Dynamic scaling invariance at low temperatures

Using thermodynamic arguments we prove that the conventional consequences of the dynamic scale hypothesis change their character in the limit as the critical temperature $T_{c}$ approaches zero. In particularly, for liquid helium-4, the critical exponent $\alpha $ associated with the heat capacity ($\alpha $ \textless 0) and other exponents related by the following new relation \begin{equation} \label{eq1} \nu (z-1)=(1+S_{I} -\alpha )/6\,\,,\,\,\,\,T_{C} =T_{\lambda } \ge 0\,\,, \end{equation} \begin{equation} \label{eq2} S_{I} =\left( {\frac{T_{C} }{T }} \right)^{n}\,,\,\,\,\,\,\,T\ge T_{C} \quad , \end{equation} where $n$ is a positive constant [1] and $z$ is the dynamic critical exponent, $\nu $ -- the critical exponent of the correlation length. It is important that now the exponent $z$ depends on $T$ and $T_{\lambda } $. If $T_{\lambda } \quad =$0 and $T$\textgreater 0, then the $S_{I} $-function [1] is zero and Eq. (\ref{eq1}) becomes \begin{equation} \label{eq3} \nu (z-1)=(1-\alpha )/6\,\,,\,\,\,\,T_{C} =0,\,\,\,(T>0,\,\,\,\alpha <0)\,\,. \end{equation} Eq. (\ref{eq3}) can be applied, for example, to a mixture of liquid $He^{3}$ and $He^{4}$. The results are valid for multi-component order parameter. 1.~Udodov V. Violating of the Essam-Fisher and Rushbrooke Relationships at Low Temperatures// World Journal of Condensed Matter Physics. --- 2015. --- Ò.5. --- ¹2. --- Ñ. 55-59. \underline {http://dx.doi.org/10.4236/wjcmp.2015.52008}.