On The Weak-Coupling Limit for Bosons and Fermions

Dario Benedetto (1) , François Castella (2) , Raffaele Esposito (3) , Mario Pulvirenti (1, 4)

1 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 DISIM - Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila]
4 UNIROMA - Università degli Studi di Roma "La Sapienza" = Sapienza University [Rome]

In this paper we consider a large system of Bosons or Fermions. We start with an initial datum which is compatible with the Bose-Einstein, respectively Fermi-Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time $t$, agrees with the analogous expansion for the solution to the Uehling-Uhlenbeck equation. This paper follows in spirit the companion work [\rcite{BCEP}], where the authors investigated the weak-coupling limit for particles obeying the Maxwell-Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

Mots clés: Uehling-Uhlenbeck equation; quantum systems; weak coupling

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Dario Benedetto, François Castella, Raffaele Esposito, Mario Pulvirenti. On The Weak-Coupling Limit for Bosons and Fermions. Mathematical Models and Methods in Applied Sciences, 2005, 15 (12), pp.1811-1843. ⟨10.1142/S0218202505000984⟩. ⟨hal-00004480⟩