Refined derivation, exact solutions, and singular limits of the Poisson- Boltzmann equation
The problem is to find the electrostatic potential in a plasma or electrolyte, allowing the presence of fixed immovable charges. The starting point of the derivation is the entropy-maximizing principle of E. T. Jaynes, namely to maximize the entropy S under the constraints of constant total energy and constant total probability. (The Lagrange multipliers are \((kT)^{-1}\) and ln (Z-1) as is well-known). The resulting equation (``Poisson-Boltzmann'') is transcendental in the potential \(\phi\). For positive temperatures the uniqueness of solutions is given and many cases of solutions are investigated in detail: the high- and low- temperature limits (and expansions), the planar solutions, and the cylindrically or spherically symmetric solutions (for single species and for two species).
scientific article; zbMATH DE number 4084708
scientific article