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arXiv:quant-ph/0312214 · Physical Review Letters

Existence of Temperature on the Nanoscale

Michael J. Hartmann, Günter Mahler A, Ortwin G. Hess

Published Aug 19, 2004• 99 citations
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Abstract

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 ${n}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ depends on the temperature $T$, contrary to the classical case. We apply our analysis to a harmonic chain and find that ${n}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ for temperatures above the Debye temperature and ${n}_{\mathrm{m}\mathrm{i}\mathrm{n}}\ensuremath{\propto}{T}^{\ensuremath{-}3}$ below.

DOI
10.1103/physrevlett.93.080402

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We consider a regular chain of quantum particles with nearest neighbor interactions in a canonical state with temperature TTT. We analyze the conditions under which the state factors into a product of canonical density matrices with respect to groups of nnn particles each and under which these groups have the same temperature TTT. In quantum mechanics the minimum group size nmin{n}_{\mathrm{m}\mathrm{i}\mathrm{n}}nmin​ depends on the temperature TTT, contrary to the classical case. We apply our analysis to a harmonic chain and find that nmin=const{n}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}nmin​=const for temperatures above the Debye temperature and nmin\ensuremath∝T\ensuremath−3{n}_{\mathrm{m}\mathrm{i}\mathrm{n}}\ensuremath{\propto}{T}^{\ensuremath{-}3}nmin​\ensuremath∝T\ensuremath−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 TTT. We analyze under what condition the state factors into a product of canonical density matrices with respect to groups of nnn subsystems each, and when these groups have the same temperature TTT. While in classical mechanics the validity of this procedure only depends on the size of the groups nnn, in quantum mechanics the minimum group size nminn_{\text{min}}nmin​ also depends on the temperature $T $! As examples, we apply our analysis to different types of Heisenberg spin chains.
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.
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 TTT. We analyse, under what condition the state factors into a product of canonical density matrices with respect to groups of nnn subsystems each, and when these groups have the same temperature TTT. In quantum systems the minimal group size nminn_{\textrm{min}}nmin​ depends on the temperature TTT, 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−3T^{-3}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 μμμm, while very low temperatures (10 Kelvin) can only exist on scales larger than 1 mm.
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.
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.