Search

New formulas are given for the grand partition function of paraboson systems of order p with n orbitals and parafermion systems of order p with m orbitals. These formulas allow the computation of statistical and thermodynamic functions for such systems. We analyze and discuss the average number of particles on an orbital, and the average number of particles in the system. For some special cases (identical orbital energies, or equidistant orbital energies) we can simplify the grand partition functions and describe thermodynamic properties in more detail. Some specific properties are also illustrated in plots of thermodynamic functions.

The software described in the article provides a method for determining the particle size distribution of the solid pulp phase, first implemented in world practice. The program for numerical analysis and representation of the simulation results of changes in the particle size distribution of the pulp solid phase under the controlling influence of high-energy ultrasound radiation pressure is developed. The particle distribution function according to the useful component disclosure degree is built.

The Boltzmann equation is a complex integral-differential equation and the basis of the kinetic theory of gases. It describes the behavior of a rarefied gas in space of time and velocity. It is used to the study of electron transport in solids and plasmas, neutron transport in nuclear reactors, in the tasks of remote sensing of the Earth from Space. The moment method is one of effective methods for solution of the Boltzmann equation. The system of Boltzmann’s moment equations is intermediate between kinetic and hydrodynamic levels of description of state of the rarefied gas and form class of nonlinear partial differential equations. If particle distribution function will be decomposed into an Fourier series on complete orthogonal system of functions, then Boltzmann’s equation will be equivalent to an infinite system of partial differential equations relative to the moments of the particle distribution function in the complete system of eigenfunctions of linearized operator. But solving infinite system of differential equations impossible. Therefore, an approximate solution of the initial and boundary value problem for the Boltzmann equation can be determined by the moment method. This article describes the one-dimensional nonlinear nonstationary Boltzmann’s moment system equations in the third approximation, which a hyperbolic system of partial differential equations and contains six equations for the moments of the particle distribution function. And formulates the statement of the initial and boundary value problem for the Boltzmann’s moment system equations in the third approximation and shows the results of the numerical solution with the Vladimirov-Marshak boundary conditions.

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.

We analyze the problem of the beam-plasma instability via the analytical treatment of the so-called Dyson equation. We first compared the prediction of the model constructed by fixing the electric field amplitude with respect to a N-body Hamiltonian numerical simulation. Then, we demonstrate that the shortcomings of such an analytical formulation must be essentially identified with the breaking-down of the self-consistent evolution of the field and the particle distribution function., Comment: 7 pages, 6 figures

Bernstein-Kruskal-Greene (or BGK) modes are ubiquitous nonlinear solutions for the 1D electrostatic Vlasov equation, with the particle distribution function $f$ given as a function of the particle energy. Here, we consider other solutions $f = f[\epsilon]$ where the particle energy is equal to the second-order velocity space Taylor expansion of the function $\epsilon(x,v,t)$ near the wave-particle resonance. This formalism allows us to analytically examine the time evolution of plasma waves with time-varying complex frequency $\omega(t) + i \gamma(t)$ in the linear and nonlinear phases. Using a Laplace-like decomposition of the electric potential, we give allowed solutions for the time-varying complex frequencies. Then, we show that $f$ can be represented analytically via a family of basis decompositions in such a system. Using a Gaussian decomposition, we give approximate solutions for contours of constant $f$ for a single stationary frequency mode, and derive the evolution equation for the nonlinear growth of a frequency sweeping mode. For this family of modes, highly nonlinear orbits are found with the effective width in velocity of the island roughly a factor of $\sqrt{2}$ larger than the width of a BGK island., Comment: 12 pages, 3 figures

