The Maxwell-Boltzmann Distribution
The equilibrium velocity distribution of a plasma species, from which reactivities and rates are computed.
The equilibrium distribution
In thermal equilibrium the velocities of particles follow the Maxwell-Boltzmann distribution, the unique stationary solution of the Boltzmann collision operator (the endpoint of the H-theorem). The speed distribution is:
f(v) = n (m / 2 pi k_B T)^(3/2) 4 pi v^2 exp(-m v^2 / 2 k_B T)It peaks at the most probable speed, has a mean thermal speed, and a long high-energy tail. Temperature is simply a measure of the width of this distribution; there is no upper speed limit, only exponentially rare fast particles.
The all-important tail
Because fusion cross sections rise so steeply with energy, the rare fast particles in the exponential tail dominate the reaction rate. Averaging the cross section over this distribution produces the reactivity
When it holds
- Requires enough collisions to thermalize, valid in the collisional bulk plasma
- Fast ions from beams or fusion form non-Maxwellian tails until they slow down
- Mirror machines have loss-cone-depleted, non-Maxwellian distributions
Numerical role
Many codes assume a local Maxwellian and evolve only its density and temperature (the fluid picture), or evolve the small deviation from it (delta-f methods). When the distribution is strongly non-Maxwellian, full kinetic (Fokker-Planck) treatment is required.
Fusion relevance
The Maxwell-Boltzmann distribution is the basis for computing reactivities used in the power balance of the Hyperion breeder. In the Kronos burner, a tandem mirror, the loss cone makes the distribution non-Maxwellian, so its reaction rate and stability need kinetic treatment rather than a simple Maxwellian average.