用Wiener路径积分解湍流中颗粒成团的随机Langevin方程
Solution of the Stochastic Langevin Equations for Clustering of Particles in Turbulent Flows in Terms of Wiener Path Integral
摘要：
我们建议利用Wiener路径积分作为正式解决方案的联合概率密度的耦合Langevin方程描述的颗粒悬浮在流体中的粘性和随机力的影响下。我们得到的形式解，给出了李雅普诺夫指数的表达式，i）将提供对这种系统的所有特征和行为的描述，例如，最近在文献中使用适当的近似研究的聚集现象，ii）可以用于确定聚集-非聚集相变的发生和性质，我们已经针对一维情况示出了该相变，以及iii）允许使用各种近似的方法适合的物理条件的问题，如瞬子的解决方案，在WKB近似的聚合阶段的一维情况下，在本文中提出的。瞬子近似的使用给出了相同的结果，为李雅普诺夫指数在聚集阶段，以前得到的其他作者使用不同的近似方法。非聚集的情况下，也被认为是在一定的近似使用的一般路径积分表达式的一维情况下。
英文摘要：
We propose to take advantage of using the Wiener path integrals as the formal solution for the joint probability densities of coupled Langevin equations describing particles suspended in a fluid under the effect of viscous and random forces. Our obtained formal solution, giving the expression for the Lyapunov exponent, i) will provide the description of all the features and the behaviour of such a system, e.g. the aggregation phenomenon recently studied in the literature using appropriate approximations, ii) can be used to determine the occurrence and the nature of the aggregation - non-aggregation phase transition which we have shown for the one-dimensional case and iii) allows the use of a variety of approximative methods appropriate for the physical conditions of the problem such as instanton solutions in the WKB approximation in the aggregation phase for the one-dimensional case as presented in this paper. The use of instanton approximation gives the same result for the Lyapunov exponent in the aggregation phase, previously obtained by other authors using a different approximative method. The case of non-aggregation is also considered in a certain approximation using the general path integral expression for the one-dimensional case.
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