A maximum entropy mean field method for driven diffusive systems

We introduce a method of generating systematic mean field (MF) approximations for the nonequilibrium steady state of ferromagnetic Ising driven diffusive systems (DDS), based on the maximum entropy principle due to Jaynes. In the phase coexistence region, MF approximations to the master equation do not provide a closed system of equations in the MF variables. This can be traced to the conservation of the order parameter by the stochastic dynamics. Our maximum entropy mean field (MEMF) approximation method is applicable to high temperatures as well to the low-temperature phase coexistence region. It is based on a derivation of a generalized variational free energy from the maximum entropy principle, with the MF evolution equations playing the role of constraints. In the phase coexistence region this free energy is nonconvex and is interpreted by means of a Maxwell construction. We use a pair-level variant of the MEMF approximation to calculate quantities of interest for the ferro-magnetic Ising DDS on a square lattice. Results of calculations with several different choices of transition rates satisfying local detailed balance are discussed and compared with those obtained by other methods.     

High precision mean-field results for lattice gauge theories: with special attention paid to the gauge dependence of mean-field perturbation theory to finite order

We do mean-field perturbation theory for U(1) lattice gauge theory in the axial gauge, and evaluate corrections from fluctuations up to fourth order for the free energy and plaquette energy. Comparing with similar results previously obtained in the Feynman gauge we find, to those orders studied, a gauge dependence of the size of the first correction term neglected with one exception. This gauge dependence decreases rapidly as the order of the approximation is increased. To any finite order, results in axial gauge are better approximations than results in the Feynman gauge. We speculate why. Assuming it to be generally true, we evaluate the first correction beyond the one-loop mean-field approximation to the free energy of SU(2) gauge theory with Wilson action in the axial gauge. This correction brings the mean-field result very close to Monte Carlo results for β > 1.6. It also makes the mean-field result identical, within a narrow margin, to ressumed strong coupling results in the interval 1.6 < β < 2.4, thus showing the absence of a phase transition.For both groups studied, we find that the asymptotic series of mean-field perturbation theory give much better approximations than do ordinary weak coupling series.     

Generalized statistics variational perturbation approximation using q-deformed calculus

A principled framework to generalize variational perturbation approximations (VPAs) formulated within the ambit of the nonadditive statistics of Tsallis statistics, is introduced. This is accomplished by operating on the terms constituting the perturbation expansion of the generalized free energy (GFE) with a variational procedure formulated using q-deformed calculus. A candidate q-deformed generalized VPA (GVPA) is derived with the aid of the Hellmann-Feynman theorem. The generalized Bogoliubov inequality for the approximate GFE are derived for the case of canonical probability densities that maximize the Tsallis entropy. Numerical examples demonstrating the application of the q-deformed GVPA are presented. The qualitative distinctions between the q-deformed GVPA model vis-á-vis prior GVPA models are highlighted.     

CALCULATION OF THE FIXED LENGTH FUNDAMENTAL su(2) HIGGS MODEL

The phase diagram of the lattice system of SU(2) gauge field coupled with the fixed length Higgs field in fundamental representation has been calculated by the variational-cumulant expansion method to the third order approximation.The method of determining the variational parameter has been improved by using the free energy to the second order approximation.Thus calculated phase diagram is in good agreement with the Monte Carlo estimation and the order of the phase transition is clearly determined in the third order approximation.     

Rayleigh-Schrödinger-Goldstone variational perturbation theory for many Fermion systems

We present a Rayleigh-Schrödinger-Goldstone perturbation formalism for many Fermion systems. Based on this formalism, variational perturbation scheme which goes beyond the Gaussian approximation is developed. In order to go beyond the Gaussian approximation, we identify a parent Hamiltonian which has an effective Gaussian vacuum as a variational solution and carry out further perturbation with respect to the renormalized interaction using Goldstones expansion. Perturbation rules for the ground state wavefunctional and energy are found, thus, opening a way for general use of the Schrödinger picture method for many Fermion systems. Useful commuting relations between operators and the Gaussian wavefunctional are also found, which could reduce the calculational efforts substantially. As examples, we calculate the first order correction to the Gaussian wavefunctional and the second order correction to the ground state of an electron gas system with the Yukawa-type interaction.     

Two-site two-electron generalized Hubbard-Holstein model: a perturbation study

The two-site two-electron generalized Hubbard-Holstein model is studied within a perturbation method based on a variational phonon basis obtained through the modified Lang-Firsov (MLF) transformation. The ground-state wave function and the energy are found including up to the seventh and eighth order of perturbation, respectively. The convergence of the perturbation corrections to the ground state energy, as well as to the correlation functions, are investigated. The kinetic energy and the correlation functions involving charge and lattice deformations are studied as a function of electron-phonon(e-ph) coupling and electron-electron interactions for different values of the adiabaticity parameter. The simultaneous effect of the e-ph coupling and Coulomb repulsion on the kinetic energy shows interesting features.     

