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In this report the problem we are going to study is the interpolation of a set of points in the plane with the use of control theory. We will discover how different systems generate different kinds of splines, cubic and exponential, and investigate the effect that the different systems have on the tracking problems. Actually we will see that the important parameters will be the two eigenvalues of the control matrix.
T-matrix computations of light scattering by polydispersions of randomly oriented nonspherical aerosols and Mie computations for equivalent spheres are compared. Findings show that even moderate nonsphericity results in suubstantial errors in the retrieved aerosol optical thickness if satellite reflectance measurements are analyzed using Mie theory. On the other hand, the use of Mie theory for nonspherical aerosols produces negligible errors in the computation of albedo and flux related quantities, provided that the aerosol size distribution and optical thickness are known beforehand. The first result can be explained by large nonspherical-spherical differences in scattering phase function, while the second result follows from small nonspherical-spherical differences in single-scattering albedo and asymmetry parameter. No cancellation of errors occurs if one consistently uses Mie theory in the retrieval algorithm and then in computing the albedo for the retrieved aerosol optical thickness.
Completed in December 2018, the Alaska North Slope Hydrate 01 stratigraphic test well provides a wealth of logging-while-drilling (LWD) data for strata to below the base of gas hydrate stability (BGHS). This well is intended to be the first of three wells drilled for a comprehensive long-term gas hydrate production test conducted by the National Energy Technology Laboratory, the Japan Oil, Gas and Metals National Corporation, and the U.S. Geological Survey (USGS). The Hydrate 01 stratigraphic test well confirmed the presence of gas hydrate in two sand reservoirs within the hydrate stability zone, indicating the suitability of this location for a long-term gas hydrate production test.<p>The USGS, using an effective-medium-theory rock-physics approach, has estimated gas hydrate saturations from compressional (P) and shear (S) wave log data acquired in the Hydrate 01 well. We assume that gas hydrate occurs as pore-filling load-bearing material (i.e., part of the grain matrix). For Unit D, approximately 500 feet above the BGHS, both P-wave and S-wave acoustic logs indicate moderate gas hydrate saturations with S-wave results slightly lower than those for P-waves. For the Unit B, located just above the BGHS, we obtain moderate to high gas hydrate saturation estimates from both sonic logs. Our P-wave saturation estimates agree well with results from electrical-resistivity-based estimates, whereas estimates from nuclear magnetic resonance LWD data generally suggest 5 to 10 percent higher saturations; our S-wave results suggest lower saturations. These differences likely indicate complexities in the form of gas hydrate occurrence within the sediment pore space, potentially including differences between hydrate occurrence in Units D and B.</p>
A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh–Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams—renormalization diagrams—whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. As a result, the normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree–Fock and first-order many-body perturbation theories.
We propose a framework to learn the time-dependent Hartree–Fock (TDHF) inter-electronic potential of a molecule from its electron density dynamics. Although the entire TDHF Hamiltonian, including the inter-electronic potential, can be computed from first principles, we use this problem as a testbed to develop strategies that can be applied to learn a priori unknown terms that arise in other methods/approaches to quantum dynamics, e.g., emerging problems such as learning exchange–correlation potentials for time-dependent density functional theory. We develop, train, and test three models of the TDHF inter-electronic potential, each parameterized by a four-index tensor of size up to 60 × 60 × 60 × 60. Two of the models preserve Hermitian symmetry, while one model preserves an eight-fold permutation symmetry that implies Hermitian symmetry. Across seven different molecular systems, we find that accounting for the deeper eight-fold symmetry leads to the best-performing model across three metrics: training efficiency, test set predictive power, and direct comparison of true and learned inter-electronic potentials. All three models, when trained on ensembles of field-free trajectories, generate accurate electron dynamics predictions even in a field-on regime that lies outside the training set. To enable our models to scale to large molecular systems, we derive expressions for Jacobian-vector products that enable iterative, matrix-free training.
