral by Blackburn (1972), the J(theta)-integral by Ainsworth et al. (1978), the J-integral by Kishimoto et al. (1980), and the Delta-T(p) and Delta T(p)-asterisk integrals by Alturi et al. (1982). The theoretical foundation of the P-I integrals is examined with an emphasis on whether or not the path independence is maintained in the presence of nonproportional loading and unloading in the plastic regime, thermal gradient, and material inhomogeneities. The simularities, difference, salient features, and limitations of the P-I integrals are discussed. Comments are also made with regard to the physical meaning, the possibility of experimental measurement, and computational aspects.

Kim, Kwang S.↗

NASA NTRS1990

Application of path-independent integrals to elevated temperature crack growth

The applicability of the J-integral in elasto-plastic fracture mechanics is limited to isothermal, monotonic loading conditions from the theoretical viewpoint, while in many applications, for instance gas turbine engines, crack growth occurs in the presence of cyclic inelastic loading, thermomechanical loading and temperature gradients. A number of path-independent (P-I) integrals have been proposed which do not have the restrictions of the J-integral. A review indicates that four of these integrals, although they are not the classical conservation integrals, are path-independent under these complex loading conditions. This paper describes a combined analytical and experimental effort to evaluate the ability of these four P-I integrals to correlate the crack growth data of Alloy 718 at elevated temperatures. Results for uniform temperature, 538 C, cases indicate that all these integrals are capable of correlating the crack growth data over a wide range of cyclic plasticity.

Kim, K. S.↗

DOE OSTI2020

On exact-WKB analysis, resurgent structure, and quantization conditions

There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

On observers in holographic maps

A straightforward gravitational path integral calculation implies that closed universes are trivial, described by a one dimensional Hilbert space. Two recent papers by Harlow-Usatyuk-Zhao and Abdalla-Antonini-Iliesiu-Levine have sought to ameliorate this issue by defining special rules to incorporate observers into the path integral. However, the proposed rules are different, leading to differing results for the Hilbert space dimension. Moreover, the former work offers a holographic map realized using a non-isometric code construction to complement their path integral result and clarify its physics. In this work, we propose a non-isometric code that implements the second construction, allowing thorough comparison. Our prescription may be thought of as simply removing the portion of the map that acts on the observer, while preserving the rest, creating an effective holographic boundary at the observer-environment interface. This proposal can be directly applied to general holographic maps for both open and closed universes of any dimension.

AdS-CFT correspondence↗

DOE OSTI2019

Lifting of states in 2-dimensional N = 4 supersymmetric CFTs

We consider states of the D1-D5 CFT where only the left-moving sector is excited. As we deform away from the orbifold point, some of these states will remain BPS while others can `lift'. The lifting can be computed by a path integral containing two twist deformations; however, the relevant 4-point amplitude cannot be computed explicitly in many cases. We analyze an older proposal by Gava and Narain where the lift can be computed in terms of a finite number of 3-point functions. A direct Hamiltonian decomposition of the path integral involves an infinite number of 3-point functions, as well the first order correction to the starting state. We note that these corrections to the state account for the infinite number of 3-point functions arising from higher energy states, and one can indeed express the path-integral result in terms of a finite number of 3-point functions involving only the leading order states that are degenerate. The first order correction to the supercharge $\bar G^{(1)}$ gets replaced by a projection $\bar G^{(P)}$; this projected operator can also be used to group the states into multiplets whose members have the same lifting.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1993

Quantum wormholes and harmonic oscillators

The quantum state of a wormhole can be represented by a path integral over all asymptotically Euclidean four-geometries and all matter fields which have prescribed values, the arguments of the wave function, on a three-surface which divides the space time manifold into two disconnected parts. Minisuperspace models which consist of a homogeneous massless scalar field coupled to a Friedmann-Robertson-Walker space time are considered. Once the path integral over the lapse function is performed, the requirement that the space time be asymptotically Euclidean can be accomplished by fixing the asymptotic gravitational momentum in the remaining path integral. It is argued that there does not exist any wave function which corresponds to asymptotic field configurations such that the effective gravitational constant is negative in the asymptotic region. Then, the wormhole wave functions can be written as linear combinations of harmonic oscillator wave functions.

