A statistical physics perspective on the evolution of Ion distribution across low-mach number quasi-perpendicular collisionless shocks

The heating of directly transmitted ions at low Mach number, perpendicular and quasi perpendicular shocks has been the subject of previous statistical physics studies. In this paper, we use a Hamiltonian formulation of the ion kinetics for a quasi-perpendicular shock model to derive the general solution to Liouville's equation as a function of six invariants, finding forms of these invariants in terms of the upstream parameters. The ion distribution is expressed as a function of one of these invariants, subject to a Maxwellian upstream boundary condition, and the evolution of the distribution through and downstream of the shock is studied. The momentum-space volume within surfaces of constant probability (related to the temperature) is shown to be inversely proportional to an average value of the canonical momentum associated with motion in the direction normal to the shock plane, generalizing a previous result to three-dimensional phase space. We also study the evolution of the distribution properties numerically, in particular noting that the “twisting” of these surfaces in phase space is the result of the unequal guiding center motion of different parts of the distribution (which is not the case for a fully perpendicular shock). This property provides insight into the damping of oscillations in the mean momentum and the temperature for a quasi-perpendicular model (as the distribution is spread about the central value through gyration) and the observed T > T anisotropy.