- Resources

maxima 工具（微积分、插值）

- 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**:

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