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We study the steady state of a stochastic particle system on atwo-dimensional lattice, with particle influx, diffusion and desorption, andthe formation of a dimer when particles meet. Surface processes are thermallyactivated, with (quenched) binding energies drawn from a \emph{continuous}distribution. We show that sites in this model provide either coverage ormobility, depending on their energy. We use this to analytically map the systemto an effective \emph{binary} model in a temperature-dependent way. Thebehavior of the effective model is well-understood and accurately describes keyquantities of the system: Compared with discrete distributions, the temperaturewindow of efficient reaction is broadened, and the efficiency decays moreslowly at its ends. The mapping also explains in what parameter regimes thesystem exhibits realization dependence.

Diffusion-limited reactions on disordered surfaces with continuous distributions of binding energies