Fastest Frozen Temperature for a Degree of Freedom

X. Y. Zhou, Z. Q. Yang, X. R. Tang, X. Wang, Q. H. Liu

Published 2020-01-09Version 1

The equipartition theorem states that the mean thermal energy for each degree of freedom at a temperature $T$ is given by $k_{B}T/2$ where $k_{B}$ is Boltzmann's constant. However, this theorem for the degree of freedom breaks down at lower enough temperature when the relevant degree of freedom is not thermally excited. The frozen condition can be \textit{qualitatively} expressed as $T\ll T_{C}$ which is the characteristic temperature defined via $k_{B}T_{C}\equiv\varepsilon_{1}-\varepsilon_{0}$ in which $\varepsilon_{0}$ and $\varepsilon_{1}$ are, respectively, the ground state and first excited energy of the degree of freedom. This characteristic temperature $T_{C}$ so defined is the temperature at which the degree of freedom is almost activated. We report that there is a well-defined temperature at which the specific heat coming from the given degree of freedom \textit{changes most dramatically} with temperature, i.e., the derivative of the specific heat with respect to temperature reaches its maximum. For some system such as a mixture of ortho- and para- hydrogen molecules, the characteristic temperature is hard to define, the fastest frozen temperature is also straightforward and well-defined. In general, the fastest frozen temperature is a \emph{quantitative}\textit{\ }criterion of the frozen temperature.