Exact analytical model of the classical Weibel instability in a relativistic anisotropic plasma

Detailed properties of the Weibel instability in a relativistic unmagnetized plasma are investigated for a particular choice of anisotropic distribution function F(p_⊥²,p_z) that permits an exact analytical solution to the dispersion relation for arbitrary energy anisotropy. The particular equilibrium-distribution function considered in the present analysis assumes that all particles move on a surface with perpendicular momentum p_⊥=p^_⊥=const and are uniformly distributed in parallel momentum from p_z=-p^_z=const to p_z=+p^_z=const. (Here, the propagation direction is the z direction.) The resulting dispersion relation is solved analytically, and detailed stability properties are determined for a wide range of energy anisotropy.