Mean-field Energy Research Articles

Mean-field and direct numerical simulations of magnetic flux concentrations from vertical field

Strongly stratified hydromagnetic turbulence has previously been found to produce magnetic flux concentrations if the domain is large enough compared with the size of turbulent eddies. Mean-field simulations (MFS) using parameterizations of the Reynolds and Maxwell stresses show a negative effective magnetic pressure instability and have been able to reproduce many aspects of direct numerical simulations (DNS) regarding the growth rate of this large-scale instability, shape of the resulting magnetic structures, and their height as a function of magnetic field strength. Unlike the case of an imposed horizontal field, for a vertical one, magnetic flux concentrations of equipartition strength with the turbulence can be reached. This results in magnetic spots that are reminiscent of sunspots. Here we want to find out under what conditions magnetic flux concentrations with vertical field occur and what their internal structure is. We use a combination of MFS, DNS, and implicit large-eddy simulations to characterize the resulting magnetic flux concentrations in forced isothermal turbulence with an imposed vertical magnetic field. We confirm earlier results that in the kinematic stage of the large-scale instability the horizontal wavelength of structures is about 10 times the density scale height. At later times, even larger structures are being produced in a fashion similar to inverse spectral transfer in helically driven turbulence. Using turbulence simulations, we find that magnetic flux concentrations occur for different values of the Mach number between 0.1 and 0.7. DNS and MFS show magnetic flux tubes with mean-field energies comparable to the turbulent kinetic energy. The resulting vertical magnetic flux tubes are being confined by downflows along the tubes and corresponding inflow from the sides, which keep the field concentrated.

Polarization corrections to single-particle energies studied within the energy-density-functional and quasiparticle random-phase approximation approaches

Background: Models based on using perturbative polarization corrections and mean-field blocking approximation give conflicting results for masses of odd nuclei.Purpose: We systematically investigate the polarization and mean-field models, implemented within self-consistent approaches that use identical interactions and model spaces, to find reasons for the conflicts between them.Methods: For density-dependent interactions and with pairing correlations included, we derive and study links between the mean-field and polarization results obtained for energies of odd nuclei. We also identify and discuss differences between the polarization-correction and full particle-vibration-coupling (PVC) models. Numerical calculations are performed for the mean-field ground-state properties of deformed odd nuclei and then compared to the polarization corrections determined using the approach that conserves spherical symmetry.Results: We have identified and numerically evaluated self-interaction (SI) energies that are at the origin of different results obtained within the mean-field and polarization-correction approaches.Conclusions: Mean-field energies of odd nuclei are polluted by the SI energies, and this makes them different from those obtained using polarization-correction methods. A comparison of both approaches allows for the identification and determination of the SI terms, which then can be calculated and removed from the mean-field results, giving the self-interaction-free energies. The simplest deformed mean-field approach that does not break parity symmetry is unable to reproduce full PVC effects.

Nuclear shape coexistence: A study of the even-even Hg isotopes using the interacting boson model with configuration mixing

Background: The Po, Pb, Hg, and Pt region is known for the presence of coexisting structures that correspond to different particle-hole configurations in the Shell Model language or equivalently to nuclear shapes with different deformation. Purpose: We intend to study the configuration mixing phenomenon in the Hg isotopes and to understand how different observables are influenced by it. Method: We study in detail a long chain of mercury isotopes, $^{172-200}$Hg, using the interacting boson model with configuration mixing. The parameters of the Hamiltonians are fixed through a least square fit to the known energies and absolute B(E2) transition rates of states up to $3$ MeV. Results: We obtained the IBM-CM Hamiltonians and we calculate excitation energies, B(E2)'s, quadrupole shape invariants, wave functions, isotopic shifts, and mean field energy surfaces. Conclusions: We obtain a fairly good agreement with the experimental data for all the studied observables and we conclude that the Hamiltonian and the states we obtain constitute a good approximation to the Hg isotopes.

