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 quantum Smoluchowski equation in the coordinate subspace. 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.
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