BBGKY Hierarchy Of Equations Research Articles

Statistical Theory of Four-Point Distribution Functions in MHD Turbulent Flow

In this paper, the four-point distribution functions for simultaneous velocity, magnetic, temperature and concentration fields in MHD turbulent flow have been studied. It is tried to derive the transport equation for four-point distribution function in MHD turbulent flow. The obtained equation is compared with the first equation of BBGKY hierarchy of equations and the closure difficulty is to be removed as in the case of ordinary turbulence.

Effect of Chemical Reaction on Statistical Theory of Dusty Fluid MHD Turbulent Flow for Certain Variables at Three- Point Distribution Functions

In this paper, an attempt is made to study the three-point distribution functions in dusty fluid MHD turbulent flow for simultaneous velocity, magnetic temperature and concentration fields in a first order chemical reaction. It has been discussed the various properties of constructed distribution functions. From beginning to end out the study, the transport equation for three-point distribution functions in dusty fluid MHD turbulent flow undergoing a first order chemical reaction has been obtained. The obtained equation is compared with the first equation of BBGKY hierarchy of equations and the closure difficulty is to be removed as in the case of ordinary turbulence.

Gravitational clustering of galaxies in an expanding universe

We inquire the phenomena of clustering of galaxies in an expanding universe from a theoretical point of view on the basis of ther- modynamics and correlation functions. The partial differential equation is developed both for the point mass and extended mass structures of a two-point correlation function by using thermodynamic equations in com- bination with the equation of state taking gravitational interaction between particles into consideration. The unique solution physically satisfies a set of boundary conditions for correlated systems and provides a new insight into the gravitational clustering problem. Galaxies interact gravitationally and the characterization of this clustering is a problem of current interest. The gravitational interaction between galaxies and galaxy clusters have played an important role in the evolution of the observed universe. Various the- ories of the cosmological many body problem have been developed mainly from a thermodynamic point of view. We have made use of the equations of state along with the correlation functions for the development of a theoretical model. This can be done by solving a system of Liouville's equation or BBGKY-hierarchy equations and have been discussed by workers like Saslaw (1972) and Peebles (1980). But BBGKY hier- archy equations are too complicated to handle for higher order correlation functions. However, the lowest order two-point correlation function is useful for discussing the phenomenon of gravitational clustering of galaxy clusters which contain information on all the higher n-particle correlations in the full BBGKY-hierarchy (Peebles 1980; Zhan & Dyer 1989; Hamilton 1993). The physical validity of the application of ther- modynamics in the clustering of galaxies and galaxy clusters has been discussed on the basis of N-body computer simulation results (Itohetal. 1993). In gravitational thermo- dynamics the value of b which is the ratio of gravitational correlation energy to twice kinetic energy measures two-point correlation function ξ2 and depends on the average number density ¯ n, temperature T and the interparticle distance r. Thus it is valuable to understand the functional form of ξ2 which depends upon the value of b(n, T ). The

Local equilibrium approach for fermi systems and quantum hydrodynamics

AbstractThe quantum distribution function is derived to reach an accordance with quantum hydrodynamics and to conserve quantum properties of the system. This distribution function, when calculating statistical averages, leads to correct local values of the fundamental physical quantities; and the local conservation laws of the microscopic quantum hydrodynamics can be obtained from such statistics by passing to mathematical expectatives. The equation for the one‐particle distribution function generates the many‐particle distribution functions. Then, the BBGKY hierarchy equations are obtained. The one‐particle statistical operator of a system of fermions in the local equilibrium at arbitrary temperatures is calculated. The dynamics of local hydrodynamic functions (chemical potential, hydrodynamic velocity, and temperature) completely determine the dynamics of the one‐particle statistical operator. The quantum hydrodynamic equations for the proton‐neutron system at low temperatures are obtained from the equation for one‐particle distribution function. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Cumulant Representation of Solutions of the BBGKY Hierarchy of Equations

We construct a cumulant representation of solutions of the Cauchy problem for the BBGKY hierarchy of equations and for the dual hierarchy of equations. We define the notion of dual nonequilibrium cluster expansion. We investigate the convergence of the constructed cluster expansions in the corresponding functional spaces.

Long-Time-Tail Effects on Lyapunov Exponents of a Random, Two-Dimensional Field-Driven Lorentz Gas

We study the Lyapunov exponents for a moving, charged particle in a two-dimensional Lorentz gas with randomly placed, nonoverlapping hard-disk scatterers in a thermostatted electric field, \(\vec E\). The low-density values of the Lyapunov exponents have been calculated with the use of an extended Lorentz–Boltzmann equation. In this paper we develop a method to extend theses results to higher density, using the BBGKY hierarchy equations and extending them to include the additional variables needed for calculation of the Lyapunov exponents. We then consider the effects of correlated collision sequences, due to the so-called ring events, on the Lyapunov exponents. For small values of the applied electric field, the ring terms lead to nonanalytic, field-dependent contributions to both the positive and negative Lyapunov exponents which are of the form ~e2ln~e, where ~e is a dimensionless parameter proportional to the strength of the applied field. We show that these nonanalytic terms can be understood as resulting from the change in the collision frequency from its equilibrium value due to the presence of the thermostatted field, and that the collision frequency also contains such nonanalytic terms.

