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The aim of this paper is to calculate the binary and triplet distribution functions for dilute relativistic plasma in terms of the thermal parameter
*μ* where
, is the mass of charge;
*c* is the speed of light;
*k* is the Boltzmann’s constant; and
*T* is the absolute temperature. Our calculations are based on the relativistic Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. We obtain classical binary and triplet distribution functions for one- and two-component plasmas. The excess free energy and pressure are represented in the forms of a convergent series expansions in terms of the thermal parameter
*μ*.

Studying the properties of plasma has received a great interest in both astrophysical and laboratory plasma application. The relativistic binary distribution function is one of the most important functions of statistical mechanics. The importance of the distribution function in statistical mechanics is due to the fact that all the thermodynamic quantities, such as the pressure, the internal energy and the free energies, can be calculated from it.

Relativistic statistical mechanics has a long story, but we may notice that, whereas the theory of relativistic ideal gases has received deep and detailed developments, little has been achieved in order to account for mutual interactions between particles [

pair-plasma distribution functions could be described by a thermal parameter

relativistic. The thermal parameter value plays a significant role for the stability of our system. Barcons and Lapiedra (1984) [

Special relativity, however, does not permit velocities greater than the speed of light

A little is known about relativistic distribution functions involving more than two particles, and in particular about the three-particle (or triplet) distribution function. This is of course due to the greater mathematical complexity of higher order correlation functions, and to a lack of a direct link with experiment. Although one can consider experimental determination of triplet distribution function from triple elastic scattering similar to that of the binary distribution function, but to our knowledge, the measurement of three-body correlation function requires the knowledge of the positions of three particles at the same time which is technically very demanding to obtain in 3D samples [

Homogeneous plasma is characterized by two parameters: the density of particles

1) Energy of the rest mass per particle

2) Kinetic energy per particle of order

3) Coulomb energy of order

The ratios of these energies give us the two main dimensionless parameters of the plasma: the thermal para-

meter of plasma

are four different regimes characterized by it: plasma with

we note that the system is safely classical if where

is the typical linear momentum of particles. A relativistic plasma with a thermal distribution function has temperatures greater than around 260 keV, or 3.0 GK (5.5 billion degrees Fahrenheit), where approximately 10% of

the electrons have

astrophysics, including gamma-ray bursts, AGN jets, and pulsar winds. In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light

In the astrophysical environment, the fraction of ionized particles varies widely from nearly no ionization in cold regions to fully ionized in regions of high temperature. This leads to a wide range of parameters where astrophysical plasmas can exist. While the astrophysical environment is frequently dominated by the presence of the plasma, this plasma is often strongly influenced by and coupled to the presence of embedded particulates (i.e., dust). These dust grains which range in size from a few nanometers to micron-sized objects can become either positively or negatively charged due to interactions with the background plasma environment and ionizing radiation sources in the astrophysical environment [

Another important point is that of the walls. We shall study plasma which is homogeneous and isotropic but the plasma must be confined or it will expand. We may assume that the plasma is confined by some kind of walls which prevent the escape of particles, but that the container is so large that the effects of the walls are negligible. Our calculations are based on the phase-space distribution function; this is defined as the number of particles per unit volume of space per unit volume of velocity space: At time t, number of particles in elementary volume of space, with velocities in range

The statistical state of a macroscopic system of N particles is in a complete―though in an intractably complex―

way described by the distribution function

tion of particle

The s-particle reduced distribution is giving by:

The relativistic BBGKY hierarchy [

where

The Einstein summation convention is only valid for Greek labels. Therefore, for each value of s we have s equations, since

According to what we done in non-relativistic case [

Let us consider the case of homogeneous plasma in equilibrium. Then

where

At first sight, the calculation of the relativistic interaction between two charged particles seems rather involved, because the force on particle 1 at time

To calculate the binary relativistic distribution function substituting from Equations (3) and (5) into (2) for

Then, by solving the integro-differential Equation (7) and by using the Fourier transform of

generalized distribution function

Let us now study the two-particle relativistic distribution for the model of two-component plasma (TCP) i.e. neutral system of point like particles of positive and negative charges such as electrons and ions. For the numerical calculation we restrict ourselves to the case of two-component plasma which anti-symmetric with respect to the charges

For two-component plasma we can use the two-particle correlation function

where

with

The standard two-body distribution function for dilute slightly relativistic plasma has been calculated previously by Kosachev and Trubnikov [

According to our knowledge few scientists studied the binary distribution function for relativistic dilute plasma [

The triplet distribution function

Substituting Equation (4) and (5) into (2) for

If we used the Kirkwood superposition approximation (KSA) [

It has been mentioned that before 2003 there is no direct measurement of three-body correlation function and this is the first study for the triplet distribution function in the case of dilute relativistic plasma. Such measurement requires the knowledge of the positions of three particles at the same time which is technically very demanding to obtain in 3D samples [

From Equation (12) we can obtain the classical TDF for two-component plasma in the following form:

And we also can used (KSA) which is given in Equation (13) to get it in the form

The triplet and quadruple distribution functions as well as binary distribution function must be incorporated for a more accurate and complete discussion of macroscopic equilibrium properties. A little is known about distribution functions involving more than two particles, and in particular about the three-particle (or triplet) distribution functions. This is of course due to the greater mathematical complexity of higher order correlation functions, and to a lack of a direct link with experiment [

In many physical systems, the description of a plasma as a Coulomb system is sufficient to reproduce most of the properties of interest. If the system is cold enough, the mean velocities of the particles are much smaller than the speed of light, and the charges may be assumed to interact via the instantaneous Coulomb potential. However, at sufficiently high temperatures, this approximation is no longer valid, and the contributions of the relativistic effects (which include, apart from the trivial kinetic corrections and of course all the retardation effects) must be incorporated when studying the equilibrium properties of the system [

In the classical (non-quantum) case, the systematic approach adopted here follows the traditional route of the relativistic Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced distribution functions by formal density expansion. Interestingly, this study is the first to display the effect of thermal parameter of plasma in the classical binary and triplet distribution function. Also the triplet relativistic distribution function for dilute plasma was calculated from the relativistic BBGKY hierarchy. We used the results to obtain the analytical forms of the classical triplet distribution functions for one- and two-component plasmas.

In

Our calculations are grounded in the classical relativistic statistical mechanics. Plasma is non-degenerate. The system is not dense, so one may neglect the contributions of higher order particle interactions.

to assume that the triplet distribution function is the product of the three radial distribution functions, and the other form is calculated by using the relativistic BBGKY hierarchy.

Finally, we can note that from Figures 11-14, the distribution function has become more concentrated to ever- smaller region of speed