V. I. Tseplyaev

EQUATIONS FOR THE CORRELATION FUNCTIONS OF AN EQUILIBRIUM SYSTEM OF COULOMB PARTICLES WITH SHORT-RANGE INTERACTION

(Presented by Academician N. N. Bogolyubov, 15 XI 1961)

In this article a method is proposed for taking into account short-range repulsive forces for a classical system of Coulomb particles in thermodynamic equilibrium. Equations are derived for corrections of various orders to the effective interaction energies of two, three, etc., particles. These equations in principle make it possible to calculate, for example, corrections to the Debye potential. At the same time the meaning is clarified of a Kramers-type potential, which is often used in considering systems of Coulomb particles.

Suppose that a system consisting of \(M\) species of Coulomb particles is in thermodynamic equilibrium. The charge of a particle belonging to species \(a_i\) is equal to \(z_{a_i}\).

At small distances between two particles, let us assume the existence of short-range repulsive forces that do not depend on the particle’s belonging to one species or another, i.e., the interaction energy between a pair of particles is

\[ U_{a_i a_j}=z_{a_i}z_{a_j}/|\mathbf r_i-\mathbf r_j|+K(|\mathbf r_i-\mathbf r_j|). \tag{1} \]

The potential of the short-range forces \(K(r)\) may, for example, be chosen in the form \(u/r^m\), with \(m\) and \(u\) chosen in a corresponding way. In this case there are three independent length parameters: \(l_0=e^2/\theta=e^2/kT\), \(l_1=(u/\theta)^{1/m}\), and \(l_2=n^{-1/3}\), where \(n\) is the mean particle density, \(e^2=4\pi\sum_{a_i} n_{a_i} z_{a_i}^2\), and \(n_{a_i}\) is the concentration of particles of species \(a_i\).

In dimensionless form the well-known Bogolyubov chain equations \((^1)\) are written as follows:

\[ \frac{\partial F_{a_1\ldots a_s}}{\partial x^\sigma} + F_{a_1\ldots a_s} \left[ \alpha\frac{\partial\Phi_{a_1\ldots a_s}}{\partial x_1^\sigma} + \beta\frac{\partial K_{a_1\ldots a_s}}{\partial x_1^\sigma} \right] + \]

\[ +\gamma\int \sum_{a_{s+1}} n_{a_{s+1}} \left( \alpha\frac{\partial\Phi_{a_1a_{s+1}}}{\partial x_1^\sigma} + \beta\frac{\partial K_{a_1a_{s+1}}}{\partial x_1^\sigma} \right) F_{a_1\ldots a_{s+1}}\,dx_{s+1} =0,\quad \sigma=1,2,3. \tag{2} \]

Here

\[ \alpha=l_0/r_0,\qquad \beta=(l_1/r_0)^m,\qquad \gamma=(r_0/l_2)^3,\qquad \mathbf x=\mathbf r/r_0, \]

\[ \Phi_{a_1\ldots a_s} = \sum_{1\le i<j\le s} \{z_{a_i}z_{a_j}/e^2|\mathbf x_i-\mathbf x_j|\}, \]

\[ K_{a_1\ldots a_s} = \sum_{1\le i<j\le s} (1/|\mathbf x_i-\mathbf x_j|)^m, \]

the remaining notation is generally accepted. We determine \(r_0\) from the condition \(\alpha\gamma=1\). In this case \(r_0=r_d=(\theta/ne^2)^{1/2}\), the Debye screening radius.

Substituting into (2) \(F_{a_1\ldots a_s}=f_{a_1\ldots a_s}\exp\{-\beta K_{a_1\ldots a_s}\}\), for \(f_{a_1\ldots a_s}\) we obtain the equation

\[ \frac{\partial f_{a_1\ldots a_s}}{\partial x_1^\sigma} +\alpha f_{a_1\ldots a_s}\frac{\partial \Phi_{a_1\ldots a_s}}{\partial x_1^\sigma} +\int \sum_{a_{s+1}} n_{a_{s+1}}\frac{\partial \Phi_{a_1a_{s+1}}}{\partial x_1^\sigma}\times \]

\[ \times \exp\left\{-\beta \sum_{1\leq i\leq s}K_{a_i a_{s+1}}\right\} f_{a_1\ldots a_{s+1}}\,dx_{s+1} \]

\[ +\frac{\beta}{\alpha}\int \sum_{a_{s+1}} n_{a_{s+1}} \frac{\partial K_{a_1a_{s+1}}}{\partial x_1^\sigma} \exp\left\{-\beta \sum_{1\leq i\leq s}K_{a_i a_{s+1}}\right\} f_{a_1\ldots a_{s+1}}\,dx_{s+1}=0 . \tag{3} \]

Assume that, for the system under consideration, \(\beta\ll\alpha\ll1\). Taking this into account, we shall neglect the last term in equation (3). This means that one considers such a system of particles for which any subsystem is closed with respect to the short-range forces.

