Creates link between microscopic and macroscopic particles Collection of microscopic particles described with a distribution function Velocity and pressure computed as momentum of distribution functions Only one unknown: distribution function Grid must be fine enough to capture all collisions: $l_{mfp}\ll l_{avg}\ll l$ Define distribution function: $f(\xi,\mathbf{x},t)$ $\xi=\frac{d\mathbf{x}}{dt}$ Defines density of molecules with: * velocity $\xi+d\xi$ * at location $\mathbf{x}+d\mathbf{x}$ * at time $t$ This allows *continuum* description at the *kinetic* level Must conserve: * mass * momentum on each collision Take time derivative of $f(\mathbf{\xi}, \mathbf{x},t)$ $ \begin{align} \frac{df(\mathbf{\xi},\mathbf{x},t)}{dt}&=\left(\frac{\partial}{\partial t}+\frac{d\mathbf{x}}{dt}\frac{\partial}{\partial\mathbf{x}}+\frac{\partial\mathbf{\xi}}{\partial t}\frac{\partial}{\partial\mathbf{\xi}}\right)f(\mathbf{\xi},\mathbf{x},t) \\ &=\left(\frac{\partial}{\partial t}+\mathbf{\xi}\frac{\partial}{\partial\mathbf{x}}+\frac{\mathbf{f}}{\rho}\frac{\partial}{\partial\mathbf{\xi}}\right)f(\mathbf{\xi},\mathbf{x},t) \\ &=:\Omega(f) \end{align} $ $\Omega(f)$ is the [[Collision Operator|collision operator]] Evolution equation: $ \left(\frac{\partial}{\partial t}+\mathbf{\xi}\frac{\partial}{\partial\mathbf{x}}+\frac{\mathbf{f}}{\rho}\frac{\partial}{\partial\mathbf{\xi}}\right)=:\Omega(f) $ Can recover [[Navier-Stokes equations]] given certain constraints