Large Deviations for Probabilistic Cellular Automata II

We obtain an upper large deviations bound which shows that for some models of probabilistic cellular automata (which are far away from the product case) the lower large deviation bound derived in Eizenberg and Kifer J. Stat. Phys. 108: 1255–1280 (2002) is sharp, and so the corresponding large deviations phenomena cannot be described via the traditional Donsker–Varadhan form of the action functional. For models which are close to the product case we derive approximate large deviations bounds using the Donsker–Varadhan functional for the product case.     

Exact and approximate results for the Anderson impurity model

This paper provides exact and rigorous upper and lower bounds to the ground state energy of the Single Impurity Anderson Model in the limit of infiniteU. The upper bound approximates the ground state energy very well and the corresponding state may be used as a starting point for further investigations. The energy spectrum of the Single Impurity Anderson Model is calculated exactly for a special non-trivial model density of states describing the bare band electrons. All approaches introduced here are compared to this exact result and to other ground state energy calculations.     

Quantum Monte Carlo assessment of density functionals for π-electron molecules: ethylene and bifuran

ABSTRACT

We perform all-electron, pure-sampling quantum Monte Carlo (QMC) calculations on ethylene and bifuran molecules. The orbitals used for importance sampling with a single Slater determinant are generated from Hartree-Fock and density functional theory (DFT). Their fixed-node energy provides an upper bound to the exact energy. The best performing density functionals for ethylene are BP86 and M06, which account for 99% of the electron correlation energy. Sampling from the π-electron distribution with these orbitals yields a quadrupole moment comparable to coupled cluster CCSD(T) calculations. However, these, and all other density functionals, fail to agree with CCSD(T) while sampling from electron density in the plane of the molecule. For bifuran, as well as ethylene, a correlation is seen between the fixed-node energy and deviance of the QMC quadrupole moment estimates from those calculated by DFT. This suggests that proximity of DFT and QMC densities correlates with the quality of the exchange nodes of the DFT wave function for both systems.     

$\mathsf{La_{0.5}Sr_{0.5}CoO_{3}}$: A ferromagnet with strong irreversibility

Large spin systems as given by magnetic macromolecules or two-dimensional spin arrays rule out an exact diagonalization of the Hamiltonian. Nevertheless, it is possible to derive upper and lower bounds of the minimal energies, i.e. the smallest energies for a given total spin S. The energy bounds are derived under additional assumptions on the topology of the coupling between the spins. The upper bound follows from “n-cyclicity", which roughly means that the graph of interactions can be wrapped round a ring with n vertices. The lower bound improves earlier results and follows from “n-homogeneity", i.e. from the assumption that the set of spins can be decomposed into n subsets where the interactions inside and between spins of different subsets fulfill certain homogeneity conditions. Many Heisenberg spin systems comply with both concepts such that both bounds are available. By investigating small systems which can be numerically diagonalized we find that the upper bounds are considerably closer to the true minimal energies than the lower ones. Received 22 October 2002 / Received in final form 4 April 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: jschnack@uos.de     

The calculation of valence shell ionization potentials by the ΔSCF method

Valence shell ionization potentials for several small molecules (HF, H₂O, CH₄, CO, H₂S, PH₃, SiH₄, SO₂ and F₂CO) are calculated by means of ab initio ΔSCF calculations on all the valence hole states of each molecule. The results are compared with the experimental low energy photoelectron spectra. In the case of the higher energy ion states, particular attention is paid to finding practical means for obtaining convergence of the ΔSCF iterations, and numerical evidence is presented, indicating that the ΔSCF procedure provides upper bounds to the energy of such states, at least for the cases studied herein. A new procedure, the ‘ combined ΔSCF ’ method, for the determination of a common set of molecular orbitals for use in the construction of wavefunctions for a number of ion states is described.     

