Bogolyubov Inequality Research Articles

Infinite-Volume Limit of Stochastic s-d System with Single-Component Impurity and s-Electron Spins

A Hamiltonian [Formula: see text], with locally smeared Ising-type s-d exchange between s-electrons and magnetic impurities, in a dilute magnetic alloy, is investigated. The Feynman-Kac theorem, Laplace expansion and Bogolyubov inequality are applied to obtain a lower and upper bound (lb and ub) on the system’s free energy per conducting electron [Formula: see text]. The two bounds differ, in the infinite-volume limit by a term [Formula: see text], linear in impurity concentration: lb[Formula: see text], ub[Formula: see text], [Formula: see text] denoting the Hamiltonian of the approximating mean-field s-d system. [Formula: see text] represents randomly positioned impurities interacting with a mean field implemented by the gas of conduction s-electrons, the latter interacting with the field of barriers and wells (according to the s-electron’s spin orientation) localized at the impurity sites. The inequality [Formula: see text] demonstrates increasing accuracy of the mean-field [Formula: see text]-theory, with decreasing impurity concentration.

Phase diagram of the J_{1}-J_{2} frustrated anisotropic antiferromagnet with spin S=1 on the quadratic lattice.

In the paper the phase diagram of J_{1}-J_{2} frustrated antiferromagnet with spin S=1 and single-ion anisotropy is studied on the planar quadratic lattice in the cluster approximation. The Bogolyubov inequality is adopted for the Gibbs energy calculation for the case of 2×2 and 4×4 clusters. On this basis, the ranges of existence of the antiferromagnetic, superantiferromagnetic, and paramagnetic phases are investigated for the antiferromagnetic nearest-neighbor (J_{1}<0) and next-nearest-neighbor (J_{2}<0) interactions. In particular, the occurrence of tricritical and triple points is discussed and a comparison between the results for 2×2 and 4×4 clusters is made. The results are also compared with the classical MFA method, adopted here for the model in question, as well as with selected literature results for particular choices of interaction parameters.

Bogolyubov inequality for the ground state and its application to interacting rotor systems

We have formulated and proved the Bogolyubov inequality for operators at zero temperature. So far this inequality has been known for matrices, and we were able to extend it to certain class of operators. We have also applied this inequality to the system of interacting rotors. We have shown that if: {\em i)} the dimension of the lattice is 1 or 2, {\em ii)} the interaction decreases sufficiently fast with a distance, and {\em iii)} there is an energy gap over the ground state, then the spontaneous magnetization in the ground state is zero, i.e. there is no LRO in the system. We present also heuristic arguments (of perturbation-theoretic nature) suggesting that one- and two-dimensional system of interacting rotors has the energy gap independent of the system size if the interaction is sufficiently small.

Equation of state, transport coefficients, and stopping power of dense plasmas from the average-atom model self-consistent approach for astrophysical and laboratory plasmas

Calculations of equation of state, transport coefficients, and stopping power of dense plasmas are presented. Theoretical results have been obtained using the first-principles average-atom model self-consistent approach for astrophysical and laboratory plasmas (SCAALP) based on the finite-temperature density-functional theory and the Gibbs–Bogolyubov inequality. Numerical results, comparisons with molecular dynamics, and Monte Carlo simulations and experiments are presented and discussed in the high energy density physics domain including part of the warm dense matter regime. Results show that the average-atom model SCAALP is well suited to describe thermodynamic and transport properties for a wide range of high energy density physics applications.

COEXISTENCE OF NON-CONVENTIONAL CONDENSATES IN TWO-LEVEL BOSE ATOM SYSTEM

In the framework of the Bogolyubov approximation and using the Bogolyubov inequalities, we give a simple proof of the coexistence of two non-conventional Bose–Einstein condensates in the case of some superstable Bose system whose atoms have an internal two-level structure, and their energy operators in the second quantized form depend on the number operators only.

Hard-Sphere variational approach using anab initiomolecular dynamics code

We present a variational method to determine the total free energy of the electron and ion system by using the Gibbs--Bogolyubov inequality and a hard-sphere reference system applied to the ab initio molecular dynamics code CPMD. Electrical conductivity and transport coefficients are calculated. Numerical results and comparisons with quantum molecular dynamics simulations and experiments are presented and discussed for dense and expanded aluminum. The variational approach using the ab initio molecular dynamics code CPMD is more than ten times faster than CPMD simulations.

Variational-method study of the spin-1 Ising model with biaxial crystal-field anisotropy

The phase diagram of the spin-1 Ising model in the presence of a biaxial crystal-field anisotropy is studied within the framework of a variational approach, based on the Bogolyubov inequality for the free energy. We have investigated the effects of a transverse crystal field Dy on the phase diagram in the T-Dx plane. Results obtained by using effective-field theory (EFT) on the honeycomb (z=3), square (z=4), and simple cubic (z=6) lattices (z is the coordination number) show only continuous phase transitions, while the variational approach presents first-order and continuous phase transitions for Dy=0. We have also used the EFT for larger values of z and we observe the presence of tricritical points in the phase diagrams, for z>or=7, in accordance with the variational approach results.

Electrical conductivity of warm expanded aluminum

The electronic and ionic structures of warm expanded aluminum are determined self-consistently using an average-atom formalism based on density-functional theory and Gibbs–Bogolyubov inequality. Ion configurations are generated using a least-square fit of the pair distribution function deduced from the average-atom model calculations. The electrical conductivity of the system is computed from the Kubo–Greenwood formula for the optical conductivity implemented in a molecular dynamics scheme based on density-functional theory. This method goes beyond the Ziman approach commonly used in the average-atom formalism. Moreover, it is faster than performing ab initio molecular dynamics simulations to obtain ion configurations for the conductivity calculation. Numerical results and comparisons with experiments are presented and discussed.

