Standard Variational Principle Research Articles

Construction of Continuous Collective Energy Landscapes for Large Amplitude Nuclear Many-Body Problems.

Several protocols are proposed to build continuous energy surfaces of many-body quantum systems, regarding both energy and states. The standard variational principle is augmented with constraints on state overlap, ensuring arbitrary precision on continuity. As an illustration, the lowest energy and excited state paths relevant for the ^{240}Pu asymmetric fission are studied. The scission is clearly signed, with a neutron excess in the neck, the ultimate glue before its rupture. Our approach can potentially connect any couple of Hilbert space states, which opens up new horizons for various applications.

Addendum: Thermodynamic quasi-equilibria in high power magnetron discharges: a generalized Poisson–Boltzmann relation (2023 Plasma Sources Sci. Technol. 32 055012)

A recent publication by Köhn et al (2023 Plasma Sources Sci. Technol. 32 055012) studied the quasi-equilibria of high power magnetron discharges through thermodynamic principles. A generalized, magnetic-field aware Poisson–Boltzmann relation for the electric potential and the electron density was established using a non-standard (multi-objective) variational principle. This addendum demonstrates that, assuming slow or quasistatic evolution, the same result can be realized via a standard (single-objective) variational principle, thereby streamlining the theoretical framework while preserving the robustness of the finding.

Exploring mean field spin models with multiple populations: A theoretical framework

This paper thoroughly investigates the analysis of a modified generalized mean-field Ising model. The thermodynamic limit of free energy is investigated, and both the upper and lower bounds are derived. The lower bound is derived using a standard entropic variational principle, whereas a decoupling method is employed to derive the upper bound. Although the two bounds differ in form, they exhibit the same local maxima. Finally, the impact of a slight external field on the magnetic order is analyzed.

Sparse linear mixed model selection via streamlined variational Bayes

Linear mixed models are a versatile statistical tool to study data by accounting for fixed effects and random effects from multiple sources of variability. In many situations, a large number of candidate fixed effects is available and it is of interest to select a parsimonious subset of those being effectively relevant for predicting the response variable. Variational approximations facilitate fast approximate Bayesian inference for the parameters of a variety of statistical models, including linear mixed models. However, for models having a high number of fixed or random effects, simple application of standard variational inference principles does not lead to fast approximate inference algorithms, due to the size of model design matrices and inefficient treatment of sparse matrix problems arising from the required approximating density parameters updates. We illustrate how recently developed streamlined variational inference procedures can be generalized to make fast and accurate inference for the parameters of linear mixed models with nested random effects and global-local priors for Bayesian fixed effects selection. Our variational inference algorithms achieve convergence to the same optima of their standard implementations, although with significantly lower computational effort, memory usage and time, especially for large numbers of random effects. Using simulated and real data examples, we assess the quality of automated procedures for fixed effects selection that are free from hyperparameters tuning and only rely upon variational posterior approximations. Moreover, we show high accuracy of variational approximations against model fitting via Markov Chain Monte Carlo sampling.

Optimal series expansion method for bound states of quantum systems

An optimal series expansion method is developed to improve variational wave functions for the bound state calculation of quantum systems. With one variational parameter residing in the exponential decay factor, the finite power series expansion of the remaining factor in the trial wave function is determined with the criterion that the Schrödinger equation be satisfied as well as possible. In this way, more and more accurate trial wave functions can be obtained without undesirably increasing the number of variational parameters. Thus, instead of the standard variational principle of minimizing the energy expectation value, we utilize an alternative variational principle with which one minimizes the deviation of the trial wave function from being a true eigenstate of the Hamiltonian. The variational principle applies to both the ground state and the excited states. We then employ the optimal series expansion method to calculate the bound states of several quantum systems, including the one-dimensional double well potential, the Morse oscillator, the two-dimensional double well potential, and the vibrational states in the lowest electronic surface of methyl iodide. Computational results indicate that this method can yield excellent results for both energy eigenvalues and eigenfunctions. Therefore, the optimal series expansion method provides an alternative approach to the bound state calculation of quantum systems.

A Generalized Variational Principle with Applications to Excited State Mean Field Theory.