Massless collective excitations in frustrated multiband superconductors

We study collective excitations in three- and four-band superconductors with interband frustration, which causes neither 0 nor $\ensuremath{\pi}$ interband phases in the superconducting state. Using a low-energy spin Hamiltonian originating from a multiband tight-binding model, we find that mass reduction of a Leggett mode occurs in a wide parameter region of this four-band system. As a limiting case, we have a massless Leggett mode. This massless mode is related to the fact that the mean-field energy does not depend on a relative phase of superconducting order parameters. In other words, we find a link of the massless mode with a degeneracy between a time-reversal-symmetry-breaking state (neither 0 nor $\ensuremath{\pi}$ phases) and a time-reversal-symmetric state (either 0 or $\ensuremath{\pi}$ phases). Therefore, the mass of collective modes characterizes well the time-reversal symmetry in frustrated multiband superconductors.

Electromagnetically induced transparency and slow light in quantum degenerate atomic gases

We systematically develop the full quantum theory for the electromagnetic induced transparency (EIT) and slow light properties in ultracold Bose and Fermi gases. It shows a very different property from the classical theory which assumes frozen atomic motion. For example, the speed of light inside the atomic gases can be changed dramatically near the Bose-Einstein condensation temperature, while the presence of the Fermi sea can destroy the EIT effect even at zero temperature. From experimental point of view, such quantum EIT property is mostly manifested in the counter-propagating excitation schemes in either the low-lying Rydberg transition with a narrow line width or in the D2 transitions with a very weak coupling field. We further investigate the interaction effects on the EIT for a weakly interacting Bose-Einstein condensate, showing an inhomogeneous broadening of the EIT profile and nontrivial change of the light speed due to the quantum many-body effects beyond mean field energy shifts.

Interacting boson model with energy density functionals

A brief overview of the recent advancement in the microscopic study of the interacting boson model (IBM) is given. A new nucleon-boson mapping method has been proposed recently, that derives the IBM Hamiltonian based on the self-consistent mean-field model with the microscopic energy density functional (EDF). The mean-field total energy surface computed with a given EDF is mapped onto the analogous energy expectation value in the boson condensate, thereby the energy spectrum and the transition rates are generated. Since the EDF framework allows an accurate global description of nuclear intrinsic properties, the IBM is derived in a unified way, basically for any situation of low-energy quadrupole collective states of nuclei. The basic notion of the new mapping technique is sketched.

Alpha-Like Four-Nucleon Correlations Viewed in Single-Particle Mean Field

The α-like four-nucleon correlations viewed in the single-particle mean field are investigated from the angle neglecting the spatially localized feature of α clustering. First, the energy gain due to the α-like correlations excluding the mean field energy is surveyed in N ≈ Z and N > Z nuclei. Second, the ground states of the N = Z f7/2-shell nuclei are described by the multiple α-like cluster model. Third, what residual interactions in the mean field cause the α-like correlations is investigated by a functional effective interaction in the f7/2-shell model. It is known that the T = 0 four-nucleon units are preferentially built up on the mean field of the ground states in the N ≈ Z f7/2-shell nuclei.

Beyond mean-field ground-state energies and correlation properties of a trapped Bose-Einstein condensate

A two-body correlated basis set is used to develop a many-body theory which is valid for any number of bosons in the trap. The formalism incorporates the van der Waals interaction and two-body correlations in an exact way. The theory has successfully been applied to Bose-Einstein condensates---dilute weakly interacting and also dilute but having a large scattering length. Even in the extreme dilute condition, we observe the breakdown of the shape-independent approximation and the interatomic correlation plays an important role in the large particle-number limit. This correlated many-body calculation can handle, within the two-body correlation approximation, the entire range of atom number of experimentally achieved condensates. Next we successfully push the basis function for large scattering lengths where the mean-field results are manifestly bad. The sharp increase in correlation energy clearly shows the beyond-mean-field effect. We also calculate one-particle densities for various scattering lengths and particle numbers. Our many-body calculation exhibits the finite-size effect in the one-body density.