This paper reports on a one-step simulation study of the generation and propagation of thermoacoustic waves in a two-dimensional enclosure using a finite-difference lattice-Boltzmann-type method (FDLBM) with a single relaxation time and one equilibrium particle distribution function (EPDF), and a direct aeroacoustic simulation (DAS) technique that solves the primitive Navier–Stokes (N–S) equations. Conventional expansion of the EPDF is not adopted; instead, it is expanded in terms of the particle velocity alone, and the expansion coefficients are determined by requiring the FDLBM to fully recover the N–S equations. The expansion coefficients are found to depend on the flow and thermal properties and their nonlinear interactions. Thus formulated, physical boundary conditions can be specified for the FDLBM. This EPDF has been validated against simple aeroacoustic problems and good agreement with DAS results is obtained. In this paper, the EPDF is further validated against three thermoacoustic cases: 1) a sudden increase of temperature on the left vertical wall of the enclosure; 2) a sudden increase of temperature on the left vertical wall coupled by a sudden decrease of temperature on the right vertical wall in the enclosure; 3) a gradually heated up left vertical wall of the enclosure. Comparisons with DAS simulations show that FDLBM and DAS give essentially identical results; hence, the validity and extent of the EPDF is ascertained for the calculation of these thermoacoustic waves. The FDLBM results are also compared with known theoretical and numerical results. These known solutions are found to be less accurate compared to the FDLBM results. The discrepancy could be attributed to the partially linearized equations solved in the theoretical analysis, and the inadequacy of the numerical scheme to replicate the nonlinear effects accurately in the generation and propagation of thermoacoustic waves.

Reconnection outflows have been under intense recent scrutiny, from in situ observations and from simulations. These regions are host to a variety of instabilities and intense energy exchanges, often even superior to the main reconnection site. We report here a number of results drawn from an investigation of simulations. First, the outflows are observed to become unstable to drift instabilities. Second, these instabilities lead to the formation of secondary reconnection sites. Third, the secondary processes are responsible for large energy exchanges and particle energization. Finally, the particle distribution function are modified to become non-Maxwellian and include multiple interpenetrating populations.

Using the linearized version of the time dependent Gross-Pitaevskii equation we calculate the dynamic response of a Bose-Einstein condensed gas to periodic density and particle perturbations. The zero temperature limit of the fluctuation-dissipation theorem is used to evaluate the corresponding quantum fluctuations induced by the elementary excitations in the ground state. In uniform conditions the predictions of Bogoliubov theory, including the infrared divergency of the particle distribution function and the quantum depletion of the condensate, are exactly reproduced by Gross-Pitaevskii theory. Results are also given for the crossed particle-density response function and the extension of the formalism to non uniform systems is discussed. The generalization of the Gross-Pitaevskii equation to include beyond mean field effects is finally considered and an explicit result for the chemical potential is found, in agreement with the prediction of Lee-Huang-Yang theory., Comment: 6 pages. To appear in the Journal of the Experimental and Theoretical Physics (JETP) dedicated to the 85th anniversary of L.P. Pitaevskiy

A model of Forbush-decrease in a magnetic cloud is presented. We calculate moments of the particle distribution function depending on time. The calculated results of cosmic ray intensity generally agree with the observed ones in events registered by ground based detectors. It is found that the magnetic flux rope is of great importance in dynamics of Forbush decrease.

Asymptotic formulas have been derived for distribution functions in cylindrical slits with finite cross-section radii. The problem has been solved based on the asymptotic principle of interbody interactions within the framework of dispersion forces. Symmetric and asymmetric slits between two identical cylinders and between a cylinder and a planar infinite solid surface have been considered. Calculations have been performed for a symmetric slit with a width equal to two cylinder radii.

Article history: We prove the existence of the global weak entropy solution to the Doi-Saintillan-Shelley model for active and passive rod-like particle suspensions, which couples a Fokker-Planck equation with the incompressible Navier-Stokes or Stokes equation, under the no-flux boundary conditions, L 2 (Ω; L 1 (S d−1 )) initial data, and finite initial entropy for the particle distribution function in two and three dimensions. Furthermore, for the model with the Stokes equation, we obtain the global L 2 (Ω × S d−1 ) weak solution in two and three dimensions and the uniqueness in two dimension.

A conserved generalized zero-range process is considered in which two sites interact such that particles hop from the more populated site to the other with a probability $p$. The steady-state particle distribution function $P(n)$ is obtained using both analytical and numerical methods. The system goes through several phases as $p$ is varied. In particular, a condensate phase appears for ${p}_{l}lpl{p}_{c}$, where the bounding values depend on the range of interaction, with ${p}_{c}l0.5$ in general. Analysis of $P(n)$ in the condensate phase using a known scaling form shows there is universal behavior in the short-range process while the infinite range process displays nonuniversality. In the noncondensate phase above ${p}_{c}$, two distinct regions are identified: ${p}_{c}lp\ensuremath{\le}0.5$ and $pg0.5$; a scale emerges in the system in the latter and this feature is present for all ranges of interaction.

