Recent laboratory experiments with laser-produced plasmas have observed and studied a number of fundamental physical processes relevant to magnetized astrophysical plasmas, including magnetic reconnection, collisionless shocks, and magnetic field generation by Weibel instability, opening up new experimental platforms for laboratory astrophysics. We develop a fully kinetic simulation model for first-principles simulation of these systems including the dynamics of magnetic fields---magnetic field generation by the Biermann battery effect or Weibel instability; advection by the ion flow, Hall effect, and Nernst effect; and destruction of the field by dissipative mechanisms. Key dimensionless parameters describing the system are derived for scaling between kinetic simulation, recent experiments, and astrophysical plasmas. First, simulations are presented which model Biermann battery magnetic field generation in plasmas expanding from a thin target. Ablation of two neighboring plumes leads to the formation of a current sheet as the opposing Biermann-generated fields collide, modeling recent laser-driven magnetic reconnection experiments. Second, we simulate recent experiments on collisionless magnetized shock generation, by expanding a piston plasma into a pre-magnetized ambient plasma. For parameters considered, the Biermann effect generates additional magnetic fields in the curved shock front and thereby increases shock particle reflection. Both cases show the importance of kinetic processes in the interaction of plasmas with magnetic fields, and open opportunities to benchmark these important processes through comparison of theory and experiments.
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