When the temperature of a trapped Bose gas is below the Bose-Einstein transition temperature and above absolute zero, the gas is composed of two distinct components: the Bose-Einstein condensate and the cloud of thermal excitations. The dynamics of the excitations can be described by quantum Boltzmann models. We establish a connection between quantum Boltzmann models and chemical reaction networks. We prove that the discrete differential equations for these quantum Boltzmann models converge to an equilibrium point. Moreover, this point is unique for all initial conditions that satisfy the same conservation laws. In the proof, we then employ a toric dynamical system approach, similar to the one used to prove the global attractor conjecture, to study the convergence to equilibrium of quantum kinetic equations.
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Keywords: Quantum Boltzmann equation, dynamical systems, bosons, Bose-Einstein condensate, rate of convergence to equilibrium, global attractor conjecture, mass-action kinetics, power law systems, biochemical networks, Petri net
@article{COCV_2021__27_1_A85_0,
author = {Craciun, Gheorghe and Tran, Minh-Binh},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {A reaction network approach to the convergence to equilibrium of quantum {Boltzmann} equations for {Bose} gases},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021079},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021079/}
}
TY - JOUR AU - Craciun, Gheorghe AU - Tran, Minh-Binh ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021079/ DO - 10.1051/cocv/2021079 LA - en ID - COCV_2021__27_1_A85_0 ER -
%0 Journal Article %A Craciun, Gheorghe %A Tran, Minh-Binh %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021079/ %R 10.1051/cocv/2021079 %G en %F COCV_2021__27_1_A85_0
Craciun, Gheorghe; Tran, Minh-Binh. A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 83. doi: 10.1051/cocv/2021079
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