Transformation of the Bogolyubov equations to an exact closed system of equations for the unary and binary distribution functions

G. A. Martynov

Teoretičeskaâ i matematičeskaâ fizika, Tome 22 (1975) no. 2, pp. 260-268

Résumé

Problem of evaluating the equlibrium distribution functions $\mathscr G_{\alpha,\dots ,\gamma}$ of systems of charged particles (plasma, soluted and melted electrolytes etc) is formulated in the most general form. Substitution of the energy values obtained from the electrostatics into the transformed Bogoliubov equations (s. the previous article by the author) leads to the exact closed system of equations for the unary $\mathscr G_{\alpha}$ and $\mathscr G_{\alpha\beta}$ distribution functions. Besides the physical solutions, the system also possesses the solutions decreasing too slowly at the infinity. In order to remove such solutions, the general and local neutrality conditions are necessary, and after imposing these conditions, all the divergent terms are removed. The system obtained splits out into the electrostatics equations, by means of which charge densities are determined from $\mathscr G_{\alpha}$ and $\mathscr G_{\alpha\beta}$ statistics equations expressing the conditions that the electro-chemical potentials of the groups of one and two particles are constant, $\mu_{(p)}$=const, $p=1, 2$. The solution of the equations obtained should be constructed in such a way that all the neutrality conditions were satisfied exactly in every approximation order. The methods of constructing the solutions by means of the expansion over the small parameter powers and by means of the successive approximations are formulated. In both cases divergences do not arise in the evaluations of leading terms.