1 Particle methods
 1.1 Brief history of particle-mesh methods
2 Phase-space sampling and markers’ weight
 2.1 Phase-space sampling and Phase space volume sampled by a marker
 2.2 Weights of markers
3 Shape function of markers: basis functions used in expanding distribution function
 3.1 Integration in velocity space
 3.2 Cell-averaged velocity moment
 3.3 Effective field on a marker
 3.4 Effective force on a marker
 3.5 Numerical implementation in codes
 3.6 Monte-Carlo integration in phase-space
 3.7 On accuracy and noise: particle methods vs. Euler-grid-based methods
 3.8 Practical comparison between particle methods and Euler-grid-based methods
 3.9 Modeling collisions in PIC simulations
4 Evolution of distribution functions
 4.1 Time evolution of the physical distribution function
 4.2 Time evolution of marker distribution function
 4.3 Time evolution of marker’s weight
5 An example: One-dimensional electrostatic simulation
 5.1 Vlasov equation
 5.2 Poisson’s equation
 5.3 Equilibrium state
 5.4 δf evolution
 5.5 Normalization
 5.6 Boundary condition for field
 5.7 Boundary condition for particles
 5.8 Evaluation of particle number density
 5.9 FFT solver for Poisson equation
 5.10 Finite difference solver for Poisson equation
 5.11 Interpolate the field to particle markers
 5.12 Integration of orbit and weight of markers
 5.13 Initial perturbations
 5.14 Verification of the code by using analytic results of Landau damping
 5.15 Methods of identifying resonant particles
 5.16 Energy conservation (check!)
 5.17 Numerical results for two-stream instability
6 Summary
7 Random number
 7.1 Uniformly distributed random number
 7.2 Non-uniformly distributed random number
8 On the noise of PIC simulation
 8.1 Choices of sampling probability function
9 Finite element theory of particle-in-cell method
 9.1 Finite element expansion of distribution function
 9.2 Basis functions: particle shape
 9.3 Moment equations
A From discrete microscopic distribution function to statistic (continuum) distribution function
References