We studied statistical properties of Wigner functions of 1D quantum maps on compact phase space of finite area . For this purpose we defined a Wigner function probability distribution , which has, by definition, fixed first and second moment. In particular, we concentrate on relaxation of time evolving quantum state in terms of , starting from a coherent state.
We have shown that for a classically chaotic quantum counterpart the distribution becomes a Gaussian distribution that is determinated by the first two moments. The numerical studies were done on the quantum sawtooth map and the quantized kicked top. In a quantum system with Hilbert space dimension the transition of to a Gaussian distribution was observed at times . In addition, it has been shown, that the statistics of Wigner functions of propagator eigenstates is a Gaussian as well in classically fully chaotic regime.