Refined derivation, exact solutions, and singular limits of the Poisson- Boltzmann equation
type: Item
instance of: scholarly article
title: Refined derivation, exact solutions, and singular limits of the Poisson- Boltzmann equation
zbMATH Open document ID: 0663.60077
DOI: 10.1016/0003-4916(88)90106-6
author: A. J. M. Garrett
author: Leon Poladian
published in: Annals of Physics
publication date: 1988-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B05
Mathematics Subject Classification ID: 82D10
zbMATH DE Number: 4084708
zbMATH Keywords: Boltzmann factor
zbMATH Keywords: high-temperature expansion
zbMATH Keywords: infinite plasma
zbMATH Keywords: electrostatic potential
zbMATH Keywords: entropy-maximizing principle
zbMATH Keywords: Lagrange multipliers
zbMATH Keywords: spherically symmetric solutions
Wikidata QID: Q62003437
MaRDI profile type: Publication
full work available at URL: 0003-4916(88)90106-6
OpenAlex ID: W2047808346
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description: scientific article; zbMATH DE number 4084708