The non-equilibrium dynamics of fluctuations is studied in the vicinity of the QCD critical point. The fluctuations are described by a system of coupled Bolzmann and Langevin equations for a particle distribution function and a critical field, or “soft mode”, respectively. As an example, the formalism is applied to derive the relationship between the fluctuations observed at kinetic freezeout and the critical fluctuations at chemical freezeout in the vicinity of the QCD critical point.

We construct an approximation to the measure valued, global in time solutions to the (Patlak-)Keller-Segel model in 2D, based on systems of stochastic interacting particles. The advantage of our approach is that it reproduces the well-known dichotomy in the qualitative behavior of the system and, moreover, captures the solution even after the (possible) blow-up events. We present a numerical method based on this approach and show some numerical results. Moreover, we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the Keller-Segel model with a regularized interaction potential. The proof is based on a BBGKY-like approach for the corresponding particle distribution function.

The components of the dielectric tensor for the distribution function given by Leubner and Schupfer have been obtained. The effect of the loss-cone index appearing in the particle distribution function in a hot magnetized plasma has been studied. A case study has been performed to calculate temporal growth rates of Bernstein waves using the distribution function given by Summers and Thorne and Leubner and Schupfer. The effect of the loss-cone index on growth rates is found to be quite different for the two distribution functions.

A one-dimensional, collisionless plasma given by the Vlasov-Poisson system is considered and the stability properties of periodic steady state solutions known as Bernstein-Greene-Kruskal (BGK) waves are investigated. Sufficient conditions are determined under which BGK waves are linearly unstable under perturbations that share the same period as the equilibria. It is also shown that such solutions cannot support a monotonically decreasing particle distribution function., Comment: 8 pages; PACS codes 52.25.Dg, 02.30.Jr, 52.35.-g

The origin of radio emission from plerions is considered. Recent observations suggest that radio emitting electrons are presently accelerated rather than having been injected at early stages of the plerion evolution. The observed flat spectra without a low frequency cutoff imply an acceleration mechanism that raises the average particle energy by few orders of magnitude but leaves most of the particles at the energy less than about few hundred MeV. It is suggested that annihilation of the alternating magnetic field at the pulsar wind termination shock provides the necessary mechanism. Toroidal stripes of opposite magnetic polarity are formed in the wind emanated from an obliquely rotating pulsar magnetosphere (the striped wind). At the termination shock, the flow is compressed and the magnetic field annihilates by driven reconnection. Jump conditions are obtained for the shock in a striped wind. It is shown that postshock MHD parameters of the flow are the same as if the energy of alternating field has already been converted into the plasma energy upstream the shock. Therefore the available estimates of the ratio of the Poynting flux to the matter energy flux, $\sigma$, should be attributed not to the total upstream Poynting flux but only to that associated with the average magnetic field. A simple model for the particle acceleration in the shocked striped wind is presented., Comment: 9 pages, 2 figures, accepted MNRAS

When weakly collisional plasmas have locally trapped particle populations, perturbations to the plasma equilibrium (such as waves or static field-errors) can induce phase-space discontinuities in the particle distribution function that strongly enhance entropy production, plasma loss, and wave damping via superbanana transport. This paper presents a simple version of this superbanana transport process, wherein a plasma is heated as it is slowly forced back and forth across a squeeze potential (at a frequency ω that is small compared with the particle bounce frequency). The squeeze potential traps low-energy particles on either side of the squeeze, but particles with higher energy can pass through it. Trapped and passing particles have different responses to the forcing, causing a collisionless discontinuity in the distribution function at the separatrix between the trapped and passing particles. Expressions for both the adiabatic and non-adiabatic distribution functions are presented, and the heating rate...

In this article, a discrete effect in the thermal Lattice BGK two-speed model is studied. These effects are due to the non-equilibrium state in the particle distribution function, and the non-equilibrium occurs near walls. The mechanism of the LBM counterpart of the thermal creep flow, which appears due to the temperature gradient of the boundary in rarefied gases, is clarified analytically and numerical calculations are performed for some cases. A technique for eliminating this effect is also shown.

