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maxima 工具（微积分、插值）

# 通用的 PIC 方法的过程（把这个调研清楚）

• References
• Problems

Particle-In-Cell (PIC) is a technique commonly used to simulate motion of charged particles, or plasma.

plasma（等离子体）: a gas of ions and atoms

• At low density, plasma behaves more like a collection of discrete particles, than a single continuous fluid. High-density plasmas are simulated using the extension of computational fluid dynamics into electromagnetics, magnetohydrodynamics.

Basic Physics: Coulomb force（库仑力）, Lorentz force（洛伦兹力）

Motivation: we could simulate plasma by computing the storm of particles directly, but plasma simulations generally require at least 1M particles to reduce numerical errors (n²-problem - not feasible), especially when a typical simulation requires Ks of steps

PIC: Compute an electric field from the particles and then compute forces from the electric field (size-n problem). Group a large number of real particles into one computational particle (let ratio be “specific weight”, a.k.a. sp_wt)

$\nabla$ (nabla operator - generates one of gradient $\nabla f$, divergence $\nabla\cdot f$ or curl $\nabla\times f$)

Fundamental equations: See below

• Particles move at very high velocities, need extremely small simulation time steps to reduce numerical errors (again not feasible). Simplify the analysis by considering as a fluid.

Steps:

1. Compute charge density (电荷密度): distribute charges to adjacent nodes on the grid mesh
2. Compute electric potential (电势): Solve Poisson equation (?)
3. Compute electric field: (equals to) the gradient of potential
4. Move particles (compute motion) using offset-by-half time frames for velocity and displacement (position/location)