The scientific inferential relationships between statistical estimation, decision theory, and modern filter theory.
Scientific inferential relationships between statistical estimation, decision theory and modern filter theory
DECISION THEORY↗
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Finetuned Cancellations and Improbable Theories
It is argued that the x–y cancellation model (XYCM) is a good proxy for discussions of finetuned cancellations in physical theories. XYCM is then analyzed from a statistical perspective, where it is argued that a finetuned point in the parameter space is not abnormal, with any such point being just as probable as any other point. However, landing inside a standardly defined finetuned region (i.e., the full parameter space of finetuned points) has a much lower probability than landing outside the region, and that probability is invariant under assumed ranges of parameters. This proposition requires asserting also that the finetuned target region is a priori established. Furthermore, it is surmised that highly finetuned theories are generally expected to be highly improbable. An actionable implication of this moderate naturalness position is that the search for a non-finetuned explanation to supplant an apparently finetuned theory is likely to be a valid pursuit, but not guaranteed to be. A statistical characterization of this moderate position is presented, as well as those of the extreme pro-naturalness and anti-naturalness positions.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Theory of the energy fluctuation of multicomponent alloys
Not Available
Pei, Zongrui (ORCID:0000000307484629)↗
Instability in dynamic fracture and the failure of the classical theory of cracks
Cracks, the major vehicle for material failure, undergo a micro-branching instability at ~40% of their sonic limiting velocity in three dimensions. Recent experiments showed that in thin systems cracks accelerate to nearly their limiting velocity without micro-branching, until undergoing an oscillatory instability. Despite their fundamental importance, these dynamic instabilities are not explained by the classical theory of cracks, which is based on linear elasticity and an extraneous local symmetry criterion to predict crack paths. We develop a two-dimensional theory for predicting arbitrary paths of ultrahigh-speed cracks, which incorporates elastic nonlinearity without extraneous criteria. We show that cracks undergo an oscillatory instability controlled by small-scale, near crack-tip, elastic nonlinearity. This instability occurs above an ultrahigh critical velocity and features an intrinsic wavelength proportional to the ratio of the fracture energy to the elastic modulus, in quantitative agreement with experiments. As a result, this ratio emerges as a fundamental scaling length assumed to play no role in the classical theory of cracks, but shown here to strongly influence crack dynamics.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Self-consistent field theory of tethered polymers: One dimensional, three dimensional, strong stretching theories and the effects of excluded-volume-only interactions
We examine end-tethered polymers in good solvents, using one- and three-dimensional self-consistent field theory, and strong stretching theories. We also discuss different tethering scenarios, namely, mobile tethers, fixed but random ones, and fixed but ordered ones, and the effects and important limitations of including only binary interactions (excluded volume terms). We find that there is a “mushroom” regime in which the layer thickness is independent of the tethering density, σ, for systems with ordered tethers, but we argue that there is no such plateau for mobile or disordered anchors, nor is there one in the 1D theory. In the other limit of brushes, all approaches predict that the layer thickness scales linearly with N. However, the σ{sup 1/3} scaling is a result of keeping only excluded volume interactions: when the full potential is included, the dependence is faster and more complicated than σ{sup 1/3}. In fact, there does not appear to be any regime in which the layer thickness scales in the combination Nσ{sup 1/3}. We also compare the results for two different solvents with each other, and with earlier Θ solvent results.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
First-order derivative couplings between excited states from adiabatic TDDFT response theory
We present a complete derivation of derivative couplings between excited states in the framework of adiabatic time-dependent density functional response theory. Explicit working equations are given and the resulting derivative couplings are compared with derivative couplings from a pseudo-wavefunction ansatz. For degenerate excited states, i.e., close to a conical intersection (CI), the two approaches are identical apart from an antisymmetric overlap term. However, if the difference between two excitation energies equals another excitation energy, the couplings from response theory exhibit an unphysical divergence. This spurious behavior is a result of the adiabatic or static kernel approximation of time-dependent density functional theory leading to an incorrect analytical structure of the quadratic response function. Numerical examples for couplings close to a CI and for well-separated electronic states are given.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Representing the thermal state in time-dependent density functional theory