A simple 1D model is presented for the heating caused by cylindrically-symmetric plasma waves in a non-neutral plasma column due to the addition of a symmetric squeeze potential applied to the center of the column. We study this model by using analytical techniques and by using a numerical grids method solution, and we compare the results of this model to previous work (Ashourvan and Dubin (2014)). squeeze divides the plasma into passing and trapped particles; the latter cannot pass over the squeeze potential. In collisionless theory, enhanced heating is caused by additional bounce harmonics induced by the squeeze in the particle distribution, leading to Landau resonances at energies En for which the bounce frequency ωb(E) and wave frequency ωm satisfy ωm = nωb(En). As a result, heating is substantially higher than the case with no squeeze, even when ωm is much greater than the thermal bounce frequency ωb(T). Adding collisions to the theory creates a boundary layer at the separatrix between trapped and passing particles that further enhances the heating at small ωm/kmvs, where km is the axial wavenumber and vs is the velocity at the separatrix. However, at large ωm/vs, the heating from the separatrix boundary layer is only a small correction to the heating from collisionless resonances in the trapped particle distribution function.

A detailed comparison between the Landau and the Dougherty collision operators has been performed by means of Eulerian simulations, in the case of relaxation toward equilibrium of a spatially homogeneous field-free plasma in three-dimensional velocity space. Even though the form of the two collisional operators is evidently different, we found that the collisional evolution of the relevant moments of the particle distribution function (temperature and entropy) are similar in the two cases, once an ‘ad hoc’ time rescaling procedure has been performed. The Dougherty operator is a nonlinear differential operator of the Fokker-Planck type and requires a significantly lighter computational effort with respect to the complete Landau integral; this makes self-consistent simulations of plasmas in presence of collisions affordable, even in the multi-dimensional phase space geometry.

Analysis of closed circuit fine grinding of minerals is carried out with a view to solving the main technological problem – provision of interspersed mineral liberation with minimal expense of energy. By studying the particle distribution function according to fineness class, it is established that the minimum power input will occur at the minimum possible specific surface of the ground product with the target fineness class content.

International audience; The one-particle distribution function is of importance both in non-relativistic and relativistic statistical physics. In the relativistic framework, Lorentz invariance is possibly its most fundamental property. The present article on the subject is a contrastive one: we review, discuss critically, and, when necessary, complete, the treatments found in the standard literature.

We consider the conformal Einstein equations for massless collisionless gas cosmologies which admit an isotropic singularity. It is shown that the Cauchy problem for these equations is well-posed with data consisting of the limiting particle distribution function at the singularity.

A renormalized quasilinear evolution equation for the particle distribution function in a weakly turbulent plasma is presented. The interaction of both resonant and nonresonant waves with the plasma particles is included. The nonresonant waves are crucial for the overall energy balance of the system and contribute significantly to the turbulent process. The velocity-space diffusion coefficients and turbulence levels of existing quasilinear theories are significantly modified.

Results of theoretical studies of the interaction between optical radiation and partially ordered disperse media are reported. In terms of the amplitude-phase screen model consideration is given to the concentration effects of whitening and darkening in random close-packed systems of optically soft particles. The concentration dependence of transmission of close-packed systems of coarse particles is described with the use of a small-angle solution of the stochastic finite-difference transfer equation. The effects of coherent reirradiation occurring in close-packed monolayers of highly refracting particles are analyzed using a quasicrystalline approximation of the theory of multiple wave scattering and the radial particle distribution function obtained from a solution of the Percus-Yevick equation. This approach extended to multilayer systems is used to describe formation of forbidden photon zones in transmission spectra of one- and three-dimensional disperse systems with a high degree of ordering. Results of quantitative calculations are shown to agree well with experimental data. The possibility of using established regularities for optimization of spectral characteristics of selective elements based on spatially ordered disperse systems with different structural organization is discussed.

Using ground-level observations of cosmic-ray (CR) intensities from a worldwide network of stations during the ground-level enhancement (GLE) of 22–23 October 1989, variations of the particle distribution function in all phases of the event were investigated.

The Ensemble Model takes advantage of an approach to express the phase space particle distribution function in terms of the first, second and higher order moments instead of considering individual particles. Based on a new flexible implementation, an arbitrary number of orders can be processed and automatically converted into proper update equations for the simulation program V-Code. In this paper the influence of the introduction of higher order moments on the beam dynamics simulation is investigated. The achievable accuracy and the numerical efforts are compared with the ones obtained from the lower order calculations.

In this paper we propose a refined definition of a cluster to be used in a theory for the rate of homogeneous nucleation. It contains n particles in a volume v (n/v cluster). Starting from the N-particle distribution function, we define the free energy of an n/v cluster interacting with its vapor. The expression obtained for the free energy is the same as that obtained previously using a different approach. However, the method of derivation proposed in this paper has the advantage of clarifying the nature of any approximation involved in obtaining this expression. We also discuss methods for evaluating this free energy using Monte Carlo simulation.

