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) proportional to T(-3) below.