发表机构
School of the Gifted Young, University of Science and Technology of China; Duke University（中国科学技术大学少年班学院; 杜克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对空间均匀量子玻尔兹曼方程，在无初始低能浓度假设下，通过引入从平衡松弛估计获凝聚的方法，证明了有限时间玻色-爱因斯坦凝聚的存在性及相关收敛性质。
AI 中文摘要
对于低动量散射简并指数满足0≤η<1的空间均匀量子玻尔兹曼方程的保守各向同性测度解，我们证明：低于临界温度时向玻色-爱因斯坦平衡的半强收敛，意味着有限时间内会发生玻色-爱因斯坦凝聚（BEC）。对于所有满足初始平均温度与临界温度之比T̄/T̄c<1的容许初始测度，无需任何零能量附近初始浓度的假设，就存在一个保守解，其在有限时间后会出现正的零能量原子。我们还证明该原子会持续存在并收敛到平衡凝聚质量，同时整个解会强收敛。该证明引入了一种从玻色-爱因斯坦平衡的松弛估计中获取凝聚的方法：松弛估计首先给出小能量水平以下的固定质量，随后我们利用越来越小的能量尺度上的碰撞估计，并将其影响以加权和泛函的形式相加，该加权和泛函有固定的上界。如果零能量处未形成原子，那么正能量壳层中会剩余足够的质量，迫使该和在有限时间内超过其上界，从而导致有限时间内发生凝聚。
英文摘要
We prove that semi-strong convergence to a Bose--Einstein equilibrium below the critical temperature implies finite-time Bose--Einstein condensation (BEC) for conservative isotropic measure solutions of the spatially homogeneous quantum Boltzmann equation with low-momentum scattering degeneracy exponent $0\leqη<1$. For every admissible initial measure with $\overline T/\overline T_c<1$, there exists a conservative solution that has a positive zero-energy atom after a finite time without any assumption of initial concentration near zero energy. We also show that it persists and converges to the equilibrium condensate mass, while the full solution converges strongly. The proof introduces a method to obtain condensation from relaxation estimate to Bose--Einstein equilibrium. The relaxation estimate first gives a fixed amount of mass below a small energy level. We then use collision estimates at smaller and smaller energy scales and add their effects in a weighted sum functional with a fixed upper bound. If no atom forms at zero energy, enough mass remains in the positive-energy shells to force this sum to exceed its upper bound in finite time, which leads to condensation in finite time.
Comments28 pages
