Collisional relaxation of a strongly magnetized two-species pure ion plasma

Abstract

The collisional relaxation of a strongly magnetized pure ion plasma that is composed of two species with slightly different masses is discussed. We have in mind two isotopes of the same singly ionized atom. Parameters are assumed to be ordered as Ω1,Ω2≫|Ω1−Ω2|≫v¯ij/b¯ and v¯⊥j/Ωj≪b¯, where Ω1 and Ω2 are two cyclotron frequencies, v¯ij=T∥/μij is the relative parallel thermal velocity characterizing collisions between particles of species i and j, and b¯=2e2/T∥ is the classical distance of closest approach for such collisions, and v¯⊥j/Ωj=2T⊥j/mj/Ωj is the characteristic cyclotron radius for particles of species j. Here, μij is the reduced mass for the two particles, and T∥ and T⊥j are temperatures that characterize velocity components parallel and perpendicular to the magnetic field. For this ordering, the total cyclotron action for the two species, I1=∑i∈1m1v⊥i2/(2Ω1) and I2=∑i∈2m2v⊥i2/(2Ω2) are adiabatic invariants that constrain the collisional dynamics. On the timescale of a few collisions, entropy is maximized subject to the constancy of the total Hamiltonian H and the two actions I1 and I2, yielding a modified Gibbs distribution of the form exp[−H/T∥−α1I1−α2I2]. Here, the αj’s are related to T∥ and T⊥j through T⊥j=(1/T∥+αj/Ωj)−1. Collisional relaxation to the usual Gibbs distribution, exp[−H/T∥], takes place on two timescales. On a timescale longer than the collisional timescale by a factor of (b¯2Ω12/v¯112)exp{5[3π(b¯|Ω1−Ω2|/v¯12)]2/5/6}, the two species share action so that α1 and α2 relax to a common value α. On an even longer timescale, longer than the collisional timescale by a factor of the order exp{5[3π(b¯Ω1/v¯11)]2/5/6}, the total action ceases to be a good constant of the motion and α relaxes to zero.