Potential of Average Force in a Plasma

The potential of average force $W_{1,2}^{}{}_{}{}^{q{q}^{\prime }}$ experienced by a charge $q^{\prime }$ at a distance|${\mathbf{r}}_1$-${\mathbf{r}}_2$| from a charge $q$ is calculated from the Bogoliubov-Born-Green-Kirkwood-Yvon equations of classical statistical mechanics without linearization or equivalent approximations. Diverging integrals are eliminated by the condition that bound-particle states with negative internal energy, e.g., atoms, be excluded from the partition function. The 3-particle distribution functions required for calculating $W_{1,2}^{}{}_{}{}^{q{q}^{\prime }}$ are obtained as solutions of a nonlinearized Poisson-Boltzmann equation for the average potential in the neighborhood of two charges fixed at ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$. For this latter calculation the difference between average potential and potential of average force is neglected. With the help of $W_{1,2}^{}{}_{}{}^{q{q}^{\prime }}$ the average thermal energy of the plasma is computed and compared with the result of the linearized Debye-Hückel theory. Numerical corrections to the latter theory are presented and it is shown that linearization is a far more significant source of errors than identification of average potential with potential of average force.