Temperature inversion in a non-homogeneous collisionless plasma with a thermal boundary: the temporal coarse-graining method

Prompted by the relevant problem of temperature inversion (i.e. gradient of density anti-correlated to the gradient of temperature) in solar physics, we introduce a novel method to model a gravitationally confined multi-component collisionless plasma in contact with a fluctuating thermostat. The dynamics is described via a set of effective partial differential equations for the coarse-grained versions of the distribution functions of the plasma components and a temporally coarse-grained energy reservoir. We derive a stationary solution of this system naturally predicting the inverted density-temperature profiles of the two-species as observed in systems of astrophysical interest such as the solar corona. We validate our method by comparing the analytical results with kinetic numerical simulations of the plasma dynamics in the context of the two-species Hamiltonian mean-field model (HMF). Finally, we apply our theoretical framework to the problem of the temperature inversion in the solar corona obtaining density and temperature profiles in remarkably good agreement with the observations.