A new variational perturbation method for double-well oscillators

We propose a variational perturbation method based on the observation that eigenvalues of each parity sector of both the anharmonic and double-well oscillators are approximately equi-distanced. The generalized deformed algebra satisfied by the invariant operators of the systems provides well defined Hilbert spaces to both of the oscillators. There appears a natural expansion parameter defined by the ratio of length scales of the trial wavefunctions. The energies of the ground state and the first order excited state, in the zeroth order variational approximation, are obtained with errors <10⁻²% for vast range of the coupling strength for both oscillators. An iterative formula is presented which perturbatively generates higher order corrections from the lower order invariant operators and the first order correction is explicitly given.     

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

Analytic Study of O(3) Nonlinear σ Model

With the variational-cumulant expansion method, the internal energy and the specific heat of O(3) nonlinear a model is calculated up to the fourth order. The variational parameter is determined by the variational method of first order free energy and accumulation point method respectively. It is shown that a more agreeable internal energy curve and specific heat curve with the MC results can be obtained by the latter.     

Extension of the local curvature approximation to third order and improved tilt invariance

The second-order local curvature approximation (LCA2) is a theory of rough surface scattering that reproduces fundamental low and high frequency limits in a tilted frame of reference. Although the existing LCA2 model provides agreement with the first order small perturbation method up to the first order in surface tilt, results reported in this paper produce a new formulation of the model that achieves consistency with perturbation theory to first order in surface height and arbitrary order in surface tilt. In addition, extension of the modified LCA to third order is presented, and allows the theory to match the second-order small perturbation method to arbitrary order in surface tilt. Crucial to the development of the theory are a set of identities involving relationships among the small perturbation method (i.e. low frequency) and Kirchhoff approximation (i.e. high frequency) kernels; a set of new identities obtained in our derivations is also presented. Sample results involving 3D electromagnetic scattering from penetrable rough surfaces, as well as 2D scattering from Dirichlet sinusoidal gratings, are provided to compare the new results with the existing LCA2 model and with other rough surface scattering theories.     

Variat ional-Cumulant Expansion on the Five-State Clock Model

The phase transition of the 5-state dock model has been studied by standard variationalcumulant expansion (VCE) method. We calculated the fiee energy (F) and the internal energy (E) to the fourth order approximation and the specific heat (C) to the third order approximation with the trial action of one variational parameter. The position of the phase transition point given above is in agreement with the results of the Monte Carlo (MC). We also calculated the model to the third order approximation with the trial action of two variational parameters. The comparison of the results for one variational parameter with that for two is given. From this, we can see how the choice of the trial action affects the result and the trial action must be equivalent to the action of the system. All above has shown that the VCE is convergent in the calculation of the 5-state clock model.     

Theoretical investigations of the vapour-liquid equilibrium and dielectric properties of dipolar Yukawa fluids in an external field

In a low field approximation, using the dipolar Yukawa fluid model (in mean spherical approximation as a reference system) a consistent field-dependent free energy expression is proposed for the calculation of the vapour-liquid equilibrium of polar fluids in an applied electric field. A perturbation theory high field approximation expression of the free energy is also proposed to study the field-dependent properties of fluids. In the high field approximation, equations for the field-dependent polarization and for the nonlinear dielectric constant (or Piekara constant) are also predicted. It has been discussed that our approximations are appropriate to describe the vapour-liquid-like phase equilibria and the magnetization curves of magnetic fluids.     

Local and non-local curvature approximation: a new asymptotic theory for wave scattering

Abstract

We present a new asymptotic theory for scalar and vector wave scattering from rough surfaces which federates an extended Kirchhoff approximation (EKA), such as the integral equation method (IEM), with the first and second order small slope approximations (SSA). The new development stems from the fact that any improvement of the ‘high frequency’ Kirchhoff or tangent plane approximation (KA) must come through surface curvature and higher order derivatives. Hence, this condition requires that the second order kernel be quadratic in its lowest order with respect to its Fourier variable or formally the gradient operator. A second important constraint which must be met is that both the Kirchhoff approximation (KA) and the first order small perturbation method (SPM-1 or Bragg) be dynamically reached, depending on the surface conditions. We derive herein this new kernel from a formal inclusion of the derivative operator in the difference between the polarization coefficients of KA and SPM-1. This new kernel is as simple as the expressions for both Kirchhoff and SPM-1 coefficients. This formal difference has the same curvature order as SSA-1 + SSA-2. It is acknowledged that even though the second order small perturbation method (SPM-2) is not enforced, as opposed to the SSA, our model should reproduce a reasonable approximation of the SPM-2 function at least up to the curvature or quadratic order. We provide three different versions of this new asymptotic theory under the local, non-local, and weighted curvature approximations. Each of these three models is demonstrated to be tilt invariant through first order in the tilting vector.     

