Density Expansions of the Reduced Distribution Functions and the Excess Free Energy for Plasma with Coulomb and Short‐Range Interactions

Abstract

A b s t r a c tThe equilibrium properties of a multi-component plasma of charged particles that interact through the Coulomb potential and general short-range potentials are investigated at low density within the framework of classical statistics. In particular, this article presents density expansions of (1) the reduced distribution functions for two and more particles up to second order as obtained from the screened Bogoliubov-BornGreen-Kirkwood-Yvon hierarchy and (2) the interaction contribution to the Helmholtz free energy including a new closed representation of the fourth duster integral derived with the help of a usual charging procedure. '1 IntroductionThe calculation of the equilibrium properties of many-particle systems for a given type of microscopic interaction is one of the most fundamental problems in statistical physics. In this article I consider a classical multi-component plasma at low density as a fluid mixture of N charged particles belonging to different species and interacting through the two-body Coulomb potential and additional short-range potentials between two and more particles.The systematic approach adopted here follows the traditional route of first solving the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced distribution functions by formal density expansion where Debye-Hiickel screening is consistently introduced, and then determining thermodynamic quantities as, e. g., the Helmholtz free energy from both the distribution functions and the interaction potentials by a charging process. In this way I calculate in Sec. 2 the first three coefficients in the expansion of the s-particle distribution function and in Sec. 3 the excess free energy up to the fourth cluster integral, thereby correcting and completing previous work of SCHMITZ [l, 21 from the late sixties on the same subject. All results are expressed in terms of the Mayer functions for two-, three-, and four-particle interaction potentials which makes it clear that the present method may be regarded as closely related to the diagrammatical derivation of cluster expansions from the ionic solution theory developed by MAYER and FRIEDMAN (for adetailed review see [3]). Density expansion of t h e reduced distribution functionThe reduced distribution function f51...a, for s particles, s < N , of species {a;} at positions {r;} (i = 1,. . . , s) from an N-particle system in the volume V at temperature T is given by where the configuration integral QN insures normalization of f5,...5, to Vd, and P = l/kwT.For a shorter notation I drop the dependence on r; always implying that, e. g., 351 = 3a, (rl).The central quantity in Eq. (1) is the total interaction potential, or direct potential, w , ,~. . .~~

The Reduced Distribution Functions for Charged Particles With Many-Body Interactions

“…(1) a 1 ...as was obtained by Schmitz, equation (3.10) in [3], more than thirty years ago, but in contrast he failed in getting an equivalent result for C (2) a 1 ...as , which was given by the present author only recently, see equation (6) and the discussion in [7]. We, therefore, regard the derivation of C (2) a 1 ...as , equation (47), in the last section as the first complete approach to the distribution function at second order for the described physical system based on the BBGKY hierarchy.…”

“…For the sole purpose of obtaining the corresponding Helmholtz free energy, however, a formal treatment of the screened BBGKY hierarchy in terms of Mayer functions is quite sufficient, and hence we made no efforts to explicitly evaluate the integrals in the virial coefficients for a special choice of the short-range potentials as, e. g., hard-core interaction [6]. A brief account of the present results has been already published [7].…”

Quantum Binary and Triplet Distribution Functions of Plasma by using Green's Function

“…All the authors who have used the method of Slater Sum are near to the classical limit in their calculations. T. Kahlbaum [7] has calculated density expansions of the reduced distribution functions and the excess free energy for plasma. T. Kahlbaum [7] has calculated density expansions of the reduced distribution functions and the excess free energy for plasma.…”

“…Hussein et al [6] have calculated second approximation coefficient for the binary distribution function of ions for the primitive model of an electrolyte solution. T. Kahlbaum [7] has calculated density expansions of the reduced distribution functions and the excess free energy for plasma. Also, many others have studied the binary and triplet distribution functions [8][9][10][11][12].…”