Some considerations on the derivation of the nonlinear quantum Boltzmann equation

被引:37
作者
Benedetto, D
Castella, F
Esposito, R
Pulvirenti, M
机构
[1] Univ Roma La Sapienza, Dipartimento Matemat, I-00185 Rome, Italy
[2] Univ Rennes 1, IRMAR, F-35042 Rennes, France
[3] Ctr Linceo Interdisciplinare Beniamino Segre, I-00165 Rome, Italy
[4] Univ Aquila, Dipartimento Matemat Pura & Applicata, I-67100 Laquila, Italy
关键词
kinetic equations; quantum systems; weak coupling;
D O I
10.1023/B:JOSS.0000037205.09518.3f
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
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 dgreater than or equal to2.
引用
收藏
页码:381 / 410
页数:30
相关论文
共 50 条
[31]   GLOBAL DIFFUSIVE EXPANSION OF BOLTZMANN EQUATION IN EXTERIOR DOMAIN [J].
Jung, Junhwa .
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2025, 57 (02) :1781-1831
[32]   The Boltzmann Equation with Soft Potentials Near a Local Maxwellian [J].
Xin, Zhouping ;
Yang, Tong ;
Yu, Hongjun .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2012, 206 (01) :239-296
[33]   Diffusive expansion for solutions of the Boltzmann equation in the whole space [J].
Liu, Shuangqian ;
Zhao, Huijiang .
JOURNAL OF DIFFERENTIAL EQUATIONS, 2011, 250 (02) :623-674
[34]   Hydrodynamic solutions of the generalized Boltzmann-Enskog equation [J].
Inozemtseva, N. G. ;
Maslennikov, I. I. .
MOSCOW UNIVERSITY PHYSICS BULLETIN, 2013, 68 (03) :201-204
[35]   Passage from the Boltzmann equation with diffuse boundary to the incompressible Euler equation with heat convection [J].
Cao, Yunbai ;
Jang, Juhi ;
Kim, Chanwoo .
JOURNAL OF DIFFERENTIAL EQUATIONS, 2023, 366 :565-644
[36]   The Hilbert expansion of the Boltzmann equation in the incompressible Euler level in a channel [J].
Huang, Feimin ;
Wang, Weiqiang ;
Wang, Yong ;
Xiao, Feng .
SCIENCE CHINA-MATHEMATICS, 2025, 68 (01) :39-88
[37]   Accelerating the Convergence of the Moment Method for the Boltzmann Equation Using Filters [J].
Fan, Yuwei ;
Koellermeier, Julian .
JOURNAL OF SCIENTIFIC COMPUTING, 2020, 84 (01)
[38]   THE INCOMPRESSIBLE EULER LIMIT OF THE BOLTZMANN EQUATION WITH ACCOMMODATION BOUNDARY CONDITION [J].
Bardos, Claude ;
Golse, Francois ;
Paillard, Lionel .
COMMUNICATIONS IN MATHEMATICAL SCIENCES, 2012, 10 (01) :159-190
[39]   Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation [J].
Li, Qin ;
Yang, Xu .
COMMUNICATIONS IN COMPUTATIONAL PHYSICS, 2014, 15 (04) :996-1011
[40]   MICROSCOPIC SOLUTIONS OF THE BOLTZMANN-ENSKOG EQUATION IN THE SERIES REPRESENTATION [J].
Pulvirenti, Mario ;
Simonella, Sergio ;
Trushechkin, Anton .
KINETIC AND RELATED MODELS, 2018, 11 (04) :911-931