Derivation of the Landau equation for weakly coupled or Coulomb particles
In plain words
Charged particles in a plasma interact through many weak, long-range Coulomb nudges, described by the Landau collision equation. This equation has never been derived from Newton's laws.
Precise statement
$N$ classical particles in $R^{3}$ with pair potential $\epsilon^{(1/2)} \phi(x/\epsilon)$, $\phi$ smooth and short-range, $N \epsilon^{3} = 1$, $\epsilon \to 0$ (weak-coupling limit); the Coulomb case $e^{2}/\mid x\mid$ with a logarithmic cutoff is the harder variant. Prove that the one-particle marginal converges, for a positive time interval, to a solution of the Landau equation. Answer: a proof for a time interval of positive length.
What would settle it
A rigorous propagation-of-chaos proof yielding the Landau equation for a positive time interval.
Status in the literature
Only formal derivations and partial results on truncated hierarchies exist (e.g. Bobylev, Pulvirenti and Saffirio, 2013).