Binomial moment equations for stochastic reaction systems

A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations.     

The Forbes 400, the Pareto power-law and efficient markets

Statistical regularities at the top end of the wealth distribution in the United States are examined using the Forbes 400 lists of richest Americans, published between 1988 and 2003. It is found that the wealths are distributed according to a power-law (Pareto) distribution. This result is explained using a simple stochastic model of multiple investors that incorporates the efficient market hypothesis as well as the multiplicative nature of financial market fluctuations.     

Analysis of molecular hydrogen formation on low-temperature surfaces in temperature programmed desorption experiments

The study of the formation of molecular hydrogen on low-temperature surfaces is of interest both because it enables the exploration of elementary steps in the heterogeneous catalysis of a simple molecule and because of its applications in astrochemistry. Here, we report results of experiments of molecular hydrogen formation on amorphous silicate surfaces using temperature-programmed desorption (TPD). In these experiments, beams of H and D atoms are irradiated on the surface of an amorphous silicate sample. The desorption rate of HD molecules is monitored using a mass spectrometer during a subsequent TPD run. The results are analyzed using rate equations, and the energy barriers of the processes leading to molecular hydrogen formation are obtained from the TPD data. We show that a model based on a single isotope provides the correct results for the activation energies for diffusion and desorption of H atoms. These results are used in order to evaluate the formation rate of H2 on dust grains under the actual conditions present in interstellar clouds. It is found that, under typical conditions in diffuse interstellar clouds, amorphous silicate grains are efficient catalysts of H2 formation when the grain temperatures are between 9 and 14 K. This temperature window is within the typical range of grain temperatures in diffuse clouds. It is thus concluded that amorphous silicates are good candidates to be efficient catalysts of H2 formation in diffuse clouds.     

Efficient simulations of gas-grain chemistry in interstellar clouds

Chemical reactions on dust grains are of crucial importance in interstellar chemistry because they produce molecular hydrogen and various organic molecules. Because of the submicron size of the grains and the low flux, the surface populations of reactive species are small and strongly fluctuate. Under these conditions rate equations fail and the master equation is needed for modeling these reactions. However, the number of equations grows exponentially with the number of reactive species, severely limiting its feasibility. Here we present a method which dramatically reduces the number of equations, thus enabling the incorporation of the master equation in models of interstellar chemistry.     

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.