We consider a quantum system of fixed size consisting of a regular chain of n -level subsystems, where n is finite. Forming groups of N subsystems each, we show that the strength of interaction between the groups scales with N −1/2 . As a consequence, if the total system is in a thermal state with inverse temperature β, a sufficient condition for subgroups of size N to be approximately in a thermal state with the same temperature is √ N ≫ β Ē , where Ē is the width of the occupied level spectrum of the total system. These scaling properties indicate on what scale local temperatures may be meaningfully defined as intensive variables. This question is particularly relevant for non-equilibrium scenarios such as heat conduction, etc.
We consider a quantum system of fixed size consisting of a regular chain of n-level subsystems, where n is finite. Forming groups of N subs-sterns each, We Show that the strength of interaction between the groups scales with N-1/2. AS a consequence. if the total system is in a thermal state with inverse temperature beta, a sufficient. condition for subgroups of size N to be approximately in a thermal state with the same temperature is root(N) over bar much greater than 3 (δE) over bar. where (δE) over bar is the width of the. occupied level spectrum of the total system. These scaling properties indicate on what scale local temperatures may be meaningfully defined as intensive variables.. This question is particularly relevant for non-equilibrium scenarios such as heat conduction, etc.
We consider a quantum system consisting of a regular chain of elementary subsystems with nearest neighbor interactions and assume that the total system is in a canonical state with temperature T . We analyze 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. While in classical mechanics the validity of this procedure only depends on the size of the groups n, in quantum mechanics the minimum group size n(min) also depends on the temperature T! As examples, we apply our analysis to a harmonic chain and different types of Ising spin chains. We discuss various features that show up due to the characteristics of the models considered. For the harmonic chain, which successfully describes thermal properties of insulating solids, our approach gives a 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 consider a regular chain of elementary quantum systems with nearest neighbor interactions and assume that the total system is in a canonical state with temperature
T. We analyze 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. While in classical mechanics the validity of this procedure only depends on the size of the groups
n, in quantum mechanics the minimum group size
nmin also depends on the temperature $T $! As examples, we apply our analysis to different types of Heisenberg spin chains.
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 depends on the temperature, 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 1/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. The technique of the approach is based on a quantum central limit theorem, which should prove usefull in different settings, too.
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
nmin 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.
We consider blocks of quantum spins in a chain at thermal equilibrium, focusing on their properties from a thermodynamical perspective. In a classical system the temperature behaves as an intensive magnitude, above a certain block size, regardless of the actual value of the temperature itself. However, a deviation from this behavior is expected in quantum systems. In particular, we see that under some conditions the description of the blocks as thermal states with the same global temperature as the whole chain fails. We analyze this issue by employing the quantum fidelity as a figure of merit, singling out in detail the departure from the classical behavior. As it may be expected, we see that quantum features are more prominent at low temperatures and are affected by the presence of zero-temperature quantum phase transitions. Interestingly, we show that the blocks can be considered indeed as thermal states with a high fidelity, provided an effective local temperature is properly identified. Such a result may originate from typical properties of reduced subsystems of energy-constrained Hilbert spaces. Finally, the relation between local and global temperatures is analyzed as a function of the size of the blocks and the system parameters.