TY - JOUR

T1 - Not to normal order-notes on the kinetic limit for weakly interacting quantum fluids

AU - Lukkarinen, Jani

AU - Spohn, Herbert

N1 - Funding Information:
Acknowledgements We would like to thank László Erdo˝s for many illuminating discussions on the subject. The research was supported by a DFG grant and by the Academy of Finland. The work was partially done at the Erwin Schrödinger Institute for Mathematical Physics (ESI), Vienna, Austria, whom we thank for their hospitality.

PY - 2009/3

Y1 - 2009/3

N2 - The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrodinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on normal order in the Duhamel expansion. We discuss the term by term convergence of the expansion and the equilibrium time correlation a(t)*a(0).

AB - The derivation of the Nordheim-Boltzmann transport equation for weakly interacting quantum fluids is a longstanding problem in mathematical physics. Inspired by the method developed to handle classical dilute gases, a conventional approach is the use of the BBGKY hierarchy for the time-dependent reduced density matrices. In contrast, our contribution is motivated by the kinetic theory of the weakly nonlinear Schrodinger equation. The main observation is that the results obtained in the latter context carry over directly to weakly interacting quantum fluids provided one does not insist on normal order in the Duhamel expansion. We discuss the term by term convergence of the expansion and the equilibrium time correlation a(t)*a(0).

KW - Boltzmann-nordheim equation

KW - Kinetic theory

KW - Quantum bbgky hierarchy

KW - Time-dependent perturbation theory

KW - Uehling-uhlenbeck equation

KW - Weakly interacting bosons

KW - Weakly interacting fermions

UR - http://www.scopus.com/inward/record.url?scp=67349146432&partnerID=8YFLogxK

U2 - 10.1007/s10955-009-9682-8

DO - 10.1007/s10955-009-9682-8

M3 - Article

AN - SCOPUS:67349146432

SN - 0022-4715

VL - 134

SP - 1133

EP - 1172

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5-6

ER -