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

Abstract

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.

Graphic abstract)

Appendix: A brief account of the background of the theoretical approach

In [7], based upon the first two members of BBGKY hierarchy an equation of pair correlation function has been developed in case of a weakly correlated inhomogeneous plasma system in steady state. This equation is reduced to the following form

$$\begin{aligned} \chi _{12} =-\frac{1}{k_{B}T}\phi _{12} - \frac{1}{k_{B}T}\int d\mathbf{{x_{3}}}\phi _{13} n(\mathbf{x_3}) \chi _{23} \end{aligned}$$

(30)

in the limit $\chi _{12}\ll 1$. In [8], this particular equation has been solved in the weakly inhomogeneous limit ($n(\mathbf{x} _{3})= n_{0}+B\cos (\mathbf{p} \cdot \mathbf{x} _{3}), n_{0}\gg B$) to obtain the pair correlation function

$$\begin{aligned} \chi _{12}=-A\frac{e^{-k_{D}r}}{r}+AB\frac{k_{D}}{2n_{0}}e^{-k_{D}r}\cos \left[ \frac{\mathbf {p}}{2}\cdot \left( \mathbf {x}_{1}+\mathbf {x}_{2}\right) \right] \nonumber \\ \end{aligned}$$

(31)

$$\begin{aligned} =-A\frac{e^{-k_{D}r}}{r}\left( 1-B\frac{k_{D}r}{2n_{0}}\cos \left[ \frac{\mathbf {p}}{2}\cdot \left( \mathbf {x}_{1}+\mathbf {x}_{2}\right) \right] \right) ,\nonumber \\ \end{aligned}$$

(32)

where A=$\frac{e^{2}}{k_{B}T}$ and $r=\mid \mathbf{x}_1-\mathbf{x}_2 \mid$.

In this article, this pair correlation function has been utilized in the first member of BBGKY hierarchy to calculate the single particle distribution function for weakly coupled weakly inhomogeneous plasma system in steady state.

Obtaining the single particle distribution function from the first two members of BBGKY hierarchy had been attempted earlier by the same author in [6] but in that case we assumed that the pairing particles are close enough to avoid inhomogeneity in the expression of $\chi _{12}$ and the inhomogeneity in $g_{12}$ was only reflected through the single particle distribution functions. This restriction on the closeness of the pairing particles has been removed in this article and we have been able to incorporate the inhmogeneity factor more correctly and that would lead to a more correct single particle distribution function which has been discussed in detail in the result and discussion section.

This research was supported by the Department of Science and Technology and Biotechnology, West Bengal.