发表机构
ETH Zurich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对麦克斯韦分子，开发了与表示无关的对称福克形式化的非线性玻尔兹曼碰撞算子，证明了提升-融合互缠定理，推导了相关分级性质，验证了其在非齐次问题中的适用性及普朗特数结果。
AI 中文摘要
我们针对麦克斯韦分子，开发了与表示无关的对称福克（symmetric-Fock）形式化的非线性玻尔兹曼碰撞算子。从标准速度空间算子出发，我们简要推导了博布雷夫（Bobylev）傅里叶恒等式，并分离出其特征结构：两个线性相关的傅里叶自变量随后相乘。接着，我们独立于碰撞几何和任何坐标实现，证明了一个典范提升-融合互缠定理（lift--fusion intertwining theorem）：单粒子线性映射提升到两个入射福克支，对称代数乘法将它们融合为单个出射态。巴尔格曼（Bargmann）坐标使这一内在构造格外清晰，但并不定义它；博布雷夫碰撞映射随后从已分级的提升-融合类中选出麦克斯韦顶点。所得双线性映射服从精确分级\\( \widehat{\mathcal Q}(\mathscr H_p,\mathscr H_q)\subseteq\mathscr H_{p+q} \\)，这为麦克斯韦动力学的三角结构提供了内在起源。其单真空限制给出完整的王-张-乌伦贝克（Wang Chang--Uhlenbeck）谱，而矩适配使非平衡层级严格三角化，每个有限相容福克喷流都是具有全局指数弛豫的精确自主因子。对于非齐次问题，我们将实现协方差扩展到非线性双支顶点，并通过厄米函数和傅里叶实现独立评估运动框架联络。两条路径给出相同的抽象福克传播和相同的第一查普曼-恩斯科格（Chapman--Enskog）源；其无迹2阶和3阶矢量部分由精确麦克斯韦速率弛豫，给出\\( \Pr=2/3 \\)。这提供了一个明确的计算例证，表明可选择坐标实现以实现经济性，而无需进入结构性福克形式化。
英文摘要
We develop a representation-independent symmetric-Fock formulation of the nonlinear Boltzmann collision operator for Maxwell molecules. Starting from the standard velocity-space operator, we give a short derivation of Bobylev's Fourier identity and isolate its characteristic structure: two linearly related Fourier arguments followed by multiplication. We then prove, independently of collision geometry and of any coordinate realization, a canonical lift--fusion intertwining theorem: one-particle linear maps lift to the two incoming Fock legs and symmetric-algebra multiplication fuses them into a single outgoing state. Bargmann coordinates make this intrinsic construction especially transparent, but do not define it; Bobylev's collision maps subsequently select the Maxwell vertex from the already graded lift--fusion class. The resulting bilinear map obeys the exact grading \[ \widehat{\mathcal Q}(\mathscr H_p,\mathscr H_q)\subseteq\mathscr H_{p+q}, \] which gives an intrinsic origin for the triangular structures of Maxwell kinetics. Its one-vacuum restriction yields the complete Wang Chang--Uhlenbeck spectrum, while moment adaptation makes the nonequilibrium hierarchy strictly triangular and every finite compatible Fock jet an exact autonomous factor with global exponential relaxation. For the inhomogeneous problem, we extend realization covariance to the nonlinear two-leg vertex and evaluate the moving-frame connection independently through Hermite-function and Fourier realizations. The two routes give the same abstract Fock propagation and the same first Chapman--Enskog source; its traceless level-$2$ and vector level-$3$ sectors are relaxed by the exact Maxwell rates, giving $\Pr=2/3$. This provides an explicit computational demonstration that coordinate realizations may be chosen for economy without entering the structural Fock formulation.
Comments53 pages, no figures