Application of TDGL Model to B2 Type Ordering with Two Step Phase Separation in Fe-Ni-Al Alloys

ABSTRACTFormation of domain structures in two-step phase separation in Fe-based Fe-Ni-Al alloys are investigated by applying a time-dependent Ginzburg-Landau (TDGL) model. The present authors recently developed a TDGL formulation for ordering processes of B2 and D03 in binary alloys, taking into account the symmetrical relationships between these ordered phases. In this formulation, multiple types of variants of the structures are represented by three order parameters which can be measured independently through crystal structure factors. Mean-field free energies are defined in a form of Landau type expansion with the order parameters and a composition parameter. Interfacial energies due to local variations of degrees of order and composition are given in a gradient square approximation. Kinetic equations for time-evolution of the order parameters and the composition one are derived from the Ginzburg-Landau type potential consisting of the mean-field free energies and the interfacial energy terms. On the other hand, coauthors have investigated domain structures in two-step phase separation of Fe-based Fe-Ni-Al alloys. The evolution of three-dimensional domain structures and composition profiles has been analyzed by electron tomography imaging and energy-dispersive X-ray spectroscopy. In this work the authors performed three-dimensional numerical simulations assuming the thermal processing. The results of the simulations well reproduced the characteristics of the micro-structures obtained from the observations.

An analytic mean-field model for the magnetic response of a ferrofluid monolayer

We extend the analysis of the mean-field model introduced in a previous paper [Piastra and Virga, Soft Matter, 2012, 8, 10969] for a monolayer of spherical colloidal particles with shifted point dipoles studied in previous work [Kantorovich et al., Soft Matter, 2011, 7, 5217]. This system was predicted to exhibit a spontaneous magnetization in the absence of any magnetic field, provided that the temperature is sufficiently low or the particles' density is sufficiently high. A ferromagnetic behavior is confirmed here by a solution in closed form for the mean-field free energy, reminiscent of (but not identical to) the classical solutions of Langevin and Weiss; the closed-form solution also applies in the presence of an external magnetic field, however oriented relative to the layer's normal. Such an analytic solution has perhaps its most telling application to the case where the external magnetic field is applied normal to the layer, where a mixed ferromagnetic and paramagnetic response is predicted. In that case, if the field's strength is not too high, then for high enough temperatures (or low enough densities) the magnetic response is paramagnetic (and with the magnetization aligned to the field), whereas for low enough temperatures (or high enough densities) an extra magnetization grows in the plane of the layer (the vertical component remaining frozen), a remnant of the ferromagnetic transition in the absence of a field. If instead the field's strength exceeds a critical value, any ferromagnetic behavior is suppressed, and the layer becomes paramagnetic at all temperatures and densities.

Evidence from adiabatic scanning calorimetry for the Halperin-Lubensky-Ma effect at the N-SmA phase transitions in mixtures of 7OCB+heptane with an injected SmA phase

The high-resolution adiabatic scanning calorimetric technique has been used to investigate the nematic-smectic A transition (N-SmA in binary mixtures of the non-smectogenic liquid crystal heptyloxycyanobiphenyl (7OCB) and heptane, exhibiting a so-called injected smectic A phase. With the exception of a mixture with the lowest heptane mole fraction for which only an upper limit of 0.2 ± 0.2 J kg(-1) for a possible latent heat could be obtained, for all other mixtures finite latent heats were obtained. The mole fraction dependence of the latent heat could be well fitted with a crossover function consistent with a mean-field free energy expression with a non-zero cubic term arising from the Halperin-Lubensky-Ma (HLM) coupling between the SmA order parameter and the orientational director fluctuations. The mole fraction dependence of the heat capacity effective critical exponents is similar to that observed in mixtures of the two liquid crystals octyloxycyanobiphenyl (8OCB) and nonylcyanobiphenyl (9OCB). The thermal behavior observed along the N-SmA phase transition line yields further strong evidence for the HLM coupling effect.

