Solution of the stochastic Langevin equations for clustering of particles in random flows in terms of the Wiener path integral.

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 behavior 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-nonaggregation 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 nonaggregation is also considered in a certain approximation using the general path integral expression for the one-dimensional case.