The Solar-Interior Equation of State with the Path-Integral Formalism I. Domain of Validity

This is the first paper in a series that deals with solar-physics applications of the equation-of-state formalism based on the formulation of the so-called "Feynman-Kac (FK) representation". Here, the FK equation of state is presented and adapted for solar applications. Its domain of validity is assessed. The practical application to the Sun will be dealt with in Paper II. Paper III will extend the current FK formalism to a higher order. Use of the FK equation of state is limited to physical conditions for which more than 90% of helium is ionized. This incudes the inner region of the Sun out to about .98 of the solar radius. Despite this limitation, in the parts of the Sun where it is applicable, the FK equation of state has the power to be more accurate than the equations of state currently used in solar modeling. The FK approach is especially suited to study physical effects such as Coulomb screening, bound states, the onset of recombination of fully ionized species, as well as diffraction and exchange effects. The localizing power of helioseismology allows a test of the FK equation of state. Such a test will be beneficial both for better solar models and for tighter solar constraints of the equation of state.Comment: Completely rewritten revised version. Accepted for publication in Astronomy & Astrophysic

Analysis and Hermite spectral approximation of diffusive-viscous wave equations in unbounded domains arising in geophysics

The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.Comment: 32 pages, 27 figure

Diamagnetic response and phase stiffness for interacting isolated narrow bands

A platform that serves as an ideal playground for realizing ``high'' temperature superconductors are materials where the electrons' kinetic energy is completely quenched, and interactions provide the only energy scale in the problem for $T_c$. However, when the non-interacting bandwidth for a set of isolated bands is small compared to the scale of the interactions, the problem is inherently non-perturbative and requires going beyond the traditional mean-field theory of superconductivity. In two spatial dimensions, $T_c$ is controlled by the superconducting phase stiffness. Here we present a general theoretical framework for computing the electromagnetic response for generic model Hamiltonians, which controls the maximum possible superconducting phase stiffness and thereby $T_c$, without resorting to any mean-field approximation. Importantly, our explicit computations demonstrate that the contribution to the phase stiffness arises from (i) ``integrating out'' the remote bands that couple to the microscopic current operator, and (ii) the density-density interactions projected onto the isolated narrow bands. Our framework can be used to obtain an upper bound on the phase stiffness, and relatedly the superconducting transition temperature, for a range of physically inspired models involving both topological and non-topological narrow bands with arbitrary density-density interactions. We discuss a number of salient aspects of this formalism by applying it to a specific model of interacting flat bands and compare it against the known $T_c$ from independent numerically exact computations.Comment: 7 + 4 pages, 3 figures, new results adde

Adjusting for Spatial Effects in Genomic Prediction

This paper investigates the problem of adjusting for spatial effects in genomic prediction. Despite being seldomly considered in genome-wide association studies (GWAS), spatial effects often affect phenotypic measurements of plants. We consider a Gaussian random field (GRF) model with an additive covariance structure that incorporates genotype effects, spatial effects and subpopulation effects. An empirical study shows the existence of spatial effects and heterogeneity across different subpopulation families while simulations illustrate the improvement in selecting genotypically superior plants by adjusting for spatial effects in genomic prediction