Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation

Abstract

Abstract.In this paper we analyze a system of N identical quantum particles in a weakcoupling 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 ≥ 2.

From the N-body Schrödinger Equation to the Quantum Boltzmann Equation: a Term-by-Term Convergence Result in the Weak Coupling Regime

“…This basic observation is the common ingredient in [BCEP1], [BCEP2], [BCEP3], and the present text. Now, the situation is as follows.…”

“…In particular, the delocalization effect gives rise to various highly oscillatory phase factors which have to be analyzed. Due to these new analytical difficulties, the result we proved in [BCEP1] is not a complete convergence result: we prove a subseries (of the complete series expansion expressing the state of the system at time t) converges, along the weak coupling limit, towards the solution of the desired quantum Boltzmann equation. The convergence is uniform for short times.…”

“…The convergence is uniform for short times. We also present in [BCEP1] arguments of heuristic nature which tend to establish the terms we neglect at once, when passing from the complete series expansion to the retained subseries, indeed go to zero along the weak coupling limit.…”

“…In [BCEP1], we consider a quantum N particles system in the weak coupling regime (see below for the definition). The key point is, we prove the Wigner transforms of the reduced density matrices satisfy a hierarchy of equations that is similar to the so-called BBGKY hierarchy encountered in the context of classical interacting particles.…”

“…Last, while the works [BCEP1] and [BCEP3] tackle the case of statistically independent particles, the analysis is also extended in [BCEP2] to handle the physically realistic case of bosons or fermions: such particles are not statistically independent (they obey the Bose-Einstein, resp. Fermi-Dirac statistics), and the limiting Boltzmann equation needs to be modified accordingly [UU] (see also [HL], [ESY], [H])).…”