Collisional Relaxation of a Strongly Magnetized, Two-Isotope, Pure Ion Plasma

The collisional relaxation of a strongly magnetized pure ion plasma\footnote{P.J. Hjorth and T.M. O'Neil, Phys. Fluids \textbf{26}, 2128(1983); M.E. Glinsky, {\it et al.}, Phys. Fluids B \textbf{4}, 1156 (1992).} that is composed of two species with slightly different mass is discussed. We assume the ordering $\Omega_{C1},\Omega_{C2}\ll|\Omega_{C1}-\Omega_{C2}|\ll v / b $, where $\Omega_{C1}$ and $\Omega_{C2}$ are the two cyclotron frequencies, $ v $ is the thermal velocity, and $ b $ is the classical distance of closest approach. We find that the total cyclotron action for the two species $I_1$ and $I_2$ are adiabatic invariants conserved on the timescale of a few collisions, so the Gibbs distribution relaxes to the form $\exp[-H/T-\alpha_1 I_1-\alpha_2 I_2]$, where $\alpha_1$ and $\alpha_2$ are thermodynamic variables like the temperature $T$. On a timescale longer than the collisional timescale, the two species share action so that $\alpha_1$ and $\alpha_2$ relax to a common value $\alpha$. During this process, $+$ remains constant. On an even longer timescale, the total action ceases to be a good constant of the motion and $\alpha$ relaxes to zero, yielding the usual Gibbs distribution $\exp [ - H/T].$