A novel method to construct stationary solutions of the Vlasov-Maxwell system : the relativistic case

A method to derive stationary solutions of the relativistic Vlasov-Maxwell system is explored. In the non-relativistic case, a method using the Hermite polynomial series to describe the deviation from the Maxwell-Boltzmann distribution is found to be successful in deriving a few stationary solutions including two dimensional one. Instead of the Hermite polynomial series, two special orthogonal polynomial series, which are appropriate to expand the deviation from the Maxwell-J\"uttner distribution, are introduced in this paper. By applying this method, a new two-dimensional equilibrium is derived, which may provide an initial setup for investigations of three-dimensional relativistic collisionless reconnection of magnetic fields.