论文标题
Smoluchowski凝血方程的异常尺寸
Anomalous dimensions of the Smoluchowski coagulation equation
论文作者
论文摘要
Smoluchowski在1916年引入了凝血（或聚集）方程，以通过扩散来描述胶体颗粒的团结，但在许多不同的情况下都使用了类似物理化学，化学工程，大气物理学，行星科学和经济学的不同情况。结块的有效性由内核$ k（x，y）$描述，该$取决于碰撞粒子$ x，y $的大小。我们认为内核$ k =（xy）^γ$，但是可以使用我们的方法来处理任何均匀功能。对于足够有效的团结$ 1 \geγ> 1/2 $，凝结方程在有限的时间内会产生一个无限大的簇（一种称为凝胶过渡的过程）。使用分析方法和数字的组合，我们计算了主要群集生长的异常缩放尺寸，呼吁在文献中大量使用的问题结果。除了源自确切可解决的情况$γ= 1 $的解决方案分支外，我们还发现了一个接近$γ= 1/2 $的新的解决方案分支，这违反了比例关系，人们普遍认为可以保持通用。
The coagulation (or aggregation) equation was introduced by Smoluchowski in 1916 to describe the clumping together of colloidal particles through diffusion, but has been used in many different contexts as diverse as physical chemistry, chemical engineering, atmospheric physics, planetary science, and economics. The effectiveness of clumping is described by a kernel $K(x,y)$, which depends on the sizes of the colliding particles $x,y$. We consider kernels $K = (xy)^γ$, but any homogeneous function can be treated using our methods. For sufficiently effective clumping $1 \ge γ> 1/2$, the coagulation equation produces an infinitely large cluster in finite time (a process known as the gel transition). Using a combination of analytical methods and numerics, we calculate the anomalous scaling dimensions of the main cluster growth, calling into question results much used in the literature. Apart from the solution branch which originates from the exactly solvable case $γ= 1$, we find a new branch of solutions near $γ= 1/2$, which violates scaling relations widely believed to hold universal.