Elastic displacement field of point defects in anisotropic cubic crystals

The elastic displacement field of point defects in cubic crystals is calculated for weak anisotropy by second order perturbation theory and by a variational procedure. The results are compared with numerical calculations for Cu. Further analytical approximations are given for the volume change in an infinite crystal and for the interaction energy of two point defects.     

New Variational Perturbation Theory Based on q−Deformed Oscillator

A new variational perturbation theory is developed based on the q−deformed oscillator. It is shown that the new variational perturbation method provides 200 and 10 times better accuracy for the ground state energy of anharmonic oscillator than the Gaussian and the post Gaussian approximation, respectively, for weak coupling.     

Andersen–Weeks–Chandler Perturbation Theory and One-Component Sticky-Hard-Spheres

We apply second order Andersen–Weeks–Chandler perturbation theory to the one-component sticky-hard-spheres fluid. We compare the results with the mean spherical approximation, the Percus–Yevick approximation, two generalized Percus–Yevick approximations, and the Monte Carlo simulations.     

Variational principle for regular and random Ising models on the cactus tree or on the usual lattice in the “cactus approximation”

The distribution functions and the free energy are expressed in terms of the effective fields for the regular and random Ising models of an arbitrary spin S on the generalized cactus tree. The same expressions apply to systems on the usual lattice in the “cactus approximation” in the cluster variation method. For an ensemble of random Ising models of an arbitrary spin S on the generalized cactus tree, the equation for the probability distribution function of the effective fields is set up and the averaged free energy is expressed in terms of the probability distribution. The same expressions apply to the system on the usual lattice in the “cactus approximation”. We discuss the quantities on the usual lattice when the system or the ensemble of random systems has the translational symmetry. Variational properties of the free energy for a system and of the averaged free energy for an ensemble of random systems are noted. The “cactus approximations” are applicable to the Heisenberg model as well as to the Ising model of an arbitrary spin, and to ensembles of random systems of these models.     

The penetrable sphere and related models of liquid—vapour equilibrium

The equation of state of the penetrable sphere model of liquid—vapour equilibrium is calculated by three different sequences of approximations; the first is based on the virial expansion of the equivalent two-component model in powers of the densities, the second on expansion in powers of the activity, and the third on a cumulant expansion of the configurational energy in powers of the reciprocal temperature. These sequences are examined both with the inclusion of all coefficients and with the sub-sets of coefficients appropriate to the first and second Percus-Yevick (PY) approximations. The first PY approximation gives a classical critical point whose density and temperature are accurately determined. The second PY and the complete set of coefficients yield badly-behaved series from which few conclusions can be drawn.

The penetrable sphere model is generalized to a wider class of potentials and one of these, in which the configurational energy is expressed in terms of gaussian functions is related to a two-component model of Helfand and Stillinger. It is more tractable than the original model and is examined by the same sequences of approximations. They have shown that the complete series leads to a non-classical critical point in their version of the model; here we show that the first PY approximation is classical but the second nonclassical.     

The quantum theory of nucleon correlation in finite nuclei. II. Higher order correlation effects and additive pair approximation

In the first part of this article it was shown that the variational solution of the Schroedinger equation of a finite Fermion system can be written as a finite sum of A terms (for A particles) the first of which is the Hartree-Fock energy, while the rest represent the correlation effects. In the first part explicit formulas for the 2-particle correlation energy were given. In this paper explicit formulas are given for the higher order correlation energies. It is shown that two different models can be developed depending on the orthogonality condition used. Beginning with the 4th order effects the “linked” and “unlinked” correlation terms are separated. An exact formula is given for the case in which only the 2-particle effects, linked and unlinked are taken into account. The “additive pair approximation” in which the correlation energy is given as the sum of 2-particle energies is investigated and it is shown to be related to the exact formula by a clearly defined set of approximations. Various possible applications of the model are discussed.     

Variational approximations for couplers and Y junctions

A simple variational method is used to determine the modal properties of single-mode couplers and Y junctions when the cores are close together or coalesce. The results complement the well-known perturbation expressions for large core separation. Together the two approximation methods provide accurate approximations over the complete range of separation from infinite separation to complete coalescence.