We examine a stochastic particle system with influx, emission, diffusion, and reaction of particles, inspired by heterogeneous catalysis on surfaces.  The system has a steady state, whose key quantity is the efficiency: What fraction of incoming particles reacts before leaving the system by emission?

Different adsorption sites have different binding energies for the particles, drawn from a distribution—such a disordered system cannot be treated exactly anymore.  For the binary case, however, where each site is either shallow or deep, the physics and relevant limiting cases have been well-understood [1].

This knowledge enables us to map the much more general case of a continuous distribution to an effective binary model, in which sites of different binding energies have been lumped together into two effective types—the shallow and the deep ones.  We motivate in detail this mapping to the effective model, and compare to microscopic Monte Carlo simulations, showing excellent agreement.  As expected, the window of conditions for which the reaction is efficient is significantly broadened compared to the discrete binary case.

[1] A. Wolff, I. Lohmar, J. Krug, Y. Frank, O. Biham, Phys. Rev. E 81 061109 (2010).