Generalized von Smoluchowski model of reaction rates, with reacting particles and a mobile trap

TL;DR

This paper extends the von Smoluchowski reaction rate model to include a mobile trap and particle interactions, providing exact long-time asymptotic descriptions and revealing an equivalence principle in the presence of particle birth.

Contribution

It introduces a generalized reaction model with a mobile trap and particle interactions, and derives exact asymptotic solutions using the IPDF method.

Findings

Exact long-time asymptotic description of the system.

Discovery of a shielding property in reversible coalescence.

An equivalence principle relating trapping, diffusion, and particle birth.

Abstract

We study diffusion-limited coalescence, A+A<-->A, in one dimension, in the presence of a diffusing trap. The system may be regarded as a generalization of von Smoluchowski's model for reaction rates, in that: (a) it includes reactions between the particles surrounding the trap, and (b) the trap is mobile -- both considerations which render the model more physically relevant. As seen from the trap's frame of reference, the motion of the particles is highly correlated, because of the motion of the trap. An exact description of the long -time asymptotic limit is found using the IPDF method, and exploiting a "shielding" property of reversible coalescence that was discovered recently. In the case where the trap also acts as a source -- giving birth to particles -- the shielding property breaks down, but we find an "equivalence principle": Trapping and diffusion of the trap may be compensated by an appropriate rate of birth, such that the steady state of the system is identical with the equilibrium state in the absence of a trap.