We analyze the spin and charge responses induced in a spin-polarized electron gas by a weak electromagnetic field. The coupled spin-charge response is derived from the equation of motion of the particle distribution function in the presence of the perturbation. To obtain the correct frequency and the wave-vector dependence we introduce the spin-dependent local-field factors, ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}{=G}_{\ensuremath{\sigma}}^{x}\ifmmode\pm\else\textpm\fi{}{G}_{\ensuremath{\sigma}}^{c}$, which give the exchange $(x)$ and correlation $(c)$ corrections to the random phase approximation. For an arbitrary degree of polarization of the electron gas, we derive the exact analytical expressions for ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(q\ensuremath{\rightarrow},\ensuremath{\omega})$ in the limit of high frequency or large wave vectors. The results for $q\ensuremath{\rightarrow}\ensuremath{\rightarrow}\ensuremath{\infty}$ are expressed in terms of the two-particle correlation function, $g(r\ensuremath{\rightarrow})$ at $r=0$.

A theory is proposed that describes in a self-consistent manner damping and particle acceleration within cavitons. The framework of the approach makes it possible to formulate a set of self-consistent equations taking into account the change in the particle distribution function due to the interaction with cavitons and the damping of the waves within the cavitons due to the interaction with resonant particles. The spatial inhomogeneity of the distribution function of resonant particles in the vicinity of a caviton produces a new type of damping. The new damping mechanism is present for any inhomogeneity produced both by the other cavitons (or empty cavitons) in a strongly turbulent plasma or by any other source. This new type of damping is important for arresting wave collapse.

The maximum entropy formalism is developed to implement a complete description of a dense gas in non-equilibrium situations. Two variational schemes are discussed in this context. The kinetic equations describing the system are consistently closed with the N -particle distribution function obtained via this formalism. The equilibrium state obtained on this basis is consistent with the description of the theory of liquids in the equilibrium state.

The antihydrogen formation via two-body recombination in a cylindrical trap is studied and a very crude estimation of the formation rate is presented. The possibility of using the thermal wave model to describe the particle distribution function in a trap has been checked in some simple cases.

The constraints on the moments of a distribution function satisfying an outflow condition are given. When a kinetic equation is approximated by a finite system of equations for the moments, the appropriate number of boundary conditions is found by analysis of the characteristics of the system.

The velocity distribution function, f1, for a (linear) shear flow of a system of rigid inelastically colliding disks in a plane is measured by applying a novel algorithm to results of (MD) simulations involving 200 000 particles. The need to consider such a relatively large system is explained. It is found that f1 is well fitted by an exponent of a second‐order polynomial in the norm of the fluctuating velocities with angle‐dependent coefficients (which also depend on the density and the granular temperature). Other characterizations of the system studied in this paper are presented as background material. A hitherto unnoticed property of systems with Lees–Edwards boundary conditions has been discovered and its origin is briefly explained.

A procedure for modeling some morphological characteristics of carbon black aggregates is presented. The method uses a standard reaction limited aggregation algorithm to predict the location of the centers of primary particles within an aggregate. The shape of the aggregate is determined by drawing simple linear links between extreme points of primary particles centered on these loci. Analysis of the particle distribution function computed from the simulated structure leads to definitions of the interior and periphery of the aggregate structure. Direct comparisons between simulated aggregates and two commercial carbon blacks are presented to illustrate the usefulness of the approach. The perimeter fractal dimension of simulated aggregates can be made to match the perimeter fractal dimension of the commercial carbon black aggregates through proper choice of the sticking probability in the simulation algorithm. With this proper choice of sticking probability, the relative amounts of voids in the periphery and interior of the simulated aggregate correspond to relative amounts of filled and unfilled voids in actual aggregates that have been soaked for extended periods of time with processing fluids.

The motion of finely dispersed particles is described statistically with the use of the Fokker-Planck equation. An expression is obtained for the particle distribution function with respect to residence time. Results of the calculation illustrate the dependence of the average particle residence time in the apparatus on the process parameters.

