# American Institute of Mathematical Sciences

September  2015, 8(3): 443-465. doi: 10.3934/krm.2015.8.443

## On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation

 1 Department of Mathematics, University of Rochester, Rochester, NY 14627, United States 2 Division of Applied Mathematics, Brown University, Providence, RI 02912

Received  December 2014 Revised  February 2015 Published  June 2015

The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum $N$ -particle dynamics. We assume the integral of the microscopic interaction is zero and we assume $W^{4,1}$ per-particle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below $W^{4,1}$.
Citation: Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic & Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443
##### References:
 [1] D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, Some considerations on the derivation of the nonlinear quantum boltzmann equation,, J. Stat. Phys., 116 (2004), 381. doi: 10.1023/B:JOSS.0000037205.09518.3f. Google Scholar [2] D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, On the weak-coupling limit for bosons and fermions,, Math. Mod. Meth. Appl. Sci., 15 (2005), 1811. doi: 10.1142/S0218202505000984. Google Scholar [3] D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, From the N-body Schroedinger equation to the quantum Boltzmann equation: A term-by-term convergence result in the weak coupling regime,, Commun. Math. Phys., 277 (2008), 1. doi: 10.1007/s00220-007-0347-7. Google Scholar [4] L. Erdös, M. Salmhofer and H. T. Yau, On the quantum Boltzmann equation,, J. Stat. Phys., 116 (2004), 367. doi: 10.1023/B:JOSS.0000037224.56191.ed. Google Scholar [5] I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-range Potentials,, Zürich Lectures in Advanced Mathematics, (2013). Google Scholar [6] F. King, BBGKY Hierarchy for Positive Potentials,, Ph.D thesis, (1975). Google Scholar [7] O. E. Lanford III, Time Evolution of Large Classical Systems,, Lecture Notes in Physics, 38 (1975), 1. Google Scholar [8] E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases,, Phys. Rev., 43 (1933), 552. doi: 10.1103/PhysRev.43.552. Google Scholar

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##### References:
 [1] D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, Some considerations on the derivation of the nonlinear quantum boltzmann equation,, J. Stat. Phys., 116 (2004), 381. doi: 10.1023/B:JOSS.0000037205.09518.3f. Google Scholar [2] D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, On the weak-coupling limit for bosons and fermions,, Math. Mod. Meth. Appl. Sci., 15 (2005), 1811. doi: 10.1142/S0218202505000984. Google Scholar [3] D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, From the N-body Schroedinger equation to the quantum Boltzmann equation: A term-by-term convergence result in the weak coupling regime,, Commun. Math. Phys., 277 (2008), 1. doi: 10.1007/s00220-007-0347-7. Google Scholar [4] L. Erdös, M. Salmhofer and H. T. Yau, On the quantum Boltzmann equation,, J. Stat. Phys., 116 (2004), 367. doi: 10.1023/B:JOSS.0000037224.56191.ed. Google Scholar [5] I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-range Potentials,, Zürich Lectures in Advanced Mathematics, (2013). Google Scholar [6] F. King, BBGKY Hierarchy for Positive Potentials,, Ph.D thesis, (1975). Google Scholar [7] O. E. Lanford III, Time Evolution of Large Classical Systems,, Lecture Notes in Physics, 38 (1975), 1. Google Scholar [8] E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases,, Phys. Rev., 43 (1933), 552. doi: 10.1103/PhysRev.43.552. Google Scholar
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