Jérémie Bec
CNRS Laboratoire Lagrange, Observatoire de la Côte d'Azurd

Effective rates in dilute advection-reaction systems

Many natural and industrial settings involve dilute systems of reacting particles transported by an unsteady fluid flow. We consider the simple case of an annihilation process $A + A \to \varnothing$ with a given rate when two particles are within a finite radius of interaction. The system is described in terms of the joint n-point number spatial density that it is shown to obey a hierarchy of transport equations. An analytic solution is obtained in either the dilute or the long-time limit by using a Lagrangian approach where statistical averages are performed along non-reacting trajectories. In this limit, we show that the moments of the number of particles have an exponential decay rather than the algebraic prediction of standard mean-field approaches. The effective reaction rate is then related to Lagrangian pair statistics by a large-deviation principle. A phenomenological model is introduced to study the qualitative behavior of the effective rate as a function of the interaction length, the degree of chaoticity of the dynamics and the compressibility of the carrier flow. Exact computations, obtained via a Feynman-Kac approach, in a smooth, compressible, random delta-correlated-in-time Gaussian velocity field support the proposed heuristic approach.
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