We consider a regular chain of quantum particles with nearest neighbour interactions in a canonical state with temperature T . We analyse 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 nmin depends on the temperature T , contrary to the classical case. We apply our analysis to a harmonic chain and find that nmin = const. for temperatures above the Debye temperature and nmin ∝ T −3 below.PACS numbers: 05.30. 05.70.Ce, 65.80.+n, Recent progress in the synthesis and processing of materials with structures on nanometer length scales calls for better understanding of thermal properties of nanoscale devices, individual nanostractures and nanostructured materials. Experimental techniques have improved to such an extent that the measurement of thermodynamic quantities like temperature with a spatial resolution on the nanometer scale seems within reach [1,2,3,4]. These techniques have already been applied for a new type of scanning microscopy, using a temperature sensor [5] that shows resolutions below 100nm.To provide a basis for the interpretation of present day and future experiments in this field, it is indispensable to clarify the applicability of the concepts of thermodynamics to systems on small length scales. In this context, one question appears to be particularly important and interesting: Can temperature be meaningfully defined on nanometer length scales?The standard procedure to show that the thermodynamical limit and therefore temperature exists are based on the idea that, as the spatial extension increases, the surface of a region in space grows slower than its volume [6]. If the coupling potential is short-ranged enough, the interactions between one region and another become negligible in the limit of infinite size.However, the full scaling behavior of these interactions with respect to the size of the parts has, to our knowledge, not been studied in detail yet [7,8,9].For standard applications of thermodynamics this might not be very important since the number of particles is typically so large that deviations from infinite systems may safely be neglected. Nevertheless, these differences could become significant as the considered systems approach nanoscopic scales. Here it is of special interest to determine the "grain-size" needed to ensure the existence of local temperature.Since a quantum description becomes imperative at nanoscopic scales, the following approach appears to be reasonable: Consider a large quantum system, brought into a thermal state via interaction with its environment, divide this system into subgroups and analyse for what subgroup-size the concept of temperature is applicable.We adopt here the convention, that a local temperature exists, if the considered part of the system is in a canonical state, i.e. the distribution is an exponentially decaying function of energy characterised by one single pa...
