Explicit Solutions of the One-dimensional Vlasov-Poisson System with Infinite Mass and Energy

TL;DR

This paper derives explicit solutions for the one-dimensional Vlasov-Poisson system with infinite total charge and energy, showing the charge density is compactly supported and providing explicit formulas for electric field and density at large distances.

Contribution

It provides explicit solutions for a Vlasov-Poisson system with infinite mass and energy, a case not previously explicitly solved.

Findings

Charge density is shown to be compactly supported.

Explicit formulas for electric field and number density at large x.

Demonstrates behavior of plasma with infinite total charge.

Abstract

A collisionless plasma is modeled by the Vlasov-Poisson system in one-dimension. A fixed background of positive charge, dependent only upon velocity, is assumed and the situation in which the mobile negative ions balance the positive charge as x tends to positive or negative infinity. Thus, the total positive charge and the total negative charge are infinite. In this paper, the charge density of the system is shown to be compactly supported. More importantly, both the electric field and the number density are determined explicitly for large values of x.