Recent advances in selected configuration interaction methods have made them competitive with the most accurate techniques available and, hence, creating an increasingly powerful tool for solving quantum Hamiltonians. In this work, we build on recent advances from the adaptive sampling configuration interaction (ASCI) algorithm. We show that a useful paradigm for generating efficient selected CI/exact diagonalization algorithms is driven by fast sorting algorithms, much in the same way iterative diagonalization is based on the paradigm of matrix vector multiplication. We present several new algorithms for all parts of performing a selected CI, which includes new ASCI search, dynamic bit masking, fast orbital rotations, fast diagonal matrix elements, and residue arrays. The ASCI search algorithm can be used in several different modes, which includes an integral driven search and a coefficient driven search. The algorithms presented here are fast and scalable, and we find that because they are built on fast sorting algorithms they are more efficient than all other approaches we considered. After introducing these techniques, we present ASCI results applied to a large range of systems and basis sets to demonstrate the types of simulations that can be practically treated at the full-CI level with modern methods and hardware, presenting double- and triple-ζ benchmark data for the G1 data set. The largest of these calculations is Si$_2$H$_6$ which is a simulation of 34 electrons in 152 orbitals. We also present some preliminary results for fast deterministic perturbation theory simulations that use hash functions to maintain high efficiency for treating large basis sets.
Quantum hydrodynamics is the emergent classical dynamics governing transport of conserved quantities in generic strongly interacting quantum systems. Here, recent matrix product operator methods [1,2] have made simulations of quantum hydrodynamics in 1+1D tractable, but they do not naturally generalize to 2+1D or higher, and they offer limited guidance as to the difficulty of simulations on quantum computers. Near-Clifford simulation algorithms are not limited to one dimension, and future error-corrected quantum computers will likely be bottlenecked by non-Clifford operations. We therefore investigate the non-Clifford resource requirements for simulation of quantum hydrodynamics using mana, a resource theory of non-Clifford operations. For infinite-temperature starting states, we find that the mana of subsystems quickly approaches zero, while for starting states with energy above some threshold the mana approaches a nonzero value. Surprisingly, in each case the finite-time mana is governed by the subsystem entropy, not the thermal state mana; we argue that this is because mana is a sensitive diagnostic of finite-time deviations from canonical typicality.
A search for climatic change predicted by climate models can easily yield unconvincing results because of 'climatic noise,' the inherent, unpredictable variability of time-average atmospheric data. A weighted average of data that maximizes the probability of detecting predicted climatic change is presented. To obtain the optimal weights, an estimate of the covariance matrix of the data from a prior data set is needed. This introduces additional sampling error into the method. This is presently taken into account. A form of the weighted average is found whose probability distribution is independent of the true (but unknown) covariance statistics of the data and of the climate model prediction.
Despite the promise of TiFe-based alloys as low-cost solid-state hydrogen storage materials with mild operating conditions and reasonable hydrogen capacity, their initial hydrogenation process is difficult, hindering broad utilization. The effect of alloying element on the initial hydrogenation kinetics of TiFe alloys, TiFe 0.9 M 0.1 (M = V, Cr, Fe, Co and Ni), was evaluated by analyzing changes to the passivating surface oxide layer that inhibits hydrogen permeation, as well as the ease of initial-stage hydrogen absorption into the underlying matrix. X-ray photoelectron spectroscopy and atom probe tomography revealed key variations in surface oxide compositions and thinning of the passivating oxide layer compared to pure TiFe, which suggests suppressed oxide growth by alloying-induced elemental redistribution. At the same time, density functional theory calculations predicted exothermic formation of hydride nuclei when alloying with V or Cr, as well as a reduced nucleation barrier when alloying with Co or Ni. Overall, these results are consistent with the observed experimental trend of the activation kinetics. We propose that improvements in activation kinetics of TiFe with alloying arises from the combined effect of reduced passivating oxide thickness and easier hydride nucleation, offering a starting point for alloy design strategies towards further improvement.
