We present a rigorous thermodynamic treatment of irreversible binary aggregation. We construct the Smoluchowski ensemble as the set of discrete finite distributions that are reached in fixed number of merging events and define a probability measure on this ensemble, such that the mean distribution in the mean-field approximation is governed by the Smoluchowski equation. In the scaling limit this ensemble gives rise to a set of relationships identical to those of familiar statistical thermodynamics. The central element of the thermodynamic treatment is the selection functional, a functional of feasible distributions that connects the probability of distribution to the details of the aggregation model. We obtain scaling expressions for general kernels and closed-form results for the special case of the constant, sum and product kernel. We study the stability of the most probable distribution, provide criteria for the sol-gel transition and obtain the distribution in the post-gel region by simple thermodynamic arguments.

中文翻译：

---

Smoluchowski 合奏——聚合的统计力学

我们提出了对不可逆二元聚集的严格热力学处理。我们将 Smoluchowski 系综构造为在固定数量的合并事件中达到的离散有限分布的集合，并在该系综上定义概率度量，使得平均场近似中的平均分布由 Smoluchowski 方程控制。在标度极限中，这个集合产生了一组与熟悉的统计热力学相同的关系。热力学处理的核心要素是选择泛函，这是一个可行分布的泛函，它将分布概率与聚合模型的细节联系起来。我们获得了一般核的标度表达式和常数核、和核和乘积核的特殊情况的封闭形式结果。