Differential equations of electrodiffusion: constant field solutions, uniqueness, and new formulas of Goldman-Hodgkin-Katz type

TL;DR

This paper derives exact constant field solutions and new Goldman-Hodgkin-Katz formulas for steady-state electrodiffusion with multiple ionic species, analyzing solution uniqueness and providing novel mathematical equations related to electrodiffusion.

Contribution

It introduces exact formulas of Goldman-Hodgkin-Katz type and analyzes the uniqueness of solutions using new differential equations for electrodiffusion with multiple ions.

Findings

Exact constant field solutions are derived.

New formulas of Goldman-Hodgkin-Katz type are established.

Uniqueness of solutions is proven in specific cases.

Abstract

The equations governing one-dimensional, steady-state electrodiffusion are considered when there are arbitrarily many mobile ionic species present, in any number of valence classes, possibly also with a uniform distribution of fixed charges. Exact constant field solutions and new formulas of Goldman-Hodgkin-Katz type are found. All of these formulas are exact, unlike the usual approximate ones. Corresponding boundary conditions on the ionic concentrations are identified. The question of uniqueness of constant field solutions with such boundary conditions is considered, and is re-posed in terms of an autonomous ordinary differential equation of order $n+1$ for the electric field, where $n$ is the number of valence classes. When there are no fixed charges, the equation can be integrated once to give the non-autonomous equation of order $n$ considered previously in the literature including, in the case $n=2$, the form of Painlev\'e's second equation considered first in the context of electrodiffusion by one of us. When $n=1$, the new equation is a form of Li\'enard's equation. Uniqueness of the constant field solution is established in this case.