Left-right correlation energy

We first attempt to determine a local exchange functional E_x[p] which accurately reproduces the Hartree-Fock (HF) energies of the 18 first and second row atoms. E_x[p is determined from p and |δp|, and we find that we can improve significantly upon Becke's original generalized gradient approximation functional (commonly called B88X) by allowing the coefficient of the Dirac exchange term to be optimized (it is argued that molecules do not behave like the uniform electron gas). We call this new two parameter exchange functional OPTX. We find that neither δ p or t = Σ δ _i |² improve the fit to these atomic energies. These exchange functionals include not only exchange, but also left-right correlation. It is therefore proposed that this functional provides a definition for exchange energy plus left-right correlation energy when used in Kohn-Sham (KS) calculations. We call this energy the Kohn-Sham exchange (or KSX) energy. It is shown that for nearly all molecules studied these KSX energies are lower than the corresponding HF energies, thus giving values for the non-dynamic correlation energy. At stretched geometries, the KSX energies are always lower than the HF energies, and often substantially so. Furthermore all bond lengths from the KSX calculations are longer than HF bond lengths and experimental bond lengths, which again demonstrates the inclusion of left-right correlation effects in the functional. For these reasons we prefer to split the correlation energy into two parts: left-right correlation energy and dynamic correlation energy, arguing that the usage of the words ‘non-dynamic’ or ‘static’ or ‘near-degeneracy’ is less meaningful. We recognize that this definition of KSX is not precise, because the definition of a local E_x[p] can never be precise. We also recognize that these ideas are not new, but we think that their importance has been insufficiently recognized in functional determination. When we include third row atoms in our analysis, we are unable to find a local exchange functional which is a substantial improvement over B88X for the reproduction of HF energies. This must arise from the effects of the core orbitals, and therefore we do not consider that this detracts from the improved accuracy of OPTX. We report some MCSCF calculations constructed from bonding-antibonding configurations, from which we attempt to calculate ab initio left-right correlation. There is only moderate agreement between the two approaches. Finally we combine the OPTX functional with established correlation functionals (LYP, P86, P91) to form OLYP, OP86 and OP91; OLYP is a great improvement on BLYP for both energy and structure, and OP86, OP91 are an improvement over BP86, BP91 for structure. The importance of the exchange functional for molecular structure is therefore underlined.     

Estimates of the number of quantal bound states in one and three dimensions

Using the relation between the number of bound states and the number of zeros of the radial eigen-functionψ(r), or equivalently, that ofφ(r)=rψ(r) in the range 0⩽r⩽∞, the upper bounds on the number of bound states generated by potentialV(r) in different angular momentum channels are obtained in three dimension. Using a similar procedure, the upper bound on the number of bound states in one dimension is also deduced. The analysis is restricted to a class of potentials for whichE=0 is the threshold. By taking a number of specific examples, it is demonstrated that both in one and three dimensions, the estimate of the upper bound obtained by this procedure is very close to or equal to the exact number of bound states. The correlation of the present method with the Levison’s theorem and WKB approximation is discussed.     

Using the local density approximation and the LYP,BLYP and B3LYP functionals within reference-state one-particle density-matrix theory

For closed-shell systems, the local density approximation (LDA) and the LYP, BLYP and B3LYP functionals are shown to be compatible with reference-state one-particle density-matrix theory, where this recently introduced formalism is based on Brueckner-orbital theory and an energy functional that includes exact exchange and a non-universal correlation-energy functional. The method is demonstrated to reduce to a density functional theory when the exchange-correlation energy-functional has a simplified form, i.e. its integrand contains only the coordinates of two electrons, say r ₁ and r ₂, and it has a Dirac delta function δ(r ₁ - r ₂ as a factor. Since Brueckner and Hartree–Fock orbitals are often very similar, any local exchange functional that works well with Hartree–Fock theory is a reasonable approximation with reference-state one-particle density-matrix theory. The LDA approximation is also a reasonable approximation. However, the Colle–Salvetti correlation-energy functional and the LYP variant are not ideal for the method, since these are universal functionals. Nevertheless, they appear to provide reasonable approximations. The B3LYP functional is derived using a linear combination of two functionals: one is the BLYP functional; the other uses exact exchange and a correlation-energy functional from the LDA.     

RVB theory of the Hubbard model on the triangular lattice

We propose a Green's function technique, to investigate finite-temperature properties of the Hubbard model on the triangular lattice. The lattices are covered by dimers. The method is exact in two limits:U=0 or decoupled dimers. We apply this approximate method to calculate the ground state energy, the specific heat and the single-particle spectral weight for the 1/2-filled case. The largest lattice considered has 16×16 sites. The approximate ground state energy as a function of the on-site interactionU oscillates around the exact energyin the 1/2-filled case. We find two peaks in the specific heat. ForU5t the single-particle spectral weight splits into upper and lower Hubbard bandasymmetrically. Thus in the 1/2-filled case the chemical potential is placed in the upper band leading to a metallic state. The approximate technique yields a finite zero-point entropy for mediumU. All the investigations signal a RVB state in the range of mediumU as formerly proposed by Callaway.     

