We consider contribution to the phonon scattering, in the temperature range of 1 K, 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 [1]. The phonon thermal transport anomaly at low temperature was demonstrated by experiments in the deformed (bent) superconducting lead samples [2] and in helium-4 crystals [3,4] 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 [5,6]. 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 [2]. 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 - 5.

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