Classical molecular dynamics (MD) provides a powerful and widely used approach to determining thermodynamic properties by integrating the classical equations of motion of a system of atoms. Time-Dependent Density Functional Theory (TDDFT) provides a powerful and increasingly useful approach to integrating the quantum equations of motion for a system of electrons. TDDFT efficiently captures the unitary evolution of a many-electron state by mapping the system into a fictitious non-interacting system. In analogy to MD, one could imagine obtaining the thermodynamic properties of an electronic system from a TDDFT simulation in which the electrons are excited from their ground state by a time-dependent potential and then allowed to evolve freely in time while statistical data are captured from periodic snapshots of the system. For a variety of systems (e.g., many metals), the electrons reach an effective state of internal equilibrium due to electron-electron interactions on a time scale that is short compared to electron-phonon equilibration. During the initial time-evolution of such systems following electronic excitation, electron-phonon interactions should be negligible, and therefore, TDDFT should successfully capture the internal thermalization of the electrons. However, it is unclear how TDDFT represents the resulting thermal state. In particular, the thermal state is usually represented in quantum statistical mechanics as a mixed state, while the occupations of the TDDFT wavefunctions are fixed by the initial state in TDDFT. We work to address this puzzle by (A) reformulating quantum statistical mechanics so that thermodynamic expectations can be obtained as an unweighted average over a set of many-body pure states and (B) constructing a family of non-interacting (single determinant) TDDFT states that approximate the required many-body states for the canonical ensemble.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Perspective: Fundamental aspects of time-dependent density functional theory
In the thirty-two years since the birth of the foundational theorems, time-dependent density functional theory has had a tremendous impact on calculations of electronic spectra and dynamics in chemistry, biology, solid-state physics, and materials science. Alongside the wide-ranging applications, there has been much progress in understanding fundamental aspects of the functionals and the theory itself. This Perspective looks back to some of these developments, reports on some recent progress and current challenges for functionals, and speculates on future directions to improve the accuracy of approximations used in this relatively young theory.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Perspective: Kohn-Sham density functional theory descending a staircase
This article presents a perspective on Kohn-Sham density functional theory (KS-DFT) for electronic structure calculations in chemical physics. This theory is in widespread use for applications to both molecules and solids. We pay special attention to several aspects where there are both concerns and progress toward solutions. These include: The treatment of open-shell and inherently multiconfigurational systems (the latter are often called multireference systems and are variously classified as having strong correlation, near-degeneracy correlation, or high static correlation; KS-DFT must treat these systems with broken-symmetry determinants). The treatment of noncovalent interactions. The choice between developing new functionals by parametrization, by theoretical constraints, or by a combination. The ingredients of the exchange-correlation functionals used by KS-DFT, including spin densities, the magnitudes of their gradients, spin-specific kinetic energy densities, nonlocal exchange (Hartree-Fock exchange), nonlocal correlation, and subshell-dependent corrections (DFT+U). The quest for a universal functional, where we summarize some of the success of the latest Minnesota functionals, namely MN15-L and MN15, which were obtained by optimization against diverse databases. Time-dependent density functional theory, which is an extension of DFT to treat time-dependent problems and excited states. Furthermore, the review is a snapshot of a rapidly moving field, and—like Marcel Duchamp—we hope to convey progress in a stimulating way.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Projected Hartree-Fock theory as a polynomial of particle-hole excitations and its combination with variational coupled cluster theory
Projected Hartree-Fock theory provides an accurate description of many kinds of strong correlations but does not properly describe weakly correlated systems. Coupled cluster theory, in contrast, does the opposite. It therefore seems natural to combine the two so as to describe both strong and weak correlations with high accuracy in a relatively black-box manner. Combining the two approaches, however, is made more difficult by the fact that the two techniques are formulated very differently. In earlier work, we showed how to write spin-projected Hartree-Fock in a coupled-cluster-like language. Here in this work, we fill in the gaps in that earlier work. Further, we combine projected Hartree-Fock and coupled cluster theory in a variational formulation and show how the combination performs for the description of the Hubbard Hamiltonian and for several small molecular systems.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Rate Theory of Ion Pairing at the Water Liquid-Vapor Interface: A Case of Sodium Iodide
Studies of ion pairing at interfaces have been intensified recently because of their importance in many chemical reactive phenomena such as ion–ion interactions that are affected by interfaces and their influence on kinetic processes. In this study, we performed simulations to examine the thermodynamics and kinetics of small polarizable sodium iodide ions in the bulk and near the water liquid–vapor interface. Using classical transition state theory, we calculated the dissociation rates, and corrected them with transmission coefficients obtained from the reactive flux formalism and Grote-Hynes theory. Our results show that in addition to affecting the free energy of ions in solution, the interfacial environments significantly influence the kinetics of ion pairing. Results on the relaxation time obtained using the reactive flux formalism and Grote-Hynes theory present an unequivocal picture that the interface suppresses ion dissociation. The effects of the use of molecular models on the ion interactions as well as the ion-pair configurations at the interface also are quantified and discussed. Acknowledgments This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences. The calculations were carried out using computer resources provided by the Office of Basic Energy Sciences.
