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Fully kinetic analytic calculations of an initially Maxwellian distributionwith arbitrary density and temperature gradients exhibit the development oftemperature anisotropies and magnetic field growth associated with the Biermannbattery. The calculation, performed by taking a small order expansion of theratio of the Debye length to the gradient scale, predicts anisotropies andmagnetic fields as a function of space given an arbitrary temperature anddensity profile. These predictions are shown to qualitatively match the valuesmeasured from particle-in-cell simulations, where the development of the Weibelinstability occurs at the same location and with a wavenumber aligned with thepredicted temperature anisotropy.

General kinetic solution for the Biermann battery with an associated pressure anisotropy generation