Generalized approach to global renormalization-group theory for fluids

The global renormalization-group theory (GRGT) for fluids is derived starting with the square-gradient approximation for the Helmholtz free energy functional such that any mean-field free energy density and direct correlation function can be employed. The new derivation uses Wilson's functions for representing density fluctuations, thereby relaxing the assumption of cosine variation of density fluctuations used in earlier approaches. The generality of the present approach is shown by deriving the relationships to the earlier developments. A qualitative way to infer the free parameters in the present form of GRGT is also suggested. The new theory is applied to square-well fluids of ranges 1.5 and 3.0 (in units of hard-sphere diameter) and Lennard-Jones fluids. It is shown that the present theory produces a flat isotherm in the two-phase region. Thus the theory accounts for fluctuations at all length scales and avoids the use of Maxwell's construction. An analysis of the liquid-vapor phase diagrams and the critical constants obtained for different potentials shows that, with a mean-field free energy density that is accurate away from the critical region and an appropriate coarse graining length for the mean-field theory, GRGT can provide results in good agreement with the simulation and experimental results.

Phase structure of monolayer graphene from effective U(1) gauge theory on honeycomb lattice

Phase structure of monolayer graphene is studied on the basis of a U(1) gauge theory defined on the honeycomb lattice. Motivated by the strong coupling expansion of U(1) lattice gauge theory, we consider on-site and nearest-neighbor interactions between the fermions. When the on-site interaction is dominant, the sublattice symmetry breaking (SLSB) of the honeycomb lattice takes place. On the other hand, when the interaction between nearest-neighboring sites is relatively strong, there appears two different types of spontaneous Kekul\'e distortion (KD1 and KD2), without breaking the sublattice symmetry. The phase diagram and phase boundaries separating SLSB, KD1, and KD2 are obtained from the mean-field free energy of the effective fermion model. A finite gap in the spectrum of the electrons can be induced in any of the three phases.

Poisson-Helmholtz-Boltzmann model of the electric double layer: Analysis of monovalent ionic mixtures

In the classical mean-field description of the electric double layer, known as the Poisson-Boltzmann model, ions interact exclusively through their Coulomb potential. Ion specificity can arise through solvent-mediated, nonelectrostatic interactions between ions. We employ the Yukawa pair potential to model the presence of nonelectrostatic interactions. The combination of Yukawa and Coulomb potential on the mean-field level leads to the Poisson-Helmholtz-Boltzmann model, which employs two auxiliary potentials: one electrostatic and the other nonelectrostatic. In the present work we apply the Poisson-Helmholtz-Boltzmann model to ionic mixtures, consisting of monovalent cations and anions that exhibit different Yukawa interaction strengths. As a specific example we consider a single charged surface in contact with a symmetric monovalent electrolyte. From the minimization of the mean-field free energy we derive the Poisson-Boltzmann and Helmholtz-Boltzmann equations. These nonlinear equations can be solved analytically in the weak perturbation limit. This together with numerical solutions in the nonlinear regime suggests an intricate interplay between electrostatic and nonelectrostatic interactions. The structure and free energy of the electric double layer depends sensitively on the Yukawa interaction strengths between the different ion types and on the nonelectrostatic interactions of the mobile ions with the surface.

Modeling the influence of adsorbed DNA on the lateral pressure and tilt transition of a zwitterionic lipid monolayer

Certain lipid monolayers at the air-water interface undergo a second-order transition from a tilted to an untilted liquid-crystalline state of their lipid hydrocarbon chains at sufficiently large lateral pressure. Recent experimental observations demonstrate that in the presence of divalent cations DNA adsorbs onto a zwitterionic lipid monolayer and decreases the tilt transition pressure. Lowering of the tilt transition pressure indicates that the DNA condenses the lipid monolayer laterally. To rationalize this finding we analyze a theoretical model that combines a phenomenological Landau approach with an extension of the Poisson-Boltzmann model to zwitterionic lipids. Based on numerical calculations of the mean-field electrostatic free energy of a zwitterionic lipid monolayer-DNA complex in the presence of divalent cations, we analyze the thermodynamic equilibrium of DNA adsorption. We find that adsorbed DNA induces a 10% reduction of the electrostatic contribution to the lateral pressure exerted by the monolayer. This result implies a small but notable decrease in the tilt transition pressure. Additional mechanisms due to ion-ion correlations and headgroup reorientations are likely to further enhance this decrease.

