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_min depends on the temperature T, contrary to the classical case. We apply our analysis to a harmonic chain and find that n_min=const for temperatures above the Debye temperature and n_min∝T^-3 below.