Physical Model

Vlasov-Poisson System

MiniPIC.jl simulates collisionless plasmas by solving the Vlasov-Poisson system in one spatial dimension.

Vlasov Equation

The evolution of the distribution function $f(x, v, t)$ of a charged species is governed by the Vlasov equation:

\[\frac{\partial f}{\partial t} + v \frac{\partial f}{\partial x} + \frac{q}{m} E \frac{\partial f}{\partial v} = 0\]

where:

• $q$ is the particle charge
• $m$ is the particle mass
• $E$ is the electric field

Poisson Equation

The electric field is determined self-consistently through the electrostatic Poisson equation:

\[\frac{\partial^2 \varphi}{\partial x^2} = -\frac{\rho}{\varepsilon}\]

\[E = -\frac{\partial \varphi}{\partial x}\]

where $\varphi$ is the electric potential and $\varepsilon$ is the permittivity.

Charge Density

The charge density is obtained by integrating the distribution function over velocity space:

\[\rho = q \int f \, \mathrm{d}v - Q\]

where $Q$ is a uniform neutralizing background charge:

\[Q = \frac{q}{L} \iint f \, \mathrm{d}x \, \mathrm{d}v\]

This background ensures quasi-neutrality of the plasma.

Boundary Conditions

The simulation uses periodic boundary conditions in both position and field quantities. Particles leaving the domain on one side re-enter from the opposite side.