The closure hypothesis in the maximum entropy formalism

Closure hypotheses in kinetic theory constitute a point of comparison between different approaches to describe the behavior of a macroscopic system. The maximum entropy formalism (MEF-I) gives us a way to close the BBGKY hierarchy equations and its discussion is the main objective of this paper.

STATIC STRUCTURE FACTOR OF HARD-SPHERE YUKAWA SYSTEMS

We have applied the Singwi, Tosi, Land and Sjölander approximation for the two-particle distribution function in the BBGKY hierarchy equations to investigate the properties of the hard-sphere Yukawa systems. The static structure factor and the radial distribution function are evaluated and compared with other approximations of the theory of liquids and computer simulations.

Perturbative Growth of Cosmological Clustering. II. The Two-Point Correlation

We use the BBGKY hierarchy equations to calculate, perturbatively, the lowest order nonlinear correction to the two point correlation and the pair velocity for Gaussian initial conditions in a critical density matter dominated cosmological model. We compare our results with the results obtained using the hydrodynamic equations which neglect pressure and we find that the two match, indicating that thare are no effects of multistreaming at this order of perturbation. We analytically study the effect of small scales on the large scales by calculating the nonlinear correction for a Dirac delta function initial two point correlation. We find that the induced two point correlation has a $x^{-6}$ behaviour at large separations. We have considered a class of initial conditions where the initial power spectrum at small $k$ has the form $k^n$ with $0 < n \le 3$ and have numerically calculated the nonlinear correction to the two point correlation, its average over a sphere and the pair velocity over a large dynamical range. We find that at small separations the effect of the nonlinear term is to enhance the clustering whereas at intermediate scales it can act to either increase or decrease the clustering. At large scales we find a simple formula which gives a very good fit for the nonlinear correction in terms of the initial function. This formula explicitly exhibits the influence of small scales on large scales and because of this coupling the perturbative treatment breaks down at large scales much before one would expect it to if the nonlinearity were local in real space. We physically interpret this formula in terms of a simple diffusion process. We have also investigated the case $n=0$ and we find that it differs from the other cases in certain respects. We investigate a recently proposed scaling property of

Distribution functions in the statistical theory of convective MHD turbulence of an incompressible fluid

In this paper, we define a hierarchy of distribution functions for simultaneous velocity, magnetic, and temperature fields. Some properties of the constructed distribution functions such as reduction, separation, and coincidence are discussed. Equations for the evolution of one- and two-point distribution functions have been derived. Finally, a comparison of the equation for the single-point distribution function in case of zero viscosity, negligible diffusivity, and infinite electrical conductivity is made with first equation of BBGKY hierarchy in the kinetic theory of gases.

The Regular Truncation Scheme of the BBGKY Hierarchy Equations for the Multi‐Component Weakly Non‐Ideal Plasma

AbstractA regular truncation scheme in which the divergencies characteristic of the Debye truncation are removed by taking into account the shall‐range correlations. As a result, the two radii of screening the Debye radius D and √2 D appear in plasma which is not symmetrical as to its particle charges. The criterion of the applicability of this approach relates the plasma parameter to the potential depth of unlike‐charged particles interaction.

Equilibrium time evolutions and BBGKY hierarchy equations of large classical systems

We show the existence of a time evolution {Pt; t∈ℝ of a locally perturbed equilibrium state P of infinitely many particles in ℝ{suv}, v≧1, evolving under the action of the infinite Newtonian dynamics associated to the same smooth, finite range pair potential as the equilibrium state itself. Moreover, it is shown that {Pt; t∈ℝ solves the weak BBGKY hierarchy equations. The treatment of this problem will be done in the general setting of so called (Σρ) point processes developed in [11, 10] and [4] and will require the method of moments.

