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Description
We present analytical expressions for the evolution of a general, gyrotropic, relativistic distribution function (VDF) of a collisionless, uniform, magnetized plasma subject to external forcings in the form of density and magnetic field variations. By analytically solving the relativistic, time-dependent drift kinetic equation for a collisionless, uniform plasma, a solution is obtained in terms of an initial condition. When considering an initial nonrelativistic Maxwellian VDF, a Bi-Maxwellian form is naturally recovered. Interestingly, when considering an initial relativistic Maxwell-Jüttner VDF, we obtain a new time-dependent anisotropic form of the Maxwell-Jüttner. By taking pressure moments of this distribution, relativistic double-adiabatic equations are obtained, extending the well-known CGL equations. We validate these double-adiabatic equations and our new time-dependent VDF with PIC simulations of shearing and compressing boxes.
These distributions can be used in linear stability analysis for accurately predict instabilities beyond the bi-Maxwellian approximation and quantification of distortions in the VDF by wave-particle interactions. Similar studies can be performed for other initial VDFs like power-laws and kappa distributions.