Incorporation of stochastic chemistry on dust grains in the PDR code using moment equations

Unlike gas-phase reactions, chemical reactions taking place on interstellar dust grain surfaces cannot always be modeled by rate equations. Due to the small grain sizes and low flux,these reactions may exhibit large fluctuations and thus require stochastic methods such as the moment equations. We evaluate the formation rates of H2, HD and D2 molecules on dust grain surfaces and their abundances in the gas phase under interstellar conditions. We incorporate the moment equations into the Meudon PDR code and compare the results with those obtained from the rate equations. We find that within the experimental constraints on the energy barriers for diffusion and desorption and for the density of adsorption sites on the grain surface, H2, HD and D2 molecules can be formed efficiently on dust grains. Under a broad range of conditions, the moment equation results coincide with those obtained from the rate equations. However, in a range of relatively high grain temperatures, there are significant deviations. In this range, the rate equations fail while the moment equations provide accurate results. The incorporation of the moment equations into the PDR code can be extended to other reactions taking place on grain surfaces.

Efficient Stochastic Simulations of Complex Reaction Networks on Surfaces

Surfaces serve as highly efficient catalysts for a vast variety of chemical reactions. Typically, such surface reactions involve billions of molecules which diffuse and react over macroscopic areas. Therefore, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations, which are based on the mean-field approximation. However, in case that the surface is partitioned into a large number of disconnected microscopic domains, the number of reactants in each domain becomes small and it strongly fluctuates. This is, in fact, the situation in the interstellar medium, where some crucial reactions take place on the surfaces of microscopic dust grains. In this case rate equations fail and the simulation of surface reactions requires stochastic methods such as the master equation. However, in the case of complex reaction networks, the master equation becomes infeasible because the number of equations proliferates exponentially. To solve this problem, we introduce a stochastic method based on moment equations. In this method the number of equations is dramatically reduced to just one equation for each reactive species and one equation for each reaction. Moreover, the equations can be easily constructed using a diagrammatic approach. We demonstrate the method for a set of astrophysically relevant networks of increasing complexity. It is expected to be applicable in many other contexts in which problems that exhibit analogous structure appear, such as surface catalysis in nanoscale systems, aerosol chemistry in stratospheric clouds and genetic networks in cells.

Efficient Simulations of Interstellar Gas-Grain Chemistry Using Moment Equations

Networks of reactions on dust grain surfaces play a crucial role in the chemistry of interstellar clouds, leading to the formation of molecular hydrogen in diffuse clouds as well as various organic molecules in dense molecular clouds. Due to the sub-micron size of the grains and the low flux, the population of reactive species per grain may be very small and strongly fluctuating. Under these conditions rate equations fail and the simulation of surface-reaction networks requires stochastic methods such as the master equation. However, the master equation becomes infeasible for complex networks because the number of equations proliferates exponentially. Here we introduce a method based on moment equations for the simulation of reaction networks on small grains. The number of equations is reduced to just one equation per reactive specie and one equation per reaction. Nevertheless, the method provides accurate results, which are in excellent agreement with the master equation. The method is demonstrated for the methanol network which has been recently shown to be of crucial importance.