From the Bloch model to the rate equations

Brigitte Bidégaray-Fesquet (1) , François Castella (2) , Pierre Degond (3)

1 LMC - IMAG - Laboratoire de Modélisation et Calcul
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 MIP - Mathématiques pour l'Industrie et la Physique

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.

Mots clés: linear Boltzmann equation; rate equations; Bloch model; averaging; Diophantine estimates.; density matrix

Domaines: Equations aux dérivées partielles [math.AP]

Brigitte Bidégaray-Fesquet, François Castella, Pierre Degond. From the Bloch model to the rate equations. Discrete and Continuous Dynamical Systems - Series A, 2004, 11 (1), pp.1-26. ⟨10.3934/dcds.2004.11.1⟩. ⟨hal-00319994⟩