On the dynamics of finite temperature trapped Bose gases

Abstract

The system that describes the dynamics of a Bose-Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators C 12 and C 22 , which describe collisions of the condensate and the non-condenstate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only a collsion operator similar to C 22 , which possesses a blow-up positive radial solution with respect to the L ∞ norm (cf.[27]). On the other hand, at the very low temperature regime (only a portion of the transition temperature T BEC ), the system can be simplified into an equation of C 12 , with a different (much higher order) transition probability, which has a unique global classical positive radial solution with weighted L 1 norm (cf. [3]). In our model, we first decouple the QB, which contains C 12 + C 22 , and the NLS equations, then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in [27], due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in [27]. A new theory is then supplied.

Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons

“…Let us finally mention that other descriptions have been used both analytically and numerically to study the behaviour beyond condensation in the quantum Boltzmann equation. In some of them the kinetic equation is coupled to a nonlinear Schrödinger equation (cubic, Gross-Pitaevski) modelling the evolution of the condensate, see [46,5,4,29] and the references therein for further details.…”

Optimal local well-posedness theory for the kinetic wave equation

“…The study of (1.12) is also a subject of rapidly growing interest in the kinetic community (cf. [3,18,17,57,56,47,34,36,37,38,6,30,54,35,53] and the references therein). Thanks to the stabilization effect of the classical Boltzmann collision operator Q 0 , the classical method of moment production developed for the classical Boltzmann equation can be applied (cf.…”

Spectral method for the Boltzmann equation for gases with viscosity

“…hương trình Boltzmann mô tả chuyển động của các phân tử của một chất khí trong đó sự tương tác của các phân tử là va chạm đàn hồi nhị phân (xem [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). Việc xây dựng các phương pháp tính cho phương trình Boltzmann có tầm quan trọng trong nhiều ứng dụng, từ động lực học khí hiếm (RGD) [27], vật lý plasma [28], dòng chảy dạng hạt [17,18], chất bán dẫn [32], và lý thuyết động lượng tử [29].…”