  We investigate possible entropy production in Yang-Mills (YM) field theory by
using a quantum distribution function called Husimi function $f_{\rm H}(A, E,
t)$ for YM field, which is given by a coarse graining of Wigner function and
non-negative. We calculate the Husimi-Wehrl (HW) entropy $S_{\rm HW}(t)=-{\rm
Tr}f_H \log f_H$ defined as an integral over the phase-space, for which two
adaptations of the test-particle method are used combined with Monte-Carlo
method. We utilize the semiclassical approximation to obtain the time evolution
of the distribution functions of the YM field, which is known to show a chaotic
behavior in the classical limit. We also make a simplification of the
multi-dimensional phase-space integrals by making a product ansatz for the
Husimi function, which is found to give a 10-20 per cent over estimate of the
HW entropy for a quantum system with a few degrees of freedom. We show that the
quantum YM theory does exhibit the entropy production, and that the entropy
production rate agrees with the sum of positive Lyapunov exponents or the
Kolmogorov-Sinai entropy, suggesting that the chaoticity of the classical YM
field causes the entropy production in the quantum YM theory.
