Scalings in diffusion-driven reactionA+B→C: Numerical simulations by lattice BGK models

We are interested in applying lattice BGK models to the diffusion-driven reactive systemA+BC, which was investigated by Gálfi and Rácz with an asymptotic analysis and by Chopard and Droz with a cellular automaton model. The lattice BGK model is free from noise and flexible for various applications. We derive the general reaction-diffusion equations for the lattice BGK models under the assumption of local diffusive equilibrium. Two fourth-order terms are derived and verified by numerical simulations. The motivation of this study is to compare the lattice BGK results with existing results before we apply the models to more complicated systems. The scalings concern two exponents and appearing in the production rate ofC componentR(x, t)t ^– G(xt ^– ). We find the same values for =1/6 and =2/3 as Gálfi and Rácz found at the long time limit. AGaussian-like function forG is numerically obtained, which confirms a similar result of Gálfi and Rácz. On the one hand, when compared with the asymptotic analysis, lattice BGK models are easy to apply to cases where no analytic or asymptotic results exist; on the other hand, when compared with cellular automaton models, lattice BGK models are faster, simpler, and more accurate. The discrepancy of the results between the cellular automaton model and the lattice BGK models for the exponents comes from the role of the intrinsic fluctuation. Once the time and space correlation of stochastic stirring is given, we can incorporate a random fluctuating term in lattice BGK models. The Schlögl model is also tested, showing the ability of lattice BGK models for generating Turing patterns, which may stimulate further interesting investigations.     

Self-consistent perturbation theory for dynamics of valence fluctuations

Dynamical properties of valence-fluctuating systems are studied at absolute zero of temperature. The self-consistent perturbation theory developed for rare-earth impurity systems is used with some refinement. The theory takes account of the orthogonality catastrophe caused by hybridization of 4f and conduction electrons. Extensive numerical results are reported for the 4f-electron density of states _4f () and the dynamical magnetic susceptibility (). The results cover both the intermediate-valence and nearly integral-valence regimes of a model Ce impurity system. The present theory gives reasonable overall feature of dynamics including a sharp peak in _4f (0) in the Kondo regime. However, the low-frequency limit of dynamical quantities is not consistent with the Fermi-liquid property. It is shown that interpolation of the present results and those by the Fermi-liquid theory is possible. Hence accurate dynamical information is obtained over a wide excitation-frequency range.     

Study of the three-dimensional Ising model on film geometry with the cluster Monte Carlo method

Topological properties of clusters are used to extract critical parameters. This method is tested for the bulk properties ofd=2 percolation and thed=2, 3 Ising model. For the latter we obtain an accurate value of the critical temperatureJ/k _B T _c=0.221617(18). In the case of thed=3 Ising model with film geometry the critical value of the surface coupling at the special transitions is determined as J_1c/J=1.5004(20) together with the critical exponents ₁ ^m =0.237(5) and=0.461(15).     

The Two Dimensional Hubbard Model at Half-Filling. I. Convergent Contributions

We prove analyticity theorems in the coupling constant for the Hubbard model at half-filling. The model in a single renormalization group slice of index i is proved to be analytic in for ||c/i for some constant c, and the skeleton part of the model at temperature T (the sum of all graphs without two point insertions) is proved to be analytic in for ||c/|log T|². These theorems are necessary steps towards proving that the Hubbard model at half-filling is not a Fermi liquid (in the mathematically precise sense of Salmhofer).     

A Theoretical Description of the Photodissociation Spectrum of the H2O...HCl Complex

The effect of association of the water molecule and hydrogen chloride on the UV absorption bands is studied. Complete ab initio calculations for the H₂O...HCl complex in the S ₁ and S ₂ states are performed. A mathematical model using Airy functions is developed to describe the absorption cross-sections in a continuous spectrum. The form of the potential is determined by accurate ab initio calculations. The cross-sections of potential surfaces of lower electron states are found from ab initio calculations using the Hartree–Fock, configurational interaction, and multi-configurational interaction techniques. A complete vibrational analysis and an analysis of the change in the electron density for the S ₀ S ₁ transition on moving along the reaction coordinate allow a conclusion to be made on the feasibility of applying the model proposed to the H₂O...HCl complex. The results obtained in the framework of the model using Airy functions show reasonably good agreement with the experiment. For the H₂O...HCl heterodimer, the absorption band has the same structureless form as for the water monomer. The absorption band (peaking at 161 nm) is seen to shift towards short wavelengths as compared with the water monomer H₂O ( 167 nm).     

