Value statistics of chaotic Wigner function

We study Wigner function value statistics of classically chaotic quantum maps on compact 2D phase space. We show that the Wigner function statistics of a random state is a Gaussian, with the mean value becoming negligible compared to the width in the semi-classical limit. Using numerical example of quantized sawtooth map we demonstrate that the relaxation of time-dependent Wigner function statistics, starting from a coherent initial state, takes place on a logarithmically short log (hbar) time scale.