The dependence of the 𝐸2 matrix elements on isospin projection 𝑇 𝑧 is linked to the conservation of the isospin symmetry. To study this conjecture, we calculated the 𝐵(𝐸2 : 2 + →0 + ) rates for the even-even 𝑇=1 mirror nuclei with 42 ≤ 𝐴 ≤ 98 within nuclear density functional theory, employing the generalized Bohr Hamiltonian, and carrying out angular momentum projection. We demonstrated that collective effects are crucial for describing experimental data near the 𝑁=𝑍 line without invoking explicit beyond-Coulomb isospin symmetry-breaking corrections. We also determined the 𝐵(𝐸2↓) values for odd-odd 𝑇 𝑧 =0 nuclei 70 Br and 78 Y in doubly blocked configurations. We discussed the requirements for accurately describing isobaric analog states and emphasized how current theoretical results should be interpreted within the study of isospin symmetry across isospin triplets.
In a recent paper, “Nonlocal nucleon matrix elements in the rest frame” [1], it was observed that the next-to-leading order calculations of the renormalization factor can describe, to a few percent accuracy, the logarithm of the lattice quantum chromodynamics (QCD) rest frame matrix elements with separations up to distances of 0.6 fm on multiple lattice spacings. We argue that perturbative QCD breaks down at such a distance scale after resumming the associated large logarithms, while the Ansatz used in the analysis there is not justified in perturbation theory. Besides, we explain the observation in Ref. [1] and demonstrate that the Ansatz fails to describe the data for 𝑧 > 0.3 fm, showing an opposite trend. Finally, although Ref. [1] proposes multiplying the Ansatz by a Gaussian correction model, which is shown to reduce the discrepancy with the data, this does not legitimize the use of perturbative QCD at such distance scales.
We present a description of saturation in small 𝑥 deep inelastic scattering from power counting in a top-down effective theory derived from quantum chromodynamics (QCD). A factorization formula isolates the universal physics of the nucleus at leading power in 𝑥. The onset of saturation is then understood as a breakdown in the expansion in an emergent power counting parameter, which is defined by the matrix element of a gauge invariant operator. We identify a new radiation mode, which enables us to extend previous literature by distinguishing the appearance of the saturation scale from the transition to nonlinear evolution.
Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.
This study demonstrates how the otherwise detrimental surface wetting and coarsening in a metallic system can be put to advantage through the inclusion of aligned fibers which accrue the excess liquid II phase while passing through the range of the miscibility gap and, some lower temperature, uniformly incorporate it into the solid I plus liquid II matrix. This is initially shown through the utilization of a hypermonotectic succinonitrile-glycerol mixture, a transparent organic system which exhibits solidification behavior in a manner analogous to metals. Results are compared with the analogue study and both are reviewed in terms of relevant wetting theory and surface area considerations.
The paper covers two distinct parts: theory and application. The goal of this work was the reduction of model size with an increase in eigenvalue/vector accuracy. This method is ideal for the condensation of large truss- or beam-type structures. The theoretical approach involves the conversion of a continuum transfer matrix beam element into an 'Exact' dynamic stiffness element. This formulation is implemented in a finite element environment. This results in the need to solve a transcendental eigenvalue problem. Once the eigenvalue is determined the eigenvectors can be reconstructed with any desired spatial precision. No discretization limitations are imposed on the reconstruction. The results of such a combined finite element and transfer matrix formulation is a much smaller FEM eigenvalue problem. This formulation has the ability to extract higher eigenvalues as easily and as accurately as lower eigenvalues. Moreover, one can extract many more eigenvalues/vectors from the model than the number of degrees of freedom in the FEM formulation. Typically, the number of eigenvalues accurately extractable via the 'Exact' element method are at least 8 times the number of degrees of freedom. In contrast, the FEM usually extracts one accurate (within 5 percent) eigenvalue for each 3-4 degrees of freedom. The 'Exact' element results in a 20-30 improvement in the number of accurately extractable eigenvalues and eigenvectors.
