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

TLDR: Detailed properties of the Weibel instability in a relativistic unmagnetized plasma are investigated for a particular choice of anisotropic distribution function that permits an exact analytical solution to the dispersion relation for arbitrary energy anisotropy.

Abstract: Detailed properties of the Weibel instability in a relativistic unmagnetized plasma are investigated for a particular choice of anisotropic distribution function F(${p}_{\ensuremath{\perp}}^{2}$,${p}_{z}$) 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 ${p}_{\ensuremath{\perp}}$=p${^}_{\ensuremath{\perp}}$=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.