This work outlines an exact combinatorial approach to finite coagulating systems through recursive equations and use of generating function method. In the classic approach the mean-field Smoluchowski coagulation is used. However, the assumptions of the mean-field theory are rarely met in real systems which limits the accuracy of the solution. In our approach, cluster sizes and time are discrete, and the binary aggregation alone governs the time evolution of the systems. By considering the growth histories of all possible clusters and applying monodisperse initial conditions, the exact expression for the probability of finding a coagulating system with an arbitrary kernel in a given cluster configuration is derived. Then, the average number of such clusters and the standard deviation of these solutions can be calculated. In this work, recursive equations for all possible growth histories of clusters are introduced. The correctness of our expressions was proved based on the comparison with numerical results obtained for systems with constant, multiplicative and additive kernels. For the first time the exact solutions for the multiplicative and additive kernels were obtained with this framework. In addition, our results were compared with the results arising from the solutions to the mean-field Smoluchowski equation. Our theoretical predictions outperform the classic approach.

中文翻译：

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通过递归方程求解有限混凝系统的精确组合方法

这项工作通过递归方程和生成函数方法的使用概述了有限凝结系统的精确组合方法。在经典方法中，使用平均场 Smoluchowski 凝固。然而，在实际系统中很少满足平均场理论的假设，这限制了解决方案的准确性。在我们的方法中，集群大小和时间是离散的，二进制聚合单独控制系统的时间演变。通过考虑所有可能簇的生长历史并应用单分散初始条件，推导出在给定簇配置中找到具有任意内核的凝结系统的概率的精确表达式。然后，可以计算这些聚类的平均数量和这些解决方案的标准偏差。在这项工作中，介绍了所有可能的集群增长历史的递归方程。通过与具有常数、乘性和加性核的系统的数值结果进行比较，证明了我们表达式的正确性。使用该框架首次获得了乘法和加法核的精确解。此外，我们的结果与平均场 Smoluchowski 方程的解所产生的结果进行了比较。我们的理论预测优于经典方法。用这个框架第一次获得了乘法和加法核的精确解。此外，我们的结果与平均场 Smoluchowski 方程的解所产生的结果进行了比较。我们的理论预测优于经典方法。使用该框架首次获得了乘法和加法核的精确解。此外，我们的结果与平均场 Smoluchowski 方程的解所产生的结果进行了比较。我们的理论预测优于经典方法。