Garay, Luis J.↗

DOE OSTI2017

Equilibrium clumped-isotope effects in doubly substituted isotopologues of ethane

We combine path-integral Monte Carlo methods with a new intramolecular potential energy surface to quantify the equilibrium enrichment of doubly substituted ethane isotopologues due to clumped-isotope effects. Ethane represents the simplest molecule to simultaneously exhibit 13C–13C, 13C–D, and D–D clumped-isotope effects, and the analysis of corresponding signatures may provide useful geochemical and biogeochemical proxies of formation temperatures or reaction pathways. Utilizing path-integral statistical mechanics, we predict equilibrium fractionation factors that fully incorporate nuclear quantum effects, such as anharmonicity and rotational-vibrational coupling which are typically neglected by the widely used Urey model. The magnitude of the calculated fractionation factors for the doubly substituted ethane isotopologues indicates that isotopic clumping can be observed if rare-isotope substitutions are separated by up to three chemical bonds, but the diminishing strength of these effects suggests that enrichment at further separations will be negligible. The Urey model systematically underestimates enrichment due to 13C–D and D–D clumped-isotope effects in ethane, leading to small relative errors in the apparent equilibrium temperature, ranging from 5 K at 273.15 K to 30 K at 873.15 K. We additionally note that the rotameric dependence of isotopologue enrichment must be carefully considered when using the Urey model, whereas the path-integral calculations automatically account for such effects due to configurational sampling. These findings are of direct relevance to future clumped-isotope studies of ethane, as well as studies of 13C–13C, 13C–D, and D–D clumped-isotope effects in other hydrocarbons.

Geochemistry & Geophysics↗

DOE OSTI2015

Quantum-classical correspondence in steady states of nonadiabatic systems

We first present nonadiabatic path integral which is exact formulation of quantum dynamics in nonadiabatic systems. Then, by applying the stationary phase approximations to the nonadiabatic path integral, a semiclassical quantization condition, i.e., quantum-classical correspondence, for steady states of nonadiabatic systems is presented as a nonadiabatic trace formula. The present quantum-classical correspondence indicates that a set of primitive hopping periodic orbits, which are invariant under time evolution in the phase space of the slow degree of freedom, should be quantized. The semiclassical quantization is then applied to a simple nonadiabatic model and accurately reproduces exact quantum energy levels.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

Global Estimates of Errors in Quantum Computation by the Feynman–Vernon Formalism

The operation of a quantum computer is considered as a general quantum operation on a mixed state on many qubits followed by a measurement. The general quantum operation is further represented as a Feynman–Vernon double path integral over the histories of the qubits and of an environment, and afterward tracing out the environment. The qubit histories are taken to be paths on the two-sphere S{sup 2} as in Klauder’s coherent-state path integral of spin, and the environment is assumed to consist of harmonic oscillators initially in thermal equilibrium, and linearly coupled to to qubit operators Ŝ{sub z}. The environment can then be integrated out to give a Feynman–Vernon influence action coupling the forward and backward histories of the qubits. This representation allows to derive in a simple way estimates that the total error of operation of a quantum computer without error correction scales linearly with the number of qubits and the time of operation. It also allows to discuss Kitaev’s toric code interacting with an environment in the same manner.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Sign Learning Kink-based (SiLK) Quantum Monte Carlo for molecular systems

The Sign Learning Kink (SiLK) based Quantum Monte Carlo (QMC) method is used to calculate the ab initio ground state energies for multiple geometries of the H{sub 2}O, N{sub 2}, and F{sub 2} molecules. The method is based on Feynman’s path integral formulation of quantum mechanics and has two stages. The first stage is called the learning stage and reduces the well-known QMC minus sign problem by optimizing the linear combinations of Slater determinants which are used in the second stage, a conventional QMC simulation. The method is tested using different vector spaces and compared to the results of other quantum chemical methods and to exact diagonalization. Our findings demonstrate that the SiLK method is accurate and reduces or eliminates the minus sign problem.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