Emergent behaviour in electrodiffusion: Planck's other quanta

A well-established nonlinear continuum model of time-independent electrodiffusion describes the migrational and diffusional transport of two ionic species, with equal and opposite valences, across a liquid junction. The ionic charge densities provide the source for a static electric field, which in turn feeds back on the charges to contribute the migrational component of the ionic transport. Underpinning the model is a form of the second Painlevé ordinary differential equation (PII). When Bäcklund transformations, extended from those known in the context of PII, are applied to an exact solution of the model first found by Planck, a sequence of exact solutions emerges. These are characterized by corresponding ionic flux and current densities that are found to be quantized in a particularly simple way. It is argued here that this flux quantization reflects the underlying quantization of charge at the ionic level: the nonlinear continuum model 'remembers' its discrete roots, leading to this emergent phenomenon.