Spontaneous Central Low-Resistance Channel in a Spherical Charge Transport Medium

We study steady charge transport in a dense spherical medium that must extract current from an external boundary source and release it through a central sink. Uniform ohmic conduction is shown to be geometrically unfavorable: the radial current density diverges as the inverse square of the radius, and the total ohmic dissipation scales as the inverse of the sink radius, becoming arbitrarily large as the sink radius is reduced. We introduce a saturable conductivity feedback in which local current flow lowers the local resistance, a phenomenon independently established in resistive switching, electrochemical conditioning, and related current-induced resistance reduction phenomena. We prove that the uniform conductivity state is not a steady state of the feedback dynamics, and that the only globally stable steady state is one in which the conductivity profile increases monotonically toward the center, saturating at a finite maximum value. This central low-resistance channel removes the dissipation singularity and lowers the total ohmic dissipation relative to all uniform conductivity configurations by orders of magnitude. The channel is not imposed from outside; it is carved by the converging current itself. The result is controlled by a single dimensionless parameter, and the dependence of the channel width on this parameter is smooth. The central conclusion is independent of the specific form of the feedback law, provided it is saturable and monotonically increasing with current density.