The sweeping rate in diffusion-mediated reactions on dust grain surfaces

A prominent chemical reaction in interstellar clouds is the formation of molecular hydrogen by recombination, which essentially takes place on dust grain surfaces. Analytical approaches to model such a system have hitherto neglected the spatial aspects of the problem by employing a simplistic version of the sweeping rate of reactants. We show how these aspects can be accounted for by a consistent definition of the sweeping rate, and calculate it exactly for a spherical grain. Two regimes can be identified: Small grains, on which two reactants almost surely meet, and large grains, where this is very unlikely. We compare the true sweeping rate to the conventional approximation and find a characteristic reduction in both regimes, most pronounced for large grains. These effects can be understood heuristically using known results from the analysis of two-dimensional random walks. We finally examine the influence of using the true sweeping rate in the calculation of the efficiency of hydrogen recombination: For fixed temperature, the efficiency can be reduced considerably, and relative to that, small grains gain in importance, but the temperature window in which recombination is efficient is not changed substantially.Comment: 10 pages, 6 figure

Accurate rate coefficients for models of interstellar gas-grain chemistry

The methodology for modeling grain-surface chemistry has been greatly improved by taking into account the grain size and fluctuation effects. However, the reaction rate coefficients currently used in all practical models of gas-grain chemistry are inaccurate by a significant amount. We provide expressions for these crucial rate coefficients that are both accurate and easy to incorporate into gas-grain models. We use exact results obtained in earlier work, where the reaction rate coefficient was defined by a first-passage problem, which was solved using random walk theory. The approximate reaction rate coefficient presented here is easy to include in all models of interstellar gas-grain chemistry. In contrast to the commonly used expression, the results that it provides are in perfect agreement with detailed kinetic Monte Carlo simulations. We also show the rate coefficient for reactions involving multiple species.Comment: 4 pages, 2 figure

Diffusion-limited reactions on a two-dimensional lattice with binary disorder

Reaction-diffusion systems where transition rates exhibit quenched disorder are common in physical and chemical systems. We study pair reactions on a periodic two-dimensional lattice, including continuous deposition and spontaneous desorption of particles. Hopping and desorption are taken to be thermally activated processes. The activation energies are drawn from a binary distribution of well depths, corresponding to `shallow' and `deep' sites. This is the simplest non-trivial distribution, which we use to examine and explain fundamental features of the system. We simulate the system using kinetic Monte Carlo methods and provide a thorough understanding of our findings. We show that the combination of shallow and deep sites broadens the temperature window in which the reaction is efficient, compared to either homogeneous system. We also examine the role of spatial correlations, including systems where one type of site is arranged in a cluster or a sublattice. Finally, we show that a simple rate equation model reproduces simulation results with very good accuracy.Comment: 9 pages, 5 figure

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

We study the steady state of a stochastic particle system on a two-dimensional lattice, with particle influx, diffusion and desorption, and the formation of a dimer when particles meet. Surface processes are thermally activated, with (quenched) binding energies drawn from a \emph{continuous} distribution. We show that sites in this model provide either coverage or mobility, depending on their energy. We use this to analytically map the system to an effective \emph{binary} model in a temperature-dependent way. The behavior of the effective model is well-understood and accurately describes key quantities of the system: Compared with discrete distributions, the temperature window of efficient reaction is broadened, and the efficiency decays more slowly at its ends. The mapping also explains in what parameter regimes the system exhibits realization dependence.Comment: 23 pages, 8 figures. Submitted to: Journal of Statistical Mechanics: Theory and Experimen

Evaluation of the Multiplane Method for Efficient Simulations of Reaction Networks

Reaction networks in the bulk and on surfaces are widespread in physical, chemical and biological systems. In macroscopic systems, which include large populations of reactive species, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations. However, many physical systems are partitioned into microscopic domains, where the number of molecules in each domain is small and fluctuations are strong. Under these conditions, the simulation of reaction networks requires stochastic methods such as direct integration of the master equation. However, direct integration of the master equation is infeasible for complex networks, because the number of equations proliferates as the number of reactive species increases. Recently, the multiplane method, which provides a dramatic reduction in the number of equations, was introduced [A. Lipshtat and O. Biham, Phys. Rev. Lett. 93, 170601 (2004)]. The reduction is achieved by breaking the network into a set of maximal fully connected sub-networks (maximal cliques). Lower-dimensional master equations are constructed for the marginal probability distributions associated with the cliques, with suitable couplings between them. In this paper we test the multiplane method and examine its applicability. We show that the method is accurate in the limit of small domains, where fluctuations are strong. It thus provides an efficient framework for the stochastic simulation of complex reaction networks with strong fluctuations, for which rate equations fail and direct integration of the master equation is infeasible. The method also applies in the case of large domains, where it converges to the rate equation results

Diffusion-limited reactions and mortal random walkers in confined geometries

Motivated by the diffusion-reaction kinetics on interstellar dust grains, we study a first-passage problem of mortal random walkers in a confined two-dimensional geometry. We provide an exact expression for the encounter probability of two walkers, which is evaluated in limiting cases and checked against extensive kinetic Monte Carlo simulations. We analyze the continuum limit which is approached very slowly, with corrections that vanish logarithmically with the lattice size. We then examine the influence of the shape of the lattice on the first-passage probability, where we focus on the aspect ratio dependence: Distorting the lattice always reduces the encounter probability of two walkers and can exhibit a crossover to the behavior of a genuinely one-dimensional random walk. The nature of this transition is also explained qualitatively.Comment: 18 pages, 16 figure

Ego-Splitting and the Transcendental Subject. Kant’s Original Insight and Husserl’s Reappraisal

In this paper, I contend that there are at least two essential traits that commonly define being an I: self-identity and self-consciousness. I argue that they bear quite an odd relation to each other in the sense that self-consciousness seems to jeopardize self-identity. My main concern is to elucidate this issue within the range of the transcendental philosophies of Immanuel Kant and Edmund Husserl. In the first section, I shall briefly consider Kant’s own rendition of the problem of the Egosplitting. My reading of the Kantian texts reveals that Kant himself was aware of this phenomenon but eventually deems it an unexplainable fact. The second part of the paper tackles the same problematic from the standpoint of Husserlian phenomenology. What Husserl’s extensive analyses on this topic bring to light is that the phenomenon of the Ego-splitting constitutes the bedrock not only of his thought but also of every philosophy that works within the framework of transcendental thinking