Bogoliubov–Born–Green–Kirkwood–Yvon Research Articles

Generalization of the Gross–Pitaevskii equation for multi-particle cases

A new method is proposed to obtain Gross–Pitaevskii equation by the chain of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) quantum kinetic equations. In that sense, we investigate the dynamics of a quantum system including infinite number of identical particles which interact via a (special) pair potential on the form of Dirac delta-function.

Determination of the single particle distribution function in a weakly correlated weakly inhomogeneous plasma

Single particle distribution function of plasma particles has been derived from the first member of the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy utilising the pair correlation function evaluated in [Bose, Phys. Plasmas 26, 064501 (2019)] from the second member of the BBGKY hierarchy. This distribution function may be employed to probe the thermodynamic properties of the weakly inhomogeneous plasma systems.

Evolution of a quantum system of many particles interacting via the generalized Yukawa potential

We study the evolution of a system of N particles that have identical masses and charges and interact via the generalized Yukawa potential. The system is placed in a bounded region. The evolution of such a system is described by the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) chain of quantum kinetic equations. Using semigroup theory, we prove the existence of a unique solution of the BBGKY chain of quantum kinetic equations with the generalized Yukawa potential.

Self-consistent RPA and the time-dependent density matrix approach

The time-dependent density matrix (TDDM) or BBGKY (Bogoliubov, Born, Green, Kirkwood, Yvon) approach is decoupled and closed at the three-body level in finding a natural representation of the latter in terms of a quadratic form of two-body correlation functions. In the small amplitude limit an extended RPA coupled to an also extended second RPA is obtained. Since including two-body correlations means that the ground state cannot be a Hartree-Fock state, naturally the corresponding RPA is upgraded to Self-Consistent RPA (SCRPA) which was introduced independently earlier and which is built on a correlated ground state. SCRPA conserves all the properties of standard RPA. Applications to the exactly solvable Lipkin and the 1D Hubbard models show good performances of SCRPA and TDDM.

A nonperturbative solution of the nonlinear BBGKY hierarchy for marginal correlation operators

We consider the problem of the rigorous description of nonequilibrium quantum correlations. Within the framework of an alternative approach to the description of the evolution of states of finitely many particles in terms of correlation operators governed by the von Neumann hierarchy, we derive the nonlinear quantum Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy for marginal correlation operators that is adopted to the description of quantum infinite‐particle systems as well. A nonperturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed. Copyright © 2013 John Wiley & Sons, Ltd.

Symmetries of a mean-field spin model

Thermodynamic limit evolution of a closed quantum Heisenberg-type spin model with mean-field interactions is characterized by classifying all the symmetries of the equations of motion. It is shown that parameters of the model induce a structure in the Hilbert space by partitioning it into invariant subspaces, decoupled by the underlying Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. All possible partitions are classified in terms of a 3 × 3 matrix of effective, thermodynamic limit coupling constants. It is found that there are either 1, 2, 4, or O(N) invariant subspaces. The BBGKY hierarchy decouples into the corresponding number of anti-Hermitian operators on each subspace. These findings imply that equilibration and the equilibrium in this model depend on the initial conditions.

Zero-Collision Term Problem in Kinetic Theory of One-Dimensional Systems

In the kinetic theory of one-dimensional (1d) systems, the collision terms, such as the Boltzmann, Landau, and Balescu–Lenard collision terms, are identically zero, when the masses of all the particles are the same. This apparently implies that there is no relaxation in 1d systems. However, relaxations are observed in numerical simulations. The problem is due to an insufficient approximation in the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. Zubarev and Novikov's theory [Teor. Mat. Fiz. 18 (1974) 78; ibid. 19 (1974) 237] for summing up all the contributions of all the orders of the BBGKY hierarchy is applied to 1d systems. As a result, the zero-collision term problem is solved. The unique properties of 1d systems are revealed.

Non-equilibrium stationary states from the equation of motion of open systems

Non-equilibrium stationary states are solved from the equation of motion of open systems via a Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY)-like hierarchy. As an example, we demonstrate this approach for the Redfield equation, which is derived from the whole dynamic equation of motion of a central system coupled to two baths through the projector technique. Generalization to other equations of motion is straightforward. For non-interacting central systems, the first equation out of the hierarchy is closed, whereas for interacting systems it is coupled to higher-order equations in the hierarchy. In the case of interacting systems, two systematic approximations, in the form of perturbation theories, are proposed to truncate and solve the hierarchy. A non-equilibrium Wick's theorem is proved to provide a basis for the perturbation theories. As a test of reliability of the proposed methods, we apply them to small systems, where it is also possible to apply other exact direct methods. Consistent results were found.

Kinetic theory of instability-enhanced collisional effects

The Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy is used to derive a generalization of the Lenard–Balescu plasma kinetic equation that accounts for wave-particle scattering due to instabilities that originate from discrete particle motion. Application to convective instabilities is emphasized for which the growing waves either propagate out of the domain of interest or modify the particle distribution to reduce the instability amplitude before nonlinear amplitudes are reached. Two such applications are discussed: Langmuir’s paradox and determining the Bohm criterion for multiple ion species plasmas. In these applications, collisions are enhanced by ion-acoustic and ion-ion two-stream instabilities, respectively. The relationship between this kinetic theory and quasilinear theory is discussed.