Equilibrium polymerization in a solvent: Solution on the Bethe lattice

The lattice model for equilibrium polymerization in a solvent proposed by Wheeler and Pfeuty is solved exactly on a Bethe lattice (core of a Caylay tree) with general coordination numberq. Earlier mean-field results are reobtained in the limitq, but the phase diagrams show deviations from them for finiteq. Whenq=2, our results turn into the solution of the one-dimensional problem. Although the model is solved directly, without the use of the correspondence between the equilibrium polymerization model and the diluten0 model, we verified that the latter model may also be solved on the Bethe lattice, its solution being identical to the direct solution in all parameter space. As observed in earlier studies of the puren0 vector model, the free energy is not always convex. We obtain the region of negative susceptibility for our solution and compare this result with mean field and renormalization group (-expansion) calculations.     

Phase uniqueness and correlation length in diluted-field Ising models

The diluted-field Ising model, a random nonnegative field ferromagnetic model, is shown to have a unique Gibbs measure with probability I when the field mean is positive. Our methods involve comparisons with ordinary uniform field Ising models. They yield as a corollary a way of obtaining spontaneous magnetization through the application of a vanishing random magnetic field. The correlation lengths of this model defined as (lim _n-(1/n) log ₀; _n)^-1, wheren is the site on the first coordinate axis at distancen from the origin and ₀; _n is the origin ton two-point truncated correlation function, is non-random. We derive an upper bound for it in terms of the correlation length of an ordinary nonrandom model with uniform field related to the field distribution of the diluted model.     

Kinetic theory of granular shear flow: Constitutive relations for the hard-disk model

A kinetic theory for the constitutive Theological relations of rapid granular shear flow of hard circular disks, characterized by a coefficient of restitutione and a surface roughness coefficient, is formulated. From a set of general constitutive equations for single-particle dynamical variables, the approximate expressions for the limit of small and large dimensionless dissipative parameterR _t are obtained. HereR _t is defined as the ratio /, where is the fluctuation of translational velocity from the mean flow velocity, is the diameter of a disk, and is the shear rate. At smallR _t the theoretical predictions can be compared with exact computer simulation results of granular dynamics that are also reported. The agreement between theory and simulation is better than expected; the present theory is accurate up to high packing density in this region ofR _t .     

Thermodynamic Properties of a Simple Model of Like-Charged Attracting Rods

We study the thermodynamic properties of a simple model for the possible mechanism of attraction between like charged rod-like polyions inside a polyelectrolyte solution. We consider two polyions in parallel planes, with Z charges each, in a solution containing multivalent counterion of valence . The model is solved exactly for Z13 for a general angle between the rods and supposing that n counterions are condensed onto each polyion. The free energy has two minima, one at =0 (parallel rods) and another at =/2 (perpendicular rods). The stability of the parallel and perpendicular configurations is analyzed.     

Neural Network Models for Finline Discontinuities

The radial basis network is used as the finline discontinuities electromagnetic artifical neural network(EMANN) models. EM software analysis is employed to characterize finline discontinuities. EMANN models are then trained using physical parameters and frequency as inputs and equivalent electric circuit element parameters of finline discontinuities as outputs. Once trained , the EMANN models can simulate equivalent electric circuit element parameters of finline step, notch and strip very fast and efficiently.     

Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators

The finite-element approach to lattice field theory is both highly accurate (relative errors 1/N ², whereN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this Letter, we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian isH=p ²/2+q ^2k /2k. Construction of such matrix elements does not require solving the implicit equations of motion. Low-order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator groundstate gives a result for thek=2 anharmonic oscillator groundstate energy accurate to better than 1% while a two-state approximation reduces the error to less than 0.1%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions.     