Our contribution is devoted to development of hybrid modelling techniques. We investigate sheath structures in the vicinity of solids immersed in low temperature argon plasma of different pressures by means of particle and fluid computer models. We discuss the differences in results obtained by these methods and try to propose a way to improve the results of fluid models in the low pressure area. There is a possibility to employ Chapman-Enskog method to find appropriate closure relations of fluid equations in a case when particle distribution function is not Maxwellian. We try to follow this way to enhance fluid model and to use it in hybrid plasma model further.

We present the design and implementation of a spectral code, called SpectralPlasmaSolver (SPS), for the solution of the multi-dimensional Vlasov-Maxwell equations. The method is based on a Hermite-Fourier decomposition of the particle distribution function. The code is written in Fortran and uses the PETSc library for solving the non-linear equations and preconditioning and the FFTW library for the convolutions. SPS is parallelized for shared- memory machines using OpenMP. As a verification example, we discuss simulations of the two-dimensional Orszag-Tang vortex problem and successfully compare them against a fully kinetic Particle-In-Cell simulation. An assessment of the performance of the code is presented, showing a significant improvement in the code running-time achieved by preconditioning, while strong scaling tests show a factor of 10 speed-up using 16 threads.

In order to realize which approximations that are made in the descriptions of plasmas that we generally use, it is instructive to start from the most general description which includes all individual particles and their correlations in the six dimensional phase space (r,v). I the absence of particle sources or sinks we must have a continuity equation for the delta function density N: $$ N(X,t) = \sum\limits_{{i = 1}}^N {(X - {{X}_i}} (t))\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,X = ({\mathbf{r}},{\mathbf{v}}) $$ $$ \frac{\partial }{{\partial t}}N + \sum\limits_i {\frac{\partial }{{\partial {{r}_i}}}} \left( {N\frac{{\partial {{r}_i}}}{{\partial t}}} \right) + \sum\limits_i {\frac{\partial }{{\partial {{\hbox{v}}_i}}}} \left( {N\frac{{\partial {{\text{v}}_i}}}{{\partial t}}} \right) = 0, $$

International audience; The validity of quasilinear (QL) theory describing the weak warm beam-plasma instability has been a controversial topic for several decades. This issue is tackled anew, both analytically and by numerical simulations which benefit from the power of modern computers and from the development in the last decade of Vlasov codes endowed with both accuracy and weak numerical diffusion. Self-consistent numerical simulations within the Vlasov-wave description show that QL theory remains valid in the strong chaotic diffusion regime. However, there is a non-QL regime before saturation, which confirms previous analytical work and numerical simulation, but contradicts another analytical work. We show analytically the absence of mode coupling in the saturation regime of the instability where a plateau is present in the tail of the particle distribution function. This invalidates several analytical works trying to prove or to contradict the validity of QL theory in the strongly nonlinear regime of the weak warm beam-plasma instability.

The Burnett transport equations for a dense hard-sphere gas as given by the standard (SET) and revised (RET) Enskog theories are discussed. It is shown that beyond the Navier-Stokes approximation, the expressions for the collisional transfer contributions to the molecular fluxes are formally different in both theories. In contrast, by using the Chapman-Enskog method, the Burnett approximation to the single particle distribution function is found to be indentical. The differences between the SET and the RET in this hydrodynamic order appear only through two linearized Burnett transport coefficients in the momentum flux. The numerical evaluation of these differences is explicitly carried out.

The particle distribution function that describes two interpenetrating plasma streams is re-investigated. It is shown how, based on the Maxwell-Boltzmann-J\"uttner distribution function that has been derived almost a century ago, a counterstreaming distribution function can be derived that uses velocity space. Such is necessary for various analytical calculations and numerical simulations that are reliant on velocity coordinates rather than momentum space. The application to the electrostatic two-stream instability illustrates the differences caused by the use of the relativistic distribution function., Comment: Accepted for publication in Astrophysics & Space Science

A new turbulent collision integral is derived from the first and second BBGKY equations, in which the effects of well-developed turbulent field are taken into account. The turbulent collision integral contains turbulent frequency-wave number spectrum and up to fourth order differential of particle distribution function.

The kinetic Freeze Out process of a pion gas through a finite layer with time-like normal is considered. The pion gas is described by a Boltzmann gas with elastic collisions among the pions. Within this model, the impact of the in-medium pion mass modification on the Freeze Out process is studied. A marginal change of the Freeze Out variables temperature and flow velocity and an insignificant modification of the frozen out particle distribution function has been found., European Physical Journal A (2009), in press