A stochastic step flow model with growth in 1+1 dimensions

Mathematical implications of adding Gaussian white noise to the Burton–Cabrera–Frank model for N terraces (‘gaps’) on a crystal surface are studied under external material deposition for large N. The terraces separate straight, non-interacting line defects (steps) with uniform spacing initially (t = 0). As the growth tends to vanish, the gaps become uncorrelated. First, simple closed-form expressions for the gap variance are obtained directly for small fluctuations. The leading-order, linear stochastic differential equations are prototypical for discrete asymmetric processes. Second, the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy for joint gap densities is formulated. Third, a self-consistent ‘mean field’ is defined via the BBGKY hierarchy. This field is then determined approximately through a terrace decorrelation hypothesis. Fourth, comparisons are made of directly obtained and mean-field results. Limitations and issues in the modeling of noise are outlined.

Flowing with time: a new approach to non-linear cosmological perturbations

Non-linear effects are crucial for computing the cosmological matter power spectrum to theaccuracy required by future generation surveys. Here, a new approach is presented,in which the power spectrum, the bispectrum and higher order correlations areobtained—at any redshift and for any momentum scale—by integrating a systemof differential equations. The method is similar to that of the familiar BBGKY(Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy. Truncating at the level ofthe trispectrum, the solution of the equations corresponds to the summation ofan infinite class of perturbative corrections. Compared to other resummationframeworks, the scheme discussed here is particularly suited to cosmologies other thanΛCDM (CDM: cold dark matter), such as those based on modifications of gravity and those containingmassive neutrinos. As a first application, we compute the baryonic acoustic oscillation featureof the power spectrum, and compare the results with perturbation theory, the halo model, andN-body simulations. The density–velocity and velocity–velocity power spectra are alsocomputed, revealing that they are much less contaminated by non-linearities than thedensity–density one. The approach can be seen as a particular formulation of therenormalization group, in which time is the flow parameter.

A formal method for solving the stationary BBGKY hierarchy for a classical inhomogeneous fluid

A methodology is developed for finding a formal solution to the entire stationary Bogoliubov, Born, Green, Kirkwood, Yvon (BBGKY) hierarchy for a classical inhomogeneous fluid in thermal equilibrium. The method is unique in that neither a closure relationship nor a restriction on the density or the temperature of the fluid is required. The decisive step in the analysis is the creation of a more general system of coupled integrodifferential equations through the introduction of a coupling parameter. Recognizing that the coupling parameter can also act as an expansion parameter, series solutions to the generalized hierarchy can be obtained through the application of regular perturbation theory. At the end of the calculation, a formal solution to the original BBGKY hierarchy can be recovered by setting the coupling parameter to unity.

A quantum kinetic theory of moderately dense gases. I. Wigner triplet distribution function

The quantum mechanical BBGKY (Bogoliubov−Born−Green−Kirkwood−Yvon) hierarchy is truncated and solution of the third BBGKY equation effected, within the ternary collision approximation, to yield an approximate Wigner triplet distribution function. This triplet distribution function is obtained as a functional of the pair and singlet distribution functions by the imposition of a generalization of the molecular chaos hypothesis. The potential is assumed to be pair−wise additive, but the discussion is not restricted to purely repulsive interactions. The resultant triplet distribution function may be used in the development of a quantum mechanical kinetic theory of moderately dense gases.

Statistical Frameworks for Weak Plasma Turbulence

A hierarchy is constructed describing the time evolution of correlations due to collective interactions in a weakly turbulent, spatially uniform ensemble of Vlasov plasmas. It is shown that this ``Vlasov hierarchy'' is identical to the Bogoliubov—Born—Green—Kirkwood—Yvon (BBGKY) hierarchy for irreducible correlations in a spatially homogeneous plasma, provided the effects of single particle encounters are deleted from the BBGKY framework. In addition, the random phase approximation used in conventional treatments of weak plasma turbulence is shown to be automatically embodied by such a formalism. Upon ordering the system of equations in the weak turbulence parameter, 〈E2〉/nmvav2, the questions of closure and time validity are examined.

Correlations in plasmas: III. Green's function solutions for correlation functionals in Bogoliubov theory

We use a set of eigendistributions in constructing the formal solution of tho set of integrodifferential equations for the correlation functionals in the context of the Bogoliubov formalism. The equations apply to fully ionized systems interacting through the Coulomb potential, and the method is displayed to an arbitrary order in the interaction strength for the homogeneous case. It is indicated that the method is equally applicable to the slightly inhomogeneous systems for which a perturbation series solution can be developed formally. It is also possible to apply this method in solving the initial-value problem of the BBGKY (Bogoliubov, Born, Green, Kirkwood, Yvon) hierarchy. The eigendistributions employed are those developed by Van Kampen and Case in connection with the linearized Vlasov equation. Although the resulting Green's function is quite complicated, it exhibits the general structure of the solutions.