The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.     

The steady spectra of particles in dispersible systems with coagulation and fragmentation

The formation of a steady dimensional distribution of particles (particle spectra) is dispersible systems with coagulation and fragmentation is considered. The relation between versions of the kinetic equation that defines these processes is traced. An analytical solution is obtained for the parametric set of coagulation coefficients and the velocities of paired fragmentation. The steady spectrum of particles is investigated in the case when the fragmentation is of the multiple type.     

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.     

Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: II. The Covariant Mean-Field Approximation

A covariant kinetic equation for the matrix Wigner function is derived in the mean-field approximation from a general kinetic equation for the fermionic subsystem of a quantum electrodynamic plasma. We show that in the semiclassical limit, the equations for the components of the Wigner function can be transformed into closed kinetic equations for the Lorentz-invariant distribution functions of particles and antiparticles.     

Derivation of the particle dynamics from kinetic equations

The microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov are considered. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, the so called approximate microscopic solutions are constructed. These solutions are continuous and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the timereversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.     

Distribution Functions in Quantum Mechanics and Wigner Functions

We formulate and solve the problem of finding a distribution function F(r,p,t) such that calculating statistical averages leads to the same local values of the number of particles, the momentum, and the energy as those in quantum mechanics. The method is based on the quantum mechanical definition of the probability density not limited by the number of particles in the system. The obtained distribution function coincides with the Wigner function only for spatially homogeneous systems. We obtain the chain of Bogoliubov equations, the Liouville equation for quantum distribution functions with an arbitrary number of particles in the system, the quantum kinetic equation with a self-consistent electromagnetic field, and the general expression for the dielectric permittivity tensor of the electron component of the plasma. In addition to the known physical effects that determine the dispersion of longitudinal and transverse waves in plasma, the latter tensor contains a contribution from the exchange Coulomb correlations significant for dense systems.     

Kinetic Equations and Integrable Hamiltonian Systems

A survey of interrelations between kinetic equations and integrable systems is presented. We discuss common origin of special classes of solutions of the Boltzmann kinetic equation for Maxwellian particles and special solutions for integrable evolution equations. The thermodynamic limit and soliton kinetic equation for the integrable Korteweg-de Vries equation are considered. The existence of decaying and degenerate dispersion laws and the appearance of additional integrals of motion for the interacting waves is discussed. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 731–741, June, 2005.     

Principal bundles and topological quantization of charges

The space of admissible particle velocities is assumed to be a four-dimensional nonholonomic distribution on a principal or associated bundle. Equations for the horizontal geodesics of this distribution coincide with the equations of motion of charged particles in general relativity theory. It is proved that, if the Lie group of the standard model of elementary particle physics is augmented by the 4-torus, then the wave functions are eigenfunctions of charge operators and the horizontal lift does not depend on the coupling constants. These wave functions satisfy the well-known Dirac equation and its generalizations. For such wave functions, the topological quantization of electric, lepton, and baryon charges takes place.     

Coloured noise for dispersion of contaminants in shallow waters

In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.     

Coagulation, diffusion and the continuous Smoluchowski equation

The Smoluchowski equations are a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers or by positive reals, these corresponding to the discrete or the continuous form of the equations. For dimension d≥3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a method similar to that used to derive the discrete form of the equations in [A. Hammond, F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles, Arch. Ration. Mech. Anal. 185 (2007) 1–67]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of the cited work. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.     

Conservative numerical method for solving the averaged Boltzmann equation

A method is proposed for averaging the Boltzmann kinetic equation with respect to transverse velocities. A system of two integro-differential equations for two desired functions depending only on the longitudinal velocity is derived. The gas particles are assumed to interact as absolutely hard spheres. The integrals in the equations are double. The reduction in the number of variables in the desired functions and the low multiplicity of the integrals ensure the computational efficiency of the averaged equations. A numerical method of discrete ordinates is developed that effectively solves the gas relaxation problem based on the averaged equations. The method is conservative, and the number of particles, momentum, and energy are conserved automatically at every time step.     

Bubble dispersion patterns in bubbly-flow released from a porous plug into a gas-stirred ladle

Bubbles released from a porous plug into a gas-stirred ladle present different bubbly dispersion patterns that can be studied by CFD means. Recent modeling (Alexiadis et al.) has, in fact, led to numerical results in accord with experimental data. However, due to the high number of equations involved, it is not easy to understand the physical reasons of the transitions between these patterns. In the present paper, attention was focused on the role of fragmentation and coalescence and the pattern-transition modeled by means of intersection points between these functions.     

