Leyvraz F
Instituto de Ciencias Físicas-Universidad Nacional Autónoma de México, Cuernavaca, Morelos 62210, México.
Phys Rev E. 2022 Aug;106(2-1):024133. doi: 10.1103/PhysRevE.106.024133.
In a recent series of papers, an exact combinatorial solution was claimed for a variant of the so-called Marcus-Lushnikov model of aggregation. In this model, a finite number of aggregates are initially assumed to be present in the form of monomers. At each time step, two aggregates are chosen according to certain size-dependent probabilities and irreversibly joined to form an aggregate of higher mass. The claimed result given an expression for the full probability distribution over all possible size distributions in terms of the so-called Bell polynomials. In this paper, we develop the asymptotics of this solution in order to check whether the exact solution yields correct expressions for the average cluster size distribution as obtained from the Smoluchowski equations. The answer is surprisingly involved: For the generic case of an arbitrary reaction rate, it is negative, but for the so-called classical rate kernels, constant, additive, and multiplicative, the solutions obtained are indeed exact. On the other hand, for the multiplicative kernel, a discrepancy is found in the full solution between the combinatorial solution and the exact solution. The reasons for this puzzling pattern of agreement and disagreement are unclear. A better understanding of the combinatorial solution's derivation is needed, the better to understand its range of validity.
在最近的一系列论文中，有人声称找到了所谓聚合的马库斯 - 卢什尼科夫模型一个变体的精确组合解。在这个模型中，最初假设存在有限数量的以单体形式存在的聚集体。在每个时间步，根据某些与尺寸相关的概率选择两个聚集体，并不可逆地结合形成一个质量更高的聚集体。所声称的结果给出了根据所谓的贝尔多项式对所有可能尺寸分布的全概率分布的表达式。在本文中，我们研究这个解的渐近性质，以检验精确解是否能给出从斯莫卢霍夫斯基方程得到的平均簇尺寸分布的正确表达式。答案出人意料地复杂：对于任意反应速率的一般情况，答案是否定的，但对于所谓的经典速率核，即常数、加法和乘法速率核，得到的解确实是精确的。另一方面，对于乘法核，在组合解和精确解的全解中发现了差异。这种令人困惑的一致和不一致模式的原因尚不清楚。需要更好地理解组合解的推导过程，以便更好地理解其有效性范围。
