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I propound a non-linear generalization of the Poisson equation describing a"medium" in D dimensions with a "dielectric constant" proportional to the fieldstrength to the power D-2. It is the only conformally invariant scalar theorythat is second order, and in which the scalar $phi$ couples to the sources$\rho$ via a $\phi\rho$ contact term. The symmetry is used to generatesolutions for the field for some non-trivial configurations (e.g. for twooppositely charged points). Systems comprising N point charges afford furtherapplication of the symmetry. For these I derive e.g. exact expressions for thefollowing quantities: the general two-point-charge force; the energy functionand the forces in any three-body configuration with zero total charge; thefew-body force for some special configurations; the virial theorem for anarbitrary, bound, many-particle system relating the time-average kinetic energyto the particle charges. Possible connections with an underlying conformalquantum field theory are mentioned.Comment: Revtex, 16 pages. To be published in Phys. Rev.

physical review. e, statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonlinear conformally invariant generalization of the Poisson equation to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi><mml:mo>&gt;</mml:mo><mml:mn /><mml:mn>2</mml:mn><mml:mn /></mml:math>dimensions

Nonlinear conformally invariant generalization of the Poisson equation toD>2dimensions