  The previously derived exact evolution equations for density matrix of
electron (quantum particle) in phonon field (boson thermostat) are
qualitatively analysed. Their statistical interpretation is explained in
detail, and their main symmetry and spectral properties are expounded. In
application to the electron's random walk, it is shown that these properties
certaimly forbid conventionally assumed Gaussian long-range asymptotic of the
walk statistics. Instead, the exact equations imply super-linear dependence of
fourth-order cumulant of total electron's path on observation time, which
signifies existence of 1/f-type low-frequency fluctuations in electro's
diffusivity and mobility. Physical meaning of this result is discussed, along
with general origin of 1/f-noise in classical and quantum Hamiltonian
many-particle systems.