Two-fluid theory of coupled kinetic equations for a test particle and a background fluid

The generalized Langevin equation formalism of Mori and Zwanzig and Akcasu and Dudenstadt is used to derive coupled kinetic equations for the distribution functions of a tagged test particle and of the untagged members of the surrounding background fluid. The interaction between any two of these particles is taken to be the sum of a long-ranged continuous potential and a short-ranged, hard-core contribution. The distribution function of the background fluid is chosen to depend on the location of a solvent particle measured relative to the instantaneous position of the test particle. Because of this choice, the frequency matrix associated with the pair of distribution functions incorporates much of the coupling between the tagged and untagged particles which otherwise would be imbedded within their separate memory functions. Consequently, the mean-field approximation obtained by neglecting the memory function of this two-fluid theory is expected to be quite accurate. Explicit forms are derived for the coupled mean-field kinetic equations gotten in this way. In regions remote from the test particle, the kinetic equation for a member of the solvent reduces to previously derived kinetic equation of the type called revised or generalized Enskog equations. Finally, it is shown that the kinetic equations of this two-fluid, mean-field theory also can be gotten by truncating the appropriate BBGKY hierarchy of equations, using a plausible, easily interpretable connection between pair-space distribution functions and the singlet distribution functions specific to the tagged and untagged particles.

A.C. conductivity in a homogeneous plasma from a convergent collision integral

Recently a spatially uniform expression for the time asymptotic binary correlation function of a Landau-stable plasma was derived from the BBGKY hierarchy equations. In this paper, the high-frequency (Vei ≪ ‖w‖ ≪ ωpe) electrical conductivity of a spatially homogeneous, unmagnetized Lorentz plasmais calculated from this expression; the result is compared with those obtained from other convergent collision integrals.

Binary correlation and convergent collision integral for a plasma

The BBGKY hierarchy equations for a homogeneous plasma are used to derive an explicit expression for the time-asymptotic binary correlation function. Closure of the hierarchy equations is shown possible, mainly due to a difference in scaling behavior, as functions of impact parameter(s), between the driving terms in the second and third hierarchy equations. The final result is shown to be a uniform approximation for interparticle distances much smaller than the minimum of the mean free path and the density inhomogeneity length. The resulting expressions for the free energy and the collision integral are convergent.

Kinetic Equations of Dilute Electron Plasmas in the Coherent Region

On the basis of the scale covariance of the correlation functions under a space-time coarse graining, kinetic equations are studied in the coherent region where the space-time cutoff (b, t,) satisfies l..n;pb;;,r,, 1/wp;;,t,;;,r, with 1-n being the Debye length, Wp the plasma frequency, ro the Landau cutoff and ro the corresponding time scale. A generalized Vlasov equation with a collision term is derived from the BBGKY hierarchy equations by using Mori's scaling method in an extended form. This collision term consists of three terms; the Boltzmann term, the Landau term and a new collision term which differs from the Balescu-Lenard term. An evolution equation for the two-particle correlation function is also derived. The kinetic equations in the kinetic region where l..n«:_b«:_lr, 1/wp«:_t,«:_rr with lr and rr being the mean free path and time, respec­ tively, are derived from these equations by introducing a further space-time coarse graining. Thus it is shown that the generalized Vlasov equation derived here reproduces the Balescu­ Lenard-Boltzmann-Landau equation in the kinetic region correctly.

A scaling method for deriving kinetic equations from the BBGKY hierarchy

On the basis of the scale covariance of correlation functions under a coarsegraining in space and time, the Boltzmann equation for neutral gases, the Balescu-Lenard-Boltzmann-Landau equation for dilute plasmas, and linear equations for the variances of fluctuations are derived from the BBGKY hierarchy equations with no short-range correlations at the initial time. This is done by using Mori's scaling method in an extended form. Thus it is shown that the scale invariance of macroscopic features affords a useful principle in nonequilibrium statistical mechanics. It is also shown that there existtwo kinds of correlation functions, one describing the interlevel correlations of the kinetic level with its sublevels and the other representing the fluctuations in the kinetic level.

A scaling method for obtaining kinetic equations from the BBGKY hierarchy

Kinetic equations for the one-particle distribution function of dilute gases and plasmas and evolution equations for the variance of their fluctuations are obtained from the BBGKY hierarchy equations by using Mori's scaling method in an extended form.

Magnetized plasma kinetic theory. I. Derivation of the kinetic equation for a uniform magnetized plasma

A kinetic equation for a multi-species plasma in an external uniform magnetic field is derived from the BBGKY hierarchy of equations. The equation generalizes the equation of Rostoker (1960), which assumes that the distribution function is independent of the azimuthal angle, and all previous results can be derived from it. An additional advantage is that the collision integral is obtained in a form which is free from infinite sums of Bessel functions, and this greatly facilitates calculations based on it.

A kinetic theory of spectral line shapes

The methods of kinetic theory are used to describe the radiation from an atom immersed in a gas of perturbing particles. It is shown that the line shape can be expressed in terms of a one-particle distribution function. The appropriate BBGKY hierarchy of equations is derived. This hierarchy is then truncated by assuming that only two-body collisions are important. The resulting equations are solved to obtain a non-Markovian kinetic equation which describes the combined effects of Doppler and pressure broadening. When the Markovian assumption is applied, a generalized linear Boltzmann equation is obtained which describes the line shape in the region where the impact limit is valid and which also describes the phenomenon of collisional narrowing.