Metric Properties of Bogoliubov Trajectories in Statistical Equilibrium Theory

We investigate some properties of the Bogoliubov measure that appear in statistical equilibrium theory for quantum systems and establish the nondifferentiability of the Bogoliubov trajectories in the corresponding function space. We prove a theorem on the quadratic variation of trajectories and study the properties implied by this theorem for the scale transformations. We construct some examples of semigroups related to the Bogoliubov measure. Independent increments are found for this measure. We consider the relation between the Bogoliubov measure and parabolic partial differential equations.     

A modified Bogoliubov method applied to a simple boson model

We use a non-gauge-invariant modification of the exact Hamiltonian to obtain a new Hamiltonian-like operator for a simple exactly solvable boson model. The eigenvalues of the new operator are close to those of the original Hamiltonian. We make a one-body approximation of the new two-body operator in the spirit of the Bogoliubov approximation. Because only the number operator appears, the c-number approximation is not required individually for the creation or annihilation operators in the ground state. For the simple model, the results using the new approximation are closer to the exact results than the usual Bogoliubov results over a wide range of parameters. The improvement increases dramatically as the model interaction strength increases.     

Some asymptotic formulas for the Bogoliubov gaussian measure

We consider problems of integrating over the Bogoliubov measure in the space of continuous functions and obtain asymptotic formulas for one class of Laplace-type functional integrals with respect to the Bogoliubov measure. We also prove related asymptotic results concerning large deviations for the Bogoliubov measure. For the basic functional, we take the Lp norm and establish that the Bogoliubov trajectories are Höldercontinuous of order γ < 1/2.     

Bogoliubov Quasiaverages: Spontaneous Symmetry Breaking and the Algebra of Fluctuations

We present arguments supporting the use of the Bogoliubov method of quasiaverages for quantum systems. First, we elucidate how it can be used to study phase transitions with spontaneous symmetry breaking (SSB). For this, we consider the example of Bose–Einstein condensation in continuous systems. Analysis of different types of generalized condensations shows that the only physically reliable quantities are those defined by Bogoliubov quasiaverages. In this connection, we also solve the Lieb–Seiringer–Yngvason problem. Second, using the scaled Bogoliubov method of quasiaverages and considering the example of a structural quantum phase transition, we examine a relation between SSB and critical quantum fluctuations. We show that the quasiaverages again provide a tool suitable for describing the algebra of critical quantum fluctuation operators in both the commutative and noncommutative cases.     

Scale-invariant asymptotic (automodel) behavior in quantum field theory

We summarize the results of our 40-year investigations of scale-invariant (automodel) behavior of form factors in hadron-lepton deep-inelastic scattering processes at high energies and large transferred momenta within the Bogoliubov axiomatics of quantum field theory. This approach was conducive to the emergence of the notion of quark color in quantum chromodynamics.     

Bogoliubov group variables in relativistic field theory

We define the Bogoliubov variables for strongly coupled systems that are invariant under the Poincaré group in (1+1)-dimensional space-time. This allows us to achieve a compatibility between taking the conservation laws into account exactly and developing a regular perturbation theory. We perform the secondary quantization in terms of the Bogoliubov variables and discuss the problem of reducing the number of states of the field. We also discuss the conditions for validity of the perturbation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 2, pp. 242–251, May, 1997.     

Gaussian functional integrals and Gibbs equilibrium averages

We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. This measure arises in the Bogoliubov T-product approach and is non-Wiener. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2, pp. 345–352, May, 1999.     

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the L^p functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the L^p norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p ₀ = 2+4π ² /β ² ω ₂ , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p ₀ and p > p ₀ . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.     

Ultratertiary Quantization of Thermodynamics

We show that if the Dirac–Bogoliubov rule for replacing the bosonic creation and annihilation operators with the c-numbers is used, then the ultratertiary quantization allows obtaining the Bardeen–Cooper–Schrieffer–Bogoliubov formulas.     

