We study how to numerically simulate quantum fermions out of thermal equilibrium, in the context of electroweak baryogenesis. We find that by combining the lattice implementation of Aarts and Smit  with the "low cost" fermions of Borsanyi and Hindmarsh , we are able to describe the dynamics of a classical bosonic system coupled to quantum fermions, that correctly reproduces anomalous baryon number violation. To demonstrate the method, we apply it to the 1+1 dimensional axial U(1) model, and perform simulations of a fast symmetry breaking transition. Compared to solving all the quantum mode equations as in , we find that this statistical approach may lead to a significant gain in computational time, when applied to 3+1 dimensional physics.
Saffin, P. M., & Tranberg, A. (2011). Real-time fermions for baryogenesis simulations. Journal of High Energy Physics, 2011(7), https://doi.org/10.1007/JHEP07%282011%29066