Europhys. Lett.
Volume 65, Number 5, March 2004
Page(s) 613 - 619
Section General
Published online 01 February 2004
Europhys. Lett., 65 (5) , pp. 613-619 (2004)
DOI: 10.1209/epl/i2003-10166-3

Scaling behavior of interactions in a modular quantum system and the existence of local temperature

M. Hartmann1, 2, J. Gemmer2, G. Mahler2 and O. Hess3

1  Institute of Technical Physics, DLR Stuttgart Pfaffenwaldring 38-40, D-70569 Stuttgart, Germany
2  Institute of Theoretical Physics I, University of Stuttgart Pfaffenwaldring 57, D-70550 Stuttgart, Germany
3  Advanced Technology Institute, University of Surrey Guilford, Surrey, GU2 7XH, UK

(Received 19 June 2003; accepted in final form 15 December 2003)

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 $\beta$, a sufficient condition for subgroups of size N to be approximately in a thermal state with the same temperature is $\sqrt{N}\gg\beta\overline{\delta E}$, where $\overline{\delta E}$ 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.

05.30.-d - Quantum statistical mechanics.
05.70.Ce - Thermodynamic functions and equations of state.
65.80.+n - Thermal properties of small particles, nanocrystals, nanotubes.

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