On the justification of the Poisson–Boltzmann equation in the context of technological plasmas

Abstract

Abstract The Poisson–Boltzmann (PB) equation is a nonlinear differential equation for the electric potential that describes equilibria of conducting fluids. Its standard justification is based on a variational principle which characterizes the thermodynamic equilibrium of a system in contact with a heat reservoir as a minimum of the Helmholtz free energy. The PB equation is also employed in the context of technological plasmas. There, however, the standard justification is inapplicable: technological plasmas are neither in thermodynamic equilibrium nor in contact with heat reservoirs. This study presents an alternative variational principle which is based on the functionals of entropy, particle number, and electromagnetic enthalpy. It allows to justify the PB equation for a wide class of technological plasmas under realistic assumptions.