We present a generalization of the variational principle that is compatible with any Hamiltonian eigenstate that can be specified uniquely by a list of properties. This variational principle appears to be compatible with a wide range of electronic structure methods, including mean field theory, density functional theory, multireference theory, and quantum Monte Carlo. Like the standard variational principle, this generalized variational principle amounts to the optimization of a nonlinear function that, in the limit of an arbitrarily flexible wave function, has the desired Hamiltonian eigenstate as its global minimum. Unlike the standard variational principle, it can target excited states and select individual states in cases of degeneracy or near-degeneracy. As an initial demonstration of how this approach can be useful in practice, we employ it to improve the optimization efficiency of excited state mean field theory by an order of magnitude. With this improved optimization, we are able to demonstrate that the accuracy of the corresponding second-order perturbation theory rivals that of singles-and-doubles equation-of-motion coupled cluster in a substantially broader set of molecules than could be explored by our previous optimization methodology.

The Master Equation and the Pauli Equation in the Fuzzy Time Model

The approach proposed in the study is based on the revision of the concept of time as a point on the real axis. It uses the concept of fuzzy time as the set of real numbers with a finite, but not equal to one, function of membership to the time set, i.e. the fuzzy time concept. It is postulated that in fuzzy time t the system dynamics follows from the standard variational principle of the least action and is ordinary Hamilton-Jacobi mechanics. This validates the passage to the limit from fuzzy mechanics to ordinary variational conservative mechanics. The Liouville equation is solved by the method of successive approximations in the time domain of a much larger characteristic scale of fuzziness, using interaction as a small parameter. A standard diagram technique is used. It can be shown that the defuzzification of the Liouville equation inevitably reduces the reversible part in the description to the irreversible evolutionary equation. The latter leads to the second law of thermodynamics. Generalization to the quantum case is possible, i.e. the so-called fuzzy Pauli equation can be drawn.

High Order Schemes for Hyperbolic Problems Using Globally Continuous Approximation and Avoiding Mass Matrices

When integrating unsteady problems using globally continuous representation of the solution, as for continuous finite element methods, one faces the problem of inverting a mass matrix. In some cases, one has to recompute this mass matrix at each time steps. In some other methods that are not directly formulated by standard variational principles, it is not clear how to write an invertible mass matrix. Hence, in this paper, we show how to avoid this problem for hyperbolic systems, and we also detail the conditions under which this is possible. Analysis and simulation support our conclusions, namely that it is possible to avoid inverting mass matrices without sacrificing the accuracy of the scheme. This paper is an extension of Abgrall et al. (in: Karasözen B, Manguoglu M, Tezer-Sezgin M, Goktepe S, Ugur O (eds) Numerical mathematics and advanced applications ENUMATH 2015. Lecture notes in computational sciences and engineering, vol 112, Springer, Berlin, 2016) and Ricchiuto and Abgrall (J Comput Phys 229(16):5653–5691, 2010).

Variational principle of stationary action for fractional nonlocal media and fields

Derivatives and integrals of non-integer orders have a wide application to describe complex properties of physical systems and media including nonlocality of power-law type and long-term memory. We suggest an extension of the standard variational principle for fractional nonlocal media with power-law type nonlocality that is described by the Riesz-type derivatives of non-integer orders. As examples of application of the suggested variational principle, we consider an N-dimensional model of waves in anisotropic fractional nonlocal media, and a one-dimensional continuum (string) with power-law spatial dispersion. The main advantage of the suggested fractional variational principle is that it is connected with microstructural lattice approach and the lattice fractional calculus, which is recently proposed.

Variational principles for multisymplectic second-order classical field theories

We state a unified geometrical version of the variational principles for second-order classical field theories. The standard Lagrangian and Hamiltonian variational principles and the corresponding field equations are recovered from this unified framework.

Frame Element with Mixed Formulations for Composite and RC Members with Bond Slip. I: Theory and Fixed-End Rotation

AbstractThis paper proposes a composite frame element for the simulation of the inelastic response of structural members made up of two or more components with different materials, such as reinforced concrete (RC), steel-concrete composite members, prestressed members, and members with fiber-reinforced polymer (FRP) reinforcement. The element accounts for the relative slip at the interface between the components. Nonlinear geometry effects are accounted for through the corotational formulation, which permits the response simulation of composite frame elements under large displacements. The element formulation enhances the standard Hu-Washizu variational principle with fields describing the bond-slip behavior between components. Three alternatives for the mixed formulation of the element are derived in this paper, which focuses on theory and implementation: mixed-displacement, mixed-force, and mixed-mixed. The paper presents the benefits and shortcomings of the formulation alternatives for modeling the pul...

