The Weibel instability for an electron–positron and an electron–ion plasma is investigated in a linear analysis of the Vlasov–Maxwell equations for temporal growth. Comparison with a computer simulation of Suzuki and Shigeyama confirms that a single exponentially temporally growing mode for the magnetic field dominates. A quasi-equilibrium one-particle distribution function for an electron–positron plasma is expressed as an expansion of even order Hermite polynomials multiplied by a Maxwellian. A quasi-equilibrium with current density linear in the vector potential is constrained by conservation of particle number, total energy and the wave equation for the vector potential. The sinh-Poisson equation, a nonlinear partial differential equation, is introduced for a more general quasi-equilibrium. An analytic expression is found for the vector potential in terms of Jacobian elliptic functions and the expansion coefficients for the expansion of the distribution function in Hermite polynomials are found.
Fokker-Planck equation for a test particle weakly coupled to a magnetized plasma