Macdonald polynomials from Sklyanin algebras: A conceptual basis for thep-adics-quantum group connection

We establish a previously conjectured connection betweenp-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which interpolate between the zonal spherical functions of related real andp-adic symmetric spaces. The elliptic quantum algebras underlie theZ _n -Baxter models. We show that in then limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZ _n -Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA ₁. The partition function of theZ ₂-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic regimes in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of q-deforming Euler products.Work supported in part by the NSF: PHY-9000386     

The free energy of a class of Hopfield models

We show that in the limitp ,N 0,=p/N 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite .     

A Universal Amplitude Ratio for the q≤4 Potts Model from a Solvable Lattice Model

The universal amplitude ratio R for the (q4)-state Potts model in two dimensions is determined by using results for the dilute A model in regime 1. The nature of the relationship between the Potts model and the dilute A model is discussed.     

Infrared bounds,phase transitions and continuous symmetry breaking

We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (·) ₃ ² quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k ^–2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).Research supported by USNSF under grants GP-38048 and MPS-74-13252A. Sloan Fellow; also in the Department of Physics     

Polymer models and generalized Potts-Kasteleyn models

The well-established relation between Potts models withv spin values and random-cluster models (with intracluster bonding favored over intercluster bonding by a factorv) is explored, but with the random-cluster model replaced by a much generalized polymer model, implying a corresponding generalization of the Potts model. The analysis is carried out in terms a given defined functionR(), an entropy/free-energy density for the polymer model in the casev=1, expressed as a function of the density of units. The aim of the analysis is to determine the analogR _v () ofR() for general nonnegativev in terms ofR(), and thence to determine the critical value of density _vg at which gelation occurs. This critical value is independent ofv up to a valuev _P, the Potts-critical value. What is principally required ofR() is that it should show a certain given concave/convex behavior, although differentiability and another regularizing condition are required for complete conclusions. Under these conditions the unique evaluation ofR _v () in terms ofR() is given in a form known to hold for integralv but not previously extended. The analysis is carried out in terms of the Legendre transforms of these functions, in terms of which the phenomena of criticality (gelation) and Potts criticality appear very transparently andv _P is easily determined. The value ofv _P is 2 under mild conditions onR. Special interest attaches to the functionR ₀(), which is shown to be the greatest concave minorant ofR(). The naturalness of the approach is demonstrated by explicit treatment of the first-shell model.     

On the dynamics and ergodic properties of theXY model

The return to equilibrium is investigated for one-dimensional (one-sided) chain of theXY model. The initial state is taken to be the Gibbs state for the sum of the Hamiltonian for theXY model of lengthN and a perturbation by a uniform magnetic field acting on the firstn sites. The time evolution under the unperturbedXY model Hamiltonian is studied for the expectation value of the average magnetization of the same firstn sites in the infinitely extended system (i.e., after taking the limitN). It is found that the return to equilibrium occurs for a finite-size perturbation (i.e., for a fixedn), while it does not occur for an infinite-size perturbation (i.e., the limit n is taken simultaneously as N). A certain twisted asymptotic Abelian property of theXY model is shown and used as a technical tool.     

Random surfaces with two-sided constraints: An application of the theory of dominant ground states

We consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states. Our models are related to the Gibbs ensemble for the multidimensional SOS model with symmetric constraints _x m/2. The main result is that for ₀, where ₀ does not depend onm, the structure of thermodynamic phases in the model is determined by dominant ground states: for an evenm a Gibbs state is unique and for an oddm the number of space-periodic pure Gibbs states is two.     

Computer simulation of driven diffusive systems with exchanges

The field-driven Kawasaki model with a fractionp admixture of Glauber dynamics is studied by computer simulation:p=0 corresponds to the order-parameter-onserving driven diffusive system, whilep=1 is the equilibrium Ising model. Forp=0.1 our best estimates of critical exponents based on a system of size 4096×128 are0.22, ^RS0.45, andv v 1. These exponents differ from both the values predicted by a field-theoretic method forp=0 and those of the equilibrium Ising model. Anisotropic finite-size scaling analyses are carried out, both for subsystems of the large system and for fully periodic systems. The results of the latter, however, are inconsistent, probably due to the complexity of the size effects. This leaves open the possibility that we are in a crossover regime fromp=0 top0 and that our critical exponents are effective ones. Forp=0 our results are consistent with the predictionsv >v .