We shall seek the solution of equation (3) in the form of an expansion of \(\ln f_{a_1\ldots a_s}\) in a series in \(\alpha\):

\[ \ln f_{a_1\ldots a_s}=-\sum_n \alpha^n\Psi^{(n)}_{a_1\ldots a_s}, \tag{4} \]

i.e., we shall expand in a series the effective interaction energy \(\Psi_{a_1\ldots a_s}\) of the subsystem of \(s\) particles, the first of which belongs to species \(a_1\), the second to \(a_2\), and so on. The principal contribution to \(\Psi_{a_1\ldots a_s}\) is given by one-particle energies; the contribution of first order in \(\alpha\) is given by the interactions of all possible pairs, etc.* These energies, in turn, are expanded in a series in \(\alpha\), so that

\[ \Psi^{(n)}_{a_1\ldots a_s} = \sum_{1\leq i\leq s}\varphi^{(n)}_{a_i} + \sum_{1\leq i<j\leq s}\varphi^{(n-1)}_{a_i a_j} +\ldots+ \varphi^{(n-s+1)}_{a_1\ldots a_s}; \tag{5} \]

\(\varphi^{(k)}_{a_1\ldots a_r}\) is the term of order \(k\) in the expansion of the interaction energy of a subsystem of \(r\) particles. With this taken into account, the equations for \(\Psi^{(n)}_{a_1\ldots a_s}\) take the form

\[ \frac{\partial \Psi^{(0)}_{a_1\ldots a_s}}{\partial x_1^\sigma} = \int \sum_{a_{s+1}} n_{a_{s+1}} \frac{\partial \Phi_{a_1a_{s+1}}}{\partial x_1^\sigma} \exp\left\{-\beta \sum_{1\leq i\leq s}K_{a_i a_{s+1}}\right\} [f_{a_{s+1}|a_1\ldots a_s}]^{(0)}dx_{s+1}; \tag{6} \]

\[ \frac{\partial \Psi^{(1)}_{a_1\ldots a_s}}{\partial x_1^\sigma} - \frac{\partial \Phi_{a_1\ldots a_s}}{\partial x_1^\sigma} = \]

\[ = \int \sum_{a_{s+1}} n_{a_{s+1}} \frac{\partial \Phi_{a_1a_{s+1}}}{\partial x_1^\sigma} \exp\left\{-\beta \sum_{1\leq i\leq s}K_{a_i a_{s+1}}\right\} [f_{a_{s+1}|a_1\ldots a_s}]^{(1)}dx_{s+1}; \tag{7} \]

\[ \frac{\partial \Psi^{(n)}_{a_1\ldots a_s}}{\partial x_1^\sigma} = \int \sum_{a_{s+1}} n_{a_{s+1}} \frac{\partial \Phi_{a_1a_{s+1}}}{\partial x_1^\sigma} \exp\left\{-\beta \sum_{1\leq i\leq s}K_{a_i a_{s+1}}\right\} [f_{a_{s+1}|a_1\ldots a_s}]^{(n)}dx_{s+1}. \tag{8} \]

Here \([f_{a_{s+1}|a_1\ldots a_s}]^{(n)}\) is defined as the coefficient of \(\alpha^n\) in the expansion of \(f_{a_{s+1}|a_1\ldots a_s}=f_{a_1\ldots a_{s+1}}/f_{a_1\ldots a_s}\) in a series in \(\alpha\).

In what follows, in the system (6)—(8) we neglect terms of the form

\[ \int y_{a_1\ldots a_{s+1}}(x_1,\ldots x_{i-1},x_{i+1}\ldots x_{s+1}) \{\exp(-\beta K_{a_i a_{s+1}})-1\}\,dx_{s+1}, \]

which are of order \(\beta^{3/m}\) in comparison with

\[ \int y_{a_1\ldots a_{s+1}}(x_1\ldots x_{i-1},x_{i+1}\ldots x_{s+1})\,dx_{s+1}. \]

* A similar expression for \(\Phi_{a_1\ldots a_s}\) was used in Ref. \((^2)\).

