The Boltzmann Equation
The founding kinetic transport equation, evolving a distribution function under streaming, forces, and a collision integral.
The general form
The Boltzmann equation is the parent of nearly all kinetic theory. It evolves f(x, v, t) under free streaming and external forces, with a collision integral on the right that accounts for particles scattered into and out of each velocity cell:
df/dt + v . grad_x f + (F/m) . grad_v f = (df/dt)_collisionsThe collision term is a nonlinear integral over pairs of colliding particles weighted by the scattering cross section. When collisions are dropped it becomes the Vlasov equation; when they are approximated as small-angle it becomes Fokker-Planck; for neutral-particle transport it keeps the full integral.
The H-theorem
Boltzmann's H-theorem proves the collision operator drives any distribution monotonically toward the Maxwell-Boltzmann equilibrium, defining the arrow of time and connecting kinetics to thermodynamic entropy.
Fluid equations from moments
Multiplying the Boltzmann equation by 1, v, and v^2 and integrating over velocity yields the continuity, momentum, and energy conservation laws. This moment hierarchy is how fluid and MHD models are formally derived; each moment depends on the next, and a closure is needed to truncate the chain.
How it is solved numerically
- Direct Simulation Monte Carlo for neutral gas and edge/divertor neutral transport
- Discrete-ordinates (S_N) and lattice-Boltzmann methods for the streaming plus collision balance
- Moment methods that solve a truncated, closed set of fluid-like equations
In fusion, the Boltzmann equation with charge-exchange and ionization operators models neutral hydrogen recycling at the wall and in the divertor, essential for predicting particle fueling and plasma-facing-component loads in devices such as the Hyperion breeder.