The Poisson-Helmholtz-Boltzmann model

We present a mean-field model of a one-component electrolyte solution where the mobile ions interact not only via Coulomb interactions but also through a repulsive non-electrostatic Yukawa potential. Our choice of the Yukawa potential represents a simple model for solvent-mediated interactions between ions. We employ a local formulation of the mean-field free energy through the use of two auxiliary potentials, an electrostatic and a non-electrostatic potential. Functional minimization of the mean-field free energy leads to two coupled local differential equations, the Poisson-Boltzmann equation and the Helmholtz-Boltzmann equation. Their boundary conditions account for the sources of both the electrostatic and non-electrostatic interactions on the surface of all macroions that reside in the solution. We analyze a specific example, two like-charged planar surfaces with their mobile counterions forming the electrolyte solution. For this system we calculate the pressure between the two surfaces, and we analyze its dependence on the strength of the Yukawa potential and on the non-electrostatic interactions of the mobile ions with the planar macroion surfaces. In addition, we demonstrate that our mean-field model is consistent with the contact theorem, and we outline its generalization to arbitrary interaction potentials through the use of a Laplace transformation.

Adapting Poisson-Boltzmann to the self-consistent mean field theory: Application to protein side-chain modeling

We present an extension of the self-consistent mean field theory for protein side-chain modeling in which solvation effects are included based on the Poisson-Boltzmann (PB) theory. In this approach, the protein is represented with multiple copies of its side chains. Each copy is assigned a weight that is refined iteratively based on the mean field energy generated by the rest of the protein, until self-consistency is reached. At each cycle, the variational free energy of the multi-copy system is computed; this free energy includes the internal energy of the protein that accounts for vdW and electrostatics interactions and a solvation free energy term that is computed using the PB equation. The method converges in only a few cycles and takes only minutes of central processing unit time on a commodity personal computer. The predicted conformation of each residue is then set to be its copy with the highest weight after convergence. We have tested this method on a database of hundred highly refined NMR structures to circumvent the problems of crystal packing inherent to x-ray structures. The use of the PB-derived solvation free energy significantly improves prediction accuracy for surface side chains. For example, the prediction accuracies for χ(1) for surface cysteine, serine, and threonine residues improve from 68%, 35%, and 43% to 80%, 53%, and 57%, respectively. A comparison with other side-chain prediction algorithms demonstrates that our approach is consistently better in predicting the conformations of exposed side chains.

Mesoscopic TDGL Model for Formation of Domain Structures in D0<sub>19</sub> Type Ordering

The present authors recently presented the time-dependent Ginzburg-Landau (TDGL) formulation for L12 type ordering processin binary alloys, taking into account the symmetrical relationships of these ordered phases. Extending the formulation, the authors have developed the TDGL model for microstructural evolution of D019 type ordering. The D019 structure based on hcp is divided into four equivalent sublattices. The site occupation probabilities are given as a function of three order parameters and a composition parameter. Multiple types of variants of the structures are represented by the order parameters. Mean-field free energies are defined in a form of Landau type expansion with the order parameters and the composition parameter. Interfacial energies due to local variations of degrees of order and composition are given in a gradient square approximation. Kinetic equations for time-evolution of the order parameters and the composition one are derived from the Ginzburg-Landau type potential consisting of the mean-field free energies and the interfacial energy terms. Three-dimensional numerical simulations based on the kinetic equations have been performed, and the domain structures obtained are compared with a TEM image of Cu3Sn alloy.

Microscopic calculation of interacting boson model parameters by potential-energy surface mapping

A coherent state technique is used to generate an interacting boson model (IBM) Hamiltonian energy surface which is adjusted to match a mean-field energy surface. This technique allows the calculation of IBM Hamiltonian parameters, prediction of properties of low-lying collective states, as well as the generation of probability distributions of various shapes in the ground state of transitional nuclei, the last two of which are of astrophysical interest. The results for krypton, molybdenum, palladium, cadmium, gadolinium, dysprosium, and erbium nuclei are compared with experiment.

Ground State at High Density

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (in J. Phys. (USSR) IX:25–40, 1945), Kirzhnits and Nepomnyashchii (in Sov. Phys. JETP 32:1191–1197, 1971), and Likos et al. (in J. Chem. Phys. 126:224502, 2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superimposed on the sites of a close-packed lattice.