On the Minimal Length Scales, Where Temperature Exists

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 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 µm, while very low temperatures (10 Kelvin) can only exist on scales larger than 1 mm.