有限温度囚禁玻色气体的动力学
On the dynamics of finite temperature trapped Bose gases
摘要：
描述有限温度下玻色-爱因斯坦凝聚体（BEC）和热云动力学的系统由非线性薛定谔（NLS）方程和量子玻尔兹曼（QB）方程组成。在这样一个有限温度下的玻色气体系统中，QB方程对应于热云密度分布函数的演化，NLS方程对应于凝聚态方程。在这个温度范围内的量子玻尔兹曼碰撞算符是两个算符$C_{12}$和$C_{22}$的和，这两个算符描述了凝聚态原子和非凝聚态原子的碰撞以及非凝聚态原子之间的碰撞。在BEC临界温度以上，系统被简化为仅包含$C_{22}$的方程，该方程具有关于$L^\infty$范数的爆破正径向解（参见图10）。\cite{EscobedoVelazquez:2015:FTB}）。另一方面，在非常低的温度区域，系统变成具有不同（高得多的阶）跃迁概率的$C_{12}$方程，其具有唯一的具有加权$L^1$范数的全局正径向解（cf. \cite{AlonsoGambaBinh}）。在目前的温度制度下，我们首先解耦QB和NLS方程，然后证明了整体的存在性和唯一性的正径向解的空间均匀动力学系统。与\cite{EscobedoVelazquez:2015:FTB}中考虑的情况不同，由于BEC的存在，碰撞积分与复杂的能量流形而不是球体相关联，因为粒子能量近似于Bogoliubov色散定律。此外，整个系统的质量不守恒，而在\cite{EscobedoVelazquez:2015:FTB}中考虑的情况下是守恒的。提出了一种新的理论。
英文摘要：
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 $C_{22}$, which possesses a blow-up positive radial solution with respect to the $L^\infty$ norm (cf. \cite{EscobedoVelazquez:2015:FTB}). On the other hand, at the very low temperature regime, the system becomes an equation of $C_{12}$, with a different (much higher order) transition probability, which has a unique global positive radial solution with weighted $L^1$ norm (cf. \cite{AlonsoGambaBinh}). In the current temperature regime, we first decouple the QB and NLS equations, then show a global existence and uniqueness result for positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in \cite{EscobedoVelazquez:2015:FTB}, 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 \cite{EscobedoVelazquez:2015:FTB}. A new theory is then supplied.
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