Mean-field description of a dilute s– d system

The Feynman–Kac theorem and Bogolyubov inequality are applied to obtain a lower bound and an upper bound to the free energy of the s– d Hamiltonian with locally smeared interactions between electrons and impurities. The two bounds, which express in terms of the free energy of impurities in a mean field and electrons in a field of barriers and wells localized at the impurity sites, are almost equal if the impurity concentration is sufficiently small or s– d coupling sufficiently weak.

Two theorems on pseudo-spin of Hubbard models

An inequality of the eigenvalues of the reduced density ρ 2 at finite temperature of the Hubbard model is obtained by means of the Bogolyubov inequality. The quasi-average of S ̃ + in a simple symmetry-breaking perturbation of the Hamiltonian for a bipartite lattice is shown to be zero.

Generalization of the mean-field Ising model within Tsallis thermostatistics

In this study, the mean-field Ising model, using the Bogolyubov inequality which has been obtained in the framework of the generalized statistical thermodynamics (GST), suitable for non-extensive systems, has been investigated. Generalized expressions for the mean-field magnetization and free energy have been established. These new results have been verified by the fact that they transform to the well-known Boltzmann-Gibbs results in the q → 1 limiting case. For the index q which characterizes the fractal structure of the magnetic system, an interval has been established where the generalized mean-field free energy has a minimum and mean-field magnetization has a corresponding finite value. The interval of q is consistent with paramagnetic free spin systems.

Variational method in generalized statistical mechanics

Concavity properties of a recently generalized (not necessarily extensive) entropy enable, among others, the generalization of the Bogolyubov inequality, hence of the variational method in equilibrium statistical mechanics.

Density functional taking into account nuclear thermal motion of compressed matter

Bogolyubov's inequality is used to generalize the density functional method to take into account nuclear thermal motion. The results apply to a two-component system in which the nuclei satisfy classical statistics. Applications to a cell model describing dense heated states of matter are considered.

On the Thermodynamical‐Statistical Consistency of a Variational Approximation for the Free Energy

AbstractAn upper bound for the free energy is found which is more general than the Bogolyubov inequality. It generates an approximation which can accurately preserve the consistency between thermodynamic and statistical mechanics definitions.

Variational principles in quantum statistical mechanics

The principle of least information is used to derive the inequality between the arithmetic and the geometric mean. Some important variational results in quantum statistical mechanics (Peierls and Bogolyubov inequalities) are pedagogically reviewed. Emphasis is put on the distinction between mixed states describing metastable and stable equilibrium.

The Surface or Interface Polaron in Polar Crystals. III. The Phase‐Transition‐Like Transition for the Surface Polaron

AbstractThe behaviour of the surface polarons in polar crystals is taken into account by making use of the Bogolyubov inequality. The results show that the ground state energy, effective mass, and the effective potential energy of the surface polarons are dependent on the temperature. For the strong coupling surface polaron, when the temperature approaches a certain critical value T0, there exists a phase transition from the polaron to the free electron; but for the weak coupling case, such a phase transition does not exist.

Theoretical study of miscibility and glass-forming trends in mixtures of polystyrene spheres

We present a theoretical study of glass-forming trends and miscibility in mixtures of polystyrene spheres (‘‘polyballs’’) of different diameters, suspended in an aqueous solution. The polyballs are assumed to be charged and to interact via a Debye–Hückel screened Coulomb potential. The Helmholtz free energy is calculated from a variational principle based on the Gibbs–Bogolyubov inequality, in which a mixture of hard spheres of different diameters is chosen as the reference system. It is found that when the charges of the two types of polyballs are sufficiently different, the variationally determined ratio of hard-sphere diameters differs substantially, leading to packing difficulties characteristic of glass formation. The experimentally observed range of glass formation corresponds to a ratio of hard-sphere diameters of 0.8 or less. Calculations of the free energy as a function of concentration indicate that the liquid polyball mixture is stable against the phase separation, even for widely different polyball charges.

Landau theory of the ferroelastic phase transition in ionic molecular solids: A microscopic approach

Starting from Bogolyubov's inequality, we derive a variational form for the Helmholtz free energy $F$ to study the ferroelastic phase transitions in simple ionic molecular solids. The coefficients of the Landau expansion of $F$ in terms of the five orientational order parameters (${\ensuremath{\eta}}_{i},i=1,\dots{},5$) are calculated from a rotation-translation Hamiltonian which has been used with reasonable success to explain the thermoelastic anomalies of the orientationally disordered phase. Detailed calculations have been performed for CsCN to understand the first-order phase transition from the pseudocubic ($\mathrm{Pm}3m$) to trigonal ($R3m$) phase. Various aspects of this transition resulting from the interplay of direct and lattice-mediated interactions and limitations of the currently used models to describe the physical properties of alkali cyanides are discussed.

Application of the Bogolyubov inequality to the calculation of thermodynamic functions of triatomic molecules

The Bogolyubov inequality is used in the calculation of the vibrational part of of thermodynamic functions of triatomic molecules where the vibrations are anharmonic. By variation of parameters in the approximate Hamiltonian, we obtain the best agreement between the free energy of the system based on the approximate Hamiltonian, and the exact free energy-of the system. The method is applied to the calculation of the thermodynamic functions for H2O and C3.

Application of the Bogolyubov inequality for the calculation of the thermodynamic properties of diatomic molecules

The frequency of vibration and the interatomic distance are considered as variational parameters in an approximate expression for the free energy of a diatomic molecule. Application of the Bogolyubov inequality allows one to find the dependence of these parameters on temperature. Calculations are given for the molecules HF and HgH.