团簇凝聚动力学的收敛性
Convergence of cluster coagulation dynamics
AI总结：
本文研究一般团簇凝聚模型的流体动力学极限，建立轨迹向多类型 Flory 方程集中及弱大数定律的条件，并刻画凝胶解与凝胶—溶胶相互作用。
摘要：
我们研究团簇凝聚模型的流体动力学极限；该凝聚模型由 Norris [《Comm. Math. Phys.》, 209(2):407-435 (2000)] 引入。在此过程中，测度空间 E 中的粒子对 x、y 依据转移核 K(x, y, dz) 合并形成单个新粒子 z，其方式使一个可被视为系统总质量的量保持守恒。该模型具有足够的一般性，可纳入团簇演化中的各种非均匀性，例如团簇的形状或其在空间中的位置。我们推导了使与该过程相关的轨迹集中于 Flory 方程一种推广形式的解的充分准则；并且在某些特殊情形下，借助该方程解的唯一性结果，证明了弱大数定律。这种多类型 Flory 方程与该过程相关的守恒量相联系，这些守恒量可以编码不同于质量守恒的信息（例如空间模型中的质心守恒）。我们还应用该过程中凝胶化的准则，推导了该方程出现凝胶解的充分准则。当这种情况发生时，多类型 Flory 方程通过相关的守恒性质，编码凝胶与有限尺寸溶胶粒子之间的相互作用。
英文摘要：
We study hydrodynamic limits of the cluster coagulation model; a coagulation model introduced by Norris [$\textit{Comm. Math. Phys.}$, 209(2):407-435 (2000)]. In this process, pairs of particles $x,y$ in a measure space $E$, merge to form a single new particle $z$ according to a transition kernel $K(x, y, \mathrm{d} z)$, in such a manner that a quantity, one may regard as the total mass of the system, is conserved. This model is general enough to incorporate various inhomogeneities in the evolution of clusters, for example, their shape, or their location in space. We derive sufficient criteria for trajectories associated with this process to concentrate among solutions of a generalisation of the $\textit{Flory equation}$, and, in some special cases, by means of a uniqueness result for solutions of this equation, prove a weak law of large numbers. This multi-type Flory equation is associated with $\textit{conserved quantities}$ associated with the process, which may encode different information to conservation of mass (for example, conservation of centre of mass in spatial models). We also apply criteria for gelation in this process to derive sufficient criteria for this equation to exhibit gelling solutions. When this occurs, this multi-type Flory equation encodes, via the associated conserved property, the interaction between the gel and the finite size sol particles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
