From the Von-Neumann equation to the Quantum Boltzmann equation in a deterministic framework

In this paper, we investigate the rigorous convergence of the Density Matri x Equation (or Quantum Lionville Equation) towards the Quantum Boltzmann Eq uation (or Pauli Master Equation). We start from the Density Matrix Equatio n posed on a cubic box of size L with periodic boundary conditions, describ ing the quantum motion of a particle in the box subject to an external pote ntial V. The physics motivates the introduction of a damping term acting on the off-diagonal part of the density matrix, with a characteristic damping time alpha (-1). Then, the convergence can be proved by letting successive ly L tend to infinity and alpha to zero. The proof relies heavily on a lemm a which allows to control some oscillatory integrals posed in large dimensi onal spaces. The present paper improves a previous announcement [CD].