pic1d: A 1D Electrostatic Particle-in-Cell Plasma Simulation

A 1D electrostatic PIC code written from scratch in Python (NumPy-vectorized), validated against classic kinetic plasma problems and applied to self-consistent plasma sheath formation and wall heat flux estimation, a problem directly relevant to Hall thruster channel wall erosion and thermal loading.

Author: Rafael Cedeno

Physics roadmap

1. Cold plasma oscillations: verify oscillation at the plasma frequency.
2. Two-stream instability: verify linear growth rate against theory, phase-space vortex formation.
3. Plasma sheath: absorbing walls, kinetic electrons and ions, sheath potential drop vs. floating potential theory, Bohm criterion check.
4. Wall heat flux: particle and energy fluxes to the wall, transient wall conduction estimate. Ties plasma results to thermal analysis.

Normalization

All quantities are dimensionless from day one:

Quantity	Normalized by
Length	Debye length (lambda_D)
Time	Inverse plasma frequency (1/w_pe)
Velocity	Electron thermal speed (v_th)
Potential	k_B T_e / e
Density	Reference density n_0

With these choices, the normalized Poisson equation is d^2(phi)/dx^2 = -(n_i - n_e) and the electron equation of motion is dv/dt = E (sign conventions handled in pusher.py).

Repo structure

src/pic1d/
    grid.py         # Grid class: domain, spacing, field arrays
    particles.py    # Species class: positions, velocities, charge, mass
    deposition.py   # Cloud-in-cell (linear) charge weighting
    poisson.py      # 1D Poisson solvers (periodic FFT + Dirichlet tridiagonal)
    pusher.py       # Leapfrog particle push + field gather
    boundary.py     # Periodic wrap, absorbing walls, particle injection
    simulation.py   # Main PIC loop orchestration
    diagnostics.py  # Energy history, phase space, flux counters
tests/              # Unit tests for each component (run: pytest)
examples/           # Driver scripts for each physics milestone
report/             # Technical report + generated figures

Quick start

pip install -r requirements.txt
pytest                                  # unit tests
python examples/run_plasma_oscillation.py
python examples/run_two_stream.py
python examples/run_sheath.py

Development order (matches the 4-week plan)

• Week 1: implement pusher.py, poisson.py, deposition.py; all unit tests pass
• Week 2: assemble simulation.py with periodic BCs; plasma oscillation + two-stream validation
• Week 3: absorbing walls + injection in boundary.py; sheath formation
• Week 4: wall heat flux diagnostics; technical report; one-command figure reproduction

Key references

• Birdsall & Langdon, Plasma Physics via Computer Simulation
• L. Brieda, Plasma Simulations by Example (practical Python walkthrough)
• Hockney & Eastwood, Computer Simulation Using Particles