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

TL;DR

This paper introduces a Wiener path integral approach to solve stochastic Langevin equations for particles in turbulent flows, enabling analysis of aggregation phenomena and phase transitions with new approximation methods.

Contribution

It presents a formal solution using Wiener path integrals for Langevin equations, allowing detailed analysis of particle aggregation and phase transitions in turbulent flows.

Findings

Derived a formal solution for joint probability densities.

Identified conditions for aggregation and non-aggregation phases.

Validated instanton approximation for Lyapunov exponent in aggregation phase.

Abstract

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.