# Kinetic Model

## Kinetic Model

The Boltzmann equation, which describes the evolution of a velocity distribution function using binary collisions, can accurately describe the flow behavior of a dilute gas from a continuum to the free molecular regime. However, due to the high complexity of the Boltzmann collision operator, it is still challenging to solve the Boltzmann equation directly. Therefore, many studies seek to find alternative approximations of the Boltzmann equation.

## Fokker-Planck Method

Fokker-Planck (FP) equation approximates the discrete collision process as a continuous stochastic process. The FP equation can be transformed into the equivalent Langevin equation using Ito’s process, which describes the Brownian motion of the particles. The FP equation does not need to resolve collision events because the Brownian motion is a macroscopic movement resulting from many microscopic random effects. Thus, the FP equation is potentially more efficient than the Boltzmann equation for multi-scale rarefied gas flows.

## Bhatnagar-Gross-Krook method

Bhatnagar-Gross-Krook (BGK) equation approximates the discrete collision process as a relaxation time process. Due to the simple structure of a collision operator, the BGK equation can be analytically integrated. The BGK equation is solved by randomly selecting a fraction of simulated particles and assigning them with new velocities, sampled from the target distribution function. Since the equation does not require binary collision modeling, the approach is more efficient than Direct Simulation Monte Carlo (DSMC) in the near continuum regime.

### Related Works