Transformation of the Bogolyubov equations to an exact closed system of equations for the unary and binary distribution functions. I. Short-range potential

G. A. Martynov

Teoretičeskaâ i matematičeskaâ fizika, Tome 22 (1975) no. 1, pp. 85-96

Résumé

Notion of the chemical potential $\mu_{(p)}$ of the group of $p$ particles $(p=1,\dots, N)$ is introduced and it is shown that in the equilibrium state $\mu_{(p)}(\mathbf r_1,\dots,\mathbf r_p)=p\mu$, where $\mu=\mathrm{const}$ is usual chemical potential ($\mathbf r_i$ is the $i$-th particle coordinate, $N$ – complete particle number in the system). Condition $\mu_{(p)}=\mathrm{const}$ follows from the vanishing of the complete force $\mathbf F_{i(p)}(\mathbf r_1,\dots,\mathbf r_i,\dots,\mathbf r_p)=0$, acting on the particle $i$, $1\leqslant i\leqslant p$. It is shown that from all methods of evaluating the distribution functions $\mathscr G_{(p)}$ available at present time, only the virial expansion in density powers satisfies both conditions $\mu_{(p)}=\mathrm{const}$, and $\mathbf F_{i(p)}=0$; all the approximate equations of the theory of fluids satisfy one of them only. The necessary condition of the existence of the Bogoliubov equations for the equilibrium distribution functions is formulated. On the basis of these equations the expansion of distribution functions $\mathscr G_{(p)} =\sum\limits_{(k)}\lambda^k\mathscr G_{(p)}^{(k)}$ with the respect to small parameter $\lambda$, connected with the volume of the integration region, is constructed and it is shown that any fragment of this series satisfies the conditions $\mu_{(p)}^{(k)}=\mathrm{const}$, $\mathbf F_{i(p)}^{(k)}=0$. The equations obtained for $\mathscr G_{(p)}^{(k)}$, $p\geqslant 3$ are solved in general form, which makes it possible to remove all the higher distribution functions from the equations for $\mathscr G_{(1)}^{(k)}$, $\mathscr G_{(2)}^{(k)}$. The subsequent summation of these equations leads to the system of two closed exact equations for the unary and binary distribution functions. The kernel of the system has the form of an infinite series with terms depending, in their turn, upon $\mathscr G_{(1)}$ and $\mathscr G_{(2)}$. The equations themselves are of the form $\mu_{(1)}=\mu$, $\mu_{(2)}=2\mu$ and in the “fluid” region of parameter values they can be solved by means of the successive approximations method.

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