Classical density functional theory meets Monte-Carlo simulations

ABSTRACT

Typically the quality of an approximate density functional is evaluated by a direct comparison of its predictions in a given test case to exact data obtained by computer simulations. An important example for such an approach is the test of equilibrium structure of a simple fluid as measured by the pair distribution function g(r) or the cavity correlation function y(r). However, the combination of exact density profiles and the analytical structure of density functional theory allows one to determine and potentially improve the quality of a functional in a more sophisticatedway.     

Universal Bounds for the Littlewood-Paley First-Order Moments of the 3D Navier-Stokes Equations

We derive upper bounds for the infinite-time and space average of the L ¹-norm of the Littlewood-Paley decomposition of weak solutions of the 3D periodic Navier-Stokes equations. The result suggests that the Kolmogorov characteristic velocity scaling, U_k ~ e^1/3 k^-1/3{\mathbf{U}_\kappa\sim\epsilon^{1/3} \kappa^{-1/3}} , holds as an upper bound for a region of wavenumbers near the dissipative cutoff.     

A simple state-dependent exchange functional for calculating Hartree-Fock bandstructures in semiconductors and insulators

A functional which takes the nonlocal character of the exchange selfenergy fully into account is derived within the nearly-free electron model for a spherical Fermi surface. This exchange functional when combined with the Hartree bandstructure gives approximate Hartree-Fock (HF) bands which agree surprisingly well with ab-initio HF-bands for covalent semiconductors and rare gas solids.     

Bounds on size-dependent behaviour of composites

Computational homogenisation is a powerful strategy to predict the effective behaviour of heterogeneous materials. While computational homogenisation cannot exactly compute the effective parameters, it can provide bounds on the overall material response. Thus, central to computational homogenisation is the existence of bounds. Classical first-order computational homogenisation cannot capture size effects. Recently, it has been shown that size effects can be retrieved via accounting for elastic coherent interfaces in the microstructure. The primary objective of this contribution is to present a systematic study to attain computational bounds on the size-dependent response of composites. We show rigorously that interface-enhanced computational homogenisation introduces two relative length scales into the problem and investigate the interplay between them. To enforce the equivalence of the virtual power between the scales, a generalised version of the Hill–Mandel condition is employed, and accordingly, suitable boundary conditions are derived. Macroscopic quantities are related to their microscopic counterparts via extended average theorems. Periodic boundary conditions provide an effective behaviour bounded by traction and displacement boundary conditions. Apart from the bounds due to boundary conditions for a given size, the size-dependent response of a composite is bounded, too. The lower bound coincides with that of a composite with no interface. Surprisingly, there also exists an upper bound on the size-dependent response beyond which the expected ‘smaller is stronger’ trend is no longer observed. Finally, we show an excellent agreement between our numerical results and the corresponding analytical solution for linear isotropic materials which highlights the accuracy and broad applicability of the presented scheme.     

External Fields, Density Functionals, and the Gibbs Inequality

By combining the upper and lower bounds to the free energy as given by the Gibbs inequality for two systems with the same intermolecular interactions but with external fields differing from each other only in a finite region of space , we show that the corresponding equilibrium densities must also differ from each other somewhere in . We note that the basic equations of density functional theory arise naturally from a simple rearrangement and reinterpretation of the terms in the upper bound Gibbs inequality for such systems and briefly discuss some of the complications that occur when the intermolecular interactions of the two systems also differ.     

Improved rigorous upper bounds for transport due to passive advection described by simple models of bounded systems

The work of J. A. Krommes and R. A. Smith on rigorous upper bounds for the turbulent transport of a passively advected scalar is extended in two directions: (1) For their reference model, improved upper bounds are obtained by utilizing more sophisticated two-time constraints which include the effects of cross-correlations up to fourth order. Numerical solutions of the model stochastic differential equation are also obtained; they show that the new bounds compare quite favorably with the exact results, even at large Reynolds and Kubo numbers. (2) The theory is extended to take account of afinite spatial autocorrelation lengthL _c. As a reasonably generic example, the problem of particle transport due to statistically specified stochastic magnetic fields in a collisionless turbulent plasma is revisited. A bound is obtained which reduces for smallL _c to the quasilinear limit and for largeL _c to the strong turbulence limit, and which provides a reasonable and rigorous interpolation for intermediate values ofL _c.     

