  We consider contribution to the phonon scattering, in the temperature range
of 1K, by the dislocation kinks pinned in the random stress fields in a
crystal. The effect of electron-kink scattering on the thermal transport in the
normal metals was considered much earlier \cite{Muk86}. The phonon thermal
transport anomaly at low temperature was demonstrated by experiments in the
deformed (bent) superconducting lead samples \cite{Mez79} and in helium-4
crystals \cite{Mez82, Mez84} and was ascribed to the dislocation dynamics.
Previously, we had discussed semi-qualitatively the phonon-kink scattering
effects on the thermal conductivity of insulating crystals in a series of
papers \cite{mezmuk, ostmukmez}. In this work it is demonstrated explicitly
that exponent of the power low in the temperature dependence of the phonon
thermal conductivity depends, due to kinks, on the distribution of the random
elastic stresses in the crystal, that pin the kinks motion along the
dislocation lines. We found that one of the random matrix distributions of the
well known Wigner-Dyson theory is most suitable to fit the lead samples
experimental data \cite{Mez79}. We also demonstrate that depending on the
distribution function of the oscillation frequencies of the kinks, the power
low temperature dependences of the phonon thermal conductivity, in principle,
may possess exponents in the range of $2\div 5$.