A thermodynamically-based work potential theory for modeling progressive damage for laminated, unidirectional composites assuming plane stress (2D Schaperystheory) is extended to three dimensioanl (3D). An internal state variable, S, is defined to account for the dissipated energy due to damage evolution in the form of microstructure changes in the matrix. With the stationary of the total work potential with respect to the internal state variable, a thermodynamically-consistent set of evolution equationsis derived. The internal state variable is related to the transverse and shear modulithrough microdamage functions. In the first part of this work, coupon specimensare prepared to conduct experiments to characterize the relations between the internalstate variable and the transverse modulus as well as shear modulus. The informationis subsequent used for the prediction of three point bending test. In the second partof this work, objectivity is studied. Three separate methods utilizing different definitions of a reduced internal state variable or of the order of the polynomials are used to represent the matrix micro-damage functions are employed. The three methods are implemented in a user defined subroutine within a commercial finite element method software package. Results from numerical simulations of a center-notched composites panel are compared. The agreement in the maximum stress predictions among the three methods indicates that objectivity, with respect to the functional form of themicrodamage functions, is satisfied.
Supercritical CO 2 (scCO 2 ) plays a crucial role as a solvent in separation processes, advanced power cycles, and materials processing. Nonetheless, the atomistic comprehension of how the dense scCO 2 matrix influences the fundamental reaction of carbon monoxide (CO) is still insufficiently explored. Experimental studies and molecular dynamics (MD) simulations frequently fail to detect the highly reactive, transient intermediates, such as atomic oxygen (O), that drive these reactions. Here, to address this issue, we have developed a novel ReaxFF reactive force field for the CO 2 /CO/O system. The force field parameters were calibrated using density functional theory and second-order Møller-Plesset calculations to model CO 2 crystal properties, intermolecular interactions, bond dissociation curves, and reaction energy barriers. The force field reproduces the cohesive energy of the CO 2 crystal, the pressure characteristics of bulk scCO 2 , the equation-of-state behavior over a wide pressure–density range, the pressure dependence of the C–O bond length under compression, and the structural properties of liquid and scCO 2 , as documented by experiments, ab-initio MD, and prominent non-reactive models. The force field was subsequently applied to study the CO + O → CO 2 reaction. In a dilute environment, the reaction is inefficient as the newly formed CO 2 rapidly dissociates due to excess kinetic and potential energy acquired from the exothermic reaction. Conversely, in a dense scCO 2 environment, the surrounding matrix acts as an efficient third body, stabilizing the emerging CO2 product via molecular collisions. Statistical analysis confirms an average excess energy dissipation of 133.9 ± 3.6 kcal/mol over 112.4 ± 17.9 ps. Kinetic energy decomposition reveals that ∼ 92% of the excess kinetic energy is stored in internal (rotational and vibrational) degrees of freedom. This ReaxFF force field establishes a mechanistic foundation for third-body stabilization in dense reactive environments.
Hadronic vacuum polarization (HVP) and light-by-light scattering (HLBL) are crucial for evaluating the Standard Model predictions concerning the muon’s anomalous magnetic moment. However, direct first-principle lattice gauge theory-based calculations of these observables in the timelike region remain challenging. Discrepancies persist between lattice quantum chromodynamics (QCD) calculations in the spacelike region and dispersive approaches relying on experimental data parametrization from the timelike region. Here, we introduce a methodology employing 1+1-dimensional quantum electrodynamics (QED), i.e. the Schwinger Model, to investigate the HVP and HLBL. To that end, we use both tensor network techniques, specifically matrix product states, and classical emulators of digital quantum computers. Demonstrating feasibility in a simplified model, our approach sets the stage for future endeavors leveraging digital quantum computers.