Dang, Liem X.↗
Stability of hairy black holes in shift-symmetric scalar-tensor theories via the effective field theory approach
Shift-symmetric Horndeski theories admit an interesting class of Schwarzschild-de Sitter black hole solutions exhibiting time-dependent scalar hair. The properties of these solutions may be studied via a bottom-up effective field theory (EFT) based on the background symmetries. This is in part possible by making use of a convenient coordinate choice — Lemaître-type coordinates — in which the profile of the Horndeski scalar field is linear in the relevant time coordinate. We construct this EFT, and use it to understand the stability of hairy black holes in shift-symmetric Horndeski theories, providing a set of constraints that the otherwise-free functions appearing in the Horndeski Lagrangian must satisfy in order to admit stable black hole solutions. The EFT is analyzed in the decoupling limit to understand potential sources of instability. Further, we also perform a complete analysis of the EFT with odd-parity linear perturbations around general spherically symmetric space-time.
79 ASTRONOMY AND ASTROPHYSICS↗
Occupation probabilities as variables in electronic structure theory: cooper pairing, OP-NSOFT-Cs,t, and the homogeneous electron liquid
The energy functional of a novel electronic structure theory, OP-NSOFT, has as variables the natural spin orbitals (NSO) of the trial function and their joint occupation probabilities in the search for the ground state energy. When occupancy is restricted to the spin-paired NSOs of DOCI, the resulting theory, OP-NSOFT-0, scales as M 3 , with M the size of the one-electron basis set. Accurate results were obtained for small molecules, particularly near dissociation where single reference theories like DFT are inaccurate. The homogeneous electron liquid (HEL) could serve as a test bed of OP-NSOFT for condensed systems, but OP-NSOFT-0 reduces to the Hartree–Fock approximation for the HEL. Cooper pairing is introduced instead, both singlet pairing, OP-NOFT-Cs, and fully polarized triplet pairing, OP-NSOFT-Ct. The former yields 1/3 of the diffusion-Monte-Carlo correlation energy, the latter 1/2 to 1/3 with decreasing electron density for r s values between 1 and 10. Both yield the discontinuity in the single-particle occupation number required by the Luttinger theorem. Furthermore, two-state joint occupation probabilities illustrate the importance of electron–electron small-angle scattering in establishing electron correlation in the unpolarized HEL.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Some elements of a theory of multidimensional complex variables. I - General theory. II - Expansions of analytic functions and application to fluid flows
The paper introduces a new theory of N-dimensional complex variables and analytic functions which, for N greater than 2, is both a direct generalization and a close analog of the theory of ordinary complex variables. The algebra in the present theory is a commutative ring, not a field. Functions of a three-dimensional variable were defined and the definition of the derivative then led to analytic functions.
Martin, E. Dale↗
Transition operators in acoustic-wave diffraction theory. I - General theory. II - Short-wavelength behavior, dominant singularities of Zk0 and Zk0 exp -1
A formal theory of the scattering of time-harmonic acoustic scalar waves from impenetrable, immobile obstacles is established. The time-independent formal scattering theory of nonrelativistic quantum mechanics, in particular the theory of the complete Green's function and the transition (T) operator, provides the model. The quantum-mechanical approach is modified to allow the treatment of acoustic-wave scattering with imposed boundary conditions of impedance type on the surface (delta-Omega) of an impenetrable obstacle. With k0 as the free-space wavenumber of the signal, a simplified expression is obtained for the k0-dependent T operator for a general case of homogeneous impedance boundary conditions for the acoustic wave on delta-Omega. All the nonelementary operators entering the expression for the T operator are formally simple rational algebraic functions of a certain invertible linear radiation impedance operator which maps any sufficiently well-behaved complex-valued function on delta-Omega into another such function on delta-Omega. In the subsequent study, the short-wavelength and the long-wavelength behavior of the radiation impedance operator and its inverse (the 'radiation admittance' operator) as two-point kernels on a smooth delta-Omega are studied for pairs of points that are close together.
Hahne, G. E.↗
Transition operators in electromagnetic-wave diffraction theory - General theory
A formal theory is developed for the scattering of time-harmonic electromagnetic waves from impenetrable immobile obstacles with given linear, homogeneous, and generally nonlocal boundary conditions of Leontovich (impedance) type for the wave of the obstacle's surface. The theory is modeled on the complete Green's function and the transition (T) operator in time-independent formal scattering theory of nonrelativistic quantum mechanics. An expression for the differential scattering cross section for plane electromagnetic waves is derived in terms of certain matrix elements of the T operator for the obstacle.
Hahne, G. E.↗