On the accuracy of the Debye shielding

TL;DR

This paper evaluates the accuracy of the Debye shielding approximation in plasma physics by comparing the linearized analytical solution with the exact numerical one, providing criteria for its validity.

Contribution

It quantitatively assesses the Debye shielding approximation's accuracy and establishes limits of its applicability based on particle number and distance.

Findings

Linearized solution underestimates the exact potential.

Approximation remains reasonable for ND around 40.

Provides criteria for the validity of the Debye shielding approximation.

Abstract

The expression for the Debye shielding in plasma physics is usually derived under the assumptions that the plasma particles are weakly coupled, so their kinetic energy is much larger than the potential energy between them, and that the velocity distributions of the plasma species are Maxwellian. The first assumption also establishes that the plasma parameter ND, the number of particles within a sphere with a Debye radius should be greater than 1, and determines the difference between weakly and strongly coupled plasmas. Under such assumptions, Poisson's equation can be linearised, and a simple analytic expression obtained for the electrostatic potential. However, textbooks rarely discuss the accuracy of this approximation. In this work we compare the linearised solution with the exact one, obtained numerically, and show that the linearisation, which underestimates the exact solution, is reasonably good even for ND ~ 40. We give quantitative criteria to set the limit of the approximation when the number of particles is very small, or the distance to the test charge too short.