  We study numerically the wavefunction evolution of a Bose-Einstein condensate
in a Bunimovich stadium billiard being governed by the Gross-Pitaevskii
equation. We show that for a moderate nonlinearity, above a certain threshold,
there is emergence of dynamical thermalization which leads to the Bose-Einstein
probability distribution over the linear eigenmodes of the stadium. This
distribution is drastically different from the energy equipartition over
oscillator degrees of freedom which would lead to the ultra-violet catastrophe.
We argue that this interesting phenomenon can be studied in cold atom
experiments.
