  A nonlinear theory of quantum Brownian motion in classical environment is
developed based on a thermodynamically enhanced nonlinear Schrodinger equation.
The latter is transformed via the Madelung transformation into a nonlinear
quantum Smoluchowski-like equation, which is proven to reproduce key results
from quantum and classical physics. The application of the theory to a free
quantum Brownian particle results in a nonlinear dependence of the position
dispersion on time, being quantum generalization of the Einstein law of
Brownian motion. It is shown that the time of decoherence for the transition
from quantum to classical diffusion is proportional to the square of the
thermal de Broglie wavelength divided by the Einstein diffusion constant.
