D.Benedetto, F. Castella, R. Esposito, M. Pulvirenti
Some Considerations on the derivation of the nonlinear
Quantum Boltzmann Equation
(663K, Postscript, Pdf)
ABSTRACT. In this paper we analyze a system of $N$ identical quantum particles in a
weak-coupling regime. The time evolution of the Wigner transform of the
one-particle
reduced density matrix is represented by means of a perturbative
series.
The expansion is obtained upon iterating the Duhamel formula.
For short times, we rigorously prove that a subseries of the latter,
converges to the solution of the
Boltzmann
equation which is physically relevant in the context.
In particular, we recover the transition rate as it is predicted by Fermi's Golden
Rule.
However, we are not able
to prove that the quantity neglected while retaining a subseries of the complete
original
perturbative expansion, indeed vanishes in the limit: we
only give
plausibility arguments in this direction.
The present study holds in any space dimension $d\ge 2$.