Combining FFT methods and standard variational principles to compute bounds and estimates for the properties of elastic composites

In this paper, we develop a computational approach based on variational principles combined with FFT technique to determine the effective elastic properties of composite materials. The unit cell problem of the composite is recast in a weak and integral form by making use of classic variational approaches, based on the strain and the stress potential, and the Hashin–Shtrikman variational principles. The problem is discretized with Fourier series and the stationary point is computed numerically by means of an iterative scheme. These algorithms use a representation of the local elastic tensor on the double grid (twice the size of the discretization) which is introduced by the integral formulation; it advantageously accounts for the exact geometry of the microstructure and allows the derivation of rigorous bounds of the effective elasticity coefficients. In the second part of the paper, we establish a hierarchy between the different FFT solutions: the strain, the stress based solutions coming from the classic variational formulations and the polarization based solution derived from the Hashin–Shtrikman principle, which depends on the choice of the elastic moduli of the reference material. It is proved that the strain and the stress based solutions deliver optimal bounds since they are always better than those obtained from the Hashin–Shtrikman variational principle when the same discretization is used. Alternatively, when the elastic moduli of the reference material is negative, the polarization based solution provides an estimate of the effective properties which is comprised between the strain based upper bound and the stress based lower bound.

Power functional theory for Brownian dynamics

Classical density functional theory (DFT) provides an exact variational framework for determining the equilibrium properties of inhomogeneous fluids. We report a generalization of DFT to treat the non-equilibrium dynamics of classical many-body systems subject to Brownian dynamics. Our approach is based upon a dynamical functional consisting of reversible free energy changes and irreversible power dissipation. Minimization of this "free power" functional with respect to the microscopic one-body current yields a closed equation of motion. In the equilibrium limit the theory recovers the standard variational principle of DFT. The adiabatic dynamical density functional theory is obtained when approximating the power dissipation functional by that of an ideal gas. Approximations to the excess (over ideal) power dissipation yield numerically tractable equations of motion beyond the adiabatic approximation, opening the door to the systematic study of systems far from equilibrium.

Holonomy corrections in the effective equations for scalar mode perturbations in loop quantum cosmology

We study the dynamics of the scalar modes of linear perturbations around a flat, homogeneous and isotropic background in loop quantum cosmology. The equations of motion include quantum geometry effects and are expected to hold at all curvature scales so long as the wavelengths of the inhomogeneous modes of interest remain larger than the Planck length. These equations are obtained by including holonomy corrections in an effective Hamiltonian and then using the standard variational principle. We show that the effective scalar and diffeomorphism constraints are preserved by the dynamics. We also make some comments regarding potential inverse triad corrections.

Capillary rise between planar surfaces

Minimization of free energy is used to calculate the equilibrium vertical rise and meniscus shape of a liquid column between two closely spaced, parallel planar surfaces that are inert and immobile. States of minimum free energy are found using standard variational principles, which lead not only to an Euler-Lagrange differential equation for the meniscus shape and elevation, but also to the boundary conditions at the three-phase junction where the liquid meniscus intersects the solid walls. The analysis shows that the classical Young-Dupré equation for the thermodynamic contact angle is valid at the three-phase junction, as already shown for sessile drops with or without the influence of a gravitational field. Integration of the Euler-Lagrange equation shows that a generalized Laplace-Young (LY) equation first proposed by O'Brien, Craig, and Peyton [J. Colloid Interface Sci. 26, 500 (1968)] gives an exact prediction of the mean elevation of the meniscus at any wall separation, whereas the classical LY equation for the elevation of the midpoint of the meniscus is accurate only when the separation approaches zero or infinity. When both walls are identical, the meniscus is symmetric about the midpoint, and the midpoint elevation is a more traditional and convenient measure of capillary rise than the mean elevation. Therefore, for this symmetric system a different equation is fitted to numerical predictions of the midpoint elevation and is shown to give excellent agreement for contact angles between 15 degrees and 160 degrees and wall separations up to 30mm . When the walls have dissimilar surface properties, the meniscus generally assumes an asymmetric shape, and significant elevation of the liquid column can occur even when one of the walls has a contact angle significantly greater than 90 degrees . The height of the capillary rise depends on the spacing between the walls and also on the difference in contact angles at the two surfaces. When the contact angle at one wall is greater than 90 degrees but the contact angle at the other wall is less than 90 degrees , the meniscus can have an inflection point separating a region of positive curvature from a region of negative curvature, the inflection point being pinned at zero height. However, this condition arises only when the spacing between the walls exceeds a threshold value that depends on the difference in contact angles.

Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems

The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of $N$ interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or $\text{MCTDHB}(M)$, where $M$ specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number $M$ of orbitals is demonstrated in a specific example of four interacting bosons in a double well.

Modelling Elastic and Thermoelastic Thick Multilayered Composites by a New Constrained Discrete Layer Approach

A new layer by layer/discrete layer model is presented for the linearly elastic and thermoelastic responses of multilayered composite plates. The transverse variation of the displacement field is defined in terms of a one-dimensional finite element representation, using quadratic Lagrangian interpolations. The laminate thickness is subdivided into a series of one-dimensional finite elements (i.e., thickness subdivisions) whose nodes correspond to planes of constant transverse normal coordinates in the undeformed configuration of the laminate. Each of the displacements and loads is expanded in a double Fourier series in the cartesian surface coordinates. Interface and boundary conditions are exactly verified and exploited to reduce the number of independent generalized displacements, before solving the boundary value problem from the standard variational principle for displacements. This leads to a new model, whose size 3N in 3D, is less than the classic layer by layer approach requiring 3(2N + 1), also in 3D. In addition, this model allows satisfying exactly all the interface and boundary conditions. Numerical evaluations and comparisons with exact 3D elasticity solutions confirm the efficacy of the proposed approach. Finally, the proposed modelling has been extended to the thermoelastic response of laminates and the results are compared with the exact 3D solutions. Temperature distributions required for the thermoelastic analysis are determined from the constrained discrete layer approach previously applied to heat conduction.

First Order LSFEM for Fluid‐Structure Problems

AbstractThe Least‐Squares Finite Element Method (LSFEM) is an interesting alternative to the standard variational principles, which are used to solve partial differential equations. Advantages of the LSFEM are its robustness and the resulting symmetric positive definite matrices, which allow the use of robust iterative solvers like the CG method. In this paper we consider the application of the LSFEM for Fluid‐Structure Interaction (FSI) problems. Our model uses the LSFEM for the discretisation of the instationary incompressible Navier‐Stokes equations, which is coupled with a standard Galerkin FEM model for a linear elastic structure. The results for a simple model problem agree well with results obtained by other authors with different numerical schemes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A primitive model of a charged hard ellipsoidal fluid

A method is developed for the theoretical investigation of the structure of fluids comprised of hard nonspherical molecules carrying electric charge. The development is based upon the mean spherical approximation for the direct correlation function of such fluids. Since the equations based upon this approximation cannot in general be solved analytically an approximate ansatz for the direct correlation function containing a small number of free parameters is introduced. The free parameters are then determined from a standard variational principle. The ansatz itself is chosen to produce the known results in the strong coupling (large charge) limit and in the special case of hard spheres. In order to ensure that the solution in the limit of strong coupling has a tractable analytic form a new model is proposed for the charge distribution on an individual molecule. The method is applied in detail to a model of charged hard ellipsoids which is a generalization of the primitive model for ionic fluids and is found to be practical; it reduces to the primitive model in the special case of charged hard spheres. It is shown that the approximation preserves the conservation of charge. Properties investigated include the direct correlation function itself, the electrostatic energy of the fluid, the pair distribution function and the electrical potential surrounding an individual molecule. Results for these quantities are obtained for a range of densities and charge covering four orders of magnitude and for molecules with elongations (ratios of lengths of axes) from 0.5 (oblate) to 10 (prolate). The direct correlation function itself is given in an analytic form which can be used together with an appropriate density functional to investigate the structure of the electrical double layer formed by the fluid at a solid boundary.

Generalized standard media and variational principles in classical and finite strain elastoplasticity

This paper uses the concept of generalized standard media in order to develop a rational theory of small and finite strain elastoplasticity. Invariance requirements for yield function and flow rule are derived especially for the finite strain case. It is shown that isotropic and kinematic hardening can be incorporated into the general concept in a natural way. Finally the thermodynamic background is discussed and used to derive a novel unified variational principle.