  We present a microscopic theory of nonlinear damping and dephasing of
low-frequency eigenmodes in nano- and micro-mechanical systems. The mechanism
of the both effects is scattering of thermally excited vibrational modes off
the considered eigenmode. The scattering is accompanied by energy transfer of
$2\hbar\omega_0$ for nonlinear damping and is quasieleastic for dephasing. We
develop a formalism that allows studying both spatially uniform systems and
systems with a strong nonuniformity, which is smooth on the typical wavelength
of thermal modes but not their mean free path. The formalism accounts for the
decay of thermal modes, which plays a major role in the nonlinear damping and
dephasing. We identify the nonlinear analogs of the Landau-Rumer,
thermoelastic, and Akhiezer mechanisms and find the dependence of the
relaxation parameters on the temperature and the geometry of a system.
