  We have studied statistical properties of the values of the Wigner function
W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this
purpose we have defined a Wigner function probability distribution P(w) = (1/V)
int delta(w-W(x)) dx, which has, by definition, fixed first and second moment.
In particular, we concentrate on relaxation of time evolving quantum state in
terms of W(x), starting from a coherent state. We have shown that for a
classically chaotic quantum counterpart the distribution P(w) in the
semi-classical limit becomes a Gaussian distribution that is fully determined
by the first two moments. Numerical simulations have been performed for the
quantum sawtooth map and the quantized kicked top. In a quantum system with
Hilbert space dimension N (similar 1/hbar) the transition of P(w) to a Gaussian
distribution was observed at times t proportional to log N. In addition, it has
been shown that the statistics of Wigner functions of propagator eigenstates is
Gaussian as well in the classically fully chaotic regime. We have also studied
the structure of the nodal cells of the Wigner function, in particular the
distribution of intersection points between the zero manifold and arbitrary
straight lines.
