Surfaces serve as highly efficient catalysts for a vast variety of chemicalreactions. Typically, such surface reactions involve billions of moleculeswhich diffuse and react over macroscopic areas. Therefore, stochasticfluctuations are negligible and the reaction rates can be evaluated using rateequations, which are based on the mean-field approximation. However, in casethat the surface is partitioned into a large number of disconnected microscopicdomains, the number of reactants in each domain becomes small and it stronglyfluctuates. This is, in fact, the situation in the interstellar medium, wheresome crucial reactions take place on the surfaces of microscopic dust grains.In this case rate equations fail and the simulation of surface reactionsrequires stochastic methods such as the master equation. However, in the caseof complex reaction networks, the master equation becomes infeasible becausethe number of equations proliferates exponentially. To solve this problem, weintroduce a stochastic method based on moment equations. In this method thenumber of equations is dramatically reduced to just one equation for eachreactive species and one equation for each reaction. Moreover, the equationscan be easily constructed using a diagrammatic approach. We demonstrate themethod for a set of astrophysically relevant networks of increasing complexity.It is expected to be applicable in many other contexts in which problems thatexhibit analogous structure appear, such as surface catalysis in nanoscalesystems, aerosol chemistry in stratospheric clouds and genetic networks incells.
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