  In this paper the first non-linear force-free Vlasov-Maxwell equilibrium is
presented. One component of the equilibrium magnetic field has the same spatial
structure as the Harris sheet, but whereas the Harris sheet is kept in force
balance by pressure gradients, in the force-free solution presented here force
balance is maintained by magnetic shear. Magnetic pressure, plasma pressure and
plasma density are constant. The method used to find the equilibrium is based
on the analogy of the one-dimensional Vlasov-Maxwell equilibrium problem to the
motion of a pseudo-particle in a two-dimensional conservative potential. This
potential is equivalent to one of the diagonal components of the plasma
pressure tensor. After finding the appropriate functional form for this
pressure tensor component, the corresponding distribution functions can be
found using a Fourier transform method. The force-free solution can be
generalized to a complete family of equilibria that describe the transition
between the purely pressure-balanced Harris sheet to the force-free Harris
sheet.
