Indirect Exponential Coupling in the Classical Many-Body Problems

The usual coupling procedure consists of multiplying the interparticle potential $U_N\left({\mathrm{r}}^N\right)$, by a coupling parameter $\lambda$ and then expanding thermodynamic functions in powers of $\lambda$. The Kirkwood variation of this procedure couples only one particle of the system, resulting in an integrodifferential equation for distribution functions, which also can be expanded in powers of the coupling parameter. These expansions converge and are valid only for weakly coupled systems. If the Ursell $f$ bonds are coupled instead of the direct interaction potentials, we can expand certain thermodynamic functions in powers of the exponential coupling parameters; for actual physical systems these expansions are practically finite low-order polynomials in the coupling parameters. Integrodifferential equations for distribution functions are derived, and it is seen that distribution functions are given by ratios of two practically finite polynomials in the exponential coupling parameters. The coefficients in these polynomials are finite even for strongly singular (e.g., hard sphere) potentials. The method provides a well-defined expansion parameter for the Kirkwood-Salzburg hierarchy and appears related to the $f$-bond chain summation and nodal expansion methods. Present and possible future applications include: theory of fused salts and electrolytes, theory of ferroelectricity, ion pairing in semiconductors, equation of state of the high-temperature electron gas, and problems of phase transitions. The possibility of applying exponential coupling to quantum-mechanical systems is noted.