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January  2004, 11(1): 1-26. doi: 10.3934/dcds.2004.11.1

From Bloch model to the rate equations

B. Bidégaray-Fesquet 1, , F. Castella 2, and  Pierre Degond 3,

1. 

LMC - IMAG, UMR 5523 (CNRS-UJF-INPG), B.P. 53, 38041 Grenoble Cedex 9, France
2. 

IRMAR, UMR 6625 (CNRS-UR1), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
3. 

Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 4, France

Received  January 2003 Revised  December 2003 Published  April 2004

We consider Bloch equations which govern the evolution of the density matrix of an atom (or: a quantum system) with a discrete set of energy levels. The system is forced by a time dependent electric potential which varies on a fast scale and we address the long time evolution of the system. We show that the diagonal part of the density matrix is asymptotically solution to a linear Boltzmann equation, in which transition rates are appropriate time averages of the potential. This study provides a mathematical justification of the approximation of Bloch equations by rate equations, as described in e.g. [Lou91]. The techniques used stem from manipulations on the density matrix and the averaging theory for ordinary differential equations. Diophantine estimates play a key role in the analysis.
Keywords: linear Boltzmann equation, Diophantine estimates., rate equations, averaging, Bloch model, Density matrix.
Mathematics Subject Classification: Primary: 34E99; Secondary: 81V8.
Citation: B. Bidégaray-Fesquet, F. Castella, Pierre Degond. From Bloch model to the rate equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 1-26. doi: 10.3934/dcds.2004.11.1
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