Asymptotic behavior of small deviations for Bogoliubov’s Gaussian measure in the L p norm, 2 ≤ p ≤ ∞

We prove several results on exact asymptotic formulas for small deviations in the L^p-norm with 2 ～ p ～ ∞ for Bogoliubov’s stationary Gaussian process ξ(t). We prove the property of mutual absolute continuity for the conditional Bogoliubov measure and the conditional Wiener measure and calculate the Radon-Nikodym derivative.     

On the motion of a generalized van der Pol oscillator

A generalized van der Pol oscillator is considered, with positive real power nonlinearities in the restoring and damping force, including fractional powers. An analytical approach based on the Krylov–Bogoliubov method is adjusted to derive analytical expressions for the amplitude of a limit cycle for small values of the damping coefficient. These expressions are also derived for some integer power nonlinearities in the equation of motion and the results obtained compared with the existing results from the literature. Relaxation oscillations are studied for larger values of the damping coefficient. Matched asymptotic expansions are used and the influence of the powers of the restoring and damping force on the period of these oscillations is investigated. It is shown that not only can the period increase with the damping power, but it can also have a decreasing trend for some cases and the condition for this to hold is obtained.     

Interaction between subbands in a quasi-one-dimensional superconductor

Theoretical and Mathematical Physics - In the framework of the Bogoliubov–de Gennes equation, we study the spinless $$p$$ -wave superconductor in an infinite strip in the presence of some...     

The problem of phase transitions in superfluid and normal liquids

A new interpretation of the well-known spectrum obtained by Bogoliubov for boson gas and of the role of the angular velocity as a thermodynamic variable determining a phase transition of the second kind is presented. A new averaging method over clusters that results in a new spectral series and a different phase transition is suggested. This method is generalized to the classical equations, which permits conjecturing a phase transition with respect to velocity from a laminar flow to a turbulent one. This paper is published for discussion. Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 492–508, December, 1999.     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

Perturbation of Solutions with Moving Singularities

We study the Cauchy problem for a nonlinear second-order differential equation with a small parameter in the case where the exact solution has a power singularity depending on a small parameter. We propose an asymptotic method similar to the Krylov–Bogoliubov method for localizing the singularity up to the accuracy of any order and construct an asymptotic expansion of the solution in the domain of regular behavior.     

Bogoliubov equations and functional mechanics

The functional classical mechanics based on the probability approach, where a particle is described not by a trajectory in the phase space but by a probability distribution, was recently proposed for solving the irreversibility problem, i.e., the problem of matching the time reversibility of microscopic dynamics equations and the irreversibility of macrosystem dynamics. In the framework of functional mechanics, we derive Bogoliubov-Boltzmann-type equations for finitely many particles. We show that a closed equation for a one-particle distribution function can be rigorously derived in functional mechanics without any additional assumptions required in the Bogoliubov method. We consider the possibility of using diffusion processes and the Fokker-Planck-Kolmogorov equation to describe isolated particles.     

Mutual transition of Andreev and Majorana bound states in a superconducting gap

Theoretical and Mathematical Physics - Using the Bogoliubov–de Gennes Hamiltonian, we analytically study two models with superconducting order, the p-wave model with an impurity potential and...     

Bogoliubov’s causal perturbative QED and white noise. Interacting fields

Theoretical and Mathematical Physics - We present Bogoliubov’s causal perturbative QFT with a single refinement: the creation–annihilation operators at a point, i.e., for a specific...     

组合评价模型在港口物流综合服务能力评价中的应用

为了对港口物流综合服务能力进行科学评价,提出了基于组合评价方法的港口物流服务能力评价模型,并对我国15个港口的物流综合服务能力进行实证分析说明模型的有效性.首先分别采用5种评价方法进行评价,然后使用KendallW系数检验各方法评价结果的一致性,再运用4种组合评价模型进行组合评价,采用Spearman等级系数进行事后检验并得到最终结果.方法可克服单一评价方法的不足,为港口物流服务能力评价提供新的思路.