本文证明关于费米狄拉克粒子的玻尔兹曼方程空间均匀解(在一般初值条件下)强收敛和时间平均强收敛到平衡态的结果。对于碰撞核的假设包含了具有较弱角截断的库仑位势。证明的关键在于矩估计，熵扩散不等式，碰撞增长算子的正则性，以及一个新的观察，很多碰撞核都大于等于某个具有全正性的核，这使得我们不需要处理碰撞积分中的三次项收敛问题。本文的结果是在硬位情形证明任意温度下强收敛到平衡态，在软位情形下得到量子情形的矩估计结果以及平均强收敛的结果。本文的创新点在于提出全正性的概念，得到对应的四次积分的非负性，从而解决四次项难处理的问题。这也是在之前的研究中需要给出额外假设$a\le 1$ (其等价于给出了一个高温条件)来证明强收敛到平衡态的唯一理由，其中a是平衡态F中的一个系数，从而使我们避开处理四次积分的非负性问题。论文中也会对全正性给出一个明确的定义，以便将来研究者使用。当然，为了方便证明，论文中需要给碰撞核提一些更强的条件，比如进行更强的角截断。为了简化证明，论文将对一些物理常数进行归一化处理，主体部分碰撞核将采用$\sg$表示的形式，并可以通过对称化，在碰撞核中增添偶函数的假设。这些假设并不影响问题的本质，仅仅是为了使证明过程更加简化。对于$\og$表示的一些定理推论，我们也将验证其在$\sg$表示下同样成立，并给出其在$\sg$表示下的形式，以便使用。在之前的研究中，碰撞核经常被选为可分离变量的形式，我们的研究中也会通过对碰撞核添加条件的调整，来进行一定的推广，使得其包含某些不可分离变量的形式。我们也试图尽可能包含一些常见的有物理意义的碰撞核，例如库仑位势，从而证明我们的研究是存在实际意义的。
In this paper we prove the strong and time-averaged strong convergence to equilibrium for solutions (with general initial data) of the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. The assumption on the collision kernel includes the Coulomb potential with a weaker angular cutoff. The proof is based on moment estimates, entropy dissipation inequalities, regularity of the collision gain operator, and a new observation that many collision kernels are larger than or equal to some completely positive kernels, which enables us to avoid dealing with the convergence problem of the cubic collision integrals.The result of this article is to prove the strong convergence to equilibrium at any temperature in the case of hard potential.And in the case of soft potential,we obtain the moment estimate results and time-averaged strong convergence to equilibrium in the quantum case. Our innovation lies in proposing the concept of total positivity and obtaining the nonnegativeness of the corresponding quartic integral, thus solving the difficulty in handling the quartic term. This is the only reason that we add the assumption $?\le1$ (which is equivalent to a high-temperature condition) to prove strong convergence to equilibrium in previous studies, where $a$ is a coefficient in equilibrium F, which allows us to avoid dealing with the nonnegativity problem of quartic integrals. In this paper,a clear definition of total positivity will also be given for researchers to use in the future. Of course, for the convenience of proof,we propose stronger conditions for the collision kernel, such as stronger angular truncation. To simplify the proof,we normalize some physical constants, and the collision kernel in the main part will use $\sg-$representation. By symmetrization, we also assume that the collision kernel is even. These assumptions are proposed to simplify the proof process and do not affect the essence of the problem. We will give some consequences in $\og-$representation and verify that they also hold true under the $\sg-$representation. In previous studies, collision kernel is usually selected as separable variables, now we extend it to include certain forms of inseparable variables. We also try to include some common collision kernels such as Coulomb potential for physical significance.
