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We propose to take advantage of using the Wiener path integrals as the formalsolution for the joint probability densities of coupled Langevin equationsdescribing particles suspended in a fluid under the effect of viscous andrandom forces. Our obtained formal solution, giving the expression for theLyapunov exponent, i) will provide the description of all the features and thebehaviour of such a system, e.g. the aggregation phenomenon recently studied inthe literature using appropriate approximations, ii) can be used to determinethe occurrence and the nature of the aggregation - non-aggregation phasetransition which we have shown for the one-dimensional case and iii) allows theuse of a variety of approximative methods appropriate for the physicalconditions of the problem such as instanton solutions in the WKB approximationin the aggregation phase for the one-dimensional case as presented in thispaper. The use of instanton approximation gives the same result for theLyapunov exponent in the aggregation phase, previously obtained by otherauthors using a different approximative method. The case of non-aggregation isalso considered in a certain approximation using the general path integralexpression for the one-dimensional case.

Solution of the Stochastic Langevin Equations for Clustering of Particles in Turbulent Flows in Terms of Wiener Path Integral