Improved condition for the absence of bound states and converging analytic bounds to critical screening parameter

Converging lower bound to the critical screening parameterD _c associated with the ground state of a two-particle system interacting through a cut-off Coulomb potential is obtained analytically using an improved condition for the absence of bound states. The predicted numerical result for the lower bound is found to be within 10⁻³% of the exact result. On the other hand, a multi-parameter variational approach yields a tight upper bound, within 0.54% of the exact result. It is shown that the critical screening parameter for the exciteds-states can also be determined in an approximate way. We obtainD _c ^ms ≈ [0.764435n ⁻²+0.617737n ⁻³]⁻¹ wheren is the principal quantum number. The predictedD _c for various quantum states (n=1 to 8) are in good agreement with the values obtained numerically by Singh and Varshni.     

Upper bounds on the convective heat transport in a rotating fluid layer of infinite Prandtl number: case of large Taylor numbers

By means of the Howard-Busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a horizontal fluid layer heated from below and rotating about a vertical axis. We consider the interval of large Taylor numbers where the intermediate layers of the optimum fields expand in the direction of the corresponding internal layers. We consider the 1 - α-solution of the arising variational problem for the cases of rigid-stress-free, stress-free, and rigid boundary conditions. For each kind of boundary condition we discuss four cases: two cases where the boundary layers are thinner than the Ekman layers of the optimum field and two cases where the boundary layers are thicker than the Ekman layers. In most cases we use an improved solution of the Euler-Lagrange equations of the variational problem for the intermediate layers of the optimum fields. This solution leads to corrections of the thicknesses of the boundary layers of the optimum fields and to lower upper bounds on the convective heat transport in comparison to the bounds obtained by Chan [J. Fluid Mech. 64, 477 (1974)] and Hunter and Riahi [J. Fluid Mech. 72, 433 (1975)]. Compared to the existing experimental data for the case of a fluid layer with rigid boundaries the corresponding upper bounds on the convective heat transport is less than two times larger than the experimental results, the corresponding upper bound on the convective heat transport, obtained by Hunter and Riahi is about 10% higher than the bound obtained in this article. When Rayleigh number and Taylor number are high enough the upper bound on the convective heat transport ceases to depend on the boundary conditions. Received 30 January 2001 and Received in final form 28 May 2001     

Bounds on enhanced turbulent flame speeds for combustion with fractal velocity fields

Rigorous upper bounds are derived for large-scale turbulent flame speeds in a prototypical model problem. This model problem consists of a reaction-diffusion equation with KPP chemistry with random advection consisting of a turbulent unidirectional shear flow. When this velocity field is fractal with a Hurst exponentH with 0<H<1, the almost sure upper bounds suggest that there is an accelerating large-scale turbulent flame front with the enhanced anomalous propagation lawy=C _H t ^1+H for large renormalized times. In contrast, a similar rigorous almost sure upper bound for velocity fields with finite energy yields the turbulent flame propagation law within logarithmic corrections. Furthermore, rigorous theorems are developed here which show that upper bounds for turbulent flame speeds with fractal velocity fields are not self-averaging, i.e., bounds for the ensemble-averaged turbulent flame speed can be extremely pessimistic and misleading when compared with the bounds for every realization.     

Lower bounds for eigenvalues of the Dirac operator on n-spheres with SO(n)-symmetry

In this paper we derive estimates for the eigenvalues of the Dirac operator and their multiplicity on manifolds diffeomorphic to Sⁿ with an isometric SO(n)-action. Especially we prove a new lower bound for the first eigenvalue and show an example, where this new bound coincides in the limit with the known upper bounds.     

Union bounds on the symbol error probability of LoRa modulation for flat Rician block fading channels

In this paper the symbol error performance of LoRa modulation is addressed for flat Rician block fading channels. First the exact symbol error probability of the LoRa modulation on Rician fading is derived. Then the upper and lower union bounds are employed on the derived symbol error probability. The proposed bounds are compared against the exact symbol error probability, the numerical evaluation of the symbol error probability and the state-of-art approximation of the LoRa symbol error probability. Numerical results show that while the proposed upper bound is very tight to the exact symbol error probability, there is approximately a 2.5 dB gap for the lower bound.