  Recent progress in the synthesis and processing of nano-structured materials
and systems calls for an improved understanding of thermal properties on small
length scales. In this context, the question whether thermodynamics and, in
particular, the concept of temperature can apply on the nanoscale is of central
interest. Here we consider a quantum system consisting of a regular chain of
elementary subsystems with nearest neighbour interactions and assume that the
total system is in a canonical state with temperature $T$. We analyse, under
what condition the state factors into a product of canonical density matrices
with respect to groups of $n$ subsystems each, and when these groups have the
same temperature $T$. In quantum systems the minimal group size
$n_{\textrm{min}}$ depends on the temperature $T$, contrary to the classical
case. As examples, we apply our analysis to a harmonic chain and different
types of Ising spin chains. For the harmonic chain, which successfully
describes thermal properties of insulating solids, our approach gives a first
quantitative estimate of the minimal length scale on which temperature can
exist: This length scale is found to be constant for temperatures above the
Debye temperature and proportional to $T^{-3}$ below. We finally apply the
harmonic chain model to various materials of relevance for technical
applications and discuss the results. These show that, indeed, high
temperatures can exist quite locally, while low temperatures exist on larger
scales only. This has striking consequences: In quasi 1-dimensional systems,
like Carbon-Nanotubes, room temperatures (300 Kelvin) exist on length scales of
1 $\mu$m, while very low temperatures (10 Kelvin) can only exist on scales
larger than 1 mm.
