Existence of Temperature on the Nanoscale

We consider a regular chain of quantum particles with nearest neighbor interactions in a canonical state with temperature $T$. We analyze the conditions under which the state factors into a product of canonical density matrices with respect to groups of $n$ particles each and under which these groups have the same temperature $T$. In quantum mechanics the minimum group size $n_{\mathrm{m}\mathrm{i}\mathrm{n}}$ depends on the temperature $T$, contrary to the classical case. We apply our analysis to a harmonic chain and find that $n_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ for temperatures above the Debye temperature and $n_{\mathrm{m}\mathrm{i}\mathrm{n}}\propto T^{-3}$ below.

Figure 1

Log-log plot of $n_{\mathrm{m}\mathrm{i}\mathrm{n}}$ from Eq. (22) (dashed line) and $n_{\mathrm{m}\mathrm{i}\mathrm{n}}$ from Eq. (24) (solid line) for $\alpha =10$ and $\delta =0.01$ as a function of $T/\Theta$ for a harmonic chain. $\delta$ and $\alpha$ are defined in Eqs. (24, 20), respectively.