For one particle in the zeroth approximation, equation (6) is the equation of the self-consistent field:

\[ \frac{\partial \varphi_{a_1}^{(0)}}{\partial x_1^\sigma} = \int \sum_{a_2} \frac{\partial \Phi_{a_1a_2}}{\partial x_1^\sigma} \, n_{a_2}\exp\{-\beta K_{a_1a_2}-\varphi_{a_2}^{(0)}\}\,dx_2 . \tag{9} \]

If the system is homogeneous, then \(\varphi_{a_1}^{(0)}=0\). To calculate the first correction \(\varphi_{a_1}^{(1)}\), we have two equations:

\[ \frac{\partial \varphi_{a_1}^{(1)}}{\partial x_1^\sigma} + \int \sum_{a_2} n_{a_2} \frac{\partial \Phi_{a_1a_2}}{\partial x_1^\sigma} \exp\{-\beta K_{a_1a_2}-\varphi_{a_2}^{(0)}\} \left(\varphi_{a_2}^{(1)}+\varphi_{a_1a_2}^{(0)}\right)\,dx_2 =0; \tag{10} \]

\[ \frac{\partial \varphi_{a_1a_2}^{(0)}}{\partial x_1^\sigma} - \frac{\partial \Phi_{a_1a_2}}{\partial x_1^\sigma} + \int \sum_{a_3} n_{a_3} \frac{\partial \Phi_{a_1a_3}}{\partial x_1^\sigma} \exp\left\{-\beta\sum_{1\le i<2}K_{a_i a_3}-\varphi_{a_3}^{(0)}\right\} \varphi_{a_2a_3}^{(0)}\,dx_3 =0 . \tag{11} \]

For \(\beta=0\), in the case of a homogeneous system, equation (11) is the equation for the Debye potential (see, for example, (1)).

For the next correction we have a system of three equations:

\[ \frac{\partial \varphi_{a_1}^{(2)}}{\partial x_1^\sigma} + \int \sum_{a_2} n_{a_2} \frac{\partial \Phi_{a_1a_2}}{\partial x_1^\sigma} \exp[-\beta K_{a_1a_2}-\varphi_{a_2}^{(0)}] \left\{ \varphi_{a_2}^{(2)} +\varphi_{a_1a_2}^{(1)} -\frac{1}{2}\left[\varphi_{a_1a_2}^{(0)}\right]^2 \right\}\,dx_2 =0; \tag{12} \]

\[ \frac{\partial \varphi_{a_1a_2}^{(1)}}{\partial x_1^\sigma} + \int \sum_{a_3} n_{a_3} \frac{\partial \Phi_{a_1a_3}}{\partial x_1^\sigma} \exp\left[-\beta\sum_{1\le i<2}K_{a_i a_3}-\varphi_{a_3}^{(0)}\right] \left\{ \varphi_{a_2a_3}^{(1)} +\varphi_{a_1a_2a_3}^{(0)} -\varphi_{a_1a_3}^{(0)}\varphi_{a_2a_3}^{(0)} -\frac{1}{2}\left[\varphi_{a_2a_3}^{(0)}\right]^2 \right\}\,dx_3 =0; \tag{13} \]

\[ \frac{\partial \varphi_{a_1a_2a_3}^{(0)}}{\partial x_1^\sigma} + \int \sum_{a_4} n_{a_4} \frac{\partial \Phi_{a_1a_4}}{\partial x_1^\sigma} \exp\left[-\beta\sum_{1\le i<s}K_{a_i a_4}-\varphi_{a_4}^{(0)}\right] \left\{ \varphi_{a_2a_3a_4}^{(0)} -\varphi_{a_2a_4}^{(0)}\varphi_{a_3a_4}^{(0)} \right\}\,dx_4 =0 . \tag{14} \]

Analogously, one can write equations for any correction to the interaction energy of any cluster of particles. The presence in the integral terms of these equations of the factor \(\exp\{-\beta\sum_{1\le i<s}K_{a_i a_{s+1}}\}\) removes the problem of divergence as \(|\mathbf{x}_i-\mathbf{x}_j|\to0\).

If \(\exp\{-\beta K_{a_1a_{s+1}}\}\) is approximated by the expression \([1-\exp\{-b|\mathbf{x}_1-\mathbf{x}_{s+1}|\}]\), and in the remaining exponentials one sets \(\beta=0\), then this is equivalent to considering a system of particles interacting according to the law

\[ U(r)=\{1-A(r)\exp[-br]\}/r, \tag{15} \]

where \(A(r)\to1\) as \(r\to0\).

Further, it can be shown that for a system neutral as a whole, representing a binary mixture of particles such that \(z_1=-z_2\), all \(\varphi_{a_1\ldots a_s}^{(n)}\) for which \(n+s=2k+1\) \((k=0,1,\ldots)\) are equal to zero, i.e., in the expansion of \(\Psi_{a_1\ldots a_s}\) there are only even powers of \(\alpha\). For any other system, all powers of \(\alpha\) will be present in the expansion.

In conclusion I express my gratitude to K. P. Stanyukovich and Yu. L. Klimontovich for discussions.

Moscow State University
named after M. V. Lomonosov

Received
14 XI 1961

References

1. N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics, 1946.
2. V. P. Galaiko, L. E. Pargamanik, DAN, 123, No. 6, 999 (1958).