Ordinary differential equation system for population of individuals and the corresponding probabilistic model

The key model for particle populations in statistical mechanics is the Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) equation chain. It is derived mainly from the Hamilton ordinary differential equation (ODE) system for the particle states in the position-momentum phase space. Many problems beyond physics or chemistry, for instance, in the living-matter sciences (biology, medicine, ecology, and sociology) make it necessary to extend the notion of a particle to an individual, or active particle. This challenge is met by the generalized kinetic theory. The corresponding dynamics of the state vector can also be regarded to be described by an ODE system. The latter, however, need not be the Hamilton one. The question is how one can derive the analogue of the BBGKY paradigm for the new settings. The present work proposes an answer to this question. It applies a very limited number of carefully selected tools of probability theory and common statistical mechanics. It also uses the well-known feature that the maximum number of the individuals which can mutually interact directly is bounded by a fixed value of a few units. The proposed approach results in the finite system of equations for the reduced many-individual distribution functions thereby eliminating the so-called closure problem inevitable in the BBGKY theory. The thermodynamic-limit assumption is not needed either. The system includes consistently derived terms of all of the basic types known in kinetic theory, in particular, both the “mean-field” and scattering-integral terms, and admits the kinetic equation of the form allowing a direct chemical-reaction reading. The approach can deal with Hamilton’s model which is nonmonogenic. The results may serve as the basis of the generalized kinetic theory and contribute to stochastic mechanics of populations of individuals.     

Kinetic equations in the method of two-time finite-temperature Green's functions. I. Renormalization of the collision integral

A system of equations that includes a generalized kinetic equation and equations for the static correlation functions is constructed for a normal quantum system of interacting Bose and Fermi particles with two-body interaction on the basis of the method of two-time finite-temperature Green's functions. The equations are in general valid for systems with arbitrary density of the particles. A method of successive approximation that makes it possible to go beyond the usual low-density expansion is discussed. The proposed method leads to a renormalization of the collision integral and makes it possible to obtain correlation functions for the total energy density, including its potential part.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 3, pp. 351–372, September, 1993.     

New insight into classical equilibrium solutions of kinetic equations

Summary. The equilibrium solutions to kinetic equations of weak turbulence (weakly nonlinear wave systems) are analyzed in a systematic manner. The study is performed for kinetic equations involving any number of interacting waves of an arbitrary dimension. Conditions for the equilibrium solutions are reduced to generalized Cauchy functional equations defined at specific hypersurfaces. The analysis proves that, among differentiable functions, the formal equilibrium solutions correspond to equipartition of a linear combination of the energy, the components of momentum, and the number of particles. The structure of the set of equilibrium solutions does not depend on the dimension of wave systems, nor does it explicitly depend on the particular dispersion relation. It depends only on the number of wave components in resonance sets, whereas the number of particles does not enter into the equilibrium distribution in interactions of an odd number of harmonics. The most important consequence of this difference is that wave systems with four-wave resonance conserve the number of particles (wave action), whereas systems with three-wave or five-wave resonance violate this law. All the formal solutions, notwithstanding their physical realizability, are proved to be linearly stable with respect to small disturbances. Received October 5, 2000; accepted August 17, 2001 Online publication November 5, 2001     

Two-time temperature Green’s functions in kinetic theory and molecular hydrodynamics: II. Equations for pair-interaction systems

We consider two approaches to the calculation of correlation functions for a system of particles with direct pair interaction. The first is based on a chain of equations that determines a Boltzmann-type kinetic equation; the second is based on a chain of molecular hydrodynamic equations. We demonstrate that the two approaches are equivalent in the sense that they completely describe the system under consideration. We discuss the advantages of the approach based on the molecular hydrodynamic equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 142–166, April, 1999.     

Phase trajectory portrait of the vibro-impact forced dynamics of two heavy mass particles motions along rough circle

Forced vibro-impact dynamics of the two heavy mass particle motions, in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is considered in combinations of applied analytical and numerical methods. System of two differential double equations, each for one of two heavy mass particle motions along same rough circle are composed with corresponding initial conditions as well as impact conditions. By use software package tools differential double equations are numerically integrated for obtaining phase portrait of phase trajectory branches for different mass particles initial kinetic states. By series of the phase trajectory branches for each mass particle motion between two impacts or between impact and alternation of the Coulomb’s friction force direction, two phase trajectory graphs of the system vibro-impact non-linear dynamics are composed. Different software tools are used as helping tools for calculate time moments of the series of the impacts between mass particles, as well as positions of the impacts, necessary for calculations of the impact velocities of the mass particles before and after impacts. Some comparison between forced and free vibro-impact dynamics of the two heavy mass particles in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is done. Trigger of coupled one side singularities in phase portraits are identified.     

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.