Prompted by the relevant problem of temperature inversion (i.e. gradient of density anti-correlated to the gradient of temperature) in astrophysics, we introduce a novel method to model a gravitationally confined multi-component collisionless plasma in contact with a fluctuating thermal boundary. The dynamics of the plasma is analytically described via the coupling of an appropriated coarse-grained distribution function and temporally coarse-grained Vlasov equations. We derive a stationary solution of the system and predict the inverted density and temperature profiles. We present applications of our theoretical framework to the problem of the temperature inversion in the solar corona obtaining density and temperature profiles in a good agreement with the observations. 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 the model to main-sequence stars, showing that it predicts the presence of a solar-like hot and rarefied corona for all such stars, regardless of their mass. We also discuss the role of stellar mass in determining the shape of the temperature and density profiles.
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