*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. Hartmann**

^{1, 2}, J. Gemmer^{2}, G. Mahler^{2}and O. Hess^{3}^{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)

** Abstract **

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
, a sufficient condition for subgroups
of size
*N* to be approximately in a thermal state with the same
temperature is
, where
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.

**PACS**

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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*EDP Sciences 2004*