  A nonlinear master equation is derived, reflecting properly the entropy of
open quantum systems. In contrast to linear alternatives, its equilibrium
solution is exactly the canonical Gibbs density matrix. The corresponding
nonlinear equation for the Wigner function accounts rigorously for the
thermo-quantum entropy. It reduces at large friction to the Smoluchowski-Bohm
equation in the coordinate subspace, which reflects the stochastic
Langevin-Bohm dynamics. The previously derived Maxwell-Heisenberg relation for
the nonequilibrium momentum dispersion of quantum Brownian particles is
confirmed as well as the related quantum generalization of the classical
Einstein law of Brownian motion.
