  The nonlinear optical Galton board (NLOGB), a quantum walk like (but
nonlinear) discrete time quantum automaton, is shown to admit a complex
evolution leading to long time thermalized states. The continuous limit of the
Galton Board is derived and shown to be a nonlinear Dirac equation (NLDE). The
(Galerkin truncated) NLDE evolution is shown to thermalize toward states
qualitatively similar to those of the NLOGB. The NLDE conserved quantities are
derived and used to construct a stochastic differential equation converging to
grand canonical distributions that are shown to reproduce the (micro canonical)
NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are
thus demonstrated to exhibit spontaneous thermalization.
