Smoluchowski with interactions

In 1917 von Smoluchowski suggested a simple-minded model of diffusion-controlled binary   
reactions. It consists of an immobile spherical trap of radius R surrounded by a gas of Brownian

particles. The particle flux into the trap mimics the rate of diffusion-controlled reactions. Since

its inception, the Smoluchowski model and its extensions inspired multitudes of studies. The vast

majority of them continued to assume that the particles do not interact with other. Here we extend

this model to a whole class of diffusive gases of interacting particles. Employing the Macroscopic

Fluctuation Theory, we evaluate the probability P(T) that no gas particle hits the trap until a

long but finite time T. We also find the most likely density history of the gas conditional on the

non-hitting. The results crucially depend on the dimension of space d and on the rescaled parameter

  l = R/√D₀T where D₀ is the gas diffusivity.