Local equilibrium Coulomb interaction

As remarked in Section VI the treatment of local equilibrium must be modified for charged particles, since the long-range coulomb interaction induces long-range correlations and neighboring volume elements cannot be considered to be statistically independent. We will indicate how the coulomb interaction can be taken into account by means of a self-consistent field, as in the Debye-Huckel theory. The results are then valid under the same conditions as the Debye-Hiickel theory, namely when the Debye radius is large compared to the mean interparticle separation. 

The basic purpose of these comparisons is to see whether the configurations of points have some kind of asymptotic regularity for large values of N that is insensitive to the precise nature of the underlying interactions. If the trends in Figure 12 can be extrapolated beyond N 112, then it would be an indication that the local equilibrium states of the Coulomb law r, - and the Tammes interaction r, - ry 10720 retain a distinct character even for arbitrarily large values of N. 

Both in the Coulomb and dipole problems analytic solutions of the equilibrium equations (2.6a) and evaluation of the associated Hessians (2.6b) becomes tedious for as few as four interacting objectsAt present, the only practical way of surveying the locally stable states of the Coulomb systems for larger values of N is to use computers to find energy minima. However, since the number of minima appears to grow exponentially with N, the energy surface (, 0, ... becomes pro-... 

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